Best Approach
JEE MAIN
2011 to 2020
PAPER
By Mathematics Wizard
Manoj Chauhan Sir
No. 1 Faculty of Unacademy,
Ex. Faculty (Etoos, Bansal & Vibrant)
IIT Delhi, Exp. 11 Years
Maths IIT-JEE ‘Best Approach’
MCSIR
JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
AIEEE - 2011
1.
Let be real and z be a complex number. If If z2 + z + = 0 has two distinct roots on the line
Re z = 1, then it is necessary that :
(1) (0, 1)
(2) (–1, 0)
(3) || = 1
(4) (1, )
1
2.
8 log(1 x)
dx is :
1 x2
0
The value of
(1) log2
3.
log 2
8
(3)
log 2
2
(4) log2
d2x
equals :
dy 2
d2y
(1) 2
dx
4.
(2)
1
1
d 2 y dy
(2) 2
dx dx
3
d 2 y dy
(3) 2
dx dx
2
d 2 y dy
(4) 2
dx dx
3
Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The
value V(t) depreciates at a rate given by differential equation
dV(t)
k(T t) , where k > 0 is a
dt
constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is
2
(1) T
5.
6.
1
k
(2) I
kT 2
2
(3) I
k(T t) 2
2
The coefficient of x7 in the expansion of (1 – x – x2 + x3)6 is :
(1) 144
(2) –132
(3) –144
(4) e–kT
(4) 132
x
5
x
0,
, define f (x) t sint dt . Then f has :
For
2
0
(1) local maximum at and 2
(2) local minimum at and 2
(3) local minimum at and local maximum at 2
(4) local maximum at and local minimum at 2
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7.
The area of the region enclosed by the curves y = x, x = e, y
(1)
8.
MCSIR
1
square units
2
(2) 1 square units
(3)
JEE Main Papers
1
and the positive x-axis is :
x
3
square units
2
(4)
5
square units
2
The line L1 : y – x = 0 and L2 : 2x + y = 0 intersect the line L3 : y + 2 = 0 at
P and Q respectively. The bisector of the acute angle between L1 and L2 intersects L3 at R.
Statement-1 : The ratio PR : RQ equals 2 2 : 5
Statement-2 : In any triangle, bisector of an angle divides the triangle
into two similar triangles.
(1) Statement-1 is true, Statement-2 is true ; Statement-2 is correct
explanation for Statement-1
(2) Statement-1 is true, Statement-2 is true ; Statement-2 is not a correct
explanation for Statement-1
(3) Statement-1 is true, Statement-2 is false
(4) Statement-1 is false, Statement-2 is true
9.
The value of p and q for which the function
sin(p 1)x sin x
x
f (x)
q
x x2 x
x 3/ 2
1
3
(1) p , q
2
2
10.
, x0
, x 0 is continuous for all x in R, are :
, x0
5
1
(2) p , q
2
2
If the angle between the line x
3
1
(3) p , q
2
2
1
3
(4) p , q
2
2
y 1 z 3
and the plane
2
5
1
x + 2y + 3z = 4 is cos
, then equals
14
(1)
2
3
(2)
3
2
(3)
2
5
(4)
5
3
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11.
The domain of the function f (x)
(1) (–, )
12.
(2) (0, )
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1
is :
| x | x
(3) (–, 0)
(4) (–,) – {0}
The shortest distance between line y – x = 1 and curve x = y2 is :
(1)
3
4
(2)
3 2
8
(3)
8
3 2
(4)
4
3
13.
A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his
saving increases by Rs. 40 more than the saving of immediately previous month. His total saving fromthe
start of service will be Rs. 11040 after :
(1) 18 months
(2) 19 months
(3) 20 months
(4) 21 months
14.
Consider the following statements
P : Suman is brilliant, Q : Suman is rich, R : Suman is honest.
The negation of the statement "Suman is brilliant and dishonest if
and only if Suman is rich" can be expressed as :
(1) P (Q R) (2) (Q (P R)) (3) Q P R
15.
16.
If (1) is a cube root of unity, and (1 + )7 = A + B. Then (A, B) equals
(1) (0, 1)
(2) (1, 1)
(3) (1, 0)
(4) (–1, 1)
1
ˆ and b 1 (2iˆ 3jˆ 6k)
ˆ
(3iˆ k)
If a
, then the value of (2a b).[(a b) (a 2b)] is :
7
10
(1) –5
17.
If
(2) –3
(3) 5
(4) 3
dy
y 3 0 and y(0) = 2, then y(n2) is equal to :
dx
(1) 7
18.
(4) (P R) Q
(2) 5
(3) 13
(4) –2
Equation of the ellipse whose axes are the axes of coordinates and which passes through the point
(–3, 1) and has eccentricity
(1) 3x2 + 5y2 – 32 = 0
(3) 3x2 + 5y2 – 15 = 0
2
is :
5
(2) 5x2 + 3y2 – 48 = 0
(4) 5x2 + 3y2 – 32 = 0
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19.
20.
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If the mean deviation about the median of the numbers a, 2a, ....., 50a is 50, then |a| equals :
(1) 2
(2) 3
(3) 4
(4) 5
1 cos{2(x 2)}
lim
x 2
x2
(1) does not exist
(2) equals 2
(3) equals – 2
(4) equals 1
2
21.
Statement-1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box
is empty is 9C3.
Statement-2 : The number of ways of choosing any 3 places from 9 different places is 9C3.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(3) Statement-1 is true, Statement-2 is false.
(4) Statement-1 is false, Statement-2 is true.
22.
Let R be the set of real numbers.
Statement-1 : A = {(x, y) R × R : y – x is an integer} is an equivalence relation on R.
Statement-2 : B = {(x, y) R × R : x = y for some rational number } is an equivalence relation on R.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(3) Statement-1 is true, Statement-2 is false.
(4) Statement-1 is false, Statement-2 is true.
23.
Consider 5 independent Bernoulli.s trials each with probability of success p. If the probability of at least
one failure is greater than or equal to
1 3
(1) ,
2 4
3 11
(2) ,
4 12
31
, then p lies in the interval :
32
1
(3) 0,
2
11
(4) ,1
12
24.
The two circles x2 + y2 = ax and x2 + y2 = c2(c > 0) touch each other if :
(1) 2|a| = c
(2) |a| = c
(3) a = 2c
(4) |a| = 2c
25.
Let A and B be two symmetric matrices of order 3.
Statement-1 : A(BA) and (AB)A are symmetric matrices.
Statement-2 : AB is symmetric matrix if matrix multiplication
of A with B is commutative.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is true; Statement-2 is not a correct explanation
for Statement-1.
(3) Statement-1 is true, Statement-2 is false.
(4) Statement-1 is false, Statement-2 is true.
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26.
28.
JEE Main Papers
If Cand D are two events such that C D and P(D) 0, then the correctstatement among the following is
(1) P(C|D) = P(C)
27.
MCSIR
(2) P(C|D) P(C)
(3) P(C|D) < P(C) (4) P(C | D)
P(D)
P(C)
The vectors a and b are not perpendicular and c and d are two vectors satisfying :
b c b d and a.d 0. Then the vector d is equal to :
b.c
b.c
a.c
a.c
c
b
c
(1) b c
2)
(3) b c
(4)
b
a.b
a.b
a.d
a.b
Statement-1 : The point A(1, 0, 7) is the mirror image of the point B(1, 6, 3) in the line :
Statement-2 : The line :
x y 1 z 2
1
2
3
x y 1 z 2
bisects the line segment joining A(1, 0, 7) and B(1, 6, 3).
1
2
3
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(3) Statement-1 is true, Statement-2 is false.
(4) Statement-1 is false, Statement-2 is true.
29.
If A = sin2x + cos4x, then for all real x :
(1)
30.
3
A 1
4
(2)
13
A 1
16
(3) 1 A 2
The number of values of k for which the linear equations
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
posses a non-zero solution is :
(1) 3
(2) 2
(3) 1
(4)
3
13
A
4
16
(4) zero
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MATHEMATICS BY MANOJ CHAUHAN SIR
AIEEE - 2012
1.
2.
The equation esin x– e–sin x– 4 = 0 has
(1) infinite number of real roots
(2) no real roots
(3) exactly one real root
(4) exactly four real roots
Let â and b̂ be two unit vectors. If the vectors c aˆ 2bˆ and d 5aˆ 4bˆ are perpendicular to each
other,then the angle between â and b̂ is :
(1)
3.
6
(2)
2
(3)
3
(4)
4
A spherical balloon is filled with 4500 cubic meters of helium gas. If a leak in the balloon causes the gas
to escape at the rate of 72 cubic meters per minute, then the rate (in meters per minute) at which the
radius of the balloon decreases 49 minutes after the leakage began is
(1)
4.
9
7
(2)
7
9
(3)
2
9
(4)
9
2
Statement 1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + ...... + (361 + 380
+ 400) is 8000.
n
Statement 2:
3
k 3 k 1 n3 = for any natural number n.
k 1
(1) Statement 1 is false, statement 2 is true
(2) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(3) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(4) Statement 1 is true, statement 2 is false
5.
The negation of the statement “If I become a teacher, then I will open a school” is
(1) I will become a teacher and I will not open a school
(2) Either I will not become a teacher or I will not open a school
(3) Neither I will become a teacher nor I will open a school
(4) I will not become a teacher or I will open a school
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6.
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5 tan x dx
If the integral
x + a ln |sin x – 2 cos x| + k, then a is equal to
tan x 2
(1) –1
7.
MCSIR
(2) –2
(3) 1
(4) 2
Statement 1: An equation of a common tangent to the parabola y 2 16 3 x and the ellipse
2 x 2 y 2 4 is y 2 x 2 3 .
Statement 2: If the line y = mx +
4 3
, (m 0) is a common tangent to the parabola y 2 16 3 x and
m
the ellipse 2 x 2 y 2 4 , then m satisfies m4 + 2m2 = 24.
(1) Statement 1 is false, statement 2 is true
(2) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(3) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(4) Statement 1 is true, statement 2 is false
8.
1 0 0
0
1
2
1
0
. If u1 and u2 are column matrices such that Au1 0 and Au2 1 , then
Let A =
3 2 1
0
0
u1 + u2 is equal to
1
(1) 1
0
9.
1
(2) 1
1
If n is a positive integer, then 3 1
1
(3) 1
0
2n
3 1
2n
1
(4) 1
1
is
(1) an irrational number
(2) an odd positive integer
(3) an even positive integer
(4) a rational number other than positive integers
10.
If 100 times the 100th term of an AP with non zero common difference equals the 50 times its 50th term,
then the 150th term of this AP is
(1) –150
(2) 150 times its 50th term
(3) 150
(4) zero
11.
In a PQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to
(1)
5
6
(2)
6
(3)
4
(4)
3
4
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12.
An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is
(1) x – 2y + 2z – 3 = 0
(2) x – 2y + 2z + 1 = 0
(3) x – 2y + 2z – 1 = 0
(4) x – 2y + 2z + 5 = 0
13.
If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and
(2, 4) in the ratio 3 : 2, then k equals
(1)
29
5
(2) 5
(3) 6
(4)
11
5
14.
Let x1, x2, ......, xn be n observations, and let x be their arithematic mean and 2 be their variance.
Statement 1: Variance of 2x1, 2x2, ......, 2xn is 4 2.
Statement 2: Arithmetic mean of 2x1, 2x2, ......, 2xn is 4x .
(1) Statement 1 is false, statement 2 is true
(2) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(3) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(4) Statement 1 is true, statement 2 is false
15.
The population p(t) at time t of a certain mouse species satisfies the differential equation
dp t
0.5 p t 450 . If p(0) = 850, then the time at which the population becomes zero is
dt
(1) 2 ln 18
16.
(2) ln 9
(3)
1
ln18
2
(4) ln 18
Let a, b R be such that the function f given by f(x) = ln |x| + bx2 + ax, x 0 has extreme values at x =
–1 and x = 2.
Statement 1: f has local maximum at x = –1 and at x = 2.
Statement 2: a
1
1
and b
2
4
(1) Statement 1 is false, statement 2 is true
(2) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(3) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(4) Statement 1 is true, statement 2 is false
17.
2
The area bounded between the parabolas x
(1) 20 2
(2)
10 2
3
y
and x2 = 9y, and the straight line y = 2 is
4
(3)
20 2
3
(4) 10 2
<
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18.
19.
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Assuming the balls to be identical except for difference in colours, the number of ways in which one or
more balls can be selected from 10 white, 9 green and 7 black balls is
(1) 880
(2) 629
(3) 630
(4) 879
2x 1
If f : R R is a function defined by f(x) = [x]cos
, where [x] denotes the greatest integer
2
function, then f is
(1) continuous for every real x
(2) discontinuous only at x = 0
(3) discontinuous only at non-zero integral values of x
(4) continuous only at x = 0
20.
x 1 y 1 z 1
x 3 y k z
t and
intersect, then k is equal to
2
3
4
1
2
1
If the lines
(1) –1
21.
2
9
(3)
9
2
(4) 0
Three numbers are chosen at random without replacement from {1, 2, 3, ...... 8}. The probability that
their minimum is 3, given that their maximum is 6, is
(1)
22.
(2)
3
8
(2)
If z 1 and
1
5
(3)
1
4
(4)
2
5
z2
is real, then the point represented by the complex number z lies
z 1
(1) either on the real axis or on a circle passing through the origin
(2) on a circle with centre at the origin
(3) either on the real axis or on a circle not passing through the origin
(4) on the imaginary axis
23.
Let P and Q be 3 × 3 matrices with P Q. If P3 = Q3 and P2Q = Q2P, then determinant of
(P2 + Q2) is equal to
(1) –2
(2) 1
(3) 0
(4) –1
24.
If g(x) =
(1)
x
0 cos 4tdt , then g(x + p ) equals
g x
g
(2) g(x) + g()
(3) g(x) – g()
(4) g(x) . g()
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The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the
point (2, 3) is
(1)
26.
MCSIR
10
3
(2)
3
5
(3)
6
5
(4)
5
3
Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that
Y X, Z X and Y Z is empty, is
(1) 52
(2) 35
(3) 25
(4) 53
27.
An ellipse is drawn by taking a diameter of the circle (x – 1)2 + y2 = 1 as its semiminor axis and a
diameter of the circle x2 + ( y – 2)2 = 4 as its semi-major axis. If the centre of the ellipse is the origin and
its axes are the coordinate axes, then the equation of the ellipse is
(1) 4x2 + y2 = 4
(2) x2 + 4y2 = 8
(3) 4x2 + y2 = 8
(4) x2 + 4y2 = 16
28.
Consider the function f(x) = |x – 2| + |x – 5|, x R.
Statement 1: f '(4) = 0
Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).
(1) Statement 1 is false, statement 2 is true
(2) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
(3) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
(4) Statement 1 is true, statement 2 is false
29.
A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a
triangle OPQ, where O is the origin. If the area of the triangle OPQ is least, then the slope of the line PQ
is
(1)
30.
1
4
(2) –4
(3) –2
(4)
1
2
Let ABCD be a parallelogram such that AB q , , AD p and BAD be an acute angle. If r is the
vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by
3 p.q
(1) r 3q p
p. p
p.q
(2) r q p
p.p
p.q
(3) r q p
p.p
3 p.q
(4) r 3q p
p. p
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN - 2013
1.
Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is
(1)
2.
3
2
(2)
5
2
(3)
7
2
(4)
9
2
At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P
w.r.t. additional number of workers x is given by
then the new level of production of items is
(1) 2500
(2) 3000
dP
100 12x . If the firm employs 25more workers,
dx
(3) 3500
(4) 4500
3.
Let A and B two sets containing 2 elements and 4 elements respectively. The number of subsets of
A × B having 3 or more elements is
(1) 256
(2) 220
(3) 219
(4) 211
4.
If the lines
5.
If the vectors AB 3iˆ 4kˆ and AC 5iˆ 2ˆj 4kˆ are the sides of a triangle ABC, then the length of
the median through A is :
x 2 y 3 z 4
x 1 y 4 z 5
and
are coplanar, then k can have
1
1
k
k
2
1
(1) any value
(2) exactly one value (3) exactly two values (4) exactly three values
(1) 18
(2)
(3)
72
(4)
33
45
6.
The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1]
(1) lies between 1 and 2
(2) lies between 2 and 3
(3) lies between .1 and 0
(4) does not exist
7.
The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,....., is
(1)
8.
7
179 1020
81
(2)
7
99 1020
9
(3)
7
179 10 20
81
(4)
7
99 1020
9
A ray of light along x 3y 3 gets reflected upon reaching x-axis, the equation of the reflected ray
is
(1) y = x +
9.
3
(2) 3 y x 3
(3) y 3 x 3
The number of values of k, for which the system of equations :
(k + 1)x + 8y = 4k
kx + (k + 3)y = 3k – 1
has no solution, is
(1) infinite
(2) 1
(3) 2
(4) 3 y x 1
(4) 3
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10.
If the equations x2 + 2x + 3 = 0 and ax2 + bx + c = 0, a,b,c R, have a common root, then
a : b : c is
(1) 1 : 2 : 3
(2) 3 : 2 : 1
(3) 1 : 3 : 2
(4) 3 : 1 : 2
11.
The circle passing through (1, –2) and touching the axis of x at (3, 0) also passes through the point
(1) (–5, 2)
(2) (2, –5)
(3) (5, –2)
(4) (–2, 5)
12.
If x, y, z are in A.P. and tan–1x, tan–1y and tan–1z are also in A.P., then
(1) x = y = z
(2) 2x = 3y = 6z
(3) 6x = 3y = 2z
(4) 6x = 4y = 3z
13.
Consider :
Statement-I : (p ~ q) (~ p q) is a fallacy.
Statement-II : (p q) (~ q ~p) is a tautology.
(1) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(2) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
(3) Statement-I is true; Statement-II is false.
(4) Statement-I is false; Statement-II is true.
14.
If
15.
f x dx x , then
x 5f x 3 dx is equal to :
(1)
1 3
x x 3 x 2 x 3 dx C
3
(2)
1 3
x x 3 3 x 3 x 3 dx C
3
(3)
1 3
x x 3 x 2 x 3 dx C
3
(4)
1 3
x x 3 x 3 x 3 dx C
3
lim
1 cos 2x 3 cos x
x tan 4x
x 0
(1)
1
4
(2)
is equal to
1
2
(3) 1
(4) 2
/3
16.
dx
Statement-I : The value of the integral
1 tan x is equal to /6.
/6
b
b
Statement-II : f x dx f a b x dx
a
a
(1) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(2) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
(3) Statement-I is true; Statement-II is false.
(4) Statement-I is false; Statement-II is true.
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17.
The equation of the circle passing through the foci of the ellipse
is
(1) x2 + y2 – 6y – 7 = 0
(3) x2 + y2 – 6y – 5 = 0
18.
JEE Main Papers
x 2 y2
1 , and having centre at (0, 3)
16 9
(2) x2 + y2 – 6y + 7 = 0
(4) x2 + y2 – 6y + 5 = 0
A multiple choice examination has 5 questions. Each question has three alternative answers of which
exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing
is :
(1)
19.
MCSIR
17
(2)
5
13
(3)
11
(4)
(2) 2 2
5
10
3
3
3
35
The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0, 1)
(1, 1) and (1, 0) is :
(1) 2 2
5
(3) 1 +
(4)1 2
2
10
20.
x 1
x 1
The term independent of x in expansion of 2/3 1/3
x x 1 x x1/2
(1) 4
21.
(2) 120
The area (in square units) bounded by the curves y =
quadrant is :
(1) 9
22.
23.
(2) 36
(4) 310
x , 2y – x + 3 = 0, x-axis, and lying in the first
(3) 18
(4) 27/4
Let Tn be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If
Tn+1 – Tn = 10, then the value of n is :
(1) 7
(2) 5
(3) 10
(4) 8
1 z
If z is a complex number of unit modulus and argument , then arg
equals :
1 z
(1) –
24.
(3) 210
is :
(2)
2
(3)
(4)
ABCD is a trapezium such thatAB and CD are parallel and BC CD. If ADB = ,
BC = p and CD = q, then AB is equal to :
(1)
p2 q 2 sin
p cos q sin
p 2 q 2 cos
(2)
p cos q sin
p2 q 2
(3)
p2 cos q 2 sin
(4)
p2 q 2 sin
p cos q sin 2
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25.
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1 3
If P 1 3 3 is the adjoint of a 3 × 3 matrix A and |A| = 4, then is equal to :
2 4 4
(1) 4
(2) 11
(3) 5
(4) 0
x
26.
The intercepts on x-axis made by tangents to the curve, y = | t | dt , x R, which are parallel to the line
0
y = 2x, are equal to :
(1) ± 1
27.
(2) ± 2
(3) ± 3
(4) ± 4
Given : A circle, 2x2 + 2y2 = 5 and a parabola, y2 = 4 5 x.
Statement-I : An equation of a common tangent to these curves is y = x + 5 .
Statement-II : If the line, y mx
5
m 0 is their common tangent, then m satisfies
m
m4 – 3m2 + 2=0.
(1) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for
Statement-I.
(2) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for
Statement-I.
(3) Statement-I is true; Statement-II is false.
(4) Statement-I is false; Statement-II is true.
28.
If y = sec(tan–1x), then
(1)
29.
30.
1
2
dy
at x = 1 is equal to :
dx
(2)
1
2
(3) 1
tan A
cot A
can be written as :
1 cot A 1 tan A
(1) sinA cos A + 1
(2) sec A cosec A + 1 (3) tan A + cot A
(4)
2
The expression
(4) sec A + cosec A
All the students of a class performed poorly in Mathematics. The teacher decided to give gracemarks of
10 to each of the students. Which of the following statistical measures will not change even after the
grace marks were given ?
(1) mean
(2) median
(3) mode
(4) variance
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN - 2014
1.
2.
The image of the line
(1)
x 3 y 5 z 2
3
1
5
(2)
x 3 y 5 z 2
3
1
5
(3)
x 3 y 5 z 2
3
1
5
(4)
x 3 y 5 z 2
3
1
5
If the coefficients of x3 and x4 in the expansion of (1 + ax + bx2 )(1 – 2x)18 in powers of x are both zero,
then (a, b) is equal to
251
(1) 16,
3
3.
4.
5.
251
(2) 14,
3
272
(3) 14,
3
272
(4) 16,
3
If a R and the equation 3(x– [x])2 + 2 (x – [x]) + a2 = 0 (where [x] denotes the greatest integer x) has no
integral solution, then all possible values of a lie in the interval
(1) (–1, 0) (0, 1)
(2) (1, 2)
(3) (–2, –1)
(4) (–, –2)(2, )
2
If a b b c c a a b c , then is equal to
(1) 2
(2) 3
(3) 0
(4) 1
The variance of first 50 even natural numbers is
(1)
6.
x 1 y 3 z 4
in the plane 2x – y + z + 3 = 0 is the line
3
1
5
833
4
(2) 833
(3) 437
(4)
437
4
A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is
45°. It flies off horizontally straight away from the point O. After one second, the elevation of the bird
from O is reduced to 30°. Then the speed (in m/s) of the bird is
(1) 40 2 1
(2) 40 3 2
(3) 20 2
(4) 20 3 1
7.
x
x
1 4 sin 2 4 sin dx equals
The integral
2
2
0
(1) – 4
(2) 2 4 4 3
3
(3) 4 3 4
(4) 4 3 4
3
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8.
9.
MCSIR
The statement ~(p ~q) is
(1) equivalent to p q
(3) a tautology
JEE Main Papers
(2) equivalent to ~p q
(4) a fallacy
If A is an 3 × 3 non-singular matrix such that AA' = A'A and B = A–1A', then BB' equals
(1) I + B
(2) I
(3) B–1
(4) (B–1)'
1
10.
1 x
The integral 1 x e x dx is equal to
x
(1) x 1 e
11.
x
1
x
c
(2) xe
1
x
c
(3) x 1 e
x
1
x
c
(4) xe
If z is a complex number such that |z| 2, then the minimum value of z
(1) is equal to
5
2
x
1
x
c
1
2
(2) lies in the interval (1, 2)
(3) is strictly greater than
12.
x
5
2
(4) is strictly greater than
If g is the inverse of a function ƒ and ƒ'(x) =
1
1 x5
3
5
but less than
2
2
, then g'(x) is equal to
1
5
4
(1) 1 + x
13.
(2) 5x
(3) 1 g x 3
(4) 1 + {g(x)}5
If 0, and ƒ(n) = n + n and
3
1 f (1) 1 f (2)
1 f (1) 1 f (2) 1 f (3) = K(1 – )2 (1 – )2 ()2, then K is equal to : 1 f (2) 1 f (3) 1 f (4)
(1)
14.
Let ƒ k x
(1)
15.
(2)
1
(3) 1
(4) – 1
1 k
sin x cos k x where x R and k 1. Then ƒ 4 x ƒ 6 x equals
k
1
6
(2)
1
3
(3)
1
4
(4)
1
12
Let and be the roots of equation px2 + qx + r = 0, p 0. If p, q, r are in A. P. and
1 1
4 , then
the value of || is
(1)
61
9
(2)
2 17
9
(3)
34
9
(4)
2 13
9
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1
1
, P A B , where A stands for the
6
4
complement of the event A. Then the events A and B are :
(1) mutually exclusive and independent
(2) equally likely but not independent
(3) independent but not equally likely
(4) independent and equally likely
16.
Let A and B be two events such that P A B
17.
If f and g are differentiable functions in [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 and f(1) = 6, then for some
c ] 0, 1[
(1) 2f '(c) = g'(c)
(2) 2f '(c) = 3g'(c)
(3) f '(c) = g'(c)
(4) f '(c) = 2g'(c)
18.
Let the population of rabbits surviving at a time t be governed by the differential equation
dp t 1
p t 200 . If p(0) = 100, then p(t) equals
dt
2
(1) 400 – 300 e–t/2
(2) 300 – 200 et/2
(3) 600 – 500 et/2
19.
Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centrad at (0, y), passing through
origin and touching the circle C externally, then the radius of T is equal to
3
2
(1)
20.
(4) 400 – 300 et/2
3
2
(2)
(3)
1
2
(4)
1
4
(3)
2
2 3
(4)
2
2 3
The area of the region described by
A = {(x, y) : x2 + y2 1 and y2 1 – x} is
(1)
4
2 3
(2)
4
2 3
21.
Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx
+ 2by + d = 0 lies in the fourth quadrant and is equidistant from the two axes then
(1) 2bc – 3ad = 0
(2) 2bc + 3ad = 0
(3) 3bc – 2ad = 0
(4) 3bc + 2ad = 0
22.
Let PS be the median of the triangle with vertices P(2, 2), Q(6, –1) and R(7, 3). The equation of the line
passing through (1, –1) and parallel to PS is
(1) 4x – 7y – 11 = 0 (2) 2x + 9y + 7 = 0
(3) 4x + 7y + 3 = 0
(4) 2x – 9y – 11 = 0
23.
sin cos 2 x
is equal to
x 0
x2
lim
(1)
24.
2
(2) 1
(3) –
(4)
If X = {4n – 3n – 1 : n N} and Y = {9(n 1) : n N}, where N is the set of natural numbers, then
X Y is equal to :
(1) N
(2) Y – X
(3) X
(4) Y
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25.
The locus of the foot of perpendicular drawn from the centre of the ellipse x2 + 3y2 = 6 on any tangent to
it is
(1) (x2 – y2)2 = 6x2 + 2y2
(2) (x2 – y2)2 = 6x2 – 2y2
(3) (x2 + y2)2 = 6x2 + 2y2
(4) (x2 + y2)2 = 6x2 – 2y2
26.
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new
numbers are in A.P. Then the common ratio of the G.P. is
(1)
27.
121
10
(2)
411
100
(3) 100
(4) 110
3
(2)
4
(3)
6
(4)
2
The slope of the line touching both the parabolas y2 = 4x and x2 – 32y is
(1)
30.
(4) 2 3
The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and
l2 = m2 + n2 is :
(1)
29.
(3) 2 3
If (10)9 + 2(11)1(10)8 + 3(11)2(10)7 + ......... + 10(11)9 = k(10)9, then k is equal to
(1)
28.
(2) 3 2
2 3
1
2
(2)
3
2
(3)
1
8
(4)
2
3
If x = –1 and x = 2 are extreme points of f(x) = log |x| + x2 + x, then
(1) = – 6, =
1
2
(2) = – 6, = –
1
1
(3) = 2, = –
2
2
(4) = 2, =
1
2
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN - 2015
8
1.
1
5
2
The term independent of x in the binomial expansion of 1 3x 2x is
x
x
(1) 496
2.
(2) –496
(3) 400
(4) – 400
ex 12
, x0
x x
Let k be a non-zero real number. If f(x) = sin log 1
is a continuous function, then
4
k
12
,x 0
the value of k is :
(1) 4
(2) 1
(3) 3
(4) 2
3.
If the incentre of an equilateral triangle is (1, 1) and the equation of its one side is 3x + 4y + 3 = 0, then
the equation of the circumcircle of this triangle is :
(1) x2 + y2 – 2x – 2y – 14 = 0
(2) x2 + y2 – 2x – 2y – 2 = 0
(3) x2 + y2 – 2x – 2y + 2 = 0
(4) x2 + y2 – 2x – 2y – 7 = 0
4.
Let f : R R be a function such that f(2 – x) = f(2 + x) and f(4 – x) = f(4 + x), for all x R and
50
2
f (x)dx 5 . Then the value of f (x)dx is :
10
0
(1) 125
5.
7.
(3) 100
(4) 200
x2 x
x 1 x 2
2
3x 3 = ax – 12, then 'a' is equal to :
If 2x 3x 1 3x
2
x 2x 3 2x 1 2x 1
(1) 24
6.
(2) 80
(2) – 12
(3) –24
(4) 12
1 x 0.6
Let k and K be the minimum and the maximum values of the function f(x) =
in [0, 1] respectively,,
1 x 0.6
then the ordered pair (k, K) is equal to :
(1) (2–0.4, 1)
(2) (2–0.4, 20.6)
(3) (2–0.6, 1)
(4) (1, 20.6)
3
1
and sin + sin =
and is the arithmetic mean of and , then
2
2
sin 2 + cos 2 is equal to :
If cos + cos =
(1)
3
5
(2)
7
5
(3)
4
5
(4)
8
5
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9.
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Let PQ be a double ordinate of the parabola, y2 = – 4x, where P lies in the second quadrant. If R divides
PQ in the ratio 2 : 1 then the locus of R is :
(1) 3y2 = – 2x
(2) 3y2 = 2x
(3) 9y2 = 4x
(4) 9y2 = – 4x
In a parallelogram ABC, | AB | = a, | AD | = b and | AC | = c, then DA.AB has the value :
(1)
1 2
(a + b2 + c2)
2
(2)
1 2 2 2
(a – b + c )
2
(3)
1 2
(a + b2 – c2)
2
(4)
1 2 2 2
(b + c – a )
2
10.
If the two roots of the equation, (a – 1)(x4 + x2 + 1) + (a + 1)(x2 + x + 1)2 = 0 are real and distinct, then
the set of all values of 'a' is :
(1) 4
(2) 3
(3) 1
(4) 2
11.
The solution of the differential equation y dx – (x + 2y2)dy = 0 is x = f(y). If f(–1) = 1,
then f(1) is equal to :
(1) 4
(2) 3
(3) 1
(4) 2
12.
The shortest distance between the z-axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is :
(1) 1
(2) 2
(3) 4
(4) 3
13.
From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of
48 m/s. The greatest height (in metres) attained by the stone, assuming the value of the gravitational
acceleration g = 32 m/s2, is :
(1) 128
(2) 88
(3) 112
(4) 100
14.
Let A = {x1, x2,......, x7} and B = {y1, y2, y3} be two sets containing seven and three distinct elements
respectively. Then the total number of functions f : A B that are onto, if there exist exactly three
elements x in A such that f(x) = y2, is equal to :
(1) 14. 7C3
(2) 16. 7C3
(3) 14. 7C2
(4) 12. 7C2
15.
If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the
probability that the triangle is of maximum area given that it is an isosceles triangle, is :
(1)
16.
1
21
(2)
1
27
(3)
1
15
1
26
If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is :
(1) 12
(2) 6
(3) 10
(4) 9
sin x
17.
(4)
Let f : (–1, 1) R be a continuous function. If
f t dt
0
(1)
1
2
(2)
3
2
(3)
3
2
3
x then f 3 is equal to :
2
2
(4)
3
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18.
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2
2
If log t 1 t dt 1 g t C , where C is a constant, then g(2) is equal to :
2
1 t2
(1)
19.
MCSIR
1
5
log 2 5
(2)
1
log 2 5
2
(3) 2log 2 5
(4) log 2 5
If a circle passing through the point (–1, 0) touches y-axis at (0, 2), then the length of the chord of the
circle along the x-axis is :
(1)
3
2
(2) 3
(3)
5
2
(4) 5
20.
The sum of the 3rd and the 4th terms of a G.P. is 60 and the product of its first three terms is 1000. If the
first term of this G.P. is positive, then its 7th term is :
(1) 7290
(2) 640
(3) 2430
(4) 320
21.
A straight line L through the point (3, – 2) is inclined at an angle of 60° to the line 3x + y = 1. If L also
intersects the x-axis, then the equation of L is :
(1) y +
3x+2–3 3 =0
(2)
3y+x–3+2 3 =0
(3) y –
3x+2+3 3 =0
(4)
3y–x+3+2 3 =0
Im z 5
22.
If z is a non-real complex number, then the minimum value of
(1) – 1
23.
(3) – 2
is :
(4) – 5
h cos a sin
9sin
(2)
h sin a cos
9 sin
(3)
h cos a sin
9 cos
(4)
h sin a cos
9 cos
If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of
the ellipse is :
(1)
25.
2
Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation at
a point O on this line and all the poles are on the same side of O. If the height of the longest pole is 'h' and
the distance of the foot of the smallest pole from O is 'a' ; then the distance between two consecutive
poles, is :
(1)
24.
(2) – 4
Im z
2 2 1
2
(2)
2 1
(3)
1
2
(4)
2 1
2
x 1 y 2 z 3
also contains the point :
1
5
4
(3) (0, –3, 1)
(4) (0, 7, 10)
A plane containing the point (3, 2, 0) and the line
(1) (0, 3, 1)
(2) (0, 7, –10)
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26.
If
1
MCSIR
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k
n n 1 n 2 n 3 3 , then k is equal to :
n 1
(1)
27.
9
16
1
5
(3)
55
336
(4)
19
112
(2)
3
4
(3)
1
16
(4)
15
16
(2)
1
25
(3) ± 1
(4) ± 5
The equation of a normal to the curve, sin y = x sin y at x = 0, is :
3
(1) 2x 3y 0
30.
17
105
If A is a 3 × 3 matrix such that |5.adjA| = 5, then |A| is equal to :
(1)
29.
(2)
If the mean and the variance of a binomial variate X are 2 and 1 respectively, then the probability that X
takes a value greater than or equal to one is :
(1)
28.
1
6
(2) 2x 3y 0
(3) 2y 3x 0
(4) 2y 3x 0
Consider the following statements :
P : Suman is brilliant
Q : Suman is rich.
R : Suman is honest
the negation of the statement
"Suman is brilliant and dishonest if and only if suman is rich" can be equivalently expressed as :
(1) ~ Q ~P R
(2) ~ Q ~P R
(3) ~ Q P ~R
(4) ~ Q P ~R
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN - 2016 (OFFLINE PAPER)
1.
Two sides of a rhombus are along the lines, x – y + 1 = 0 and 7x – y – 5 = 0. If its diagonals intersect at
(–1, –2), then which one of the following is a vertex of this rhombus ?
1 8
(2) ,
3 3
(1) (–3, –8)
10 7
(3) ,
3 3
(4) (–3, –9)
2.
If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:
(1) 4/3
(2) 1
(3) 7/4
(4) 8/5
3.
Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle,
x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is
(1) x2 + y2 – x – 4y – 12 = 0
(2) x2 + y2 – x/4 + 2y – 24 = 0
(3) x2 + y2 – 4x + 9y + 18 = 0
(4) x2 + y2 – 4x + 8y + 12 = 0
4.
The system of linear equations
x + y – z = 0
x – y – z = 0
x + y – z = 0
has a non-trivial solution for :
(1) Exactly one value of .
(3) Exactly three values of .
5.
(2) Exactly two values of .
(4) Infinitely many values of .
1
If f(x) + 2f = 3x, x 0, and S : {x R: f(x) = f(–x)} then S :
x
(1) contains exactly one element
(2) contains exactly two elements.
(3) contains more than two elements.
(4) is an empty set.
1
6.
Let p = lim
x 0
1 tan 2
(1) 1
7.
then log p is equal to :
(2) 1/2
A value of for which
(1) /6
8.
x 2x
(3) 1/4
(4) 2
2 3i sin
is purely imaginary, is :
1 2i sin
1 3
(2) sin
4
1 1
(3) sin
3
(4) /3
The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its
conjugate axis is equal to half of the distance between its foci, is :
(1)
4
3
(2)
2
3
(3)
3
(4)
4
3
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10.
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If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?
(1) 3a2 – 32a + 84 = 0
(2) 3a2 – 34 a + 91 = 0
2
(3) 3a – 23 a + 44 = 0
(4) 3a2 – 26 a + 55 = 0
2x12 5x 9
dx is equal to
The integral
3
5
x x 3 1
(1)
(3)
11.
MCSIR
x10
2
2 x 5 x 3 1
x10
2
2 x 5 x 3 1
If the line,
(1) 18
C
(2)
(4)
C
x5
2
2 x 5 x 3 1
x5
2
2 x 5 x 3 1
C
C
x 3 y2 z4
lies in the plane, lx + my – z = 9, then l2 + m2 is equal to
2
1
3
(2) 5
(3) 2
(4) 26
12.
If 0 x < 2 , then the number of real values of x, which satisfy the equation
cosx + cos2x + cos3x + cos4x = 0, is
(1) 5
(2) 7
(3) 9
(4) 3
13.
The area (in sq. units) of the region {(x, y) : y2 2x and x2 + y2 4x, x 0, y 0} is :
(1)
14.
8
3
(2)
4 2
3
(3)
2 2
2
3
(4)
4
3
3
Let a, b and c be three unit vectors such that a b c
b c . If b is not parallel to c , then
2
the angle between a and b is.
(1) /2
(2) 2/3
(3) 5/6
(4) 3/4
15.
A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units
and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum,
then
(1) (4 – ) x = r
(2) x = 2r
(3) 2x = r
(4) 2x = (4 – )r
16.
The distance of the point (1, –5, 9) from the plane x – y + z = 5 measured along the line x = y = z is
(1) 10 3
17.
(2)
10
3
(3) 20/3
(4) 3 10
If a curve y = f(x) passes through the point (1, –1) and satisfies the differential equation,
1
y(1 + xy) dx = xdy, then f is equal to
2
(1) – 4/5
(2) 2/5
(3) 4/5
(4) – 2/5
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n
18.
2 4
If the number of terms in the expansion of 1 2 , x 0 , is 28, then the sum of the coefficients
x x
of all the terms in this expansion, is
(1) 2187
(2) 243
19.
(3) 729
(4) 64
1 1 sin x
Consider f(x) = tan
, x 0, . A normal to y = f(x) at x = also passes through
2
6
1 sin x
the point
2
(1) 0,
3
(2) ,
6
0
(3) ,
4
0
(4) (0, 0)
20.
For x R, f(x) = |log 2 – sin x| and g(x) = f(f(x)), then
(1) g'(0) = cos(log2)
(2) g'(0) = – cos(log2)
(3) g is differentiable at x = 0 and g'(0) = – sin (log2)
(4) g is not differentiable at x = 0
21.
Let two fair six-faced dice A and B be thrown simultaneously. If E1 is the event that die A shows up four,
E2 is the event that die B shows up two and E3 is the event that the sum of numbers on both dice is odd,
then which of the following statements is NOT True ?
(1) E2 and E3 are independent
(2) E1 and E3 are independent
(3) E1, E2 and E3 are independent
(4) E1 and E2 are independent
22.
5a b
T
If A =
and A adj A = AA , then 5a + b is equal to :
3
2
(1) 5
23.
24.
(3) 13
(4) – 1
The Boolean Expression (p ~q) q (~p q) is equivalent to :
(1) p q
(2) p q
(3) p q
The sum of all real values of x satisfying the equation x 2 5x 5
(1) – 4
25.
(2) 4
(2) 6
(3) 5
(4) ~ p q
x 2 4x 60
1
(4) 3
The centres of those circles which touch the circle, x2 + y2 – 8x – 8y – 4 = 0, externally and also touch
the x-axis, lie on :
(1) an ellipse which is not a circle
(2) a hyperbola
(3) a parabola
(4) a circle
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27.
MCSIR
If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL
and arranged as in a dictionary; then the position of the word SMALL is :
(1) 59th
(2) 52th
(3) 58th
(4) 46th
1/n
n 1 n 2 ......3n
lim
is equal to :
n
n 2n
(1)
27
e
2
(2)
9
e
2
(3) 3 log 3 – 2
2
28.
18
e4
2
2
(2) 100
(3) 99
(4) 102
If one of the diameters of the circle, given by the equation, x2 + y2 – 4x + 6y – 12 = 0, is a chord of a
circle S, whose centre is at (–3, 2), then the radius of S is :
(1) 5 3
30.
2
(4)
16
3 2 1
4
If the sum of the first ten terms of the series 1 2 3 42 4 ......., is m,
5
5 5 5
5
then m is equal to :
(1) 101
29.
JEE Main Papers
(2) 5
(3) 10
(4) 5 2
A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the
path, he observes that the angle of elevation of the top of the pillar is 30°, After walking for 10 minutes
from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar
is 60°. Then the time taken (in minutes) by him, from B to reach the pillar, is :
(1) 10
(2) 20
(3) 5
(4) 6
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN (09-04-2016) ONLINE PAPER
1.
If A and B are any two events such that P(A) =2/5 and P(A B) = 3/20, then the conditional probability,
P(A/(A' B')) where A' denotes the complement of A, is equal to :
(1) 8/17
(2) 1/4
(3) 5/17
(4) 11/20
2.
For x R, x 0, x 1, let f0(x) =
1
and fn+1 (x) = f0(f(n(X)), n = 0, 1, 2, ....... Then the value of
1 x
2
3
f100(3) + f1 + f2 is equal to :
2
3
(1)
3.
(2)
1
2
(2) 2
(2) 2
2
1
If 2 tan
1
1
0
xdx cot
3
2
If P = 1
2
1
1 x x 2 dx
0
(1) log 2
6.
(3)
5
3
(4)
8
3
(3)
2
(4) 2 2
If the equations x2 + bx – 1 = 0 and x2 + x + b = 0 have a common root different from –1, then | b | is
equal to
(1)
5.
1
3
The distance of the point (1, –2, 4) from the plane passing through the point (1, 2, 2) and perpendicular
to the planes x – y + 2z = 3 and 2x – 2y + z + 12 = 0, is
(1)
4.
4
3
(3)
3
(4) 3
1
1
2
then tan 1 x x dx is equal to :
0
(2)
+ log 2
2
(3) log 4
(4)
– log 4
2
1
2
1 1
,
A
=
APT, then PT Q2015 P is :
0 1 and Q = PAP
3
2
1
2015
(1)
2015
0
1 2015
(2)
1
0
0 2015
(3)
0
0
0
2015
(4)
2015
1
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dx
A
B
3
If
cos x 2sin 2x = (tan x ) + C(tan x) + k, where k is a constant of integration, then A + B + C
equals
(1)
8.
16
5
(2)
(3)
7
10
(4)
27
10
The point (2, 1) is translated parallel to the line L : x – y = 4 by 2 3 units. If the new point Q lies in the
third quadrant, then the equation of the line passing through Q and perpendicular to L is :
(1) 2x + 2y = 1 –
9.
21
5
6 (2) x = y = 3 – 3
6
(3) x + y = 2 –
6
(4) x + y = 3 – 2
6
x,
x 1
If the function f(x) =
is differentiable at x = 1, then a/b is equal to :
1
a cos x b , 1 x 2
(1)
2
2
(2) –1 – cos–1(2)
(3)
2
2
(4)
2
2
15
Cr
2
r
is equal to :
The value of 15
r 1 C r 1
15
10.
(1) 1085
11.
(2) 560
(3) 680
(4) 1240
In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively
ˆ ˆi 3jˆ pkˆ and 5iˆ qjˆ 4kˆ , then the point (p, q) lies on a line
3iˆ ˆj k,
(1) parallel to y-axis
(2) making an acute angle with the positive direction of x-axis
(3) parallel to x-axis
(4) making an obtuse angle with the position direction of x-axis.
12.
a 4
If lim 1 2
x
x x
(1) 2/3
13.
e3 then 'a' is equal to :
(2) 3/2
The number of x [0, 2] for which
(1) 6
14.
2x
(2) 4
(3) 2
(4) 1/2
2sin 4 x 18cos 2 x 2cos 4 x 18sin 2 x 1 is
(3) 8
(4) 2
If m and M are the minimum and the maximum values of 4 + 1/2 sin2 2x – 2 cos4x, x R, then M – m
is equal to :
(1) 7/4
(2) 15/4
(3) 9/4
(4) 1/4
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x y
x y
1 and 1 , meets the coordinate
3 4
4 3
axes at A and B, (A B), then the locus of the midpoint of AB is :
(1) 7xy = 6(x + y)
(2) 6xy = 7(x + y)
2
(3) 4(x + y) – 28(x + y) + 49 = 0
(4) 14(x + y)2 – 97(x + y) + 168 = 0
15.
If a variable line drawn through the intersection of the lines
16.
If f(x) is a differentiable function in the interval (0, ) such that f(1) = 1 and lim
t 2f x x 2f t
1,
t x
tx
3
for each x > 0, then f is equal to :
2
(1) 13/6
(2) 23/18
(3) 25/9
(4) 31/18
17.
If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3 – 1, t R, meets the curve
again at a point Q, then the coordinates of Q are :
(1) (t2 + 3, – t3 – 1)
(2) (t2 + 3, t3 – 1)
(3) (16t2 + 3, – 64t3 – 1)
(4) (4t2 + 3, – 8t3 – 1)
18.
If the tangent at a point on the ellipse
x 2 y2
1 meets the coordinate axes at A and B, and O is the
27 3
origin, then the minimum area (in sq. units) of the triangle OAB is :
(1) 9
19.
(2) 9/2
(3) 9 3
(4) 3 3
The point represented by 2+i in the Argand plane moves 1 unit eastwards, then 2 units northwards and
finally from there 2 2 units in the south-westwards direction. Then its new position in the Argand plane
is at the point represented by :
(1) 2 + 2i
(2) – 2 – 2i
(3) 1 + i
(4) – 1 – i
20.
A circle passes through (–2, 4). Which one of the following equations can represent a diameter of this
circle?
(1) 4x + 5y – 6 = 0
(2) 5x + 2y + 4 = 0
(3) 2x – 3y + 10 = 0 (4) 3x + 4y – 3 = 0
cos x
21.
The number of distinct reat roots of the equation, sin x
sin x
(1) 4
22.
(2) 1
The shortest distance between the lines
(1) (2, 3]
(2) [0, 1)
(3) 2
sin x
sin x
cos x sin x = 0 in the interval , is
4 4
sin x cos x
(4) 3
x y z
x 2 y4 z5
and
lies in the interval :
2 2 1
1
8
4
(3) (3, 4]
(4) [1, 2)
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If the four letter words (need not be meaningful) are to be formed using the letters from the word.
"MEDITERRANEAN". such that the first letter is R and the fourth letter is E, then the total number of all
such words is :
11!
(1)
2!3
(2) 59
(3) 110
(4) 56
24.
Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity
satisfies the equation 9e2 – 18e + 5 = 0. If S(5, 0) is a focus and 5x = 9 is the corresponding directrix of
hyperbola, then a2 – b2 is equal to
(1) –7
(2) – 5
(3) 5
(4) 7
25.
Consider the following two statements :
P : If 7 is an odd number, then 7 is divisible by 2.
Q : If 7 is a prime number then 7 is an odd number.
If V1 is the truth value of contrapositive of P and V2 is the truth value of contrapositive of Q, then the
ordered pair (V1, V2) equals :
(1) (F, T)
(2) (T, F)
(3) (F, F)
(4) (T, T)
26.
The minimum distance of a point on the curve y = x2 – 4 from the origin is :
(1)
15
2
19
2
(2)
(3)
15
2
(4)
19
2
27.
Let x, y, z be positive real numbers such that x + y + z = 12 and x3y4z5 = (0. 1) (600)3. Then
x3 + y3 + z3 is equal to
(1) 270
(2) 258
(3) 216
(4) 342
28.
If the mean deviation of the numbers 1, 1 + d, ..., 1 + 100d from their mean is 255, then a value of d is
(1) 10
(2) 20.2
(3) 5.05
(4) 10.1
2016
29.
2016
For x R, x = –1, if (1 + x)
2015
+ x (1 + x)
2
+ x (1 + x)
2014
2016
+ .... + x
=
a i xi , then a
i 0
17
is equal
to :
(1)
30.
2016!
16!
(2)
2017!
2000!
2017!
(3) 17! 2000!
2016!
(4) 17! 1999!
The area (in sq. units) of the region described by A = {(x, y) | y x2 – 5x + 4, x + y 1, y 0} is :
(1) 7/2
(2) 13/6
(3) 17/6
(4) 19/6
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN (10-04-2016) ONLINE PAPER
1.
Let C be a curve given by y(x) = 1 4x 3, x
3
. If P is a point on C, such that the tangent at P has
4
2
, then a point through which the normal at P passes, is
3
(1) (3, –4)
(2) (1, 7)
(3) (4, –3)
slope
2.
(4) (2, 3)
Let a, b R, a 0 . If the function f defined as
2x 2
,
a
f x
a
,
2
2b 4b ,
x 3
(1)
2,1 3
0 x 1
1 x 2
2x
(2) 2,1 3
(3)
2, 1 3
(4) 2,1 3
3.
Let a1, a2, a3,......., an,.......be in A.P. If a3 + a7+ a11+ a15 = 72 then the sum of its first 17 terms is equal
to
(1) 153
(2) 306
(3) 612
(4) 204
4.
If A > 0, B > 0 and A + B =
(1) 2 3
5.
6.
(2)
, then the minimum value of tanA + tanB is
6
2
(3) 3 2
(4) 4 2 3
3
The contrapositive of the following statement,
" If the side of a square doubles, then its area increases four times", is
(1) If the area of a square does not increase four times, then its side is not doubled.
(2) If the area of a square increases four times, then its side is not doubled.
(3) If the area of a square increases four times, then its side is doubled.
(4) If the side of a square is not doubled, then its area does not increase four times.
Let A be a 3 × 3 matrix such that A2 – 5A + 7I = 0.
1
5I A
7
Statement - II : The polynomial A3 – 2A2 – 3A + I can be reduced to 5(A – 4I).
Then
(1) Statement–I is false, but Statement-II is true.
(2) Both the statements are false.
(3) Both the statements are true.
(4) Statement–I is true, but Statement-II is false.
1
Statement - I : A
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Equation of the tangent to the circle, at the point (1, –1), whose centre is the point of intersection of the
straight lines x – y = 1 and 2x + y = 3 is
(1) 3x – y – 4 = 0
(2) x + 4y + 3 = 0
(3) x – 3y – 4 = 0
(4) 4x + y – 3 = 0
10
8.
2
The sum r 1 r! is equal to
r 1
(1) 10 × (11!)
9.
10.
(4) 11 × (11!)
Let f(x) = sin4x + cos4x. Then f is an increasing function in the interval
5 3
(2) ,
8 4
(3) 0,
4
5
(4) ,
2 8
Let z = 1 + ai be a complex number, a > 0 such that z3 is a real number. Then the sum
1 + z + z2 +....+ z11 is equal to
(1) –1250 3 i
12.
(3) (11!)
Let ABC be a triangle whose circumcentre is at P. If the position vectors of A,B,C and P are a, b, c and
a b c
respectively, then the position vector of the orthocentre of this triangle, is
4
abc
a
bc
(1) 0
(2) –
(3) a b c
(4)
2
2
(1) ,
4 2
11.
(2) 101 × (10!)
(2) 1250 3 i
(3) –1365 3 i
(4) 1365 3 i
Let P = : sin cos 2 cos and Q = : sin cos 2 sin be two sets. Then
(1) Q P
(3) PQ and Q – P
(2) P Q
(4) P = Q
13.
The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1,2 and 6, then
the mean deviation from the mean of the data is
(1) 2.5
(2) 2.8
(3) 2.6
(4) 2.4
14.
The number of distinct real values of for which the lines
x 3 y 2 z 1
2
are coplanar is :
1
2
(1) 3
(2) 2
15.
x 1 y 2 z 3
2 and
1
2
(3) 1
(4) 4
The angle of elevation of the top of a vertical tower from a point A, due east of it is 45°. The angle of
elevation of the top of the same tower from a point B, due south of A is 30°. If the distance between A
and B is 54 2 m, then the height of the tower (in metres), is
(1) 54
(2) 108
(3) 54 3
(4) 36 3
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16.
17.
1 cos2x 2
lim
is
x0 2x tan x x tan 2x
(1) 2
(2) – 1/2
MCSIR
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(3) 1/2
The solution of the differential equation
(4) – 2
dy y
tan x
sec x
, where 0 x and y(0) = 1, is given
dx 2
2y
2
by
2
(1) y 1
(3) y 1
x
sec x tan x
2
(2) y 1
x
sec x tan x
(4) y 1
x
sec x tan x
x
sec x tan x
18.
P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the
normal at p passes through Q, then the minimum value of t12 is :
(1) 4
(2) 6
(3) 8
(4) 2
19.
A hyperbola whose transverse axis is along the major axis of the conic,
x 2 y2
4 and has vertices
3
4
at the foci of this conic. If the eccentricity of the hyperbola is 3/2, then which of the following points does
NOT lie on it ?
(1)
5, 2 2
(2) 5, 2 3
(3) (0, 2)
(4)
10, 2 3
x
20.
x
For x R, x 0, if y(x) is a differentiable function such that x y t dt x 1 ty t dt , then y(x)
1
1
quals
(where C is a constant)
(1)
1
3 x
Cx e
1
C
(2) e x
x
(3)
C
x2
e
1
x
(4)
C
x3
e
1
x
21.
ABC is a triangle in a plane with vertices A(2,3,5), B(–1,3,2) and C(, 5, µ). If the median through A is
equally inclined to the coordinate axes, then the value of (3 + µ3 + 5) is
(1) 676
(2) 1130
(3) 1348
(4) 1077
22.
A ray of light is incident along a line which meets another line, 7x –y + 1 = 0, at the point (0, 1). The ray
is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the
ray of light is
(1) 41x + 38y –38 = 0
(2) 41x – 38y + 38 = 0
(3) 41x + 25y – 25 = 0
(4) 41x – 25y + 25 = 0
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A straight line through origin O meets the line 3y = 10 – 4x and 8x + 6y + 5 = 0 at points A and B
respectively. Then O divides the segment AB in the ratio
(1) 3 : 4
(2) 1 : 2
(3) 2 : 3
(4) 4 : 1
x 2 dx
10
24.
MCSIR
x 2 28x 196 x 2 , where [x] denotes the greatest integer less than or
The value of the integral
4
equal to x, is
(1) 3
25.
n2
C6
n 2
P2
If
(2) 7
If x is a solution of the equation,
18
, (x > 0), are m and n respectively,,
(3) 27
(4) 182
(3) – 175
(4) – 25
1
2x 1 2x 1 1, x , then 4x 2 1 is equal to :
2
(2) 3/4
(3) 2 2
(4) 1/2
An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials
of this experiment is
(1)
30.
(2) 4/5
(2) 2016
(1) 2
29.
(3) n2 + 3n – 108 = 0 (4) n2 + 2n – 80 = 0
4 1
If A
, then the determinant of the matrix (A2016 – 2A2015 – A2014) is :
3
1
(1) 2014
28.
(2) n2 + 5n – 84 = 0
1
x3 1
–2
–4
1
If the coefficients of x and x in the expansion of
2x 3
then m/n is equal to :
(1) 5/4
27.
(4) 1/3
11 , then n satisfies the equation
(1) n2 + n – 110 = 0
26.
(3) 6
192
729
(2)
256
729
(3)
240
729
(4)
496
729
dx
The integral
is equal to (where C is a constant of integration)
1 x x x 2
(1) 2
1 x
1 x
C
(2) 2
1 x
1 x
C
(3)
1 x
1 x
C
(4) 2
1 x
1 x
C
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN - 2017 (ONLINE PAPER)
1.
Let f(x) = 210. x + 1 and g(x) = 310. x – 1. If (f o g)(x) = x, then x is equal to :
(1)
2.
310 1
(2)
310 210
210 1
(3)
210 310
1 310
(4)
210 310
1 210
310 210
Let p(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by x – 1
and it leaves remainder 6 when divided by x + 1, then :
(1) p(–2) = 11
(2) p(2) = 11
(3) p(2) = 19
(4) p(–2) = 19
/4
3.
8 cos 2x
dx equals :
The integral
tan x cot x 3
/2
(1) 15/128
4.
(2) 13/32
0
cos x sin x
If S x 0, 2 : sin x
0
cos x 0 , then
cos x sin x
0
(1) 2 3
5.
(3) 13/256
(2) 4 2 3
(4) 15/64
tan 3 x is equal to :
xS
(3) 4 2 3
(4) 2 3
If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the centre and
1 1
subtend angles cos and sec–1(7) at the centre, respectively, then the distance between these
7
chords, is :
(1) 16/7
6.
(2)
8
(3) 8/7
7
(4)
4
7
The area (in sq. units) of the parallelogram, whose diagonals are along the vectors 8iˆ 6ˆj and
3iˆ 4ˆj 12kˆ , is :
(1) 65
7.
(2) 52
15
(3) 26
15
If y x x 2 1 x x 2 1
(1) 225 y2
(2) 224 y2
(4) 20
2
then x 2 1 d y x dy is equal to :
dx
dx 2
(3) 125 y
(4) 225 y
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The line of intersection of the planes, r. 3iˆ ˆj kˆ 1 and r. ˆi 4ˆj 2kˆ 2 , is :
4
5
z
y
7
7
2
7
13
x
(1)
(2)
4
5
z
y
7
7
2
7
13
x
(3)
6
5
y
13
13 z
2
7
13
x
6
5
y
13
13 z
2
7
13
x
(4)
9.
If the sum of the first n terms of the series 3 75 243 507 .... is 435 3 , then n equals :
(1) 29
(2) 18
(3) 15
(4) 13
10.
The proposition (~p) (p ~q) is equivalent to :
(1) p ~ q
(2) p ~ q
(3) p ~ q
(4) q p
If (27)999 is divided by 7, then the remainder is :
(1) 6
(2) 1
(3) 4
(4) 3
11.
12.
The tangent at the point(2, –2) to the curve, x2y2 –2x = 4(1 – y) does not pass through the point :
(1) (–2, –7)
13.
1
(3) 4,
3
(4) (8, 5)
If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then
the distance of P from the origin, is :
(1) 2 2 1
14.
(2) (–4, –9)
(2) 3 2 2
(3) 2 3 2 2
(4)
2 1
The integral 1 2 cot x cos ecx cot x dx 0 x is equal to
2
(where c is a constant of integration)
15.
x
(1) 2log sin C
2
x
(2) 2log cos C
2
x
(3) 4log cos C
2
x
(4) 4log sin C
2
The coordinates of the foot of the perpendicular from the point (1, –2, 1) on the plane containing the
x 1 y 1 z 3
x 1 y 2 z 3
and
is :
6
7
8
3
5
7
(1) (0, 0, 0)
(2) (2, –4, 2)
(3) (–1, 2, –1)
lines,
(4) (1, 1, 1)
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16.
Consider an ellipse, whose centre is at the origin and its major axis is along the x-axis. If its eccentricity
is 3/5 and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the
ellipse, with the vertices as the vertices of the ellipse, is :
(1) 8
(2) 80
(3) 32
(4) 40
17.
If all the words, with or without meaning, are written using the letters of the word QUEEN and are
arranged as in English dictionary, then the position of the word QUEEN is :
(1) 47th
(2) 44th
(3) 45th
(4) 46th
18.
Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are
3 1
5
, and , respectively, then the probability that the target is hit by P or Q but not by R is
4 2
8
(1)
19.
(2)
21
64
(3)
15
64
(4)
9
64
An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail
is :
(1)
20.
39
64
63
64
(2)
255
256
(3)
127
128
(4)
1
2
If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then
ab
ab
is equal to :
(1)
21.
6
2
(2)
5 6
12
(4)
7 3
12
1 1
(2) ,
4 2
1 1
(3) ,
3 3
1 1
(4) ,
3 3
The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is :
(1)
23.
(3)
The curve satisfying the differential equation, ydx –(x + 3y2)dy = 0 and passing through the point (1, 1)
also passes through the point :
1 1
(1) ,
4 2
22.
3 2
4
1
2 3
3
(2)
1
2 3
2
3
(3)
1
3
2
3
(4)
1
3
4
3
If a point P has coordinates (0, –2) and Q is any point on the circle, x2 + y2 – 5x – y + 5 = 0, then the
maximum value of (PQ)2 is :
(1)
25 6
2
(2) 8 5 3
(3) 14 5 3
(4)
47 10 6
2
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24.
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1 x 2 1 x2
1
The value of tan 1
, | x | , x 0, is equal to :
2
2
2
1 x 1 x
(1)
1
1
cos 1 x 2 (2) cos 1 x 2 (3) cos 1 x 2
4 2
4 2
4
(4)
cos 1 x 2
4
25.
The number of real values of , for which the system of linear equations :
2x + 4y – z = 0
4x + y + 2z = 0
x + 2y + 2z = 0
has infinitely many solutions, is
(1) 0
(2) 1
(3) 2
(4) 3
26.
the mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new
teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the
age (in years) of the newly appointed teacher is :
(1) 25
(2) 35
(3) 30
(4) 40
27.
lim
x 3
(1)
28.
3x 3
2x 4 2
is equal to :
3
(2)
3
2
(3)
2 2
(4)
1
2
Let z C, the set of complex numbers, then the equation 2|z + 3i| – |z – i| = 0 represents
(1) a circle with radius
8
3
(2) a circle with diameter
(3) an ellipse with length of major axis
29.
1
16
3
10
3
(4) an ellipse with length of minor axis
16
9
The locus of the point of intersection of the straight lines, tx – 2y – 3t = 0; x – 2ty + 3 = 0 (t R), is
(1) a hyperbola with the length of conjugate axis 3.
(2) an ellipse with eccentricity
2
5
(3) an ellipse with the length of major axis 6.
(4) a hyperbola with eccentricity 5
30.
Let A be any 3 × 3 invertible matrix. Then, which one of the following is not always true ?
(1) adj(adj(A)) = |A|.(adj(A))–1
(2) adj(adj(A)) = |A|2.(adj(A))–1
(3) adj(A) = |A|.A–1
(4) adj(adj(A)) = |A|.A
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN - 2018 (ONLINE PAPER)
1.
Two sets A and B are as under :
A = {(a, b) R × R : |a – 5| < 1 and |b – 5| < 1};
B = {(a, b) R × R : 4 (a – 6)2 + 9(b – 5)2 36}.
Then :
(1) B A
(2) A B
(3) A B = (an empty set)
(4) neither A B nor B A
2.
Let S = {x R: x 0 and 2 x 3 x x 6 6 0 }. Then S :
(1) is an empty set.
(2) contains exactly one element.
(3) contains exactly two element.
(4) contains exactly four elements
3.
If c are the distinct roots, of the equation x2 – x + 1 = 0 then 101 + 107 is equal to :
(1) – 1
(2) 0
(3) 1
(4) 2
x4
4.
If 2x
2x
2x
2x
x 4 2x = (A + Bx)(x – A)2,
2x x 4
then the ordered pair (A, B) is equal to :
(1) (–4, –5)
(2) (–4, 3)
5.
(3) (–4, 5)
(4) (4, 5)
If the system of linear equations
x + ky + 3z = 0
3x + ky – 2z = 0
2x + 4y – 3z = 0
xz
has a non - zero solution (x, y, z), then
(1) – 10
(2) 10
y2
is equal to :
(3) – 30
(4) 30
6.
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and
arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements
is :
(1) at least 1000
(2) less than 500
(3) at least 500 but less than 750
(4) at least 750 but less than 1000
7.
The sum of the co-efficients of all odd degree terms in the expansion of
x
5
5
x 3 1 x x 3 1 , (x > 1) is :
(1) – 1
(2) 0
(3) 1
(4) 2
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12
8.
Let a 1 , a 2, a 3 ,......,a 49 be in A. P. such t hat
a 4k 1 416
k 0
and a 9 + a 43 = 66. If
2
a12 a 22 .... a17
140m, then m is equal to :
(1) 66
9.
(2) 68
(3) 34
(4) 33
Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
12 + 2. 22 + 32 + 2.42 + 52 + 2.62 + .... If B-2A=100 , then is equal to:
(1) 232
(2) 248
(3) 464
(4) 496
10.10. For each t R, let [t] be the greatest int eger less than or equal to t . Than
1 2
15
lim x ......
x 0 x x
x
(1) is equal to 0
(2) is equal to 50
(3) is equal to 120
(4) does not exist (in R)
11.
Let S = {t R : f(x) = |x – |· (e|x| – 1) sin |x| is not differentiable at t}. Then the set S is equal to :
(1) (an empty set) (2) {0}
(3) {}
(4) {0, }
12.
If the curve y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :
(1) 6
(2) 7/2
(3) 4
(4) 9/2
13.
2
Let f(x) = x
1
x2
of value of h(x) is :
(1) 3
14.
f x
1
, x R – {–1, 0, 1}. If h(x) = , then the local minimum value
g x
x
(2) –3
The integral
(3) 2 2
sin 2 x cos 2 x
sin 5 x cos3 x sin 2 x sin 3 x cos 2 x cos5 x 2
1
(1)
and g(x) x
3 1 tan x
3
c (2)
1
3 1 tan x
3
c (3)
(4)
2
2
dx is equal to :
1
3
1 cot x
c
(4)
1
1 cot 3 x
c
2
15.
sin 2 x
dx is :
the value of
1 2x
2
(1)
8
(2)
2
(3) 4
(4)
4
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Let g(x) = cosx2, f(x) = x , and ) be the roots of the quadratic equation 18 x2 – 9x + 2 = 0,
Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = , x = and y = 0, is :
(1)
17.
MCSIR
1
3 1
2
(2)
1
3 1
2
(3)
1
3 2
2
(4)
1
2 1
2
Let y = y(x) be the solution of the differential equation
sin x
(1)
dy
+ y cos x = 4x, x (0, ). If
dx
4
9 3
2
(2)
8
9 3
2
y 0 , then
2
y is equal to :
6
8 2
(3)
9
4 2
(4)
9
18.
A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O
is the origin and the rectangle OPRQ is completed, then the locus of R is :
(1) 3x + 2y = 6
(2) 2x + 3y = xy
(3) 3x + 2y = xy
(4) 3x + 2y = 6xy
19.
Let the orthocenter and centroid of a triangle be A(–3, 5) and B(3, 3) respectively. If C is the circumcentre
of this triangle, than the radius of the circle having line segment AC as diameter, is :
(1) 10
(2) 2 10
(3)
3
5
2
(4)
3 5
2
20.
If the tangent at (1, 7) to the curve x2 = y – 6 touches the circle x2 + y2 + 16x + 12y + c = 0 than the value
of c is :
(1) 195
(2) 185
(3) 85
(4) 95
21.
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and
CPB = , then a value of tan is :
(1) 1/2
(2) 2
(3) 3
(4) 4/3
22.
Tangents are drawn to the hyperbola 4x2 – y2 = 36 at the points P and Q. If these tangents intersects at
the point T(0, 3) then the area (in sq. units) of PTQ is :
(1) 45 5
23.
(2) 54 3
(3) 60 3
(4) 36 5
If L1 is the line of intersection of the planes 2x – 2y + 3z = 0, x – y + z = 0 and L2 is the line of intersection
of the planes x + 2y – z – 3 = 0, 3x – y + 2z = 0 then the distance of the origin from the plane, containing
the lane L1 and L2, is :
(1)
1
4 2
(2)
1
3 2
(3)
1
2 2
(4)
1
2
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The length of the projection of the line segment joining the points (5, –1, 4) and
(4, –1, 3) on the plane x + y + z = 7 is :
2
(1)
25.
MCSIR
3
(2) 2/3
(3) 1/3
(4)
2
3
Let u be a vector coplanar with the vectors a 2iˆ 3j k and b j k . If u is perpendicular to a
and u .b 24 , than equal to :
(1) 336
(2) 315
(3) 256
(4) 84
26.
A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed
and this ball along with two additional balls of the same colour is returned to the bag. If now a ball is
drawn at radom from the bag, then probability that this drawn ball is red, is :
(1) 3/10
(2) 2/5
(3) 1/5
(4) 3/4
27.
If x i 5 9 and x i 5
9
i1
(1) 9
28.
45 , then the standard deviation of the 9 items x , x ........x is :
1
2
9
(2) 4
(3) 2
(4) 3
1
(2) 13/9
(3) 8/9
(4) 20/9
PQR is a triangular park with PQ = PR = 200 m. AT. V. tower stands at the mid-point of QR. If the angle
of elevation of the top of the tower at P, Q and R are respectively 45°, 30° and 30° then the height of
tower (in m) is :
(1) 100
30.
i1
2
If sum of all the solution of equation 8cos x. cos x .cos x 1 in [0, ] is k, then k is
6
6
2
equal to :
(1) 2/3
29.
9
(2) 50
(3) 100 3
The Boolean expression ~(p q) (~p q) is equivalent to :
(1) ~ p
(2) p
(3) q
(4) 50 2
(4) ~ q
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (9/01/2019) SHIFT-1
1.
3
The value of | cos x | dx is :
0
(A) 0
2.
(B)
(C)
2
3
(D)
4
3
The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :
(A) 6
3.
4
3
(B) 3 3
(C)
4
3
For x n + 1, n N (the set of natural numbers), the integral x.
2
(D) 2 3
2sin(x 2 1) sin 2(x 2 1)
2sin(x 2 1) sin 2(x 2 1)
dx is
equal to (where c is a constant of integration):
(A) log e
1 2 2
sec (x 1) c
2
2
1
2 x 1
c
(C) log e sec
2
2
4.
(B)
1
log e sec(x 2 1) c
2
x2 1
c
(D) log e sec
2
If y = y(x) is the solution of the differential equation x
dy
1
2y x 2 satisfying y(1) = 1, then y is
2
dx
equal to :
(A)
5.
7
64
1
4
(C)
49
16
(D)
13
16
Axis of a parabola lies along x – axis. If its vertex and focus are at distance 2 and 4 respectively from the
origin, on the positive x – axis then which of the following points does not lie on it ?
(A) 5, 2 6
6.
(B)
(B) (8, 6)
(C) 6, 4 2
(D) (4, –4)
x2
y2
1 is greater than 2, then the length
Let 0 . If the eccentricity of the hyperbola
2
cos 2 sin 2
of its latus rectum lies in the interval :
(A) (3, )
3
(B) , 2
2
(C) (2, 3]
3
(D) 1,
2
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7.
For x R – [0, 1], let f1 (x)
MCSIR
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1
1
, f 2 (x) 1 x and f3 (x)
be three given functions. If a function,
x
1 x
J(x) satisfies (f 2 oJof1 ) (x) = f3(x) then J(x) is equal to :
(A) f3(x)
8.
10.
19
2
13.
14.
(D) f1(x)
(B) 9
(C) 8
(D)
17
2
3
2
3
If cos 1 cos 1 ; x then x is equal to :
4x 2
4
3x
145
12
(B)
145
10
(C)
146
12
(D)
145
11
Equation of a common tangent to the circle , x2 + y2 – 6x = 0 and the parabola. y2 = 4x is :
(A) 2 3y 12x 1
12.
(C) f2(x)
If a, b and c be three distinct real numbers in G.P. and a + b + c = xb, then x cannot be:
(A) 4
(B) –3
(C) –2
(D) 2
(A)
11.
1
f3 (x)
x
2
Let a ˆi ˆj, b ˆi ˆj kˆ and c be a vector such that a c b 0 and a c 4 , then c is equal
to:
(A)
9.
(B)
(B) 3y x 3
(C) 2 3y x 12
(D)
3y 3x 1
The system of linear equations x + y + z = 2, 2x + 3y + 2z = 5, 2x + 3y + (a2 – 1)z = a + 1
(A) is inconsistent when a = 4
(B) has a unique solution for | a | 3
(C) has infinitely many solutions for a = 4
(D) is inconsistent when | a | 3
k
2403
If the fractional part of the number
is , then k is equal to :
15
15
(A) 6
(B) 8
(C) 4
(D) 14
The equation of line passing through (–4, 3, 1) parallel to the plane x + 2y – z – 5 = 0 and intersecting the
line
x 1 y 3 z 2
is :
3
2
1
(A)
x 4 y 3 z 1
2
1
4
(B)
x 4 y 3 z 1
1
1
3
(C)
x 4 y 3 z 1
3
1
1
(D)
x 4 y 3 z 1
1
1
1
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Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following
statement is true ?
3 1
(A) The lines are concurrent at the point , (B) Each line passes through the origin
4 2
(C) The lines are all parallel
(D) The lines are not concurrent
16.
1 1 y4 2
lim
y4
y 0
(A) exists and equals
(C) exists and equals
1
1
(B) exists and equals
4 2
1
2 2 2 1
(D) does not exist
2 2
17.
The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to
y – axis also passes through the point :
(A) (–3, 0, – 1)
(B) (–3, 1, 1)
(C) (3, 3, – 1)
(D) (3, 2, 1)
18.
If denotes the acute angle between the curves, y = 10 x2 and y = 2 + x2 at a point of their intersection,
then tan is equal to
(A)
19.
(B)
8
15
(C)
7
17
(D)
8
17
cos sin
If A sin cos , then the matrix A50 when , is equal to
12
(A)
20.
4
9
1
3
2
2
3 1
2 2
(B)
3 1
2
2
1
3
2 2
(C)
3
2
1
2
1
2
3
2
(D)
1
2
3
2
3
2
1
2
If the Boolean expression (p q) (~pq) is equivalent to p q, where , , then the
ordered pair ( , ) is:
(A)
(B)
(C)
(D)
21.
5 students of a class have an average height 150 cm and variance 18 cm2. A new student, whose height
is 156 cm, joined them. The variance (in cm2) of the height of these six students is :
(A) 16
(B) 22
(C) 20
(D) 18
22.
For any 4 , 2 , the expression 3(sin cos )4 6 sin cos 2 4sin 6 equals:
(A) 13 4 cos 2 6 sin 2 cos 2
(B) 13 4 cos6
(C) 13 4 cos 2 6 cos4
(D) 13 4 cos 4 2 sin 2 cos2
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MCSIR
JEE Main Papers
23.
The area (in sq. units) bounded by the parabola y = x2 – 1, the tangent at the point (2, 3) to it and the
y-axis is :
(A) 8/3
(B) 32/3
(C) 53/3
(D) 14/3
24.
Let a1, a2 ,......, a30 be an A.P., S a i and T a (2i 1). If a5 = 27 and S –2T = 75, then a10 is equal
30
i 1
to:
(A) 52
25.
(B) 57
Let f : R R be a function defined as f x
(A) continuous if a = 5 and b = 5
(C) continuous if a = 0 and b = 5
26.
15
i 1
(C) 47
5,
a bx,
b 5x,
30,
(D) 42
if
x 1
if 1 x 3
. Then f is :
if 3 x 5
if
x 5
(B) continuous if a = –5 and b = 10
(D) not continuous for any values of a and b
3 2isin
Let A 0 , :
is purely imaginary . Then the sum of the elements in A is
2 1 2i sin
(A)
6
(B)
(C)
3
4
(D)
2
3
27.
Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that
can be formed from this class, if there are two specific boys A and B, who refuse to be the members of
the same team, is :
(1) 500
(2) 200
(3) 300
(4) 350
28.
Let and be two roots of the equation x 2 2x 2 0 then 15 15 is equal to
(A) 256
(B) 512
(C) 512
(D) 256
29.
Three circles of radii a, b, c(a < b < c) touch each other externally. If they have x-axis as a common
tangent, then :(A)
1
a
1
b
1
(B)
c
(C) a, b, c are in A.P.
30.
(D)
1
b
1
a
1
c
a , b, c are in A.P..
Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. Let X denote
the random variable of number of aces obtained in the two drawn cards. Then P( X = 1) + P ( X = 2)
equals :
(A)
49
169
(B)
52
169
(C)
24
169
(D)
25
169
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (9/01/2019) SHIFT-2
1.
3
Let f be a differentiable function from R to R such that f x f y 2 x y 2 , for all x, y R.
1
2
If f(0) = 1 then f x dx is equal to
0
(1) 0
3
2.
If
0
(2) 1/2
tan
2k sec
d 1
(1) 2
1
2
(3) 2
(4) 1
, (k > 0) then the value of k is :
(2) 1/2
(3) 4
(4) 1
3
3.
1 t6
4
The coefficient of t in the expansion of
is
1 t
(1) 12
4.
(2) 15
6.
(4) 14
For each x R, let [x] be the greatest integer less than or equal to x. Then lim
x 0
equal to
(1) – sin1
5.
(3) 10
(2) 0
(3) 1
x x x sin x
is
x
(4) sin1
If both the roots of the quadratic equation x2 – mx + 4 = 0 are real and distinct and they lie in the interval
[1, 5], then m lies in the interval :
(1) (4, 5)
(2) (3, 4)
(3) (5, 6)
(4) (–5, –4)
e t
t
If A = e
e t
e t cos t
e t cos t e t sin t
2e t sin t
e t sin t
t
t
e sin t e cos t
2e t cost
Then A is :
2
(3) invertible for all t R
(1) Invertible only if t =
7.
(2) not invertible for any t R
(4) invertible only if t =
The area of the region
A = {(x, y) : 0 y x |x| + 1 and –1 x 1}
in sq. units, is :(1) 2/3
(2) 1/3
(3) 2
(4) 4/3
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8.
4
(2)
(2) 4
22
(3) 0
(4)
6
(3)
32
(4) 6
Let A(4, –4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the
parabola, where O is the origin, such that the area of ACB is maximum. Then the area (in sq. units) of
ACB, is :(1) 31
11.
3
ˆ b b ˆi b ˆj 2 kˆ and c 5iˆ ˆj 2 kˆ be three vectors such that the
Let a ˆi ˆj 2 k,
1
2
projection vector of b on a is a . If a b is perpendicular to c , then | b | is equal to :(1)
10.
JEE Main Papers
93
Let z0 be a root of the quadratic equation, x2 + x + 1 = 0. If z = 3 + 6iz81
0 3iz0 , then arg z is equal to
(1)
9.
MCSIR
3
4
(2) 32
(3) 30
1
2
(4) 31
1
4
The logical statement ~ ~ p q p r ~ q r is equivalent to :(1) p r ~ q
(2) ~ p ~ q r
(3) ~ p r
(4) p ~ q r
12.
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is
green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn;
the original ball is not returned to the urn. Now a second ball is drawn at random from it. The probability
that the second ball is red, is :
(1) 26/49
(2) 32/49
(3) 27/49
(4) 21/49
13.
If 0 x <
(1) 2
14.
, then the number of values of x for which sin x – sin 2x + sin 3x = 0, is :
2
(2) 1
(3) 3
(4) 4
The equation of the plane containing the straight line
x y z
and perpendicular to the plane containing
2 3 4
x y z
x y z
and is :3 4 2
4 2 3
(1) x + 2y – 2z = 0
(2) x – 2y + z = 0
(3) 5x + 2y – 4z = 0
the straight lines
15.
(4) 3x + 2y – 3z = 0
Let the equations of two sides of a triangle be 3x – 2y + 6 = 0 and 4x + 5y – 20 = 0. If the orthocentre
of this triangle is at (1, 1) then the equation of its third side is :(1) 122 y – 26 x – 1675 = 0
(3) 122 y + 26 x + 1675 = 0
(2) 26 x + 61 y + 1675 = 0
(4) 26 x – 122 y – 1675 = 0
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16.
If x = 3 tan t and y = 3 sec t, then the value of
(1)
17.
3
(2)
2 2
1
3 2
MCSIR
d2 y
dx 2
(3)
at t
JEE Main Papers
, is :4
1
6
(4)
If x = sin–1(sin10) and y = cos–1(cos10), then y – x is equal to :(1)
(2) 7
(3) 0
1
6 2
(4) 10
18.
If the lines x = ay + b, z = cy + d and x = a'z + b', y = c'z + d' are perpendicular, then :
(1) cc' + a + a' = 0
(2) aa' + c + c' = 0
(3) ab' + bc' + 1 = 0 (4) bb' + cc' + 1 = 0
19.
The number of all possible positive integral values of for which the roots of the quadratic equation,
6x2 – 11x + = 0 are rational numbers is :(1) 2
(2) 5
(3) 3
(4) 4
20.
A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length
4 along the x-axis. Then the eccentricity of the hyperbola is :
(1)
21.
2
(2)
3
3
2
3
(4) 2
Let A = {xR:x is not a positive integer}. Define a function f: A R as f(x) =
(1) injective but not surjective
(3) surjective but not injective
22.
(3)
If f(x) =
(1)
1
2
5x 8 7x 6
x
2
1 2x
7
2
(2)
2x
then f is:x 1
(2) not injective
(4) neither injective nor surjective
dx, x 0 and f(0) = 0, then the value of f(1) is :-
1
2
(3)
1
4
(4)
1
4
23.
If the circles x2 + y2 – 16x – 20 y + 164 = r2 and (x – 4)2 + (y – 7)2 = 36 intersect at two distinct points,
then :
(1) 0 < r < 1
(2) 1 < r < 11
(3) r > 11
(4) r = 11
24.
Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two
vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then
the number of elements in the set S is :
(1) 9
(2) 18
(3) 32
(4) 36
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25.
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JEE Main Papers
The sum of the following series
9 12 22 32 12 12 22 32 42 15 12 22 .... 52
....... up to 15 terms, is :
7
9
11
(1) 7820
(2) 7830
(3) 7520
(4) 7510
1 6
26.
Let a, b and c be the 7th, 11th and 13th terms respectively of a non-constant A. P. If these are also the
three consecutive terms of a G.P., then
(1) 1/2
27.
28.
(2) 4
a
is equal to :c
(3) 2
If the system of linear equations
x – 4y + 7z = g
3y – 5z = h
–2x + 5y – 9z = k is consistent, then :(1) g + h + k = 0
(2) 2g + h + k = 0
(4) 7/13
(3) g + h + 2k = 0
(4) g + 2h + k = 0
Let ƒ : [0, 1] R be such that ƒ(xy) = ƒ(x). ƒ(y) for all x, y, [0, 1], and ƒ(0) 0. If y = y(x) satisfies
the differential equation,
(1) 4
dy
ƒ x with y(0) = 1, then
dx
(2) 3
1
3
y y is equal to :
4
4
(3) 5
(4) 2
n
29.
A data consists of n observations : x1, x2, ......, xn. If
2
x i 1
n
9n and
i1
xi 1
2
5n , then
i1
the standard deviation of this data is :
(1) 5
30.
(2)
5
(3)
7
(4) 2
The number of natural numbers less than 7,000 which can be formed by using the digits 0,1,3,7,9
(repitition of digits allowed) is equal to :
(1) 250
(2) 374
(3) 372
(4) 375
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (10/01/2019) SHIFT-1
1.
An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of
the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well
shuffled pack of nine cards numbered 1,2,3,......., 9 is randomly picked and the number on the card is
noted. The probability that the noted number is either 7 or 8 is:
(A)
2.
13
36
15
72
(C)
19
72
(D)
19
36
3
The shortest distance between the point , 0 and the curve y x , x 0 , is:
2
5
2
(A)
3.
(B)
(B)
3
2
(C)
3
2
(D)
5
4
The plane passing t hrough t he point (4, – 1, 2) and parallel to the lines
x 2 y 2 z 1
x 2 y 3 z 4
and
also passes through the point:
3
1
2
1
2
3
(A) (1, 1, – 1)
(B) (1, 1, 1)
(C) (–1, –1, –1)
(D) (–1, –1, 1)
4.
The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are
1,3 and 8, then a ratio of other two observations is:
(A) 10 : 3
(B) 4 : 9
(C) 5 : 8
(D) 6 : 7
5.
If 5, 5r, 5r2 are the lengths of sides of a triangle, then r can not be equal to:
(A)
3
2
7.
(C)
5
4
(D)
7
4
3
(C) 200
(D) 100
3
2
4
The sum of all values of 0, satisfying sin 2 cos 2 is:
4
2
(A)
8.
3
4
20
Ci 1
k
If 20
, then k equals
20
21
Ci Ci 1
i 1
(A) 400
(B) 50
20
6.
(B)
2
(B)
(C)
3
8
(D)
4
Consider the quadratic equation (c–5)x2 – 2 cx + (c – 4) = 0, c 5. Let S be the set of all integral values
of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval
(2, 3). Then the number of elements in S is:
(A) 11
(B) 18
(C) 10
(D) 12
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9.
If
MCSIR
JEE Main Papers
dy
3
1
4
y
,
x
,
y
y
and
,
then
equals:
dx cos 2 x
cos 2 x
3 3
4 3
4
(A)
1 6
e
3
(B)
1
3
(C)
4
3
(D)
1 3
e
3
10.
In a class of 140 students numbered 1 to 140, all even numbered students opted mathematics course,
those whose number is divisible by 3 opted physics course and those whose number is divisible by 5
opted chemistry course. Then the number of students who did not opt for any of the three courses is:
(A) 102
(B) 42
(C) 1
(D) 38
11.
If the third term in the binomial expansion of 1 x log 2 x
(A)
12.
1
4
14.
(C)
5
equals 2560, then a possible value of x is:
1
8
(D) 2 2
If the parabolas y2 = 4b (x – c) and y2 = 8ax have a common normal, then which one of the following is
a valid choice for the ordered triad (a, b, c)?
1
(A) , 2,3
2
13.
(B) 4 2
(B) (1, 1,3)
1
(C) , 2, 0
2
If the system of equations
x+y + z=5
x + 2y + 3z = 9
x + 3y +z =
has infinitely many solutions, then equals:
(A) 21
(B) 8
(C) 18
(D) (1, 1, 0)
(D) 5
For each t R , let [t] be the greatest integer less than or equal to t.
(1 | x | sin |1 x |)sin [1 x]
2
Then, lim
x 1
|1 x |[1 x]
(A) equals 1
15.
(B) equlas 0
(C) equals –1
(D) does not exist
Let d R and
4d
(sin ) 2
2
, [0, 2]
A1
(sin ) 2
d
5 (2sin ) d ( sin ) 2 2d
. If the minimum value of det (A) is 8, then a value of d is:
(A) –7
(B) 2 2 2
(C) –5
(D) 2 2 1
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16.
18.
(B) | z |
5
2
(C) | z |
1 17
2 2
(D) Im(z) = 0
b
x 4 2x 2 dx . If I is minimum then the ordered pair (a, b) is:
(A) 0, 2
(B) 2, 0
(C) 2, 2
(D)
Let I
a
2, 2
A point P moves on the line 2x – 3y + 4 = 0. If Q (1, 4) and R (3,– 2) are fixed points, then the locus of
the centroid of PQR is line:
(A) with slope
19.
JEE Main Papers
3z1 2z 2
Let z1 and z2 be any two non - zero complex numbers such that 3|z1| = 4 |z2|. If z 2z 3z , then:
2
1
(A) Re(z) = 0
17.
MCSIR
3
2
(B) parallel to x axis
(C) with slope
2
3
(D) parallel to y - axis
max | x |, x 2 ,
| x | 2
f
(x)
Let
. Let S be the set of points in the interval (–4, 4) at which f is
8 2 | x |,
2 | x | 4
not differentiable. Then S:
(A) is an empty set
(B) equals {–2, –1, 0, 1, 2}
(C) equals {–2, –1, 1, 2}
(D) equals {–2, 2}
20.
If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the
point (1, –1), then the radius of C is
(A) 57
(B) 4
(C) 2 5
(D) 5
21.
Let f :R R be a function such that f (x) x 3 x 2f '(1) xf "(2) f '"(3), x R. Then f(2) equals:
(A) –4
(B) 30
(C) –2
(D) 8
22.
ˆ b 4iˆ (3 )ˆj 6kˆ and c 3iˆ 6ˆj ( 1)kˆ be three vectors such that
Let a 2iˆ 1ˆj 3k,
2
3
b 2a and a is perpendicular to c . Then a possible value of () is :
(A) (1, 3, 1)
23.
1
(B) , 4, 0
2
1
(C) , 4, 2
2
(D) (1, 5, 1)
Let A be a point on the line r 1 3 iˆ 1 ˆj 2 5 kˆ and B(3, 2, 6) be a point in the space.
Then the value of for which the vector AB is parallel to the plane x – 4y + 3z = 1 is:
(A)
1
4
(B)
1
8
(C)
1
2
(D)
1
4
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24.
MCSIR
Let n 2 be a natural number and 0 < < Then
sin
JEE Main Papers
n
sin
sin
n 1
1
n
cos
d is equal to :
(where C is a constant of integration)
(A)
(C)
25.
n 1
n
n
1
1 n 1
n 1 sin
2
n
1
n 1
n
1
n 1 sin n 1
2
C
C
(B)
(D)
n 1
n
n
1
1 n 1
n 1 sin
2
n
1
n 1
n
1
n 1 sin n 1
2
C
C
Consider a triangular plot ABC with sides AB = 7m, BC = 5m and CA = 6m. A vertical lamp - post at
the mid point D of AC subtends an angle 30° at B. The height (in m) of the lamp - post is:
(A) 7 3
(B)
2
21
3
(C)
3
21
2
(D) 2 21
26.
The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is:
(A) x – y + 1 = 0
(B) x – y + 7 = 0
(C) x – y + 9 = 0
(D) x – y – 3 = 0
27.
If the line 3x + 4y – 24 = 0 intersects the x - axis at the point A and the y-axis at the point B, then the
incentre of the triangle OAB, where O is the origin, is
(A) (3, 4)
(B) (2, 2)
(C) (4, 4)
(D) (4, 3)
28.
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:
(A) 1256
(B) 1465
(C) 1365
(D) 1356
29.
If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is:
(A)
30.
3
2
(B)
1
3
(C)
3
(D)
2
3
Consider the statement: "P(n): n2 - n + 41 is prime." Then which one of the following is true?
(A) P(5) is false but P (3) is true
(B) Both P(3) and P (5) are false
(C) P(3) is false but P(5) is true
(D) Both P(3) and P (5) are true
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (10/01/2019) SHIFT-2
1.
The value of such that sum of the squares of the roots of the quadratic equation,
x2 + (3 – ) x + 2 = has the least value is:
15
4
(A)
(B) 1
(C)
(D) 2
8
9
2.
The value of cos
(A)
3.
4.
2
2
1
256
.cos
3
2
. ........ .cos
(B)
10
2
1
2
.sin
210
is:
(C)
1
512
(D)
1
1024
The curve amongst the family of curves represented by the differential equation, (x2 – y2) dx + 2xy dy = 0
which passes through (1, 1), is:
(A) a circle with centre on the x - axis
(B) an ellipse with major axis along the y - axis
(C) a circle with centre on the y - axis
(D) a hyperbola with transverse axis along the x - axis
2
Let f : (–1, 1) R be a function defined by f (x) = max | x |, 1 x . If K be the set of all points
at which f is not differentiable, then K has exactly:
(A) five elements
(B) one element
(C) three elements
(D) two elements
10
5.
The positive value of for which the co - efficient of x in the expression x x 2 is 720, is:
x
2
(A) 4
6.
(C)
5
(D) 3
2
The tangent to the curve, y xe x passing, through the point (1, e) also passes through the point :
(A) (2, 3e)
7.
(B) 2 2
2
4
(B) , 2e
3
5
(C) , 2e
3
(D) (3, 6e)
Let N be the set of natural numbers and two functions f and g be defined as f, g : N N such that
n 1
if n is odd
2
f n
and g (n) = n – (– 1)n. Then fog is:
n
if n is even
2
(A) onto but not one - one
(B) one - one but not onto
(C) both one - one and onto
(D) neither one one nor onto
8.
The number of values of 0, for which the system of linear equations
x + 3y + 7z = 0
– x + 4y + 7z = 0
(sin 3)x + (cos 2) y + 2z = 0 has a non - trivial solution, is:
(A) three
(B) two
(C) four
(D) one
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9.
(B) –3
1
24
25
(B)
18
25
5
(C)
(D)
6
25
5
(B) R (z) > O and I (z) >0
(D) R (z) = – 3
1
If the probability of hitting a target by a shooter, in any shot, is , then the minimum number of independent
3
shots at the target required by him so that the probability of hitting the target at least once is greater than
5
, is:
6
(A) 3
(B) 6
3
14.
4
5
3 i 3 i
. If R (z) and I(z) respectively denote the real and imaginary parts of z,
Let z
2 2 2 2
then:
(A) I (z) = 0
(C) R (z) < 0 an I (z) > 0
13.
(D) 3
2
2
If f (t)dt x t f (t)dt, then f ' 1 2 is:
0
x
(A)
12.
(C) 4
Two sides of a parallelogram are along the lines, x + y = 3 and x – y + 3 = 0. If its diagonals intersect at
(2,4), then one of its vertex is:
(A) (2, 6)
(B) (2, 1)
(C) (3, 5)
(D) (3, 6)
x
11.
JEE Main Papers
Let 2 a b and 4 2 a 3b be two given vectors where a and b are non collinear..
The value of for which vectors and are collinear, is:
(A) – 4
10.
MCSIR
5 4x
If x e dx
(A) – 2x3 – 1
(C) 5
(D) 4
1 4x3
e f (x) C , where C is a constant of integration, then f (x) is equal to :
48
(B) – 4x3 – 1
(C) – 2x3 + 1
(D) 4x3 + 1
15.
If the area of an equilateral triangle inscribed in the circle, x2 + y2 + 10x + 12y + c = 0 is 27 3 sq. units
then c is equal to:
(A) 20
(B) 25
(C) 13
(D) –25
16.
Consider the following three statements:
P : 5 is a prime number.
Q : 7 is a factor of 192
R : L.C. M. of 5 and 7 is 35.
Then the truth value of which one of the following statements is true?
(A) ~ P Q R
(B) P Q ~ R
(C) ~P ~ Q R (D) P (~ Q R)
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17.
(C) 8 2
(B) 2 11
(B) 2 3
(C) 3
(D) 3
y2 x2
Let S x, y R2 :
1 , where r 1 . Then S represents:
1 r 1 r
(A) a hyperbola whose eccentricity is
(B) an ellipse whose eccentricity is
If
50
C r .50 r C 25r K
50
2
, when 0 < r < 1
r 1
1
, when r > 1.
r 1
(D) an ellipse whose ecentricity is
25
2
, when 0 < r < 1.
1 r
2
, when r > 1.
r 1
(C) a hyperbola whose eccentricity is
20.
(D) 6 3
b
1
2
det A
2
Let A b b 1 b where b > 0. Then the minimum value of
is:
b
1
b
2
(A) 2 3
19.
JEE Main Papers
The length of the chord of the parabola x2= 4y having equation x 2y 4 2 0 is:
(A) 3 2
18.
MCSIR
C25 , then K is equal to:
r 0
(A) (25)2
21.
22.
(C) 224
(D) 225
The plane which bisects the line segment joining the points (–3, – 3, 4) and (3, 7, 6) at right angles,
passes through which one of the following points?
(A) (– 2, 3, 5)
(B) (4, – 1, 7)
(C) (2, 1,3)
(D) (4, 1, – 2)
n
19
1
cot
1
2p
The value of cot
p 1 is :
n 1
(A)
23.
(B) 225 – 1
21
19
(B)
19
21
(C)
22
23
(D)
23
22
If mean and standard deviation of 5 observations x1, x2, x3, x4, x5 are 10 and 3, respectively, then the
variance of 6 observations x1, x2....., x5 and – 50 is equal to:
(A) 509.5
(B) 586.5
(C) 582.5
(D) 507.5
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24.
1
3 f (x)
, x 0 and f (1) 4 . Then lim xf :
x 0
4 x
x
(A) exists and equals
4
7
(B) exists and equals 4
(C) does not exist
The value of
(A)
27.
(D) exists and equals 0.
Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies
in which quadrant?
(A) Fourth
(B) Second
(C) Third
(D) First
26.
2
dx
, where [t] denotes the greatest integer less than or equal to t, is:
x sin x 4
2
1
7 5
12
(B)
1
7 5
12
(D)
3
4 3
10
x 4 y5 z 3
and the
2
2
1
x 4 y 5 z 5
1
1
1
x 2 y3 z 3
(D)
2
2
3
(B)
log e a1r a 2k
log e a 2r a 3k
log e a 3r a 4k
log e a 4r a 5k
log e a 5r a 6k
log e a 6r a 7k 0 . Then the number of elements in S, is :
log e a 7r a 8k
log e a 8r a 9k
k
log e a 9r a10
(B) 4
(C) 10
(D) 2
With the usual notation, in ABC , if A B 120, a 3 1 and b 3 1 then the ratio A : B , is:
(A) 7 : 1
30.
3
4 3
20
Let a1, a2, a3, ......., a10 be in G.P. with ai > 0 for i = 1, 2, ...., 10 and S be the set of pairs (r,k), r, k N
(the set of natural numbers) for which
(A) infinitely many
29.
(C)
On which of the following lines lies the point of intersection of the line,
plane , x + y + z = 2 ?
x 3 4 y z 1
(A)
3
3
2
x 1 y 3 z 4
(C)
1
2
5
28.
JEE Main Papers
Let f be differentiable function such that
f 'x 7
25.
MCSIR
(B) 5 :3
(C) 9 : 7
A helicopter is flying along the curve given by y x
3
2
(D) 3 : 1
7 , (x > 0). A soldier positioned at the point
1
, 7 wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:
2
(A)
5
6
(B)
1 7
3 3
(C)
1 7
6 3
(D)
1
2
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (11/01/2019) SHIFT-1
1.
If the system of linear equations
2x + 2y + 3z = a
3x – y + 5z = b
x – 3y + 2z = c
Where a, b, c are non zero real numbers, has more than one solution. then:
(A) b – c + a = 0
(B) b – c – a = 0
(C) a + b + c = 0
(D) b + c – a = 0
2.
Let fk (x) 1 (sin k x cosk x) for k = 1, 2, 3,... Then for all x R, the value of f 4 (x) f6 (x) is equal to :k
(A)
5
12
(B)
1
12
(C)
1
4
(D)
1
12
3.
A square is inscribed in the circle x2 + y2 – 6x + 8y – 103 = 0 with its sides parallel to the coordinate
axes. Then the distance of the vertex of this square which is nearest to the origin is :
(A) 13
(B) 137
(C) 6
(D) 41
4.
If q is false and p q r is true, then which one of the following statements is a tautology?
(A) p r p r
(C) p r
5.
(B) p r p r
(D) p r
The area (in sq. units ) of the region bounded by the curve x2 = 4y and the straight line x = 4y –2 is :
(A) 5/4
(B) 9/8
(C) 7/8
(D) 3/4
2
6.
sin 2 x
dx (where [x] denotes the greatest integer less than or equal to x)
The value of the integral
x 1
2
2
is :
(A) 0
7.
(B) sin 4
(C) 4
(D) 4 – sin 4
The outcome of each of 30 items was observed; 10 items gave an outcome
outcome
data is
(A)
2
3
1
d each, 10 items gave
2
1
1
each and the remaining 10 items gave outcome d each. If the variance of this outcome
2
2
4
then d equals:
3
(B) 2
(C)
5
2
(D)
2
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JEE Main Papers
x 3 y 2 z 1
and also containing its projection on the plane
2
1
3
2x + 3y – z = 5, contains which one of the following points ?
(A) (2, 2, 0)
(B) (–2, 2, 2)
(C) (0, –2, 2)
(D) (2, 0, –2)
8.
The plane containing the line
9.
Two integers are selected at random from the set {1, 2, ........,11}. Given that the sum of selected
numbers is even, the conditional probability that both the numbers are even is :
7
1
2
3
(A)
(B)
(C)
(D)
10
2
5
5
10.
Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point
(0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the
centres of these circles is :
(A) 1
(B) 2
(C) 2 2
(D) 2
11.
The value of r for which 20Cr 20C0 + 20Cr–1 20C1 + 20Cr–2 20C2 + ...... + 20C0 20Cr is maximum is :
(A) 15
(B) 20
(C) 11
(D) 10
12.
If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then
the mid points of the tangents intercepted between the coordinate axes lie on the curve:
1
(A)
13.
14.
4x
2
1
2y
2
1
(B)
If xloge(logex) – x2 + y2 = 4(y > 0), then
(1 2e)
2x
2
1
4y
2
1
(D)
x 2 y2
1
2
4
(2e 1)
(B)
2 4 e2
1 x2
dy
at x = e is equal to :
dx
dx A(x)
2 4 e2
1 x2
(1 2e)
(C)
4 e2
e
(D)
4 e2
m
C , for a suitable chosen integer m and a function A(x), where C
x4
is a constant of integration, then (A(x))m equals :
If
(A)
16.
1
(C)
1, 2 x 0
Let f (x) 2
and g(x) = |f(x)| + f(|x|). Then, in the interval (–2, 2), g is :
x 1, 0, x 2
(A) differentiable at all points
(B) not continuous
(C) not differentiable at two points
(D) not differentiable at one point
(A)
15.
x 2 y2
1
4
2
1
(B)
27x 9
1
3x 3
(C)
1
27x 6
(D)
1
9x 4
Let [x] denote the greatest integer less than or equal to x. Then :
lim
tan( sin 2 x) (| x | sin(x[x])) 2
x 0
(A) does not exist
x2
(B) equals
(C) equals + 1
(D) equals 0
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17.
The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = {x R : x2 + 30 11x}
is :
(A) –122
(B) –222
(C) 122
(D) 222
18.
Let f : R R be defined by f (x)
1 1
(A) ,
2 2
19.
x
1 x2
x R . Then the range of f is :
(B) R – [–1, 1]
1 1
(C) R ,
2 2
(D) (–1, 1) – {0}
The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is
27
. Then the common ratio of this series is :
19
(A)
20.
2
3
(C)
2
9
(D)
4
9
5
2
(B) 2 5
(C)
5
4
(D) 4 5
In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is
y. If x2 - c2 = y , where c is the length of the third side of the triangle, then the circumradius of the triangle
is:
(A)
22.
(B)
The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the
origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin
is:
(A)
21.
1
3
y
3
(B)
c
3
(C)
c
3
(D)
3
y
2
Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is :
(A) x + y + 1 = 0
(B) x – 2y + 4 = 0
(C) x + 2y + 4 = 0
(D) 4x + 2y + 1 = 0
8
23.
24.
25.
x3 3
The sum of the real values of x for which the middle term in the binomial expansion of equals
3 x
5670 is :
(A) 0
(B) 6
(C) 4
(D) 8
a3
a9
Let a1, a2, ..... a10 be a G.P. If a 25 , then a equals :
1
5
4
2
(A) 5
(B) 4(5 )
(C) 53
(D) 2(52)
0 2q r
Let A p q r . If AAT = I3, then |p| is :
p q r
(A)
1
5
(B)
1
3
(C)
1
2
(D)
1
6
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26.
MCSIR
If y (x) is the solution of the differential equation
JEE Main Papers
dy 2x 1
1 2
2x
y e , x > 0, where y(1) = e ,
dx x
2
then :
27.
log e 2
4
(A) y(loge 2) = loge 4
(B) y(log e 2)
1
(C) y(x) is decreasing in ,1
2
(D) y(x) is decreasing in (0, 1)
The direction ratios of normal to the plane through the points (0, –1, 0) and (0, 0, 1) and making an angle
with the plane y – z + 5 = 0 are :
4
(A) 2, –1, 1
28.
(B) 2, 2, 2
2,1, 1
(D) 2 3,1, 1
ˆ b ˆi ˆj 4kˆ and c 2iˆ 4jˆ (2 1)kˆ be coplanar vectors. Then the non Let a ˆi 2ˆj 4k,
zero vector a c is :
(A) 10iˆ 5jˆ
29.
(C)
(B) 14iˆ 5jˆ
(C) 14iˆ 5ˆj
(D) 10iˆ 5ˆj
If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of
k is :
(A) –81
(B) 100
(C) 144
(D) –300
3
30.
1
x iy
i 1 , where x and y are real numbers, then y – x equals :
Let 2 i
3
27
(A) 91
(B) –85
(C) 85
(D) –91
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (11/01/2019) SHIFT-2
1.
x cot(4x)
lim
x 0 sin 2
x cot 2 (2x)
(A) 0
2.
is equal to:
(B) 2
All x satisfying the inequality cot 1 x
(A) (– , cot5) (cot4, cot2)
(C) (– , cot5) (cot2, )
3.
2
(D) 1
7 cot 1 x 10 0 , lie in the interval:
(B) (cot2, )
(D) (cot5, cot4)
If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :
(A)
4.
(C) 4
13
12
(B) 2
(C)
13
6
(D)
13
8
If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the
other two vertices are the points of intersection of the parabola and y – axis, is 250 sq. units, then a value
of 'a' is :
(B) 5(21/3)
(A) 5 5
(C) (10)2/3
(D) 5
x 3 y 1 z 6
x 5 y 2 z 3
and
intersect at the point R. The reflection of R
1
3
1
7
6
4
in the xy - plane has coordinates :
(A) (2, –4, –7)
(B) (2, 4, 7)
(C) (2, –4, 7)
(D) (–2, 4, 7)
5.
Two lines
6.
Contrapositive of the statement
"If two numbers are not equal, then their squares are not equals" is :
(A) If the squares of two numbers are not equal, then the numbers are equal
(B) If the squares of two numbers are equal, then the numbers are not equal
(C) If the squares of two numbers are equal, then the numbers are equal
(D) If the squares of two numbers are not equal, then the numbers are not equal
7.
If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then
the equation of the diagonal AD is :
(A) 5x – 3y + 1 = 0 (B) 5x + 3y – 11 = 0 (C) 3x – 5y + 7 = 0 (D) 3x + 5y – 13 = 0
8.
The integral /6
/4
dx
equals :
sin 2x(tan 5 x cot 5 x)
(A)
1
1
tan 1
20
9 3
1
1 1
(B) 10 4 tan
9 3
(C)
40
1
1 1
(D) 5 4 tan
3 3
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9.
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JEE Main Papers
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression
x m yn
1 x 2m 1 y 2n is:
(A)
1
2
(B)
1
4
mn
6mn
(C)
(D) 1
2
10.
11.
n
q 1 q 1
q 1
Let Sn = 1 + q + q + .... + q and Tn 1
....
where q is a real number
2 2
2
and q 1. If 101C1 + 101C2.S1 + .... + 101C101.S100 = T100 then is equal to :
(A) 299
(B) 202
(C) 200
(D) 2100
2
n
Let and be the roots of the quadratic equation x 2 sin x(sin cos 1) cos 0
0 45 , and .
n 1 n
Then n is equal to:
n 0
12.
(A)
1
1
1 cos 1 sin
(B)
1
1
1 cos 1 sin
(C)
1
1
1 cos 1 sin
(D)
1
1
1 cos 1 sin
A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag
mean of X
with replacement. If X be the number of white balls drawn, then standard deviation of X is equal
to :
(A) 4
13.
(C) 3 2
(D)
4 3
3
Let z be a complex number such that |z| + z = 3 + i where i 1 . Then |z| is equal to :
(A)
14.
(B) 4 3
34
3
(B)
5
3
(C)
41
4
(D)
5
4
a bc
2a
2a
2b
bca
2b
If
= (a + b + c) (x + a + b + c)2, x 0 and a + b + c 0, then x is equal
2c
2c
ca b
to :
(A) abc
(B) – (a + b + c)
(C) 2(a + b + c)
(D) –2(a + b + c)
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15.
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JEE Main Papers
Let 3iˆ ˆj, ˆi 3jˆ and ˆi (1 )ˆj respectively be the position vectors of the points A, B and C with
respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is
3
, then the sum of all possible values of is :
2
(A) 4
(B) 3
(C) 2
(D) 1
16.
If 19th terms of non - zero A.P. is zero, then its (49th term) : (29th term) is :
(A) 4 : 1
(B) 1 : 3
(C) 3 : 1
(D) 2 : 1
17.
If
(A)
18.
x 1
dx f (x) 2x 1 C , where C is a constant of integration, then f(x) is equal to :
2x 1
1
(x 1)
3
(B)
2
(x 2)
3
(C)
2
(x 4)
3
(D)
1
(x 4)
3
1
. Then f is :
x
(B) injective only
(D) both injective as well as surjective
Let a function f : (0, ) (0, ) be defined by f (x) 1
(A) not injective but it is surjective
(C) neither injective nor surjective
19.
Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2 (x – ) cos|x| is not
differentiable. Then the set K is equal to :
(A) (an empty set) (B) {}
(C) {0}
(D) {0, }
20.
The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the
point (2, 5) and the coordinate axes is :
(A)
21.
8
3
Given
(B)
bc ca a b
11
12
13
37
24
The solution of the differential equation
(A) log e
2x
xy
2y
(C) log e
187
24
(D)
14
3
for a ABC with usual notation. If cos A cos B cos C , then the ordered
triad , , has a value:(A) (3, 4, 5)
(B) (19, 7, 25)
22.
(C)
1 x y
xy2
1 x y
(C) (7, 19, 25)
(D) (5, 12, 13)
dy
(x y) 2 when y(1) = 1, is :
dx
(B) log e
(D) log e
1 x y
2(x 1)
1 x y
2y
2(y 1)
2x
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23.
26.
(B) 4 3, 2 2
(C) 4 3, 2 3
(D) 4 2, 2 3
Let S = {1, 2, ..... 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then
the probability that a randomly chosen subset of S is "nice" is :
(A)
25.
JEE Main Papers
Let the length of the latus rectum of an ellipse with its major axis along x - axis and center at the origin, be
8. If the distance between the foci of this ellipse is equal to the length of the length of its minor axis, then
which one of the following points lies on it ?
(A) 4 2, 2 2
24.
MCSIR
7
220
(B)
5
220
(C)
4
220
(D)
6
220
If the point (2, , ) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is
perpendicular to the plane 2x – 5y = 15, then 2 – 3 is equal to :
(A) 12
(B) 7
(C) 5
(D) 17
a2
Let (x + 10)50 + (x – 10)50 = a0 + a1x + a2x2 + ... + a50x50, for all x R ; then a is equal to :
0
(A) 12.50
(B) 12.00
(C) 12.25
(D) 12.75
27.
The number of functions f from {1, 2, 3, ....... , 20} onto {1, 2, 3, ..........., 20} such that f(k) is a multiple
of 3, whenever k is a multiple of 4, is :
(A) 65 × (15)!
(B) 5! × 6!
(C) (15)! × 6!
(D) 56 × 15
28.
A circle cuts a chord of length 4a on the x - axis and passes through a point on the y - axis, distance 2b
from the origin. Then the locus of the centre of this circle, is :
(A) A hyperbola
(B) A parabola
(C) A straight line
(D) An ellipse
29.
Let f (x)
x
2
a x
2
dx
2
b (d x) 2
, x R , where a, b and d are non - zero real constant. Then :
(A) f is an increasing function of x
(B) f is a decreasing function of x
(C) f is not a continuous function of x
(D) f is neither increasing nor decreasing function of x
30.
Let A and B be two invertible matrices of order 3 × 3. If det (ABAT) = 8 and det(AB–1) = 8, then
det(BA–1 BT) is equal to :
(A)
1
4
(B) 1
(C)
1
16
(D) 16
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JEE MAIN-2019 (12/01/2019) SHIFT-1
1.
An ordered pair for which the system of linear equations
1 x y z 2
x 1 y z 3
x y 2z 2
has a unique solution, is :
(A) (2, 4)
(B) (3, 1)
(C) (4, 2)
(D) (1, 3)
2.
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second
of these terms, the three terms now form an A. P. Then the sum of the original three terms of the given G.
P. is :
(A) 36
(B) 32
(C) 24
(D) 28
3.
The Boolean expression ((p q) (p ~ q)) ~ p ~ q is equivalent to:
(A) p q
(B) p (~ q)
(C) (~ p) (~ q)
(D) p (~ q)
4.
Consider three boxes, each containing 10 balls labelled 1, 2, ....., 10. Suppose one ball is randomly
drawn from each of the boxes. Denote by ni, the label of the ball drawn from the ith box, (i = 1, 2, 3 ).
Then, the number of ways in which the balls can be chosen such that n1 < n2 < n3 is:
(A) 120
(B) 82
(C) 240
(D) 164
5.
A tetrahedron has vertices P (1, 2, 1), Q(2, 1, 3), R(1, 1, 2) and O(0, 0, 0). The angle between the
faces OPQ and PQR is :
1 17
(A) cos
31
6.
1 9
(C) cos
35
1 7
(D) cos
31
The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices
on the parabola, y = 12 x2 such that the rectangle lies inside the parabola, is :
(A) 36
7.
1 19
(B) cos
35
(B) 20 2
(C) 32
(D) 18 3
If the straight line, 2x 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and
(15, ), then equals :
(A)
35
3
(B) 5
(C)
35
3
(D) 5
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8.
11.
(B) 0
(C) 1
(D) 2
Let P(4, 4) and Q(9, 6) be two points on the parabola, y2 = 4x and let X be any point on the arc POQ
of this parabola, where O is the vertex of this parabola, such that the area of PXQ is maximum. Then
this maximum Area (in sq. units) is :
(A)
10.
JEE Main Papers
ˆ ˆi ˆj k,
ˆ ˆi ˆj kˆ are coThe sum of the distinct real values of for which the vectors, ˆi ˆj k,
planar, is :
(A) 1
9.
MCSIR
75
2
(B)
125
4
(C)
625
4
(D)
125
2
1 0 0
q 21 q 31
Let P 3 1 0 and Q = [qij] be two 3 3 matrices such that Q P5 = I3. Then
is equal
q 32
9 3 1
to
(A) 10
(B)135
(C) 15
(D) 9
Let y = y(x) be the solut ion of the differential equation, x
If 2y(2) = loge 4 then y(e) is equal to :
(A) – e/2
(B) – e2/2
(C) e/4
dy
y x log e x, (x 1).
dx
(D) e2/4
12.
The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0
and x = 3, is:
15
21
17
15
(A)
(B)
(C)
(D)
4
2
4
2
13.
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that
the experiment will end in the fifth throw of the die is equal to:
(A)
200
5
6
(B)
150
6
(C)
5
225
6
5
(D)
175
65
14.
Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2–6x– 6y+14 =0
respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the
quadrilateral PC1QC2 is:
(A) 8
(B) 6
(C) 9
(D) 4
15.
The maximum value of 3cos 5sin 6 for any real value of is :
(A) 19
(B)
79
2
(C) 31
(D) 34
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Considering only the principal values of inverse functions, the set
A x 0 : tan 1 (2x) tan 1 (3x)
(A) contains two elements
(C) is a singleton
17.
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4
(B) contains more than two elements
(D) is an empty set
If be the ratio of the roots of the quadratic equation in x, 3m2x2+m(m-4)x+2=0, then the least value of
m for which
1
1 , is:
(A) 2 3
(B) 4 3 2
(C) 2 2
(D) 4 2 3
18.
If a variable line, 3x + 4y – = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x
– 2y + 78 = 0 are on its opposite sides, then the set of all values of is the interval:(A) [12, 21]
(B) (2, 17)
(C) (23, 31)
(D) [13, 23]
19
For x > 1, if (2x)2y = 4e2x–2y, then (1 + loge2x)2
(A)
20.
21.
x log e 2x log e 2
(B) loge2x
x
(C)
dy
is equal to :
dx
x log e 2x log e 2
x
(D) xloge2x
The integral cos(log e x)dx is equal to : (where C is a constant of integration)
(A)
x
sin(log e x) cos(log e x) C
2
(B) x[cos(logex) + sin(logex)] + C
(C)
x
cos(log e x) sin(log e x) C
2
(D) x[cos(logex) – sin(logex)] + C
A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of
1/3
1
2
is :
2(3)1/3
(A) 1 : 2(6)1/3
(B) 1 : 4(16)1/3
(C) 4(36)1/3 : 1
(D) 2(36)1/3 : 1
1 2 3 .... k
5
A , then A is equal to :
. If S12 + S22 + ..... + S102 =
k
12
(A) 283
(B) 301
(C) 303
(D) 156
22.
Let Sk
23.
The perpendicular distance from the origin to the plane containing the two lines,
x 2 y2 z5
x 1 y 4 z 4
and
, is :
3
5
7
1
4
7
(A) 11 6
(B)
11
6
(C) 11
(D) 6 11
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24.
If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is :
(A) 30
(B) 51
(C) 50
(D) 31
25.
Let S be the set of all points in (–, ) at which the function, f(x) = min{sinx, cosx} is non-differentiable.
Then S is a subset of which of the following ?
26.
(A) , 0,
4
4
3 3
(B) , , ,
4 4 4
4
(C) , , ,
2 4 4 2
3 3
(D) , , ,
2 2 4
4
Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) and g (x) + g(a – x) = 4, then
a
f (x)g(x)dx is equal to:
0
a
a
(A) 4 f (x) dx
0
27.
0
(C) 2 f (x)dx
(D) 3 f (x)dx
0
0
cot3 x tanx
is :
x 4 cos x / 4
lim
If
(C) 8 2
(B) 4 2
(A) 4
28.
a
a
(B) f (x)dx
(D) 8
z
R is a purely imaginary number and |z| = 2, then a value of is:
z
(A) 2
(B) 1
(C)
1
2
2
(D)
29.
Let S = {1, 2, 3, ......, 100}. The number of non - empty subsets A of S such that the product of elements
in A is even is:
(A) 2100 – 1
(B) 250 (250 – 1)
(C) 250 –1
(D) 250 + 1
30.
If the vertices of a hyperbola be at (– 2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of
the of the following points does not lie on this hyperbola?
(A) 6, 2 10
(B) 2 6,5
(C) 4, 15
(D) 6,5 2
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JEE MAIN-2019 (12/01/2019) SHIFT-2
1.
lim
1
2
(B)
4.
2
(C)
2
(D)
Let f be a differentiable function such that f (1) = 2 and f '(x) = f(x) for all x R. If h (x) = f (f(x)), then
h' (1) is equal to:
(A) 2e2
(B) 4e
(C) 2e
(D) 4e2
e
3.
is equal to :-
1 x
x 1
(A)
2.
2sin 1 x
x
The integral
e
1
2x
e
x
x
log e x dx is equal to:
(A)
1
1
e 2
2
e
1 1
1
(B) 2
2 e 2e
(C)
3 1 1
2 e 2e 2
(D)
3
1
e 2
2
2e
Let a, b and c be three unit vectors, out of which vectors b and c are non- parallel. If and are the
1
angles which vector a makes with vectors b and c respectively and a b c b , then | | is
2
equal to:
(A) 30°
(B) 90°
(C) 60°
(D) 45°
5.
The integral
3x13 2x11
2x
4
2
x4
(A)
6(2x 4 3x 2 1)3
x4
(C)
6.
3x 1
(2x 4 3x 2 1)3
C
C
4
dx is equal to (where C is a constant of integration)
x12
(B)
6(2x 4 3x 2 1)3
x12
(D)
(2x 4 3x 2 1)3
C
C
If sin 4 4 cos 4 2 4 2 sin cos ; , [0, ] , then cos - cos () is equal to:
(A) 0
(B) 2
(C) –1
(D) 2
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7.
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If a curve passes through the point (1, –2) and has slope of the tangent at any point (x, y) on it as
x 2 2y
, then the curve also passes through the point:
x
(A) (3, 0)
8.
3, 0
(C) (–1, 2)
(D) 2,1
5
3
3
5
(B)
(C)
3
5
(D)
5
3
n
n
1
n
lim 2 2 2
2
.....
is equal to :2
2
n n 1
5n
n 2
n 3
(A)
10.
x 1 y 2 z 3
1 2 2
cos
,
If an angle between the line,
and the plane, x 2y kz 3 is
2
1
2
3
then a value of k is:
(A)
9.
(B)
4
(B) tan 1 3
(C)
2
(D) tan 1 2
In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any
other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum
of three throws, then his expected gain/loss (in rupees) is:
(A)
400
loss
9
(B) 0
(C)
400
gain
3
(D)
400
loss
3
11.
If a straight line passing through the point P (–3, 4) is such that its intercepted portion between the
corrdinate axes is bisected at P, then its equation is:
(A) 3x – 4y + 25 = 0
(B) 4x – 3y + 24 = 0
(C) x – y + 7 = 0
(D) 4x + 3y = 0
12.
There are m men and two women participating in a chess tournament. Each participant plays two games
with every other participant. If the number of games played by the men between themselves exceeds the
number of games played between the men and the women by 84, then the value m is:
(A) 12
(B) 11
(C) 9
(D) 7
13.
The tangent to the curve y = x2 – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point:
7 1
(A) ,
2 4
14.
1
(B) , 7
8
1
(C) , 7
8
1 7
(D) ,
4 2
Let S be the set of all real values of such that a plane passing through the points
2 ,1,1 , 1, 2 ,1 and 1,1, 2 also passes through the point (–1, –1, 1). Then S is equal to:
(A)
3
(B)
3, 3
(C) {1, –1}
(D) {3, – 3}
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15.
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Let z1 and z2 be two complex numbers satisfying |z1| = 9 and |z2 –3 – 4i| = 4. Then the minimum value of
|z1 – z2| is:
(A) 0
(B) 2
(C) 1
(D) 2
60
16.
1
1
The total number of irrational terms in the binomial expansion of 7 5 3 10
(A) 55
(B) 49
(C) 48
(D) 54
is:
17.
The equation of a tangent to the parabola, x2 = 8y, which makes an angle with the positive direction of
x - axis, is:
(A) y = x tan + 2 cot
(B) y = x tan – 2 cot
(C) x = y cot + 2 tan
(D) x = y cot – 2 tan
18.
In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS.
If one of these students is selected at random, then the probability that the student selected has opted
neighter for NCC nor for NSS is:
(A)
19.
20.
2
3
(B)
1
6
(C)
1
3
(D)
5
6
The mean and the variance of five observations are 4 and 5.20, respectively, if three of the observations
are 3, 4 and 4; then the absolute value of the difference of the other two observations, is:
(A) 7
(B) 5
(C) 1
(D) 3
sin
1
1
3 5
1
sin ; then for all , , det (A) lies in the interval:
If A sin
4 4
1
sin
1
5
(A) 1,
2
5
(B) , 4
2
3
(C) 0,
2
3
(D) ,3
2
21.
If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the
locus of the foot of perpendicular from O on AB is:
(A) (x2 +y2)2 = 4Rx2y2
(B) (x2 +y2)2 = 4R2x2y2
2
2 3
2 2 2
(C) (x +y ) = 4R x y
(D) (x2 +y2)(x + y) = R2xy
22.
The set of all values of for which the system of linear equations :
x – 2y – 2z = x
x + 2y + z = y
–x – y = z
has a non - trivial solution.
(A) contains more than two elements
(B) is a singleton
(C) is an empty set
(D) contains exactly two elements
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23.
MCSIR
If the function f given by f(x) = x3 – 3 (a – 2)x2 + 3ax + 7, for some a R is increasing in (0, 1] and
f (x) 14
decreasing in [1, 5), then a root of the equation,
(A) – 7
24.
25.
26.
JEE Main Papers
(B) 5
(x 1)2
0(x 1) is:
(C) 7
(x 2)(x
Let Z be the set of integers, if A x Z : 2
(D) 6
2
5x 6)
1 and B x Z : 3 2x 1 9 ,
then the number of subsets of the set A × B, is:
(A) 218
(B) 210
(C) 215
(D) 212
If n C4 , n C5 and n C6 are in A.P., then n can be:
(A) 9
(B) 14
(C) 11
(D) 12
Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If S'BS is
a right angled triangle with right angle at B and area S'BS =8 sq. units, then the length of a latus
rectum of the ellipse is:
(A) 4
(B) 2 2
(C) 4 2
(D) 2
27.
If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30° and the angle of
depression of reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters)
from the surface of the lake is :
(A) 42
(B) 50
(C) 45
(D) 60
28.
The expression p q) is logically equivalent to :
(A) p q
(B) p q
(C) p q
3
29.
30.
(D) p q
3
3
3
3 1 1
3 3
If the sum of the first 15 terms of the series 1 2 3 3 ...... is equal to 225
4 2 4
4
k, then k is equal to:
(A) 108
(B) 27
(C) 54
(D) 9
The number of integral values of m for which the quadratic expression,
(1 + 2m) x2 – 2 (1 + 3m) x + 4 (1 +m), x R, is always positive, is :
(A) 3
(B) 8
(C) 7
(D) 6
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JEE MAIN-2019 (8/04/2019) SHIFT-1
JEE MAIN_8 APRIL-2019 FIRST SHIFT
1.
The shortest distance between the line y = x and the curve y2 = x – 2 is :
(A)
2.
7
4 2
7
8
(C)
11
4 2
2
2
Let y = y(x) be the solution of the differential equation, (x 1)
0. If ay(1)
(A)
3.
(B)
(D) 2
dy
2x(x 2 1)y 1 such that y(0) =
dx
, then the value of ‘a’ is :
32
1
2
(B)
1
16
(C)
1
4
(D) 1
A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only in :
(A) 1st and 2nd quadrants
(B) 4th quadrant
st
nd
th
(C) 1 , 2 and 4 quadrants
(D) 1st quadrant
n
4.
5.
If and be the roots of the equation x – 2x + 2 = 0, then the least value of n for which 1 is
:
(A) 2
(B) 3
(C) 4
(D) 5
2
lim
x 0
sin 2 x
equals :
2 1 cos x
(A) 2 2
6.
(B) 4 2
2
(D) 4
x 3 y2 z
is :
10
7
1
(B) greater than 3 but less than 4
(D) greater than 2 but less than 3
The length of the perpendicular from the point (2, –1, 4) on the straight line,
(A) less thatn 2
(C) greater than 4
7.
(C)
The magnitude of the projection of the vector 2iˆ 3jˆ kˆ on the vector perpendicular to the plane
containing the vectors ˆi ˆj kˆ and ˆi 2ˆj 3kˆ , is :
(A)
3
2
(B)
3
2
(C)
6
(D) 3 6
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8.
The contrapositive of the statement “If you are born in India, then you are a citizin of India”, is :
(A) If you are born in India, then you are not a citizen of India.
(B) If you are not a citizen of India, then you are not born in India.
(C) If you are a citizen of India, then you are born in India.
(D) if you are not born in India, then you are not a citizen of India.
9.
The mean and variance of seven observations are 8 and 16, respectively. If 5 of the observations are 2,
4, 10, 12, 14, then the product of the remaining two observations is :
(A) 40
(B) 49
(C) 48
(D) 45
4
10.
If f (x)
2 x cos x
and g(x) = logex, (x > 0) then the value of the integral g(f (x))dx is :
2 x cos x
4
(A) loge 3
11.
(C) logee
(D) loge1
If the tangents on the ellipse 4x2 + y2 = 8 at the points (1, 2) and (a, b) are perpendicular to each other,
then a2 is equal to :
(A)
12.
(B) loge2
64
17
(B)
2
17
(C)
128
17
(D)
4
17
1 3
1 1
If cos , tan , where 0 , , then – is equal to :
5
3
2
9
(A) sin 1
5 10
1 9
(B) tan
14
1 9
(C) cos
5 10
1 9
(D) tan
5 10
13.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, f(x) =
9x4 + 12x3 – 36x2 + 25, x R, then :
(A) S1 = {–2, 1} ; S2 = {0}
(B) S1 = {–2, 0} ; S2 = {1}
(C) S1 = {–2} ; S2 = {0, 1}
(D) S1 = {–1} ; S2 = {0, 2}
14.
Let O(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of AOP
is 4, is :
(A) 8x2 – 9y2 + 9y = 18
(B) 9x2 + 8y2 – 8y = 16
(C) 8x2 + 9y2 – 9y = 18
(D) 9x2 – 8y2 + 8y = 16
15.
cos sin
32 0 1
Let A
, ( R) such that A
. Then a value of is :
1 0
sin cos
(A)
16
(B) 0
(C)
32
(D)
64
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16.
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2x
1 x
If f (x) log e
,| x | 1 , then f
is equal to :
1 x
1 x2
(B) 2f(x2)
(A) 2f(x)
(C) (f(x))2
(D) –2f(x)
17.
The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and
y + 2z – 4 = 0 and passing through the point (1, 1, 0) is :
(A) x + 3y + z = 4
(B) x – y – z = 0
(C) x – 3y – 2z = – 2 (D) 2x – z = 2
18.
The sum of all natural numbers 'n' such that 100 < n < 200 and H.C. F. (91, n) > 1 is:
(A) 3221
(B) 3121
(C) 3203
(D) 3303
19.
The sum of the series 2.20C0 + 5.20C1+ 8.20C2+ 11.20C3+ .... + 62.20C20 is equal to :
(A) 224
(B) 225
(C) 226
(D) 223
20.
The sum of the solutions of the equation
| x 2 | x x 4 2 0, (x 0) is equal to :
(A) 4
(B) 9
(C) 10
(D) 12
21.
Let A and B be two non-null events such that A B. Then, which of the following statements is always
correct ?
(A) P(A|B) = 1
(B) P(A|B) = P(B) – P(A)
(C) P(A|B) P(A)
(D) P(A|B) P(A)
22.
The sum of the co-efficient s of all even degree terms in x in t he expansion of
x
x3 1
(A) 32
23.
6
x
x3 1
6
(B) 26
, (x 1) is equal to :
(C) 29
(D) 24
The area (in sq. units) of the region A = {(x, y) R × R|0 x 3, 0 y 4, y x2 + 3x} is :
(A)
53
6
(B)
59
6
(C) 8
(D)
26
3
24.
Let f : [0, 2] R be a twice differentiable function such that f "(x) > 0, for all x (0, 2).
If (x) = f(x) +f(2 – x), then is :
(A) decreasing on (0, 2)
(B) decreasing on (0, 1) and increasing on (1, 2)
(C) increasing on (0, 2)
(D) increasing on (0, 1) and decreasing on (1, 2)
25.
The sum of the squares of the lengths of the chords intercepted on the circle, x2 + y2 = 16, by the lines,
x + y = n, n N, where N is the set of all natural numbers, is :
(A) 320
(B) 160
(C) 105
(D) 210
26.
All possible numbers are formed using the digits 1,1,2, 2,2,2,3,4,4 taken all at a time. The number of
such numbers in which the odd digits occupy even places is :
(A) 175
(B) 162
(C) 160
(D) 180
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5x
2 dx
is equal to : where c is a constant of integration
x
sin
2
sin
27.
(A) 2x + sinx + 2sin2x + c
(C) x + 2sinx + sin2x + c
(B) x + 2sinx + 2sin2x + c
(D) 2x + sinx + sin2x + c
2
28.
1 3 cos x sin x
dy
, x 0, then
If 2y cot
is equal to :
2
dx
cos x 3 sin x
(A) 2x
29.
(B)
x
3
(C)
x
6
(D) x
6
The greatest value of c R for which the system of linear equations :
x – cy – cz = 0
cx – y + cz = 0
cx + cy – z = 0
has a non-trivial solution, is :
(A)
30.
3
1
2
(B) –1
(C) 0
(D) 2
3
5
If cos( ) ,sin( )
and 0 , , then tan(2) is equal to :
5
13
4
(A)
21
16
(B)
63
52
(C)
33
52
(D)
63
16
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (8/04/2019) SHIFT-2
1.
The minimum number of times one has to toss a fair coin so that the probability of observing at least one
head is at least 90% is :
(A) 5
(B) 3
(C) 2
(D) 4
2.
A student scores the following marks in five tests : 45, 54, 41, 57, 43. His score is not known for the
sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is :
(A)
10
3
(B)
20
3.
The sum k
k 1
(A) 2
4.
100
3
(C)
100
3
(D)
10
3
1
is equal to :
2k
3
217
(B) 2
11
219
(C) 1
11
2 20
(D) 2
21
220
Let a 3iˆ 2ˆj xkˆ and b ˆi ˆj kˆ , for some real x. Then | a b | r is possible if :
(A) 3
3
3
r 5
2
2
(B) 0 r
3
2
(C)
3
3
r3
2
2
(D) r 5
3
2
5.
If the system of linear equation
x – 2y + kz = 1
2x + y + z = 2
3x – y – kz = 3
has a solution (x, y, z), z 0, then (x, y) lies on the straight line whose equation is :
(A) 3x – 4y – 1 = 0 (B) 3x – 4y – 4 = 0
(C) 4x – 3y – 4 = 0 (D) 4x – 3y – 1 = 0
6.
If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the
tangent to the hyperbola at (4, 6) is :
(A) 2x – y – 2 = 0
(B) 3x – 2y = 0
(C) 2x – 3y + 10 = 0 (D) x – 2y + 8 = 0
7.
If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio
of lengths of the sides of this triangle is:
(A) 5 : 9 : 13
(B) 5 : 6 : 7
(C) 4 : 5 : 6
(D) 3 : 4 : 5
8.
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd
function. Then f1(x + y) + f1(x – y) equals :
(A) 2f1(x) f1(y)
(B) 2f1(x)f1(y)
(C) 2f1(x + y) f2(x – y) (D) 2f1(x + y) f1(x – y)
6
9.
1
1
12
If the fourth term in the binomial expansion of 1 log10 x x is equal to 200, and x > 1, then the
x
value of x is :
(A) 103
(B) 100
(C) 104
(D) 10
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11.
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Let S() = {(x, y) : y2 x, 0 x } and A() is area of the region S(). If for a , 0 < < 4,
A() : A(4) = 2 : 5, then equals :
1
1
1
1
4 3
(A) 2
25
4 3
(B) 4
25
2 3
(C) 2
5
2 3
(D) 4
5
Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is
2y
. If the curve passes
x2
through the centre of the circle x2 + y2 – 2x – 2y = 0, then its equation is :
(A) x loge|y| = 2(x – 1)
(B) x loge|y| = x – 1
2
(C) x loge|y| = – 2(x – 1)
(D) x loge|y| = – 2(x – 1)
12
The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and
2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0 is :
(A) r ˆi kˆ 2 0 (B) r. iˆ kˆ 2 0 (C) r. iˆ kˆ 2 0 (D) r iˆ kˆ 2 0
13.
Which one of the following statements is not a tautology ?
(A) (p q) p
(B) (p q) (~ p) q
(C) p (p q)
(D) (p q) (p (~q))
1
14.
1 f (3 x) f (3) x
Let f : R R be a differentiable function satisfying f '(3) + f '(2)= 0. Then lim
is
x 0 1 f (2 x) f (2)
equal to :
(A) e2
15.
1 1
(B) ,
4 2
(D) 1
3 7
(C) ,
4 4
1 1
Let the numbers 2, b, c be in an A.P. and A 2 b
2
4 b
interval :
(A) [2, 3)
17.
(C) e–1
The tangent to the parabola y2 = 4x at the point where it intersects the circle x2 + y2 = 5 in the first
quadrant, passes through the point :
1 4
(A) ,
3 3
16.
(B) e
(B) (2 + 23/4, 4)
1 3
(D) ,
4 4
1
c . If det(A) [2,16], then c lies in the
c 2
(C) [3, 2 + 23/4]
(D) [4, 6]
If three distinct numbers a, b, c are in G.P. and the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0
have a common root, then which one of the following statements is correct?
(A) d,e,f are in A.P.
d e f
d e f
(B) a , b , c are in G.P.. (C) a , b , c are in A.P.. (D) d,e,f are in G.P.
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18.
The number of integral values of m for which the equation (1 + m2)x2 – 2(1 + 3m) x + (1 + 8m) = 0 has
no real root is :
(A) infinitely many
(B) 2
(C) 3
(D) 1
19.
If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the
distance of R from the origin is :
(A) 2 14
20.
(B) 6
(C)
53
3 i
i 1 . then (1 + iz + z5 + iz8)9 is equal to :
2 2
(A) –1
(B) 1
(C) 0
(D) 2 21
If z
(D) (–1 + 2i)9
x
x
21.
Let f (x) g(t)dt . where g is a non-zero even function. If f(x + 5) = g(x), then f (t)dt equals :
0
0
5
(A)
5
g(t)dt
x 5
22.
(C)
x 5
x 5
g(t)dt
5
(D) 2 g(t)dt
5
3,1 to the circle x2 + y2 = 4 and the x-axis form a
triangle. The area of this triangle (in square units) is :
The tangent and the normal lines at the point
(A)
23.
x 5
(B) 5 g(t)dt
1
3
(B)
4
3
(C)
1
3
(D)
2
3
In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10
and one of the foci is at 0,5 3 , then the length of its latus rectum is :
(A) 10
(B) 8
(C) 5
(D) 6
24.
If f(1) = 1, f '(1) = 3, then the derivative of f(f(f(x))) + (f(x))2 at x = 1 is :
(A) 12
(B) 33
(C) 9
(D) 15
1
25.
3
dx
xf (x)(1 x 6 ) C
If 3
6 2/3
x (1 x )
where C is a constant of integration, then the function f(x) is equal to :
(A)
26.
1
6x 3
(B)
3
x2
(C)
1
2x 2
(D)
1
2x 3
Suppose that the points (h, k), (1, 2) and (–3, 4) lie on the line L1. If a line L2 passing through the points
(h, k) and (4, 3) is perpendicular to L1, then
(A) 3
(B)
1
7
k
equals :
h
(C)
1
3
(D) 0
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27.
Let f : [–1, 3] R be defined as
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| x | [x] , 1 x 1
f (x) x | x | , 1 x 2
x [x] , 2 x 3,
where [t] denotes the greatest integer less than or equal to t. Then, f is discontinuous at :
(A) four or more points
(B) only one point
(C) only two points
(D) only three points
28.
Two vertical poles of heights, 20 m and 80 m stand apart on a horizontal plane. The height (in meters) of
the point of intersection of the lines joining the top of each pole to the foot of the other, from this
horizontal plane is:
(A) 12
(B) 15
(C) 16
(D) 18
29.
The number of four-digit numbers strictly greater than 4321 that can be formed using the digits
0,1,2,3,4,5 (repetition of digits is allowed) is :
(A) 288
(B) 306
(C) 360
(D) 310
30.
The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is :
(A) 2 3
(B) 3
(C)
6
(D)
2
3
3
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (9/04/2019) SHIFT-1
1.
Let 3iˆ ˆj and 2iˆ ˆj 3kˆ . If 1 2 , where 1 is parallel to and 2 is perpendicular to ,
then 1 2 is equal to :
(1) 3iˆ 9jˆ 5kˆ
2.
(3)
1
ˆ
(3iˆ 9ˆj 5k)
2
(4)
1 ˆ ˆ ˆ
(3i 9 j 5k)
2
For any two statements p and q, the negation of the expression p (~ p q) is :
(1) p q
(2) p q
(3) ~ p ~ q
(4) ~ p ~ q
/2
3.
(2) 3iˆ 9jˆ 5kˆ
The value of
0
(1)
2
4
sin 3 x
dx is :
sin x cos x
(2)
2
8
(3)
1
4
(4)
1
2
4.
If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set
S = {x R : f(x) = f(0)} Contains exactly :
(1) four irrational numbers
(2) two irrational and one rational number
(3) four rational numbers
(4) two irrational and two rational numbers
5.
If the standard deviation of the numbers –1, 0, 1, k is 5 where k > 0, then k is equal to :
(1) 2
6.
10
3
(2) 2 6
(3) 4
(4) 6
i
: R i 1 lie on a :
i
All the points in the set S
(1) circle whose radius is 1
(3) straight line whose slope is –1
7.
5
3
(2) straight line whose slope is 1
(4) circle whose radius is 2
Let S be the set of all values of x for which the tangent to the curve y = f(x) = x3 – x2 – 2x at (x, y) is
parallel to the line segment joining the points (1, f(1)) and (–1, f(–1)), then S is equal to :
1
(1) 3 , 1
1
(2) 3 , 1
1
(3) 3 ,1
1
(4) 3 ,1
8.
Let f(x) = 15 –|x – 10|; x R. Then the set of all values of x, at which the function, g(x) = f(f(x)) is not
differentiable, is :
(1) {5, 10, 15, 20}
(2) {10, 15}
(3) {5, 10, 15}
(4) {10}
9.
Let p, q R. If 2 3 is a root of the quadratic equation, x2 + px + q = 0, then :
(1) q2 + 4p + 14 = 0 (2) p2 – 4q – 12 = 0 (3) q2 – 4p – 16 = 0 (4) p2 – 4q + 12 = 0
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Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from
P, is :
5 1
(1)
11.
MCSIR
(2)
5 1
1 5
(3)
1 5
1 7
7 1
(4)
1 7
7 1
A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the
committee is formed with at least 6 males and n is the number of ways the committee is formed with at
least 3 females, then :
(1) m = n = 78
(2) n = m–8
(3) m + n = 68
(4) m = n = 68
6
12.
2
If the fourth term in the binomial expansion of x log8 x (x 0) is 20 × 87, then a value of x is :
x
(2) 82
(1) 8
13.
The solution of the differential equation x
(1) y
14.
(3) 8-2
x3
1
5 5x 2
4
(4) 83
dy
2y x 2 (x 0)
dx
1
3
(2) y 5 x 2
5x
with y (1) = 1, is :
3
1
2
(3) y 4 x 2
4x
(4) y
x2
3
4 4x 2
A plane passing through the points (0, – 1, 0) and (0, 0, 1) and making an angle
with the plane
4
y –z + 5 = 0, also passes through the point :
(1) 2,1, 4
15.
2,1, 4
(3)
2, 1, 4
(4) 2, 1, 4
The integral sec2/3 x cos ec4/3 xdx is equal to (Hence C is a constant of integration)
(1) 3 tan–1/3 x + C
16.
(2)
3
4
(2) tan 4/3 x C
(3) –3 cot–1/3 x + C
(4) –3 tan–1/3 x + C
Let the sum of the first n terms of a non-constant A.P., a1, a2, a3,... be 50n
n n 7
2
A , where A is a
constant. If d is the common difference of this A.P., then the ordered pair (d, a50) is equal to :
(A) (A, 50 + 46A) (B) (A, 50 + 45A)
(C) (50, 50 + 46A)
(D) (50, 50 + 45A)
17.
The area (in sq. units) of the region A = {(x, y): x2 < y < x + 2 } is :
(1)
18.
10
3
If the line,
(2)
9
2
x 1 y 1 z 2
2
3
4
(3)
31
6
(4)
13
6
meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from
the origin is :
(1)
9
2
(2) 2 5
(3)
5
2
(4)
7
2
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10
19.
Let
f (a k) 16 210 1 , where the function f satisfies f (x + y) = f (x) f (y) for all natural numbers x,
k 1
y and f (1) = 2, then the natural number 'a' is :
(1) 4
(2) 3
(3) 16
(4) 2
y 1
20.
Let and be the roots of the equation x2 + x + 1 = 0. Then for y 0 in R,
to :
(1) y3
(2) y3 – 1
(3) y (y2 –1)
y
1
1
y
is equal
(4) y (y2– 3)
21.
If the tangent to the curve, y = x3 + ax – b at the point (1, –5) is perpendicular to the line, – x + y + 4 =
0, then which one of the following points lies on the curve?
(1) (–2, 2)
(2) (2, – 2)
(3) (2, –1)
(4)(–2, 1)
22.
Four persons can hit a target correctly with probabilities 2 , 3 , 4 and 8 respectively. If all hit at the target
independently, then the probability that the target would be hit, is
1 1 1
(1)
23.
5
2
(3)
25
32
(4)
7
32
y2
(2)
3
5
(3)
2
5
15
2
(4)
Let S = { [2, 2] : 2cos2 3sin 0} . Then the sum of the elements of S is
13
6
(2)
(3) 2
(4)
5
3
(4)
3
2
The value of cos 2 10 cos10 cos 50 cos2 50 is :
3
2
(1) 1 cos 20
26.
1
192
x2
(1)
25.
(2)
If the line y mx 7 3 is normal to the hyperbola 1 , then a value of m is:
24 18
(1)
24.
25
192
1
(2)
3
4
(3)
3
cos 20
4
If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the
locus of the mid - point of PQ is :
(1) x2 + y2 – 2xy = 0
(2) x2 + y2 – 16x2y2 = 0
2
2
2 2
(3)x + y – 4x y = 0
(4) x2 + y2 – 2x2y2 = 0
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2 cos x 1
, x
4 is continuous then k is equal to :
If the function f defined on 6 , 3 by f (x) cot x 1
k,
x
4
(1)
28.
MCSIR
1
2
1 1 1 2 1 3
(2) 1
1 n 1 1 78
1 0 1
If 0 1 . 0 1 . 0 1 ..... 0
1 13
1
(1) 0
1
1
(3)
0
(2) 12 1
(4) 2
2
1 n
then the inverse of 0 1 is :
1 12
1
(3) 0
1
0
(4) 13 1
29.
If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is :
(1) 25
(2) 24
(3) 20
(4) 22
30.
If the function f : R – {1, –1} A defined by f (x)
(1) R – [–1, 0)
(2) R – (–1, 0)
x2
1 x2
, is surjective, then A is equal to :
(3) R– {–1}
(4) [0, )
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (9/04/2019) SHIFT-2
1.
If the tangent to the parabola y2 = x at a point (, ), ( > 0) is also a tangent to the ellipse,
x2 + 2y2 = 1, then is equal to :
(A) 2 2 1
2.
(B)
2 1
(C)
(D) 2 2 1
2 1
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one
ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total
number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square
whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains.
Then the number of balls used to form the equilateral triangle is :
(A) 190
(B) 262
(C) 225
(D) 157
f (x)
3.
If f : R R is a differentiable function and f(2) = 6, then lim
x 2
(A) 0
4.
(B) 2f '(2)
6
(C) 12f '(2)
2tdt
(x 2) is :
(D) 24f '(2)
If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution
(x, y, z), then
(A)
x y z
k is equal to :
y z x
4
(B) –4
1
2
(C)
(D)
1
4
5.
The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y – 24 =0 also passes through the
point:(A) (–4, 6)
(B) (6, –2)
(C) (–6, 4)
(D) (4, –2)
6.
If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its
11th term is :
(A) –25
(B) 25
(C) –36
(D) –35
7.
The value of the integral x cot 1 (1 x 2 x 4 )dx is :
1
0
(A)
8.
1
log e 2
4 2
(B)
log e 2
2
(C)
1
log e 2
2 2
(D)
log e 2
4
(C)
1
18
(D)
1
16
The value of sin 10º sin30º sin50º sin70º is :
(A)
1
36
(B)
1
32
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9.
Let z C be such that |z| < 1. If
(A) 5Im() < 1
MCSIR
JEE Main Papers
5 3z
, then :
5(1 z)
(B) 4Im() > 5
(C) 5Re() > 1
(D) 5Re() > 4
10.
If some three consecutive coefficients in the binomial expansion of (x + 1)n in powers of x are in the ratio
2 : 15 : 70, then the average of these three coefficient is :
(A) 964
(B) 625
(C) 227
(D) 232
11.
If cosx
(A)
12.
2
4 3
2
(B)
2
(C)
y is equal to :
6
2
2 3
(D)
2
2 3
If the two lines x + (a – 1) y = 1 and 2x + a2y = 1 (a R – {0, 1}) are perpendicular, then the distance
of their point of intersection from the origin is :
(A)
13.
dy
–y sin x = 6x, (0 < x < ) and y 0 , then
dx
2
3
2
5
(B)
2
5
(C)
2
5
(D)
2
5
1 1
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is tan .
2
Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate
(in m/min), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :
(A) 2/
(B) 1/5
(C) 1/10
(D) 1/15
14.
Two poles standing on a horizontal ground are of height 5m and 10m, respectively. The line joining their
tops makes an angle of 15° with the ground. Then the distance (in m) betweeen the poles is :
(A)
15.
5
2 3
2
(C) 5 2 3
(D) 10 3 1
x 2 y 1 z
such that BC = 5 units. The the area
3
0
4
(in sq. units) of this triangle, given that the point A(1, –1, 2), is :
The vertices B and C of a ABC lie on the line,
(A) 2 34
16.
(B) 5 3 1
(B) 34
0
The total number of matrices A 2x
2x
(A) 6
(B) 2
(C) 6
(D) 5 17
2y
1
y 1 , (x, y R, x y) for which ATA = 3I3 is :
y 1
(C) 3
(D) 4
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17.
(B) 8(3 2 2)
(C) 4(3 2)
(D) 8(2 2)
a | x | 1, x 5
If the function f (x)
is continuous at x = 5, then the value of a – b is :
b | x | 3, x 5
(A)
19.
JEE Main Papers
The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2)
and the x-axis is :
(A) 4(2 2)
18.
MCSIR
2
5
(B)
2
5
(C)
2
5
(D)
2
5
x
If f (x) [x] , x R , where [x] denotes the greatest integer function, then :
4
(A) both lim f (x) and lim f (x) exist but are not equal
x 4
x 4
f (x) exists but lim f (x) does not exist
(B) xlim
4
x 4
f (x) exists but lim f (x) does not exist
(C) xlim
4
x 4
(D) f is continuous at x = 4
20.
If esec x (sec x tan xf(x) + (sec x tan x + sec2 x))dx = esec xf (x) + C, then a possible choice of f(x) is :
(A) sec x – tan x – 1/2
(B) x sec x + tan x + 1/2
(C) sec x + x tan x – 1/2
(D) sec x + tan x + 1/2
21.
If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is
greatest, then the absolute difference of the cubes of its roots is :
(A) 8 3
(B) 4 3
(C) 10 5
(D) 8 5
22.
Two newspapers A and B are published in a city. It is known that 25% of the city population reads Aand
20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into
advertisements and 40% of those who read B but not A also look into advertisements, while 50% of
those who read both A and B look into advertisements. Then the percentage of the population who look
into advertisement is:
(A) 12.8
(B) 13.5
(C) 13.9
(D) 13
23.
Let P be the plane, which contains the line of intersection of the planes, x + y + z – 6 = 0 and
2x + 3y + z + 5 = 0 and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256)
from P is equal to :
(A) 63 5
24.
(B) 205 5
(C)
17
5
(D)
11
5
If p (q r) is false, then the truth values of p, q, r are respectively :
(A) F,T,T
(B) T,F,F
(C) T,T,F
(D) F,F,F
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25.
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JEE Main Papers
1
log10 (x 3 x) is :
2
4x
(B) (–1, 0) (1, 2) (3, )
(D) (–2, –1) (–1, 0) (2, )
The domain of the definition of the function f (x)
(A) (1, 2) (2, )
(C) (–1, 0) (1, 2) (2, )
26.
The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 + ...... upto 11th term is :
(A) 915
(B) 946
(C) 945
(D) 916
27.
The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x 42, 67,
70, y are 42 and 35 respectively, then
(A) 7/3
28.
29.
(D) 8/3
y2
The area (in sq. units) of the region A = {(x, y) :
x y + 4} is :
2
(A) 53/3
(B) 18
(C) 30
(D) 16
If a unit vector a makes angles /3 with î, with ˆj and (0, ) with k̂ , then a value of is :
4
(A)
30.
(B) 9/4
y
is equal to :
x
(C) 7/2
5
12
(B)
5
6
(C)
2
3
(D)
4
A rectangle is inscribed in a circle with a diameter lying along the line, 3y = x + 7. If the two adjacent
vertices of the rectangle are (–8, 5) and (6,5), then the area of the rectangle (in sq. units) is:(A) 72
(B) 84
(C) 98
(D) 56
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MCSIR
JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (10/04/2019) SHIFT-1
1.
If for some x R, the frequency distribution of the marks obtained by 20 students in a test is :
Marks
2
Frequency (x 1) 2
then the mean of the marks is :
(A) 2.8
(B) 3.2
x
2.
If 1 sin
cos
1
x
x
(D) 2.5
sin 2 cos 2
and 2 sin 2
cos 2
(A) 1 – 2 = x(cos2 – cos4)
(C) 1 – 2 = –2x3
3.
7
x
(C) 3.0
sin cos
x
1
3
5
2x 5 x 2 3x
x
1
1
x
, x 0; then for all 0, :
2
(B) 1 + 2 = –2x3
(D) 1 + 2 = –2(x3 + x – 1)
x4 1
x3 k 3
, then k is :
lim 2
x 1 x 1
x k x k 2
If lim
(A)
3
8
(B)
3
2
(C)
4
3
(D)
8
3
4.
If the system of linear equations
x+ y+ z= 5
x + 2y + 2z = 6
x + 3y + z = µ, (, R), has infinitely many solutions, then the value of + µ is :
(A) 12
(B) 10
(C) 9
(D) 7
5.
If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y–1 = 0, (K R), intersect at the
points P and Q, then the line 4x + 5y – K = 0 passes through P and Q for :
(A) exactly two values of K
(B) exactly one value of K
(C) no value of K
(D) infinitely many values of K
6.
Let f(x) = x2, x R. For any A R , define g(A) = {x R : f(x) A}. If S = [0, 4], then which one of
the following statements is not true ?
(A) f(g(S)) f(S)
(B) f(g(S)) = S
(C) g(f(S)) = g(S)
(D) g(f(S)) S
7.
Let f(x) = ex – x and g(x) = x2 – x, x R. Then the set of all x R, where the function h(x) = (fog)(x)
is increasing, is :
1 1
(A) 1, ,
2 2
1
(B) 0, 1,
2
1
(C) , 0 1,
2
(D) [0, )
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8.
9.
MCSIR
JEE Main Papers
Which one of the following Boolean expressions is a tautology ?
(A) p q ~ p ~ q
(B) p q p ~ q
(C) p q p ~ q
(D) p q p ~ q
2
All the pairs (x, y) that satisfy the inequality 2 sin x 2sin x 5.
(A) sin x = |siny|
(C) 2|sinx| = 3 siny
1
4sin
2
y
1 also satisfy the equation.
(B) sinx = 2 sin y
(D) 2 sinx = siny
10.
The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible
by 11 and no digit is repeated, is :
(A) 36
(B) 60
(C) 48
(D) 72
11.
Assume that each born child is equally likely to be a boy or a girl. If two families have two children each,
then the conditional probability that all children are girls given that at least two are girls is :
(A)
12.
1
11
(B)
2
5 13 23
2
1
1 2
(A) 660
(C) 680
14.
1
10
(C)
(D)
1
12
The sum
3 13
13.
1
17
2
7 13 23 33 ..... upto 10
th
2
2
2
terms, is :
1 2 3
(B) 620
(D) 600
If a directrix of a hyperbola centred at the origin and passing through the point 4, 2 3 is 5x = 4 5
and its eccentricity is e, then :
(A) 4e4 –24e2 + 35 = 0
(B) 4e4 + 8e2 – 35 = 0
(C) 4e4 – 12e2 – 27 = 0
(D) 4e4 – 24e2 + 27 = 0
sin(p 1)x sin x
x
q
If f (x)
x x2 x
x 3/2
, x0
, x0
, x0
is continuous at x = 0, then the ordered pair (p, q) is equal to :
5 1
(A) ,
2 2
3 1
(B) ,
2 2
1 3
(C) ,
2 2
3 1
(D) ,
2 2
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15.
MCSIR
If y = y(x) is the solution of the differential equation
JEE Main Papers
dy
=(tanx – y) sec2x, x , ,
dx
2 2
such that y(0) = 0, then y is equal to :
4
(A) 2
16.
1
e
(B)
1
e
2
If the line x – 2y = 12 is tangent to the ellipse
(C) e – 2
x 2 y2
1 at the point
a 2 b2
(D)
1
2
e
9
3, , then the length of the
2
latus- rectum of the ellipse is :
(A) 9
(B) 8 3
(C) 12 2
(D) 5
2
17.
The value of
sin 2x(1 cos 3x) dx , where [t] denotes the greatest integer function, is :
0
(A) –2
(B)
(C) –
(D) 2
18.
The region represented by |x – y| 2 and |x+ y| 2 is bounded by a :
(A) square of side length 2 2 units
(B) rhombus of side length 2 units
(C) square of area 16 sq. units
(D) rhombus of area 8 2 sq. units
19.
The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, –3), then
its radius is :
(A) 3 2
(B) 3
(C) 2 2
(D) 2
20.
Let A(3, 0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If
G divides BM in the ratio, 2 : 1, then cos(GOA) (O being the origin) is equal to :
(A)
21.
22.
1
30
(B)
1
6 10
(C)
1
15
(D)
1
2 15
Let f : R R be differentiable at c R and f(c) = 0. If g(x) = |f(x)|, then at x = c, g is :
(A) differentiable if f '(c) = 0
(B) not differentiable
(C) differentiable if f '(c) 0
(D) not differentiable if f '(c) = 0
If and are the roots of the quadratic equation, x2 + xsin – 2sin= 0, 0, , then
2
12 12
12 12 24
is equal to :
26
(A)
(sin 8)12
212
(B)
(sin 8)6
212
(C)
(sin 4)12
212
(D)
(sin 8)12
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23.
25.
JEE Main Papers
If the length of the perpendicular from the point (, 0, ) ( 0) to the line,
then is equal to :
(A) –1
24.
MCSIR
If
(B) 2
(C) –2
x y 1 z 1
is
1
0
1
3
,
2
(D) 1
dx
f (x)
x 1
A tan 1
2
C where C is a constant of integration, then
2
(x 2x 10)
3 x 2x 10
2
(A) A
1
and f(x) = 9(x – 1)
27
(B) A
1
and f(x) = 3(x – 1)
81
(C) A
1
and f(x) = 9(x – 1)2
54
(D) A
1
and f(x) = 3(x – 1)
54
ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid point of BC.
If the angles of elevation of the top of the tower at A and B are cot 1 3 2 and cos ec1 2 2
respectively, then the height of the tower (in metres) is:
(A) 10 5
26.
30.
(D) 25
(B) 98
(C) 76
(D) 64
4 4/3
(2)
3
(B)
3 4/3 4
(2)
2
3
(C)
3 4/3 3
(2)
4
4
(D)
4 3/4
(2)
3
If Q(0, –1, –3) is the image of the point P in the plane 3x – y +4z = 2 and R is the point (3, –1, –2), then
the area (in sq. units) of PQR is :
(A)
29.
(C) 20
(n 1)1/3 (n 2)1/3
(2n)1/3
lim
......
is equal to :
n
n 4/3
n 4/3
n 4/3
(A)
28.
100
3 3
If a1, a2, a3, ......an are in A.P. and a1 + a4 + a7 +.......+ a16 = 114, then a1 + a6 + a11 + a16 is equal to:
(A) 38
27.
(B)
65
2
(B)
91
4
(C) 2 13
(D)
91
2
If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax + bx2) (1 – 3x)15
in powers of x, then the ordered pair (a, b) is equal to :
(A) (28, 315)
(B) (–54, 315)
(C) (–21, 714)
(D) (28, 861)
(1 i)2
If a > 0 and z
, has magnitude
a i
3 1
(A) i
5 5
1 3
(B) i
5 5
2
, then z is equal to :
5
1 3
(C) i
5 5
(D)
1 3
i
5 5
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MCSIR
JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (10/04/2019) SHIFT-2
1.
The distance of the point having position vector ˆi 2ˆj 6kˆ from the straight line passing through the
point (2,3, –4) and parallel to the vector, 6iˆ 3jˆ 4kˆ is
(A) 7
(B) 4 3
(C) 2 13
(D) 6
2.
If both the mean and the standard deviation of 50 observations x1, x2 ........., x50 are equal to 16, then the
mean of (x1 – 4)2, (x2 – 4)2 ,.............(x50 – 4)2 is:
(A) 525
(B) 380
(C) 480
(D) 400
3.
A perpendicular is drawn from a point on the line
x 1 y 1 z
2
1 1
to the plane x + y + z = 3 such that the
foot of the perpendicular Q also lies on the plane x – y + z = 3. Then the co-ordinates of Q are:
(A) (2, 0, 1)
(B) (4, 0, –1)
(C) (–1, 0, 4)
(D) (1, 0, 2)
4.
The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P (2,2) meet the x - axis at Q and R,
respectively. Then the area (in sq. units) of the triangle PQR is:
14
16
(A) 3
5.
68
(B) 3
34
(C) 15
(D) 15
Letbe a real number for which the system of linear equations
x + y + z =6
4x + y – z = –2
3x + 2y – 4 z = – 5
has infinitely many solutions. Then is a root of the quadratic equation.
(A) 2 3 4 0 (B) 2 6 0
(C) 2 3 4 0
(D) 2 6 0
n
6.
1
The smallest natural number n, such that the coefficient of x in the expansion of x 2 3 is n C23 , is:
x
(A) 35
7.
(B) 38
1
5
(B) 6
1
(C) 18
1
(D) 36
If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is:
5
(A) 3 , 0
9.
(D) 58
A spherical iron ball of radius 10cm is coated with a layer of ice of uniform thickness that melts at a rate
of 50 cm3 / min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/ min)
of the ice decreases, is:
(A) 9
8.
(C) 23
The sum 1
(A) 1240
(B) (5, 0)
(C) (–5, 0)
5
(D) 3 , 0
13 23 13 23 33
13 23 33 ........ 153 1
....
1 2 3 .... 15
1 2
1 2 3
1 2 3 ..... 15
2
(B) 1860
(C) 660
(D) 620
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10.
(B) 2
(C) 2
y
2
1
(D)
2
y
2
If cos 1 x cos 1 , where 1 x 1, 2 y 2, x , then for all x, y, 4x2 – 4xy cos + y2 is equal to
(A) 4sin 2 2x 2 y2
12.
JEE Main Papers
If the line ax + y = c, touches both the curves x2 + y2 = 1 and y2 = 4 2x , then |c| is equal to:
(A) 1/2
11.
MCSIR
2
(B) 4cos 2 2x 2 y2
(C) 4 sin 2
(D) 2 sin 2
2
If x 5e x dx g(x)e x c , where c is a constant of integration, then g(–1) is equal to :
(A)
5
2
(B) 1
(C)
1
2
(D) –1
13.
The locus of the centres of the circles, which touch the circle, x2 + y2 = 1 externally, also touch the y - axis
and lie in the first quadrant, is:
(A) y 1 4x , x 0
(B) x 1 4y, y 0 (C) x 1 2y, y 0 (D) y 1 2x , x 0
14.
Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance
3
5
from the origin.
Then which one of the following points lies on any of these lines?
1 2
1 1
(A) 4 , 3
15.
2
1
1
(D) 4 , 3
The are (in sq. units) of the region bounded by the curves y = 2x and y = |x + 1|, in the first quadrant is:
3
1
(A) 2 log 2
e
16.
1
(C) 4 , 3
(B) 4 , 3
(B)
1
2
(C) loge 2
1
3
If the plane 2x – y + 2z + 3 = 0 has the distances and
2
3
3
2
(D)
3
2
units from the planes 4x – 2y + 4z + = 0 and
2x – y + 2z + =0, respectively, then the maximum value of is equal to :
(A) 15
(B) 5
(C) 13
(D) 9
17.
If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg (w) =
(A) zw i
18.
(C) zw
(B) zw i
1i
2
, then :
(D) zw
2
1 i
2
1
2
Let a, b and c be in G.P. with common ratio r, where a 0 and 0 r . If 3a, 7b and 15c are the first
three terms of an A.P., then the 4th term of this A.P. is:
7
(A) 3 a
(B) a
2
(C) 3 a
(D) 5a
/3
19.
The integral
sec
2/3
x cos ec 4/3 xdx is equal to :
/6
(A) 37/6 – 35/6
(B) 35/3 – 31/3
(C) 34/3 – 31/3
(D) 35/6– 32/3
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20.
MCSIR
JEE Main Papers
dy
2
Let y = y (x) be the solution of the differential equation, dx y tan x 2x x tan x, x 2 , 2 , such that
y (0) = 1. Then :
(A) y ' 4 y ' 4 2
(C) y 4 y 4 2
21.
2
(D) y y 2
4
4
2
Let a1, a2, a3 ,.................. be an A.P. with a6 = 2. Then the common difference of this A.P., which
maximises the product a1 a4 a5, is:
(A)
22.
(B) y ' 4 y ' 4 2
6
5
(B)
8
5
(C)
2
3
(D)
3
2
The angles A, B and C of a triangle ABC are in A.P. and a : b = 1: 3 . If c = 4 cm, then the area (in sq.
cm) of this triangle is:
(A) 4 3
(B)
2
3
(C) 2 3
(D)
4
3
23.
Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is
more than 99% is
(A) 5
(B) 6
(C) 7
(D) 8
24.
Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium.
If the top of each pillar has been connected by beams with the top of all its non adjacent pillars, then the
total number of beams is:
(A) 210
(B) 190
(C) 170
(D) 180
25.
The sum of the real roots of the equation :
x
2
6
3x
1
x 3 0 , is equal to :
3
2x
x2
(A) 6
26.
(B) 1
(C) 0
(D) –4
Let f(x) = loge(sinx), (0 < x < ) and g(x) = sin–1(e–x), (x 0). If is a positive real number such that
a = (fog)’() and b = (fog)(), then :
(A) a2 – b – a = 0
(B) a2 + b – a = –22
(C) a2 + b + a = 0
(D) a2 – b – a = 1
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27.
If the tangent to the curve y
x
2
x 3
MCSIR
29.
(C) |6 + 2| = 9
The number of real roots of the equation 5 + |2x – 1| = 2x(2x – 2) is :
(A) 2
(B) 3
(C) 4
(D) |2 + 6| = 19
(D) 1
x 2 ax b
5 , then a + b is equal to :
x 1
x 1
If lim
(A) –7
30.
, x R, x 3 , at a point (, ) (0, 0) on it is parallel to
the line 2x + 6y – 11 = 0, then :
(A) |6 + 2| = 19 (B) |2 + 6| = 11
28.
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(B) –4
(C) 5
(D) 1
The negation of the boolean expression ~ s (~ r s) is equivalent to :
(A) r
(B) s r
(C) s r
(D) ~ s ~ r
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (12/04/2019) SHIFT-1
1.
If m is the minimum value of k which the function f (x) x kx x 2 is increasing in the interval
[0, 3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :
(A) (4,3 2)
2.
(B) (4,3 3)
(C) a circle of radius
12
3
10
221
2
7
12
(C) tan
11
12
(D) tan
5
12
(B)
1
10
(C)
3
20
(D)
1
5
(B)
157
2
(C)
61
2
(D)
5 5
2
dy d 2 y
If e + xy = e, the ordered pair , 2 at x = 0 is equal to :
dx dx
y
1 1
(A) , 2
e e
7.
(B) tan
If the normal to the ellipse 3x2 + 4y2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent
to the ellipse at P passes through Q(4, 4) then PQ is equal to :
(A)
6.
(D) the line through the origin with slope 1.
If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle
formed with these chosen vertices is equilateral is :
(A)
5.
1
2
(B) a circle of radius 1.
1 x2
3
x
0,
h(x)
For
, let f (x) x , g(x) tan x and
.
If
(x)
=
((hof)og)(x),
then
2
3
1 x2
is equal to :
(A) tan
4.
(D) (5,3 6)
The equation |z – i| = |z – 1|, i 1 , represents:
(A) the line through the origin with slope –1.
3.
(C) (3,3 3)
1 1
(B) , 2
e e
1
1
(C) , 2
e e
1
1
(D) , 2
e e
x 2 y 1 z 1
intersects the plane 2x + 3y – z + 13 = 0 at a point P and the plane 3x
3
2
1
+ y + 4z = 16 at a point Q, then PQ is equal to :
If the line
(A) 2 14
(B) 14
(C) 2 7
(D) 14
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8.
1 33
(B) cos
65
(C)
56
sin 1
2
65
(D)
9
cos1
2
65
n
n
r 1
r 1
r nlim
r is equal to:If and are the roots of the equation 375x2 – 25x – 2 = 0, then nlim
(A)
10.
JEE Main Papers
12
3
The value of sin 1 sin 1 is equal to :
13
5
1 63
(A) sin
65
9.
MCSIR
If
21
346
2
0
(B)
29
358
(C)
1
12
(D)
7
116
cot x
dx m( n) , then m.n is equal to :
cot x cos ecx
(A) –1
(B) 1
(C)
1
2
(D)
1
2
11.
The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining
21 are distinct, is :
(A) 220
(B) 220–1
(C) 220+1
(D) 221
12.
If the data x1, x2 .........., x10 is such that the mean of first four of these is 11, the mean of the remaining six
is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is :
(A) 4
(B) 2
(C) 2
(D) 2 2
13.
14.
15.
5 5
4
2
The number of solutions of the equation 1 sin x cos 3x, x , is :
2 2
(A) 5
(B) 4
(C) 7
(D) 3
Let Sn denote the sum of the first n terms of an A.P. If S4 = 16 and S6 = –48, then S10 is equal to:
(A) –320
(B) –260
(C) –380
(D) –410
5 2 1
If B 0 2 1 is the inverse of a 3 × 3 matrix A, then the sum of all values of for which
3 1
det (A) + 1 = 0, is :
(A) 0
16.
(B) 2
(C) 1
(D) –1
Let a random variable X have a binomial distribution with mean 8 and variance 4. If P(x 2)
k
216
,
then k is equal to :
(A) 17
(B) 1
(C) 121
(D) 137
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17.
18.
1
2
Consider the differential equation, y dx x y dy 0 . If value of y is 1 when x = 1, then the value
of x for which y = 2, is :
1 1
2
e
f (x)
3
e
2
(C)
5 1
2
e
(D)
3 1
2
e
1
. If
48
4t 3dt (x 2)g(x) , then lim g(x) is equal to :
x 2
(A) 24
(B) 36
(C) 12
(D) 18
The coefficient of x18 in the product (1 + x)(1 – x)10(1 + x + x2)9 is :
(A) –84
21.
(B)
Let f : R R be a continuously differentiable function such that ƒ(2) = 6 and f '(2)
6
20.
JEE Main Papers
If the truth value of the statement P (~q r) is false(F), then the truth values of the statements p, q,
r are respectively :
(A) F, T, T
(B) T, F, ,F
(C) T, T, F
(D) T, F, T
(A)
19.
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(B) 84
(C) 126
(D) –126
For x R, let [x] denote the greatest integer x, then the sum of the series
1 1 1 1 2
1 99
3 3 100 3 100 ....... 3 100
is :
(A) –153
(B) –133
(C) –131
(D) –135
22.
The equation y = sinx sin(x + 2) – sin2(x + 1) represents a straight line lying in :
(A) second and third quadrants only
(B) third and fourth quadrants only
(C) first, third and fourth quadrants
(D) first, second and fourth quadrants
23.
Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the hyperbola
8x2 – y2 = 8. If S and S' denote the foci of the hyperbola where S lies on the positive
x-axis then P divides SS' in a ratio :
(A) 5 : 4
(B) 14 : 13
(C) 2 : 1
(D) 13 : 11
24.
ˆ ˆj kˆ and ˆi kˆ is minimum, then
If the volume of parallelopiped formed by the vectors ˆi ˆj k,
is equal to :
(A)
25.
3
(B)
1
3
(C)
1
3
(D) 3
A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate
25 cm/sec., then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the
horizontal ground when the top of the ladder is 1m above the ground is:
(A) 25 3
(B) 25
(C)
25
3
(D)
25
3
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26.
28.
4 2
(B)
1 4
4 2
(D)
1 4
2x 3 1
dx is equal to :
x4 x
Here C is a constant of integration
The integral
1
(x 3 1)2
| x 3 1|
1
(x 3 1)
C (C) log e
C
(B) log e
C (D) log e
2
2
| x3 |
x2
| x2 |
If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90º, then
the length (in cm) of their common chord is :
(A)
30.
4 2
(C)
1 4
Let a 3iˆ 2ˆj 2kˆ and b ˆi 2ˆj 2kˆ be two vectors. If a vector perpendicular to both the vectors
a b and a b has the magnitude 12 then one such vector is :
ˆ
ˆ
ˆ
ˆ
(A) 4(2iˆ 2ˆj k)
(B) 4( 2iˆ 2ˆj k)
(C) 4(2iˆ 2ˆj k)
(D) 4(2iˆ 2ˆj k)
x3 1
(A) log e
C
x
29.
JEE Main Papers
2 3
If A is a symmetric matrix and B is a skew symmetrix matrix such that A B
, then AB is
5
1
equal to
4 2
(A)
1 4
27.
MCSIR
60
13
(B)
120
13
(C)
13
2
(D)
13
5
If the area (in sq. units) of the region {(x, y) : y2 4x, x + y 1, x 0, y 0}is a 2 b , then a – b is
equal to
(A)
8
3
(B)
10
3
(C) 6
(D)
2
3
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2019 (12/04/2019) SHIFT-2
1.
Let A , B and C be sets such that A B C. Then which of the following statements is not true?
(A) If (A–C) B, then A B
(B) C A C B C
(C) If (A–B) C, then A C
(D) B C
2.
If 20C1 + (22) 20C2 + (32) 20C3 + ........ + (202) 20C20 = A(2), then the ordered pair (A, ) is equal to :
(A) (420, 18)
(B) (380, 19)
(C) (380, 18)
(D) (420, 19)
3.
A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes
and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then an
equation of the line L is
(A)
3 1 x
(B)
3 1 y 8 2
(C) 3x y 8
4.
3 1 x
3 1 y 8 2
(D) x 3y 8
A value of (, /3), for which
1 cos 2
2
sin 2
2
cos
1 sin
cos 2
sin 2
(A)
5.
7
24
4 cos 6
4 cos 6 0 , is :
1 4 cos 6
(B)
18
(C)
9
(D)
7
36
If [x] denotes the greatest integer x, then the system of linear equations
[sin] x + [–cos]y = 0; [cot] x + y = 0 :
2 7
(A) have infinitely many solutions if , ,
2 3 6
7
2
(B) have infinitely many solutions if , and has a unique solution if ,
2 3
6
7
2
(C) has a unique solution if , and have infinitely many solutions if ,
6
2 3
(D) has a unique solution if , 2 , 7
2 3 6
6.
lim
x 2sin x
x 2 2sin x 1 sin 2 x x 1
(A) 3
(B) 2
x 0
is :
(C) 6
(D) 1
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7.
If a1, a2, a3, ..... are in A.P. such that a1 + a7 + a16 = 40, then the sum of the first 15 terms of this A.P. is
:
(A) 200
(B) 280
(C) 120
(D) 150
8.
The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines
ˆ ˆ
r (i j) ˆi 2ˆj kˆ and r (iˆ ˆj) iˆ ˆj 2kˆ is :
(A)
9.
10.
3
(B) –108
A value of such that
(A)
1
2
(D) 3
(C) –72
(D) –36
dx
9
log e is :
(x )(x 1)
8
(C)
(B) 2
1
2
(D) –2
A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (–1, 1) and (2, 3). Then
the centroid of this triangle is :
1
(A) ,1
3
14.
1
3
Let R and the three vectors a ˆi ˆj 3kˆ , b 2iˆ ˆj kˆ and c ˆi 2ˆj 3kˆ . Then the set
S { : a, b and c are coplanar}
(A) is singleton
(B) Contains exactly two numbers only one of which is positive
(C) Contains exactly two positive numbers
(D) is empty
13.
(C)
6
1 x8
3
The term independent of x in the expansion of . 2x 2 2 is equal to :
60 81
x
1
12.
1
3
If , and are three consecutive terms of a non constant G.P. such that the equations x2 + 2x +
= 0 and x2 + x – 1 = 0 have a common root, then ( + ) is equal to:
(A)
(B) 0
(C)
(D)
(A) 36
11.
(B)
1
(B) , 2
3
7
(C) 1,
3
1 5
(D) ,
3 3
tan x tan
dx A(x)cos2 + B(x)sin2 + C, where C
tan x tan
is a constant of integration, then the functions A(x) and B(x) are respectively :
(A) x – and loge|cos(x – )|
(B) x + and loge|sin(x – )|
(C) x – and loge|sin(x – )|
(D) x + and loge|sin(x + )|
Let (0, /2) be fixed. If the integral
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15.
A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the
point
(A) (3, 10)
(B) (2, 3)
(C) (1, 5)
(D) (3, 5)
16.
For and initial screening of an admission test, a candidate is given fifty problems to solve. If the probability
4
that the candidate can solve any problem is , then the probability that he is unable to solve less than
5
two problems is :
(A)
17.
316 4
25 5
48
(B)
54 4
5 5
49
(C)
164 1
25 5
48
(D)
201 1
5 5
49
x
sin x cos x
The derivative of tan 1
, with respect to 2 , where x 0, is :
2
sin x cos x
(A)
1
2
(B)
2
3
(C) 1
(D) 2
18.
Let S be the set of all R such that the equation, cos2x + sinx = 2–7 has a solution. Then S is
equal to
(A) [2, 6]
(B) [3, 7]
(C) R
(D) [1, 4]
19.
The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at
the point
5
(A) , 1
2
5
(B) ,1
2
5
(C) , 1
2
5
(D) ,1
2
20.
A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students
can randomly be selected from this group such that there is at least one boy and at least one girl in each
team, is 1750, then n is equal to :
(A) 25
(B) 28
(C) 27
(D) 24
21.
The equation of a common tangent to the curves, y2 = 16x and xy = –4 is :
(A) x + y + 4 = 0
(B) x – 2y + 16 = 0 (C) 2x – y + 2 = 0
(D) x – y + 4 = 0
22.
Let z C with Im(z) = 10 and it satisfies
(A) n = 20 and Re(z) = –10
(C) n = 40 and Re(z) = –10
23.
2z n
2i 1 for some natural number n. Then :
2z n
(B) n = 20 and Re(z) = 10
(D) n = 40 and Re(z) = 10
The general solution of the differential equation (y2 – x3)dx – xydy = 0 (x 0) is :
(where c is a constant of integration)
(A) y2 + 2x3 + cx2 = 0
(B) y2 + 2x2 + cx3 = 0
(C) y2 – 2x3 + cx2 = 0
(D) y2 – 2x2 + cx3 = 0
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24.
(B)
x
(A) 1/2
(C)
1
loss
4
(D)
1
gain
2
(x 1)(x2 5x 6)
x 2 6x 8
(B) –3/2
is equal to :
(C) 3/2
(D) –1/2
The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45°
from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation
of the top of the tower from B be 30°. then the distance (in m) of the foot of the tower from the point A
is:
(A) 15 3 3
27.
1
loss
2
Let f(x) = 5 – |x –2| and g(x) = |x + 1|, x R. If f(x) attains maximum value at and g(x) attains minimum
value at , then lim
26.
JEE Main Papers
A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice),
wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw.
Then the expected gain/loss (in Rs.) of the person is :
(A) 2 gain
25.
MCSIR
(B) 15 3 3
(C) 15 1 3
(D) 15 5 3
The Boolean expression p ~ q is equivalent to :
(A) (~ p) q
(B) p q
(C) q ~p
(D) p q
28.
A plane which bisects the angle between the two given planes 2x y + 2z 4 = 0 and
x 2y + 2z 2 = 0, passes through the point :
(A) (2, 4, 1)
(B) (2, 4, 1)
(C) (1, 4, 1)
(D) (1, 4, 1)
29.
If the area (in sq. units) bounded by the parabola y2 = 4x and the line y = x, > 0, is
1
,then is
9
equal to
(A) 24
30.
(B) 48
(C) 4 3
(D) 2 6
An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the
following points ?
(A) (1, 2 2)
(B) (2, 2)
(C) (2, 2 2)
(D) ( 2, 2)
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (7/01/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by
the parabola y2 = x and the straight line y = x, is :
(1) (1/3)(12–1)
(2) (1/6)(12–1)
(3) (1/3)(6–1)
(4) (1/6)(24–1)
2.
Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is
(1) 56
(2) (1/2)(6!)
(3) 6!
(4) (5/2)(6!)
3.
An unbiased coin is tossed 5 times, Suppose that a variable X is assigned the value k when k consecutive
heads are obtained for k = 3, 4, 5, otherwise X takes the value –1. Then the expected value of X, is :
(1) 1/8
(2) 3/16
(3) – 1/8
(4) – 3/16
4.
z 1
If Re
1 , where z = x + iy, then the point (x, y) lies on a :
2z i
2
3
(1) straight line whose slope is –
(3) circle whose diameter is
5.
(2) straight line whose slope is
3
2
1 3
(4) circle whose centre is at ,
2 2
5
2
If f(a + b + 1 – x) = f(x), for all x, where a and b are fixed positive real numbers, then
b
1
x f (x) f (x 1) dx is equal to :
(a b) a
b 1
(1)
b 1
f x dx
(2)
a 1
6.
(3)
a 1
b 1
f x 1 dx
a 1
(4)
f x 1 dx
a 1
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the
length of its latus rectum is :
3
2
The logical statement (p q) (q ~p) is equivalent to :
(1) ~ q
(2) ~ p
(3) p
(1) 2 3
7.
b 1
f x dx
(2)
3
(3)
(4) 3 2
(4) q
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8.
9.
10.
The greatest positive integer k, for which 49k + 1 is a factor of the sum
49125 + 49124 + ......+ 492 + 49 + 1, is :
(1) 32
(2) 60
(3) 65
12.
14.
1
4
(2)
4
3
(3) 4
(4) –4
If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :
(1) – 64
(2) 128
(3) – 128
(4) – 32
Let be a root
of equation x2
1 1
1
1
+ x + 1 = 0 and the matrix A
3
2
1
(2) A2
(3) A3
5
If g(x) = x2 + x – 1 and (gof)(x) = 4x2 – 10x + 5, then f is equal to :
4
3
1
1
(1)
(2)
(3)
2
2
2
1
2 , then the matrix
4
(4) I3
(4)
3
2
Let & be two real roots of equation (k + 1)tan2x – 2 tan x = 1 – k, where k ( – 1) and are
real numbers. If tan2( + ) = 50, then a value of is :
(1) 5 2
15.
(4) 63
1
dy
5
3
tan cot
2
at =
is :
2 , , , then the value of
2
4
d
6
1 tan sin
If y() =
A31 is equal to :
(1) A
13.
JEE Main Papers
A vector a ˆi 2ˆj kˆ ( R) lies in the plane of the vectors, b iˆ ˆj and c iˆ ˆj 4kˆ . If
a bisects the angle between b and c , then :
(1) a.iˆ 3 0
(2) a.kˆ 4 0
(3) a.iˆ 1 0
(4) a.kˆ 2 0
(1)
11.
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(2) 10 2
(3) 10
(4) 5
Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6).
Then the image of R in the plane P is :
(1) (6, 5, 2)
(2) (6, 5, – 2)
(3) (4, 3, 2)
(4) (3, 4, – 2)
1/3
16.
17.
Let
xk
(1)
1
3
+
yk
=
ak,
dy y
(a, k > 0) and
dx x
2
(2)
3
0 then k is :
(3)
4
3
(4) 3/2
Let the function, f : [– 7, 0] R be continuous on [– 7, 0] and differentiable on (– 7, 0). If f(– 7) = – 3
and f '(x) 2 for all x (– 7, 0), then for all such functions f, f(–1) + f(0) lies in the interval :
(1) [–6, 20]
(2) (– , 20]
(3) (– , 11]
(4) [–3, 11]
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x
y dy
Let y = y(x) is the solution of the differential equation, e 1 e such that y(0) = 0, then y(1) is
dx
equal to :
(1) loge 2
(2) 2e
(3) 2 + loge 2
(4) 1 + loge 2
19.
Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is –1/2 ,
then the greatest number amongst them is :
(1) 16
(2) 27
(3) 7
(4) 21/2
20.
If the system of linear equations :
2x + 2ay + az = 0
2x + 3by + bz = 0
2x + 4cy + cz = 0, where a, b, c R are non-zero and distinct ; has a non-zero solution, then :
(1)
1 1 1
, , are in A.P..
a b c
(2) a, b, c are in A.P.
(3) a, b, c are in G.P.
(4) a + b + c = 0
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL VALUE with
two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto TWO decimal
places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
3x 33x 12
lim x/2 1x is equal to :
21.
x 2 3
3
22.
If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is
16, then m + n is equal to ______ .
23.
If the sum of the coefficients of all even powers of x in the product (1 + x + x2 + .... + x2n)
(1 – x + x2 – x3 + ... + x2n) is 61, then n is equal to ______ .
24.
Let S be the set of points where the function f(x) = |2 – |x – 3||, x R is not differentiable, then
f f x is equal to :
xS
25.
3
Let A(1, 0), B(6, 2) and C , 6 be the vertices of a triangle ABC. If P is a point inside the triangle
2
ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment
7 1
PQ, where Q is the point , , is ______ .
6 3
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (7/01/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
If 3x + 4y = 12
2 is a tangent to the ellipse
x2
a2
y2
= 1 for some a R, then the distance between
9
the foci of the ellipse is :
(A) 2 5
(B) 2 7
(C) 2 2
(D) 4
2.
Let A, B, C and D be four non-empty sets. The contrapositive statement of "If A B and B D, then
A C" is :
(A) If A C, then B A and D B
(B) If A C, then A B and B D
(C) If A C, then A B and B D
(D) If A C, then A B or B D
3.
The coefficient of x7 in the expression (1 + x)10 + x(1 + x)9 + x2(1 + x)8 + .... + x10
(A) 420
(B) 330
(C) 210
(D) 120
4.
In a workshop, there are five machines and the probability of any one of them to be out of service on a
3
1
3
day is . If the probability that at most two machines will be out of service on the same day is k,
4
4
then k is equal to :
(A)
5.
17
2
(B) 4
(C)
17
4
(D)
17
8
The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y
is :
(A) 2x – 3y = 0
(B) 3x – 2y = 0
(C) 5x – 7y = 0
(D) 7x – 5y = 0
2
6.
The value of for which 4
e
x
dx = 5, is :
1
(A) loge 2
7.
(B) loge 2
(C) loge 4
3
(D) loge 3
2
If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + .... is (102)m, then m
is equal to :
(A) 10
(B) 25
(C) 5
(D) 20
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8.
If
10.
11.
JEE Main Papers
3 i sin
,[0, 2], is a real number, then an argument of sinicosis :
4 i cos
(A) – tan –1 4
3
9.
MCSIR
(B) – tan –1 3
4
(C) – tan –1 3
4
(D) tan –1 4
3
Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i + j – 2)aji, where i, j = 1,2,3. If the
deteminant of B is 81, then the determinant of A is :
(A) 1/9
(B) 1/81
(C) 1/3
(D) 3
f (x)
Let ƒ(x) be a polynomial of degree 5 such that x = ± 1 are its critical points. If lim 2 3 = 4 ,
x 0
x
then which one of the following is not true ?
(A) ƒ(1) – 4ƒ(–1) = 4.
(B) x = 1 is point of maxima and x = –1 is a point of minimum of ƒ.
(C) ƒ if an odd function.
(D) x = 1 is a point of minima and x = –1 is a point of maxima of ƒ.
The number of ordered pairs (r,k) for which 6Cr = (k2 – 3)Cr + 1, where k is an integer, is :
(A) 4
(B) 6
(C) 2
(D) 3
9
12.
Let a1, a2, a3, ... be a G. P. such that a1 < 0, a1 + a2 = 4 and a3 + a4 =16. If
ai
= 4, then is equal to :
i 1
(A) 171
13.
14.
(B)
511
3
(C) –171
Let a , b and c be three unit vectors such that a + b + c = 0 . If = a b + b c + c a and
d = a × b + b × c c × a , then the ordered pair,, (,
( d ) is equal to :
3
3
3
3
(A) ,3a c
(B) ,3c b
(C) ,3a b
(D) ,3b c
2
2
2
2
Let y = y(x) be the solution curve of the differential equation, y 2 x
curve intersects the x-axis at a point whose abscissa is :
(A) 2 + e
(B) 2
(C) 2 – e
15.
(D) –513
dy
= 1 , satisfying y(0) = 1. This
dx
(D) – e
If 1 and 2 be respectively the smallest and the largest values of in (0, 2) – {} which satisfy the
equation, 2cot2–
(A)
2
3
5
+ 4 = 0, then
sin
(B)
3
2
cos
2
3 d is equal to :
1
(C)
1
3 6
(D)
9
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16.
Let and be the roots of the equation x2 – x – 1 = 0. If pk = ()k + ()k , k 1, then which one of the
following statements is not true ?
(A) (p1 + p2+ p3+ p4+ p5) = 26
(B) p5 = 11.
(C) p5 = p2 p3
(D) p3 = p5 p4
17.
The area (in sq. units) of the region {(x, y) R24x2 y x} is :
(A)
18.
(B)
128
3
(C)
124
3
(D)
127
3
The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 – 4x2 + 8x + 11, when
x is :
(A)
19.
125
3
4 7
3
(B)
2
3
7 2
3
(C)
(D)
4 5
3
Let y = y(x) be a function of x satisfying y 1 x 2 = k – x 1 y 2 where k is a constant and
dy
1
1
1
y = – . Then
at x = , is equal to :
2
dx
2
4
(A) –
20.
(B)
5
2
(C) –
5
4
(D)
2
5
Let the tangents drawn from the origin to the circle, x2 + y2 – 8x – 4y + 16 = 0 touch it at the points A
and B. The (AB)2 is equal to :
(A)
5
2
32
5
(B)
64
5
(C)
52
5
(D)
56
5
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal
places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
21.
If the system of linear equations,
x+ y+ z= 6
x + 2y + 3z = 10
3x + 2y + z =
has more than two solutions, then is equal to ________.
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If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (, 7, 1) is
5 7 17
, , , then is equal to ________.
3 3 3
23.
1 1
If the function ƒ defined on , by
3 3
1
1 3x
log e
, when x 0
x
1
2x
ƒ(x) =
k
, when x 0
is continuous, then k is equal to _________.
24.
If the mean variance of eight numbers 3, 7, 9, 12, 13, 20, x and y be 10 and 25 respectively, then x y
is equal to _______.
25.
Let X = {n N : 1 n}. If A = {n X : n is a multiple of 2} and B = {n X : n is a
multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is
_______.
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JEE Main Papers
MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (8/01/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
For which of the following ordered pairs (µ, ), the system of linear equations
x + 2y + 3z = 1
3x + 4y + 5z = µ
4x + 4y + 4z =
is inconsistent ?
(1) (4, 6)
(2) (3, 4)
(3) (1, 0)
2.
2
Let y = y(x) be a solution of the differential equation, 1 x
(4) (4, 3)
3
dy
1
,
1 y 2 0,| x | 1 . If y
dx
2 2
1
then y
is equal to :
2
(1)
3.
(2)
3
2
(3)
1
2
(4)
3
2
If a, b and c are the greatest values of 19Cp, 20Cq and 21Cr respectively, then :
(1)
4.
1
2
a
b
c
11 22 42
(2)
a
b
c
10 11 42
Which one of the following is a tautology ?
(1) (P (P Q)) Q (2) P (P Q)
(3)
a
b
c
11 22 21
(4)
a
b
c
10 11 21
(3) Q (P (P Q)) (4) P (P Q)
5.
Let f : R R be such that for all xR (21 + x + 21–x), f(x) and (3x + 3–x) are in A.P., then the minimum value
of f(x) is :
(1) 0
(2) 4
(3) 3
(4) 2
6.
The locus of a point which divides the line segment joining the point (0, – 1) and a point on the parabola,
x2 = 4y, internally in the ratio 1 : 2, is :
(1) 9x2 – 12y = 8
(2) 4x2 – 3y = 2
(3) x2 – 3y = 2
(4) 9x2 – 3y = 2
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For a > 0, let the curves C1 : y2 = ax and C2 : x2 = ay intersect at origin O and a point P. Let the line
x = b (0 < b < a) intersect the chord OP and the x-axis at points Q and R, respectively. If the line x = b
bisects the area bounded by the curves, C1 and C2, and the area of OQR
equation :
(1) x6 – 12x3 + 4 = 0
8.
The inverse function of f (x)
82x 82x
(2)
1
1 x
(log8 e) log e
4
1 x
(3)
1
1 x
log e
4
1 x
(4)
1
1 x
log e
4
1 x
(3)
1
e
1
2
2 x
3x
lim 2
x 0 7x 2
is equal to :
(2)
1
e
2
Let f(x) = (sin(tan–1 x) + sin(cot–1x))2 – 1, |x| > 1. If
(4) x6 – 6x3 + 4 = 0
, x (1,1) , is ____.
1
1 x
(log8 e) log e
4
1 x
(1) e
10.
82x 82x
(3) x6 + 6x3 – 4 = 0
(1)
2
9.
(2) x6 – 12x3 – 4 = 0
, then 'a' satisfies the
2
(4) e2
dy 1 d
(sin 1 (f (x))) and y
dx 2 dx
3 6 , then
y 3 is equal to :
(1)
3
(2)
2
3
(3)
6
(4)
5
6
11.
If the equation, x2 + bx + 45 = 0 (b R) has conjugate complex roots and they satisfy| z 1| 2 10 ,
then :
(1) b2 + b = 12
(2) b2 – b = 42
(3) b2 – b = 30
(4) b2 + b = 72
12.
The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these
10 observations is multiplied by p and then reduced by q, where p 0 and q 0. If the new mean and
new s.d. become half of their original values, then q is equal to :
(1) – 20
(2) – 5
(3) 10
(4) – 10
13.
If
cos x dx
2
3
6
sin x(1 sin x) 3
f (x)(1 sin
6
1
x)
c where c is a constant of integration, then f is
3
equal to :
(1)
9
8
(2)
9
8
(3) 2
(4) – 2
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MCSIR
Let A and B be two independent events such that P(A)
JEE Main Papers
1
1
and P(B) . Then, which of the following
3
6
is TRUE ?
(1) P(A / (A B))
15.
7
6 6
(2)
5
7
(3)
(2) 1
7
6 3
(3) – 3
The shortest distance between the lines
(1) 2 30
18.
2
3
(4) P(A '/ B')
1
3
(4)
5
3 3
Let two points be A(1, – 1) and B(0, 2). If a point P(x', y') be such that the area of PAB = 5 sq. units
and it lies on the line, 3x + y – 4 = 0, then a value of is :
(1) 4
17.
(3) P(A / B)
ˆ v iˆ ˆj 3kˆ
Let the volume of a parallelopiped whose coterminous edges are given by u ˆi ˆj k,
and w 2iˆ ˆj kˆ be 1 cu. unit. If be the angle between the edges u and w , then cos can be :
(1)
16.
1
1
(2) P(A / B')
4
3
(2)
7
30
2
(4) 3
x 3 y 8 z 3
x 3 y7 z 6
and
is :
3
1
1
3
2
4
(3) 3
(4) 3 30
Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a point P in the first quadrant. If the normal to
1
, 0 and (0, ), then is equal to :
this ellipse at P meets the co-ordinate axes at
3 2
(1)
19.
2
3
(2)
2
3
(3)
2 2
3
(4)
2
3
x2
If c is a point at which Rolle's theorem holds for the function, f (x) log e
in the interval
7x
[3, 4], where R, then f "(c) is equal to :
(1)
1
24
(2)
1
12
(3)
3
7
(4)
1
12
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Let f(x) = xcos–1 (– sin |x|), x , , then which of the following is true ?
2 2
(1) f '(0)
2
(2) f ' is decreasing in , 0 and increasing in 0,
2
2
(3) f is not differentiable at x = 0
(4) f ' is increasing in , 0 and decreasing in 0,
2
2
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal
places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
21.
22.
An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which
4 marbles can be drawn so that at the most three of them are red is _____.
3
Let the normal at a point P on the curve y2 – 3x2 + y + 10 = 0 intersect the y-axis at 0, . If m is the
2
slope of the tangent at P to the curve, then |m| is equal to _____.
23.
The least positive value of 'a' for which the equation, 2x 2 (a 10)x
24.
The sum
33
2a has real roots is ______.
2
20
(1 2 3 ...... k) is _____.
k 1
25.
The number of all 3 × 3 matrices A, with entries from the set {–1, 0, 1} such that the sum of the diagonal
elements of AAT is 3, is _______.
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (8/01/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
Let A and B be two events such that the probability that exactly one of them occurs is 2/5 and the
probability that A or B occurs is 1/2, then the probability of both of them occur together is :
(1) 0.10
(2) 0.20
(3) 0.01
(4) 0.02
2.
Let S be the set of all real roots of the equation, 3x(3x – 1) + 2 = |3x –1| + |3x – 2|. Then S :
(1) is a singleton.
(2) is an empty set.
(3) contains at least four elements.
(4) contains exactly two elements.
3.
The mean and variance of 20 observations are found to be 10 and 4, respectively. On rechecking, it was
found that an observation 9 was incorrect and the correct observation was 11. Then the correct variance is
(1) 4.01
(2) 3.99
(3) 3.98
(4) 4.02
4.
5.
Let a iˆ 2 ˆj kˆ and b iˆ ˆj kˆ be two vectors. If c is a vector such that b c b a and
c . a 0 then c . b is equal to :
(1) 1/2
(2) – 3/2
(3) – 1/2
(4) – 1
Let f : (1, 3) R be a function defined by f(x) =
x x
, where [x] denotes the greatest integer x.
1 x2
Then the range of f is :
6.
2 3 3 4
(1) , ,
5 5 4 5
2 4
(2) ,
5 5
3 4
(3) ,
5 5
2 1 3 4
(4) , ,
5 2 5 5
If and be the coefficients of x4 and x2 respectively in the expansion of
x
6
6
x 2 1 x x 2 1 , then :
(1) = – 30
(2) = – 132
7.
(3) = 60
(4) = 60
If a hyperbola passes through the point P(10, 16) and it has vertices at (±6, 0), then the equation of the
normal to it at P is :
(1) 3x + 4y = 94
(2) x + 2y = 42
(3) 2x + 5y = 100
(4) x + 3y = 58
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8.
lim
x
0
t sin 10t dt
JEE Main Papers
is equal to :
x
x 0
(1) 0
9.
MCSIR
(2) 1/10
(3) – 1/10
If a line, y = mx + c is a tangent to the circle (x – 3)2 + y2 = 1 and it is perpendicular to a line L1, where
1 1
,
L1 is the tangent to the circle, x2 + y2 = 1 at the point
; then :
2 2
(1) c2 + 7c + 6 = 0
(2) c2 – 6c + 7 = 0
(3) c2 – 7c + 6 = 0
100
10.
11.
12.
1 i 3
2k
let
. If a = (1 + ) and b =
2
k 0
equation :
(1) x2 + 101x + 100 = 0
(3) x2 – 102x + 101 = 0
(4) c2 + 6c + 7 = 0
100
3k
, then a and b are the roots of the quadratic
k 0
(2) x2 + 102x + 101 = 0
(4) x2 – 101x + 100 = 0
7 4 1
The mirror image of the point (1, 2, 3) in a plane is , , . Which of the following points lies
3 3 3
on this plane ?
(1) (1, –1, 1)
(2) (–1, –1, 1)
(3) (1, 1, 1)
(4) (–1, –1, –1)
The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point
(2, 2) is :
(2) 2 2
(1) 2
13.
(4) – 1/5
Which of the following statements is a tautology ?
(1) ~(p ~q) p q
(3) p ~q) p q
(3) 4 2
(4)
2
(2) ~(p ~q) p q
(4) ~(p ~q) p q
2
14.
dx
If I
, then :
2x 3 9x 2 12x 4
1
(1)
15.
1
1
< I2 <
6
2
2 2
If A =
and I =
9 4
(1) 6I – A
16.
(2)
1
1
< I2 <
8
4
(3)
1
1
< I2 <
9
8
(4)
1
1
< I2 <
16
9
1 0
0 1 , then 10A–1 is equal to :
(2) A – 6I
(3) 4I – A
The area (in sq. units) of the region {(x, y) R2 : x2 y 3 – 2x}, is :
(1) 31/3
(2) 32/3
(3) 29/3
(4) A – 4I
(4) 34/3
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17.
f 1 f c
f ' c
1 c
(3) |f(c) + f(1)| < (1 + c) |f '(c)|
19.
20.
(2) |f(c) – f(1)| < |f '(c)|
(4) |f(c) – f(1)| < (1 – c) |f '(c)|
The differential equation of the family of curves, x2 = 4b(y + b), b R, is :
(1) xy'' = y'
(2) x(y')2 = x + 2yy'
(3) x(y')2 = x – 2yy'
The system of linear equations
x + 2y + 2z = 5
2x + 3y + 5z = 8
4x + y + 6z = 10 has :
(1) No solution when = 2
(3) no solution when = 8
(4) x(y')2 = 2yy' – x
(2) infinitely many solutions when = 2
(4) a unique solution when = – 8
If the 10th term of an A. P. is 1/20 and its 20th term is 1/10, then the sum of its first 200 terms is :
(1) 50
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Let S be the set of all functions f : [0, 1]R,
which are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S, there exists
a c (0, 1), depending on f, such that :
(1)
18.
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1
4
(2) 100
(3) 50
(4) 100
1
2
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
21.
Let a line y = mx(m > 0) intersect the parabola, y2 =x at a point P, other than the origin. Let the tangent
to it at P meet the x-axis at the point Q. If area (OPQ) = 4
sq. units, then m is equal to _______.
22.
Let f(x) be a polynomial of degree 3 such that f(–1) = 10, f(1) = – 6, f(x) has a critical point at x = –1 and
f '(x) has a critical point at x = 1. Then f(x) has a local minima at x = _______.
23.
If
24.
The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the
word 'EXAMINATION' is ________.
2 sin
1 cos 2
1
and
7
1 cos 2
1
, 0, , then tan ( + 2) is equal to ________.
2
2
10
n n 1 2n 1
is equal to________.
4
n 1
7
25.
The sum,
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (9/01/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of
intersection of the lines x + 3y–1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and
P also passes through the point :
(1) (–9, –7)
(2) (–9, –6)
(3) (7, 6)
(4) (9, 7)
2.
The product
(1)
3.
1
24
to is equal to :
(2) 2
(3)
5
6
(2)
1
54
(1)
bc
ca
(2) 1
1
4
(2)
1
36
(4)
1
18
f (c) f (a)
is greater than :
f (b) f (c)
(3)
ca
bc
1
2 2
(3)
1
2
The number of real roots of the equation, e4x + e3x – 4e2x + ex + 1 = 0 is :
(1) 3
(2) 4
(3) 1
2
7.
(3)
(4)
ba
ba
(4)
1
2
3
3
The value of cos3 .cos sin 3 .sin is :
8
8
8
8
(1)
6.
(4) 1
Let f be any function continuous on [a, b] and twice differentiable on (a, b). If for all x (a, b),
f '(x) > 0 and f ''(x) < 0, then for any c (a, b),
5.
1
22
A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate
of 50 cm3/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of
ice decreases, is :
(1)
4.
1
1
1 1
....
16
48
128
4
2 .4 .8 .16
The value of
(4) 2
x sin8 x
sin8 x cos8 x dx is equal to :
0
(1) 2
(2) 4
(3) 2
(4)
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If for some and in R, the intersection of the following three planes :
x + 4y – 2z = 1
x + 7y – 5z =
x + 5y + z = 5
is a line in R3, then + is equal to :
(1) 0
(2) –10
(3) 10
(4) 2
x 2 y2
x 2 y2
1 and the hyperbola,
1 respectively
18 4
9
4
and (e1, e2) is a point on the ellipse, 15x2 + 3y2 = k, then k is equal to :
(1) 14
(2) 15
(3) 17
(4) 16
9.
If e1 and e2 are the eccentricities of the ellipse,
10.
sin(a 2)x sin x
x
b
If f (x)
2 1/3
1/3
(x 3x ) x
x 4/3
; x0
; x0
; x0
is continuous at x = 0, then a + 2b is equal to :
(1) –2
(2) 1
11.
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(3) 0
(4) –1
1 1 2
| adjB |
If the matrices A 1 3 4 , B = adj A and C = 3A, then
is equal to :
|C|
1 1 3
(1) 16
(2) 2
(3) 8
(4) 72
12.
A circle touches the y-axis at the point (0, 4) and passes through the point (2, 0). Which of the following
lines is not a tangent to this circle ?
(1) 4x – 3y + 17 = 0
(2) 3x + 4y – 6 = 0
(3) 4x + 3y – 8 = 0
(4) 3x – 4y – 24 = 0
13.
Let z be a complex number such that
(1) 10
14.
(2)
7
2
If f '(x) = tan–1(secx + tanx),
(1)
1
4
(2)
5
z i
1 and z . Then the value of |z + 3i| is :
2
z 2i
(3)
15
4
(4) 2 3
x , and f(0) = 0, then f(1) is equal to :
2
2
2
4
(3)
1
4
(4)
1
4
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15.
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Negation of the statement :
5 is an integer or 5 is irrational' is :
(1) 5 is irrational or 5 is an integer
(2) 5 is not an integer or 5 is not irrational
(3) 5 is an integer and 5 is irrational
(4) 5 is not an integer and 5 is not irrational
1
16.
If for all real triplets (a, b, c), f(x) = a + bx + cx2; then f (x)dx is equal to :
0
1
(1) 2 3f (1) 2f
2
(3)
1
1
f (1) 3f
2
2
(2)
1
1
f (0) f
3
2
(4)
1
1
f (0) f (1) 4f
6
2
17.
If the number of five digit numbers with distinct digits and 2 at the 10th place is 336k, then k is equal to
(1) 8
(2) 7
(3) 4
(4) 6
18.
Let the observations xi (1 i 10) satisfy the equations,
10
10
(xi 5) 10 and
(xi 5)2 40 . If µ
i 1
i 1
and are the mean and the variance of the observations, x1 – 3, x2 – 3, ....., x10 – 3, then the ordered pair
(µ, ) is equal to :
(1) (6, 3)
(2) (3, 6)
(3) (3, 3)
(4) (6, 6)
19.
dx
The integral
8/7
(x 4)
(x 3)6/7
is equal to :
(where C is a constant of integration)
x 3
(1)
x4
1/7
1 x 3
(2)
2 x4
C
1/7
x 3
(3)
x4
20.
C
(4)
3/7
1 x 3
13 x 4
C
13/7
C
In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are labelled as B.
Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained.
The probability that the second A-card appears before the third B-card is :
(1)
15
16
(2)
9
16
(3)
13
16
(4)
11
16
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SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
21.
If the vectors, p (a 1)iˆ ajˆ akˆ , q aiˆ (a 1)ˆj akˆ and r aiˆ ajˆ (a 1)kˆ (a R) are
2
2
coplanar and 3 p.q r q 0, then the value of is ______.
22.
The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the
points (–1, 2, 3) and (3, –2, 10) is ________.
23.
The number of distinct solutions of the equation, log1/2|sinx| = 2 – log1/2|cosx| in the interval [0, 2] is___
24.
If for x 0, y = y(x) is the solution of the differential equation, (x + 1)dy = ((x+1)2 + y – 3)dx, y(2) = 0,
then y(3) is equal to ______.
25.
The coefficient of x4 in the expansion of (1 + x + x2)10 is _________.
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (9/01/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
If A = x R : x 2 and
B = x R : x 2 3 ; then :
(1) A – B = [–1, 2)
(3) A B R (2,5)
2.
If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of
these boxes contain exactly 2 and 3 balls is :
(1)
3.
965
210
(2)
945
210
(3)
945
211
(4)
965
211
d2 y
If x = 2sin – sin2and y = 2cos– cos2then
at is :
dx 2
(1)
4.
(2) B – A =R –(–2, 5)
(4) A B ( 2, 1)
3
8
(2)
3
4
(3)
3
2
(4)
3
4
Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b R,
g' (a) = 5 and g(a) = b, then f ' (b) is equal to :
(1)
2
5
(2) 5
(3) 1
(4)
1
5
16
5.
1
x
In the expansion of
, if l1 is the least value of the term independent of x when
cos x sin
, then the ratio
and l2 is the least value of the term independent of x when
16
8
8
4
l2 : l1 is equal to :
(1) 16 : 1
(2) 8 : 1
(3) 1 : 8
(4) 1 : 16
6.
Let a, b R, a 0 be such that the equation, ax2 – 2bx + 5 = 0 has a repeated root , which is also a root
of the equation, x2 – 2bx – 10 = 0. If is the other root of this equation, then 2 + 2 is equal to :
(1) 24
(2) 25
(3) 26
(4) 28
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Let a function ƒ : [0, 5] R be continuous, ƒ(1) = 3 and F be defined as :
x
t
2
f(x) = t g(t)dt, where g(t) =
1
f (u ) du .
1
Then for the function F, the point x = 1 is :
(1) a point of inflection.
(3) a point of local minima.
8.
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(2) a point of local maxima.
(4) not a critical point.
Let [t] denote the greatest integer t and
4
lim x = A. Then the function,
x 0 x
ƒ(x) = [x2] sin (x) is discontinuous, when x is equal to :
(1)
9.
(2)
A 1
A5
(4)
A 21
Let a – 2b + c = 1.
xa
x2
x 1
if ƒ(x) = x b
xc
x3
x4
x 2 , then :
x3
(1) ƒ(– 50) = 501
10.
(3)
A
(2) ƒ(–50) = – 1
x ,
1
Given : ƒ(x) = ,
2
1 x,
0x
x
(3) ƒ(50) = 1
(4) ƒ(50) = –501
1
2
1
2
1
x 1
2
2
1
and g(x) = x , x R. Then the area (in sq. units) of the region bounded by the curves, y = ƒ(x)
2
and y = g(x) between the lines, 2x = 1 and 2x = 3 , is :
(1)
11.
3 1
4 3
(2)
1
3
3 4
(3)
1
3
2 4
(4)
1
3
2 4
The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
x – 2y – 6z = 0, has
(1) infinitely many solutions, (x, y, z) saisfying y = 2z.
(2) infinitely many solutions, (x, y, z) saisfying x = 2z.
(3) no solution.
(4) only the trivial solution.
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12.
13.
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If p (p ~q) is false, then the truth values of p and q are respectively :
(1) F, T
(2) T, F
(3) F, F
The length of the minor axis (along y-axis ) of an ellipse in the standard form is
(4) T, T
4
. If this ellipse
3
touches the line, x + 6y = 8; then its eccentricity is :
(1)
14.
1
2 3
(2)
1 11
2 3
(3)
5
6
(4)
1 11
3 3
If z be a complex number satisfying
Re( z ) Im( z ) = 4, then z cannot be :
(1)
17
2
(2)
7
15.
If x =
(1)
n
tan2n and y =
n 0
If
2n
, for 0 < , then :
4
(2) x(1 – y) = 1
(3) y(1 – x) = 1
3 e
(2)
1
3e
2
(3)
(4)
e
2
1
If one end of a focal chord AB of the parabola y2 = 8x is at A , 2 , then the equation of the
2
tangent to it at B is :
(1) x + 2y + 8 = 0
(2) 2x – y – 24 = 0
(3) x – 2y + 8 = 0
(4) 2x + y – 24 = 0
Let an be the n term of a G. P. of positive terms. If a 2 n 1 =200 and
th
n 1
equal to :
(1) 300
19.
(4) x(1 + y) = 1
2 e
100
18.
(4) 8
dy
xy
2
; y(1) = 1; then a value of x satisfying y(x) = e is :
dx x y 2
(1)
17.
cos
n 0
(1) y(1 + x) = 1
16.
(3) 10
(2) 175
(3) 225
100
200
a 2n = 100, then
a
n 1
n 1
n
is
(4) 150
A random variable X has the following probability distribution :
X
:
1
2
3
4
5
2
P(X) :
K
2K
K
2K
5K2
Then P(X > 2) is equal to :
(1)
7
12
(2)
23
36
(3)
1
36
(4)
1
6
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20.
If
cos
2
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d
tan 2 sec 2
tanloge ƒ + C where C is a constant of integration, then the ordered pair (ƒ()) is equal
to :
(1) (–1, 1 – tan )
(2) (–1, 1 + tan )
(3) (1, 1 + tan )
(4) (1, 1 – tan )
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
21.
Let a , b and c be three vectors such that a 3, b = 5, b c = 10 and the angle between b and c
is
. If a is perpendicular to the vector b c , then a b c is equal to ______.
3
22.
If Cr 25Cr and C0 + 5 C1 + 9 C2 + ......+ (101) C25=225 k, then k is equal to _________.
23.
If the curves , x2 – 6x + y2 + 8 = 0 and
x2 – 8y + y2 + 16 – k = 0, (k > 0 ) touch each other at a point, then the largest value of k is _______.
24.
The number of terms common to the two A.P.'s 3,7,11,.....,407 and 2,9,16, ......, 709 is ________.
25.
If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane containing the lines
x 1 y 3 z 1
x 3 y 2 z 1
R is equal to
and
2
4
3
2
6
k
, then k is equal to
633
________.
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (2/09/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Cor r ect Questions
Q.1
A line parallel to the straight line 2x-y=0 is tangent to the hyperbola
Then x12 5y12 is equal to :
(1) 6
(2) 10
Q.2
| x | 5
2
x 1
The domain of the function f x sin1
17 1
2
(1)
Q.3
(3) 8
(2)
17
2
x2 y2
1
4
2
at the point (x1,y1).
(4) 5
is (, a] [a, ). Then a is equal to :
(3)
17
1 17
2
(4) 2 1
ae x be x , 1 x 1
If a function f(x) defined by f x cx2
, 1 x 3 be continuous for some a, b,c R and
2
ax 2cx , 3 x 4
f’(0)+f’(2) =e, then the value of a is :
(1)
1
e2 3e 13
(2)
e
e2 3e 13
(3)
e
e2 3e 13
(4)
e
e2 3e 13
Q.4
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :
(1) , 9 3, (2) 3,
(3) , 9
(4) , 3 9,
Q.5
If R x, y :x, y Z, x2 3y2 8 is a relation on the set of integers Z, then the domain of R-1 is :
(1) 1, 0,1
(2) 2, 1,1,2
(4) 2, 1, 0,1, 2
(3) 0,1
3
Q.6
2
2
1 sin 9 i cos 9
The value of
2
2 is :
1
sin
i
cos
9
9
(1)
1
1i 3
2
(2)
1
1i 3
2
(3)
1
2
3 i
1
(4) 2 3 i
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Q.7
Let P(h,k) be a point on the curve y=x2+7x+2, nearest to the line, y=3x-3. Then the equation of the
normal to the curve at P is:
(1) x+3y-62=0
(2) x-3y-11=0
(3) x-3y+22=0
(4) x+3y+26=0
Q.8
Let A be a 2×2 real matrix with entries from 0,1 and | A | 0 . Consider the following two statements:
(P) If A I2, then | A | 1
(Q) If |A|=1, then tr(A) =2,
where I2 denotes 2×2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then:
(1) Both (P) and (Q) are false
(2) (P) is true and (Q) is false
(3) Both (P) and (Q) are true
(4) (P) is false and (Q) is true
Q.9
Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is
selected at random and a card is drawn from it. The number on the card is found to be a non-prime
number. The probability that the card was drawn from Box I is:
(1)
4
17
(2)
2
2
8
17
(3) 5
(4) 3
Q.10 If p(x) be a polynomial of degree three that has a local maximum value 8 at x=1 and a local minimum
value 4 at x=2; then p(0) is equal to :
(1) 12
(2) -12
(3) -24
(4) 6
Q.11
The contrapositive of the statement ”If I reach the station in time, then I will catch
(1) If I will catch the train, then I reach the station in time.
(2) If I do not reach the station in time, then I will catch the train.
(3) If I do not reach the station in time, then I will not catch the train.
(4) If I will not catch the train, then I do not reach the station in time.
the train’’ is:
Q.12 Let and be the roots of the equation, 5x2+6x-2=0. If Sn n n , n=1,2,3,....., then:
(1) 5S6+6S5+2S4=0 (2) 6S6+5S5=2S4
(3) 6S6+5S5+2S4=0 (4) 5S6+6S5=2S4
3
1
Q.13 If the tangent to the curve y=x+siny at a point (a,b) is parallel to the line joining 0, 2 and 2 ,2 , then:
(1) b
a
2
(2) |a+b|=1
(3) |b-a|=1
(4) b=a
|x| |y|
x2 y 2
1 is:
Q.14 Area (in sq. units) of the region outside 2 3 1 and inside the ellipse
4
9
(1) 3 2
(2) 6 2
(3) 6 4
(4) 3 4
Q.15 If |x| < 1,|y| < 1 and x y, then the sum to infinity of the following series
(x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3)+.... is:
x y xy
(1) 1 x 1 y
x y xy
(2) 1 x 1 y
x y xy
(3) 1 x 1 y
x y xy
(4) 1 x 1 y
Q.16 Let 0, 0 be such that 3 2 4 . If the maximum value of the term indepen
dent of x in
10
1
1
the binomial expansion of x 9 x 6 is 10k, then k is equal to:
(1) 176
(2) 336
(3) 352
(4) 84
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Q.17 Let S be the set of all R for which the system of linear equations
2x – y + 2z = 2
x – 2y + z = –4
x+ y+ z = 4
has no solution. Then the set S
(1) is an empty set.
(2) is a singleton.
(3) contains more than two elements.
(4) contains exactly two elements.
Q.18 Let X x N : 1 x 17 and Y= ax b : x X and a,b R, a 0 . If mean and
variance of elements of Y are 17 and 216 respectively then a+b is equal to:
(1) –27
(2) 7
(3)-7
(4) 9
Q.19 Let y = y(x) be the solution of the differential equation,
2 sin x dy
.
cos x, y 0, y 0 1. . If
y 1 dx
dy
y a , and dx at x is b, then the ordered pair (a,b) is equal to:
3
(1) 2, 2
(2) (1,1)
(3) (2,1)
(4) (1,-1)
Q.20 The plane passing through the points (1,2,1), (2,1,2) and parallel to the line, 2x=3y, z=1 also passes
through the point:
(1) (0,-6,2)
(2) (0,6,-2)
(3) (-2,0,1)
(4) (2,0,-1)
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 The number of integral values of k for which the line, 3x+4y=k intersects the circle,
x2 + y2 – 2x – 4y + 4 = 0 at two distinct points is................
2
2
2
2
Q.22 Let a,b and c be three unit vectors such that a b a c 8 . Then a 2b a 2c is equal to
Q.23 If the letters of the word ’MOTHER’ be permuted and all the words so formed (with or without meaning)
be listed as in a dictionary, then the position of the word ’MOTHER’ is.........
Q.24. If lim
x 1
x x2 x3 ... xn n
820, n N
x 1
then the value of n is equal to :
2
Q.25 The integral || x 1 | x | dx is equal to :
0
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JEE MAIN-2020 (2/09/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 Let f : R R be a function which satisfies f (x y) f (x) f (y) x, y R . If f(1) = 2 and
(n 1)
g(n)
f (k), n N then the value of n, for which g(n) = 20, is:
k 1
(1) 9
Q.2
(2) 5
(4) 20
If the sum of first 11 terms of an A.P., a1, a2, a3, .... is 0( a1 0 ) then the sum of the A.P., a1, a3, a5, ..., a23
is ka1, where k is equal to:
(1 )
Q.3
(3) 4
121
10
(2)
72
5
(3)
72
5
(4)
121
10
Let EC denote the complement of an event E. Let E1, E2 and E3 be any pairwise independent events with
P(E1)>0 and P(E1 E 2 E 3 ) 0 . Then P E C2 E 3C / E 1 is equal to:
Q.4
(1) P E 3C P E C2
(2) P E 3 P E 2C
(3) P E C3 P E 2
(4) P E C2 P E 3
If the equation cos 4 sin 4 0 has real solutions for , then lies in the interval:
1 1
(1) ,
2 4
Q.5
1
(2) 1,
2
5
(4) , 1
4
The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices
on the vertex of this parabola, is:
(1) 128 3
Q.6
3 5
(3) ,
2 4
(2) 192 3
1/ 2
The imaginary part of 3 2 54
(1)
6
(3) 64 3
(2) 2 6
3 2
(4) 256 3
1/ 2
54
can be :
(3) 6
(4) 6
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Q.7
Q.8
Q.9
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A plane passing through the point (3,1,1) contains two lines whose direction ratios are 1,–2,2 and 2,3,
–1 respectively. If this plane also passes through the point ( ,–3,5), then is equal to:
(1) –5
(2) 10
(3) 5
(4) –10
1 2 1
Let A={X=(x, y, z) : PX=0 and x +y +z = 1} ,where P 2 3 4 , then the set A:
1 9 1
(1) contains more than two elements
(2) is a singleton.
(3) contains exactly two elements
(4) is an empty set.
T
2
2
2
The equation of the normal to the curve y=(1+x)2y +cos2(sin–1x) at x=0 is:
(1) y+4x=2
(2) 2y+x=4
(3) x+4y=8
(4) y=4x+2
Q.10 Consider a region R {(x, y) R 2 : x 2 y 2x} . If a line y divides the area of region R into
two equal parts, then which of the following is true.?
(1) 3 6 2 16 0
(2) 3 2 8 3/ 2 8 0
(3) 3 63/ 2 16
(4) 3 2 8 8 0
Q.11
Let f : ( 1, ) R be defined by f(0)=1 and f (x)
(1) increases in (–1, )
(3) increases in (–1,0) and decreases in (0, )
Q.12 Which of the following is a tautology?
(1) (p q) (q p)
(3) (q p) ~(p q) (4) (~q) (p q) q
1
log e (1 x), x 0 . Then the function f:
x
(2) decreases in (–1,0) and increases in (0, )
(4) decreases in (–1, ).
(2) (~p) (p q) q
Q.13 Let f(x) be a quadratic polynomial such that f(–1)+f(2)=0. If one of the roots of f(x)=0 is 3, then its other
roots lies in:
(1) (0,1)
(2) (1,3)
(3) (–1,0)
(4) (–3,–1)
Q.14 Let S be the sum of the first 9 terms of the series :
{x+ka}+{x 2 +(k+2)a}+{x 3 +(k+4)a}+{x 4 +(k+6)a}+...
x10 x 45a(x 1)
, then k is equal to:
x 1
(1) 3
(2) –3
where
a0
and
a 1.
If S
(3) 1
(4) –5
Q.15 The set of all possible values of in the interval (0, ) for which the points (1,2) and sin , cos lie
on the same side of the line x+y=1 is:
(1) 0,
4
(2) 0,
2
3
(3) 0,
4
3
(4) ,
4 4
Q.16 Let n>2 be an integer. Suppose that there are n Metro stations in a city located along a circular path.
Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is
connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number
of red lines is 99 times the number of blue lines, then the value of n is:
(1) 201
(2) 199
(3) 101
(4) 200
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Q.17 If a curve y=f(x), passing through the point (1,2) is the solution of the differential equation,
2x2dy = (2xy + y2) dx, then f 1 is equal to:
2
1
(1) 1 log 2
e
(2) 1 log e 2
1
(3) 1 log 2
e
1
(4) 1 log 2
e
Q.18 For some 0, , if the eccentricity of the hyperbola, x 2 y 2 sec 2 10 is 5 times the eccentricity
2
of the ellipse, x 2 sec 2 y 2 5 , then the length of the latus rectum of the ellipse, is:
(1)
4 5
3
(2)
2 5
3
(3) 2 6
(4)
(3) 2
(4) 1
30
1/ x
tan x
Q.19 lim
x 0
4
(1) e
is equal to:
(2) e2
a b c
Q.20 Let a, b, c R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A b c a
c a b
T
satisfies A A=I, then a value of abc can be:
(1)
2
3
(2) 3
(3)
1
3
(4)
1
3
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
ˆ respectively. A point ‘P’
Q.21 Let the position vectors of points ‘A’ and ‘B’ be ˆi ˆj kˆ and 2iˆ ˆj 3k,
divides the line segment AB internally in the ratio :1 0 . If O is the origin and
2
OB.OP 3 OA OP 6 , then is equal to____
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Q.22 Let [t] denote the greatest int eger less than or equal to t . Then t he value of
2
1
Q.23
2x [3x] dx is ____
6
4
3
dy
y
k cos 1 cos kx sin kx , then
If
at x=0 is ____
5
dx
5
k 1
Q.24 If the variance of the terms in an increasing A.P., b1, b2, b3, ...., b11 is 90, then the common difference of
this A.P. is ____
n
Q.25
1
For a positive integer n, 1 is expanded in increasing powers of x. If three consecutive coefficients
x
in this expansion are in the ratio, 2:5:12, then n is equal to _____
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (3/09/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 The value of (2.1P0-3.2P1+4.3P2-..... up to 51th term) +(1!–2!+3!-...... up to 51th term) is equal to:
(1) 1-51(51)!
(2) 1+(52)!
(3) 1
(4) 1+ (51)!
Q.2
Let P be a point on the parabola, y2=12x and N be the foot of the perpendicular drawn from P on the
axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets
4
the parabola at Q. If the y-intercept of the line NQ is 3 , then:
1
(1) PN=4
(2) MQ= 3
x 2
Q.3
If =
2x 3
3x 4
2x 3 3x 4
4x 5
3x 5 5x 8 10x 17
(1) 1
Q.4
Q.5
1
4
(3) -3
(4) 9
The foot of the perpendicular drawn form the point (4,2,3) to the line joining the points (1,-2,3) and
(1,1,0) lies on the plane:
(1) x-y-2z=1
(2) x-2y+z=1
(3)2x+y-z=1
(4) x+2y-z=1
If y2+loge(cos2x)=y, x , , then
2 2
(2) y“(0)=0
(3) |y‘(0)|+|y’’(0)|=3
(4) |y“(0)|=2
4
5
16
2 sin1 sin1
sin1
is equal to:
5
13
25
(1)
Q.7
(4) MQ=
= Ax3+Bx2+Cx+D, then B+C is equal to:
(2)-1
(1) |y‘(0)|+|y“(0)|=1
Q.6
(3) PN=3
5
4
(2)
3
2
(3)
7
4
(4) 2
A hyperbola having the transverse axis of length 2 has the same foci as that of the ellipse 3x2+4y2=12,
then this hyperbola does not pass through which of the following
points ?
3
1
(1) 2 , 2
(2) 1,
1
2
1
,0
2
(3)
3
(4) 2 ,1
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Q.8
MCSIR
For the frequency distribution:
Variate(x):
x1
x2
Frequency(f):
f1
f2
JEE Main Papers
x3....x15
f3.....f15
15
where 0<x1<x2<x3<....<x15=10 and
(1) 1
Q.9
f
i
0, the standard deviation cannot be:
i1
(2) 4
(3) 6
(4) 2
A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of
4. Then the conditional probability that the score 4 has appeared atleastonce is:
1
(1) 3
(2)
1
1
1
4
(3) 8
(4) 9
n
Q.10
1
12
8
3
5
is exactly 33, then the least value of n is:
If the number of integral terms in the expansion of
(1) 128
Q.11
(2) 248
(3) 256
(4) 264
(3) 22
(4) 22
| | x || dx is equal to:
(1) 2
(2)
2
2
Q.12 Consider the two sets:
A={m R : both the roots of x2-(m+1)x+m+4=0 are real} and B=[-3,5).
Which of the following is not true ?
(1) A B , 3 5,
(2) A B {3}
(3) B-A=(-3,5)
(4) A B R
Q.13 The proposition p ~ p q is equivalent to :
(1) p ~ q
(2) ~ p q
(4) ~ p q
(3) q
Q.14 The function, f(x) = (3x-7)x2/3, x R is increasing for all x lying in:
14
14
(1) , 15 0, (2) , 1 5
3
14
, 0 ,
(3) , 0 15 , (4)
7
Q.15 If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of
terms, then the common difference of this A.P. is:
(1)
1
6
(B)
1
5
(C)
1
4
(D)
it s next 15
1
7
dy
x
2
2
Q.16 The solution curve of the differential equation, 1 e 1 y dx y , which passes through the
point (0,1), is:
1 e x
2
y
1
y
log
(1)
e
2
1 ex
2
2
(3) y 1 y loge 2
1 e x
2
y
1
y
log
2
e
(2)
2
1 ex
2
(4) y 1 y loge 2
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1
2
Q.17 The area (in sq. units) of the region x, y : 0 y x 1, 0 y x 1, 2 x 2 is
23
79
(1) 16
23
(2) 16
(3) 6
(4)
79
24
1
1
Q.18 If and are the roots of the equation x2+px+2=0 and and are the roots of the e q u a t i o n
1
1
1
1
2x2+2qx+1=0, then is equal to :
(1)
9
9 p2
4
(2)
9
9 q2
4
(3)
9
9 p2
4
Q.19 The lines r ˆi ˆj l 2iˆ kˆ and r 2iˆ ˆj m ˆi ˆj kˆ
(1) do not intersect for any values of l and m
(3) intersect when l=2 and m=
(4)
9
9 q2
4
(2) intersect when l=1 and m=2
1
2
(4) intersect for all values of l and m
Q.20 Let [t] denote the greatest integer t. if for some
lim
x 0
R 0,
1
1 x | x |
L , then L is equal to:
x x
(1) 0
(2) 2
(3)
1
2
(4) 1
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 If
lim
x 0
x2
x2
x2
x2
1
k
cos
cos
cos
8 1 cos
2 , then the vlaue of k is .......
2
4
2
4
x
Q.22 The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant
touches both the lines x=3 and y=2, is ..........
1
1
1
and which
Q.23 The value of 0.16 log2.5 3 32 32 ..........to is equal to...............
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x 1
4
Q.24 Let A
, x R and A aij . If a11=109, then a22 is equal to .............
1 0
m
Q.25 If
n
1 i 2 1 i 3
1,
1 i
i 1
(m, n N) then the greatest common divisor of the least values of m and n is
.............
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JEE MAIN-2020 (3/09/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 If x3dy+xy dx=x2dy+2y dx; y(2)=e and x>1, then y(4) is equal to:
(1)
Q.2
e
2
(2)
3
e
2
(3)
1
e
2
3
(4) e
2
2 1 1
Let A be a 3×3 matrix such that adj A 1 0 2 and B=adj(adj A).
1 2 1
T
If | A | and | B1 | , then the ordered pair,, | |, is equal to:
1
(1) 9,
81
Q.3
1
(2) 9,
9
1
(3) 3,
81
(4) (3, 81)
2
4
Let a,b,c R be such that a2+b2+c2 =1, If a cos b cos c cos
, where ,
3
3
then the angle between the vectors aiˆ bjˆ ckˆ and biˆ cjˆ akˆ is
(1)
Q.4
2
(2)
2
3
9
(4) 0
Suppose f(x) is a polynomial of degree four, having critical points at –1,0,1. If T {x R | f (x) f (0)},
then the sum of squares of all the elements of T is:
(1) 6
(2) 2
(3) 8
(4) 4
1/ 2
Q.5
(3)
If the value of the integral 0
(1) 2 3
x2
2 3/ 2
1 x
(2) 3 2
dx
is
k
, then k is equal to:
6
(3) 3 2
(4) 2 3
9
Q.6
3 2 1
If the term independent of x in the expansion of x is k, then 18 k is equal to:
3x
2
(1) 5
(2) 9
(3) 7
(4) 11
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If a ABC has vertices A(–1,7), B(–7,1) and C(5,–5), then its orthocentre has coordinates:
3 3
(2) ,
5 5
(1) (–3,3)
Q.8.
MCSIR
3 3
(3) ,
5 5
Let e1 and e2 be the eccentricities of the ellipse,
(4) (3,–3)
x 2 y2
x 2 y2
2 1 (b<5) and the hyperbola,
1
25 b
16 b 2
respectively satisfying e1e2=1. If and are the distances between the foci of the ellipse and the foci
of the hyperbola respectively, then the ordered pair (, ) is equal to:
24
(2) ,10
5
(1) (8,12)
Q.9
20
(3) ,12
3
(4) (8,10)
If z1, z2 are complex numbers such that Re(z1)=|z1–1|, Re(z2)=|z2–1| and arg (z1–z2)= , then Im(z1+z2)
6
is equal to:
(1) 2 3
(2)
2
3
(3)
1
3
(4)
3
2
Q.10 The set of all real values of for which the quadratic equations, 2 1 x 2 4x 2 0 always have
exactly one root in the interval (0,1) is:
(1) (–3,–1)
(2) (2,4]
Q.11
(3) (1,3]
(4) (0,2)
Let the latus ractum of the parabola y2=4x be the common chord to the circles C1 and C2 each of them
having radius 2 5 . Then, the distance between the centres of the circles C1 and C2 is:
(1) 8
(2) 8 5
(3) 4 5
(4) 12
Q.12 The plane which bisects the line joining the points (4,–2,3) and (2,4,–1) at right angles also passes
through the point:
(1) (0,–1,1)
(2) (4,0,1)
(3) (4,0,–1)
(4) (0,1,–1)
Q.13
lim
1
3
1
3
1
3
1
3
(a 2x) (3x)
x a
(3a x) (4x)
(a 0) is equal to :
4
4
1
1
2 3
(1)
9
2 3
(2)
3
2 2 3
(3)
3 9
2 2 3
(4)
9 3
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10
Q.14
10
2
Let x i (1 i 10) be ten observations of a random variable X. If x i p 3 and x i p 9
i 1
i 1
where 0 p R , then the standard deviation of these observations is :
(1)
7
10
(2)
9
10
3
5
(3)
(4)
4
5
Q.15 The probability that a randomly chosen 5–digit number is made from exactly two digits is :
(1)
134
104
(2)
121
104
x
1
1
Q.16 If sin 1 x dx A(x) tan
ordered pair (A(x),B(x)) can be:
(1) x 1, x
(3)
135
104
(4)
150
104
x B(x) C , where C is a constant of integration, then the
(2) x 1, x
(3) x 1, x
(4) x 1, x
3
1
4
Q.17 If the sum of the series 20 19 19 18 ... upto nth term is 488 and the nth term is negative, then:
5
5
5
(1) n=60
(2) n=41
(3) nth term is –4
(4) nth term is 4
2
5
Q.18 Let p, q, r be three statements such that the truth value of p q (~ p r) is F. Then the truth values
of p, q, r are respectively :
(1) F, T, F
(2) T, F, T
(3) T, T, F
(4) T, T, T
Q.19 If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of
change of its volume (in cm3/sec), when the lenght of a side of the cube is 10cm, is :
(1) 9
(2) 10
(3) 18
(4) 20
Q.20 Let R1 and R2 be two relations defined as follows:
R1 {(a, b) R 2 : a 2 b 2 Q} and
R 2 {(a, b) R 2 : a 2 b 2 Q} , where Q is the set of all rational numbers. Then :
(1) R1 is transitive but R2 is not transitive
(2) R1 and R2 are both transitive
(3) R2 is transitive but R1 is not transitive
(4) Neither R1 nor R2 is transitive
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SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such
that 4th A.M. is equal to 2nd G.M., then m is equal to___
Q.22 Let a plane P contain two lines r ˆi ˆi ˆj , R and r ˆj ˆj kˆ , R . If Q (, , ) is
the foot of the perpendicular drawn from the point M(1,0,1) to P,then 3 ( ) equals ____
Q.23 Let S be the set of all integer solutions, (x, y, z), of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 x 2 y 2 z 2 150 . Then, the number of elements in the set S is equal to ___
Q.24 The total number of 3–digit numbers, whose sum of digits is 10, is ____
Q.25 If the tangent to the curve, y=ex at a point (c,eC) and the normal to the parabola, y2=4x at the point (1,2)
intersect at the same point on the x-axis, then the value of c is ____
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (4/09/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
1.
Let y=y(x) be the solution of the differential equation, xy’-y=x2(xcosx+sinx),x > 0. if y , then
y'' y is equal to
2
2
(1) 2
2
2 4
(2) 2
2
(3) 1
2
(4) 1
2
2 4
20
2.
The value of
50 r
C6 is equal to:
r 0
(1)
51
C7 30 C7
(2)
51
C7 30 C7
(3)
50
C7 30 C7
(4) 50 C6
30
3.
Let [t] denote the greatest integer t. Then the equation in x,[x]2+2[x+2]-7=0 has :
(1) exactly four integral solutions.
(2) infinitely many solutions.
(3) no integral solution. (4) exactly two solutions.
4.
Let P(3,3) be a point on the hyperbola,
C6
x2 y2
1 . If the normal to it at P intersects
a2 b2
t he x-axis
at (9,0) and e is its eccentricity, then the ordered pair (a2,e2) is equal to :
9
5.
(2) 2 , 2
(1) (9,3)
9
a
b
(1) 135
Let f x
(1)
7.
3
(4) 2 ,2
2
2
Let x2 y2 1 (a>b) be a given ellipse, length of whose latus rectum is 10. If its
eccentricity is the maximum value of the function, t
6.
(3) 2 , 3
(2) 116
x
2
1 x
1
3
6 2
4
5
t t2 , then a2+b2 is equal to
12
(3) 126
(4) 145
dx x 0 . Then f(3) – f(1) is eqaul to :
(2)
1
3
6 2
4
(3)
1
3
12 2
4
1
3
(4) 12 2 4
If 1+(1–22.1)+(1–42.3)+(1-62.5)+......+(1-202.19)= 220 , then an ordered pair , is equal to:
(1) (10,97)
(2) (11,103)
(3) (11,97)
(4) (10,103)
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2
8.
x
The integral
dx is equal to
x
sin
x
cos
x
(where C is a constant of integration):
(1) tan x
x sec x
C
x sin x cos x
x tan x
(3) sec x x sin x cos x C
(2) sec x
x tan x
C
x sin x cos x
x sec x
(4) tan x x sin x cos x C
3
9.
Let f(x)= |x-2| and g(x) = f(f(x)) ,x [0,4]. Then g x f x dx is equal to:
0
(1)
10.
3
1
2
(2) 0
(3) 1
(4) 2
ˆ b 2iˆ xj
ˆ and
ˆ k
Let x0 be the point of Local maxima of f x a. b c , where a xˆi 2jˆ 3k,
. + . + .
ˆ . Then the value of a
ˆ xk
c 7iˆ 2j
b b c c a
(1) -22
11.
(2) -4
(3) -30
at x=x0 is :
(4) 14
A triangle ABC lying in the first quadrant has two vertices as A(1,2) and B(3,1) If BAC 900 , and
ar( ABC) = 5 5 s units, then the abscissa of the vertex C is :
(1) 1 5
(2) 1 2 5
(3) 2 5 1
(4) 2 5
12.
Let f be a twice differentiable function on (1,6). If f(2)=8, f’(2)=5, f’(x) 1 and f‘‘(x) 4, for all
x (1,6), then:
(1) f(5)+f‘(5) 28
(2) f’(5)+f‘‘(5) 20
(3) f(5) 10
(4) f(5)+f’(5) 26
13.
Let and be the roots of x2-3x+p=0 and and be the roots of x2-6x+q=0. If , , ,
form a geometric progression.Then ratio (2q+p): (2q-p) is:
(1) 33 :31
(2) 9 : 7
(3) 3 : 1
(4) 5 : 3
14.
Let u
2z i
,
z ki
z = x +iy and k>0. If the curve represented by Re(u) +Im(u) =1 intersects
the y-axis at the points P and Q where PQ =5, then the value of k is :
(1) 4
(2) 1/2
(3) 2
15.
cos i sin
a b
5
If A i sin cos , 24 and A c d , where i 1 , then which one of the following is
not true?
(1) a2-d2=0
16.
(4) 3/2
(B) a2-c2=1
(C) 0 a2 b2 1
(D) a2 b2
1
2
The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations
are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is:
(1) 3
(2) 9
(3) 7
(4) 5
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17.
A survey shows that 63% of the people in a city read newspaper A whereas 76% read
newspaper B. If x% of the people read both the newspapers, then a possible value of x can be:
(1) 37
(2) 29
(3) 65
(4) 55
18.
Given the following two statements:
S1 : q v p P ~ q is a tautology
S2 :~ q
~ p q is a fallacy. Then:
(1) only (S1) is correct. (2) both (S1) and (S2) are correct.
(3) only (S2) is correct (4) both (S1) and (S2) are not correct.
19.
Two vertical poles AB=15 m and CD=10 m are standing apart on a horizontal ground with points A
and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m)
above the line AC is:
(1) 5
(2) 20/3
(3) 10/3
(4) 6
20.
If a 2 b cos x a 2 b cos y a2 b2 , where a>b>0, then dy at 4 , 4 is:
dx
ab
a 2b
(1) a b
ab
(2) a 2b
2a b
(3) a b
(4) 2a b
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
21.
Suppose a differentiable function f(x) satisfies the identity f(x+y)=f(x)+f(y)+xy2+x2y,for all real x
and y. If lim
x 0
f x
x
1 , then f‘(3) is equal to............
22.
If the equation of a plane P, passing through the intersection of the planes, x+4y-z+7=0
and 3x+y+5z=8 is ax+by+6z=15 for some a, b R, then the distance of the
point
(3,2,-1) from the plane P is..........
23.
If the system of equations
x-2y+3z=9
2x+y+z=b
x-7y+az=24, has infinitely many solutions, then a-b is equal to.........
24.
Let (2x2+3x+4)10= ar x . Then a is equal to .............
r 0
13
25.
The probability of a man hitting a target is
20
r
a7
1
.
10
The least number of shots required, so that the
probability of his hitting the target at least once is greater than
1
4
, is .........
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (4/09/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax=b when the
vector b on the right side is equal to b 1 , b 2 and b 3 respectively. if
1
0
0
1
0
0
x1 1 , x 2 2 , x 3 0 , b1 0 , b 2 2 and b3 0
, then the determinant of A is equal to
1
1
1
0
0
2
(1) 2
Q.2
1
2
(3)
3
2
If a and b are real numbers such that (2 ) 4 a b , where
to:
(1) 33
Q.3
(2)
(2) 57
(4) 4
1 i 3
then a+b is
2
(3) 9
equal
(4) 24
The distance of the point (1, –2, 3) from the plane x–y+z=5 measured parallel to the line
x y z
2 3 6
is:
(1)
Q.4
1
7
(2) 7
(3)
7
5
(4) 1
Let f : (0, ) (0, ) be a differentiable function such that f(1) = e and lim
t x
t 2f 2 (x) x 2 f 2 (t)
0 . If
t x
f(x)=1,then x is equal to :
(1) e
Q.5
(2) 2e
(3)
1
e
(4)
1
2e
Contrapositive of the statement :
‘If a function f is differentiable at a, then it is also continuous at a’, is:
(1) If a function f is not continuous at a, then it is not differentiable at a.
(2) If a function f is continuous at a, then it is differentiable at a.
(3) If a function f is continuous at a, then it is not differentiable at a.
(4) If a function f is not continuous at a, then it is differentiable at a.
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Q.6
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The minimum value of 2sinx+2cosx is:
(1) 21
(2) 21
2
1
(3) 21
2
2
(4) 21
1
2
Q.7
If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept
equal to –4, then a value of k is:
(1) –2
(2) 15
(3) 14
(4) –4
Q.8
The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and
vertices C and D lie on the parabola, y=x2–1 below the x-axis, is:
(1)
Q.9
2
(2)
3 3
4
3
(3)
3 3
(4)
4
3 3
/3
The integral / 6 tan3 x.sin2 3x(2 sec2 x.sin2 3x 3 tan x.sin 6x)dx is equal to:
(1)
9
2
1
1
(2) 18
Q.10 If the system of equations
x+y+z=2
2x+4y–z=6
3x+2y+ z=
has infinitely many solutions, then
(1) 2 5
(2) 2 14
Q.11
1
7
(3) 9
(4) 18
(3) 2 14
(4) 2 5
In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of
scores on the two dice, in each throw is noted. A wins the game if he throws total a of 6 before B throws
a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops
as soon as either of the players wins. The probability of A winning the game is:
(1)
5
31
(2)
31
61
(3)
30
61
(4)
5
6
Q.12 If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of
(1+x)n+5 are in the ratio 5:10:14, then the largest coefficient in this expansion is :
(1) 792
(2) 252
(3) 462
(4)330
Q.13 The function
1
4 tan x, | x | 1
f (x )
1 (| x | 1), | x | 1
2
is :
(1) both continuous and differentiable on R–{–1}
(2) continuous on R–{–1} and differentiable on R–{–1,1}
(3) continuous on R–{1} and differentiable on R–{–1,1}
(4) both continuous and differentiable on R–{1}
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dy
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y 3x
Q.14 The solution of the differential equation dx log (y 3x) 3 0 is:
e
(where c is a constant of integration)
1
2
2
(1) x–loge(y+3x)= C
(2) x log e (y 3x) C
(3) x–2loge(y+3x)=C
(4) y 3x log e x C
1
2
2
Q.15 Let 0 be in R. If and are the roots of the equation, x 2 x 2 0 and and are the roots
of the equation, 3x 10x 27 0 , then
is equal to:
(1) 27
(2) 9
(3) 18
(4) 36
Q.16 The angle of elevation of a cloud C from a point P, 200 m above a still lake is 30°. If the angle of
depression of the image of C in the lake from the point P is 60°,then PC (in m) is equal to :
(1) 200 3
(2) 400 3
(3) 400
(4) 100
50
n
Q.17 Let i1 Xi i1 Yi T , where each Xi contains 10 elements and each Yi contains 5 elements. If each
element of the set T is an element of exactly 20 of sets Xi ’s and exactly 6 of sets Yi’s, then n is equal to
(1) 15
(2) 30
(3) 50
(4) 45
1
2
Q.18 Let x=4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is . If P(1, ), > 0
is a point on this ellipse, then the equation of the normal to it at P is :
(1) 8x–2y=5
(2) 4x–2y=1
(3) 7x–4y=1
(4) 4x–3y=2
Q.19 Let a1, a2, ..., an be a given A.P. whose common difference is an integer and Sn=a1+a2+ .... +an. If a1=1,
an=300 and 15 n 50, then the ordered pair (Sn–4, an–4) is equal to:
(1) (2480,248)
(2) (2480,249)
(3) (2490,249)
(4) (2490,248)
Q.20 The circle passing through the intersection of the circles, x2+y2–6x=0 and x2+y2–4y=0, having its centre
on the line, 2x–3y+12=0, also passes through the point:
(1) (–1,3)
(2) (1,–3)
(3) (–3,6)
(4) (–3,1)
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 Let {x} and [x] denote the fractional part of x and the greatest integer x respectively of a real number
x. If
n
0
n
{x}dx, [x]dx and 10(n2–n), n N, n 1 are three consecutive terms of a G.P., then n is equal
0
to_____
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Q.22 A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is
correct. The number of ways, in which a candidate answers all six questions such that exactly four of the
answers are correct, is ______
Q.23 If a 2iˆ ˆj 2kˆ , then the value of ˆi a ˆi ˆj a ˆj ˆk a ˆk is equal to____
2
2
2
Q.24 Let PQ be a diameter of the circle x2+y2=9. If and are the lengths of the perpendiculars from P and
Q on the straight line, x+y=2 respectively, then the maximum value of is _____
Q.25 If the variance of the following frequency
distribution :
Class
:
10–20 20–30 30–40
Frequency
:
2
x
2
is 50, then x is equal to____
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (5/09/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 If the volume of a parallelopiped, whose coterminus edges are given by the vectors
ˆ n 0 , is 158 cu. units, then:
ˆ 3k
ˆ b 2iˆ 4j
ˆ and c ˆi nj
ˆ nk
a ˆi ˆj nk,
(1) a . c =17
(2) b . c =10
(3) n=9
(4) n=7
Q.2
A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x
denotes the percentage of them, who like both coffee and tea, then x cannot be:
(1) 63
(2) 54
(3) 38
(4) 36
Q.3
The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2,4,10,12,14,
then the absolute difference of the remaining two
observations is:
(1) 1
(2) 4
(3) 3
(4) 2
Q.4
If 210+29.31+28.32+.....+2.39+310=S-211, then S is equal to:
(1) 311
(2)
311
210
2
(3) 2.311
(4) 311 —212
Q.5
If 32 sin2-1,14 and 34-2 sin2are the first three terms of an A.P. for some , then the sixth terms of this
A.P. is:
(1) 65
(2) 81
(3) 78
(4) 66
Q.6
If the common tangent to the parabolas, y2=4x and x2=4y also touches the circle, x2+y2 = c2, then c is
equal to:
(1)
Q.7
1
2
(2)
1
4
(3)
1
1
(4) 2 2
2
If the minimum and the maximum values of the function f : , R , defined by
4 2
sin2 1 sin2 1
f cos2 1 cos2 1
12
10
2
(1) (0,4)
(2) (-4,0)
are m and M respectively, then the ordered pair (m,M) is equal to :
(3) (-4,4)
(4) 0, 2 2
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Q.8
Q.9
Let R . The system of linear equations
2x1-4x2+ x3=1
x1-6x2+x3=2
x1-10x2+4x3=3
is inconsistent for:
(1) exactly two values of
(3) every value of .
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(2) exactly one negative value of .
(4) exactly one positive value of .
If the point P on the curve, 4x2+5y2=20 is farthest from the point Q(0, -4), then PQ2 is equal to:
(1) 48
(2) 29
(3) 21
(4) 36
Q.10 The product of the roots of the equation 9x2-18|x|+5=0 is :
(1)
Q.11
25
81
(2)
5
9
(3)
If y=y(x) is the solution of the differential equation
y(0)=1, then a value of y(loge13) is:
(1) 1
(2) 0
5
27
(4)
5 ex dy
.
ex 0
2 y dx
(3) 2
25
9
satisfying
(4) -1
1
1
1
1
1
1
1
1
Q.12 If S is the sum of the first 10 terms of the series tan 3 tan 7 tan 13 tan 21 .....,
then tan(S) is equal to :
(1)
5
5
11
(2) 6
2
Q.13 The value of
(1)
1
1e
sin x
dx
2
2
(2)
6
(3) 5
(4)
10
11
(3)
(4)
3
2
is:
4
Q.14 If (a, b, c) is the image of the point (1,2,-3) in the line,
(1) 2
Q.15 If the function
(2) 3
k x 2 1 , x
f x 1
, x
k 2 cos x
x 1 y 3
z
, then a+b+c is
2
2
1
(3) -1
(4) 1
is twice differentiable, then the ordered pair (k1,k2) is
equal to:
(1) (1,1)
(2) (1,0)
1
(3) 2 , 1
1
(4) 2 ,1
Q.16 If the four complex numbers z, z , z -2Re( z ) and z-2Re(z) represent the vertices of a
4 units in the Argand plane, then |z| is equal to:
(1) 2
(2) 4
(3) 4 2
(4) 2 2
square of side
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Q.17 If e2x 2ex e x 1 e
g(0) is equal to :
(1) 2
ex e x
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dx g x e
e
x
e x
(2) e
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c , where c is a constant of integration,
then
(4) e2
(3) 1
Q.18 The negation of the Boolean expression x ~ y is equivalent to :
(1) x y ~ x ~ y
(2) x y ~ x ~ y
(3) x ~ y ~ x y
(4) ~ x y ~ x ~ y
Q.19 If is positive root of the equation, p(x) =x2-x-2=0, then lim
x
(1)
1
2
(2)
3
(3)
2
3
2
1 cos p x
x4
(4)
is equal to :
1
2
Q.20 If the co-ordinates of two points A and B are 7, 0 and 7, 0 respectively and P is any point on the
conic, 9x2+16y2=144, then PA+PB is equal to :
(1) 6
(2) 16
(3) 9
(4) 8
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 The natural number m, for which the coefficient of x in the binomial expansion of
22
m 1
x 2 is 1540, is .............
x
Q.22 Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice
show up a three or a five, is ...........
x
Q.23 Let f x x. 2 , for-10<x<10, where [t] denotes the greatest integer function. Then the number of
points of discontinuity of f is equal to...........
Q.24 The number of words, with or without meaning, that can be formed by taking 4 letters at a time from
the letters of the word ’SYLLABUS’ such that two letters are distinct and
two letters are alike, is.
Q.25 If the line, 2x-y+3=0 is at a distance
1
5
and
2
5
from the lines 4x-2y+ =0 and 6x-
3y+ =0, re-
spectively, then the sum of all possible values of and is
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (5/09/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 If x=1 is a critical point of the function f(x)=(3x2+ax–2–a)ex, then:
Q.2
(1) x=1 is a local minima and x
2
3
is a local maxima of f.
(2) x=1 is a local maxima and x
2
3
is a local minima of f.
(3) x=1 and x
2
3
are local minima of f.
(4) x=1 and x
2
3
are local maxima of f.
x e
lim
x 0
1 x 2 x 4 1 / x
1
1 x2 x4 1
(1) is equal to e
(2) is equal to 1
(3) is equal to 0
(4) does not exist
Q.3
The statement (p (q p)) (p (p q)) is:
(1) equivalent to (p q) (~ p)
(2) equivalent to (p q) (~ p)
(3) a contradiction
(4) a tautology
Q.4
If L sin2 sin2 and M cos2 sin2 , then:
16
8
16
8
(1) M
1
2 2
(3) L
1
2 2
1
cos
2
8
(2) M
1
cos
2
8
(4) L
1
4 2
1
4 2
1
cos
4
8
1
cos
4
8
Q.5
If the sum of the first 20 terms of the series log7 x log7 x log7 x ... is 460, then x is equal to:
(1) 71/2
(2) 72
(3) e2
(4) 746/21
Q.6
There are 3 sections in a question paper and each section contains 5 questions. A candidate has to
answer a total of 5 questions, choosing at least one question from each section. Then the number of
ways, in which the candidate can choose the questions, is:
(1) 2250
(2) 2255
(3) 1500
(4) 3000
Q.7
If the mean and the standard deviation of the data 3,5,7,a,b are 5 and 2 respectively, then a and b are the
roots of the equation:
(1) x2–20x+18=0
(2) x2–10x+19=0
(3) 2x2–20x+19=0
(4) x2–10x+18=0
1/ 2
1/ 3
1/ 4
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Q.8
The derivative of
(1)
Q.9
If
1 x2 1
tan1
x
2 3
3
(2)
cos
5 7 sin 2 cos
(1)
5(2 sin 1)
sin 3
2
MCSIR
with respect to
2 3
5
(3)
d A loge | B() | C
5(sin 3)
JEE Main Papers
2x 1 x 2
tan1
1 2x 2
3
12
is:
(4)
3
10
where C is a constant of integration, then
B()
A
can be:
2 sin 1
2 sin 1
(2) 2 sin 1
1
2
at x
(3) sin 3
(4) 5(sin 3)
Q.10 If the length of the chord of the circle, x2+y2 = r2(r>0) along the line, y–2x=3 is r, then r2 is equal to:
(1) 12
Q.11
(2)
24
5
(3)
9
5
(4)
12
5
If and are the roots of the equation, 7x2–3x–2=0, then the value of 1 2 1 2 is equal to:
27
(1) 32
(2)
27
1
24
3
(3) 16
(4) 8
Q.12 If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth,
saventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is:
2
50
(1) 13 3 1
1
1
49
(2) 26 3 1
1
50
(3) 13 3 1
Q.13 If the line y=mx+c is a common tangent to the hyperbola
which one of the following is true?
(1) 4c2=369
(2) c2=369
50
(4) 26 3 1
x2
y2
1 and the circle x2+y2=36, then
100 64
(3) 8m+5=0
(4) 5m=4
Q.14 The area (in sq. units) of the region A {(x, y) : (x 1)[x] y 2 x , 0 x 2} where [t] denotes the
greatest integer function, is:
4
1
(1) 3 2 2
8
1
8
(2) 3 2 2
4
(3) 3 2 1
(4) 3 2 1
x a y xa
Q.15 If a+x=b+y=c+z+1, where a,b,c,x,y,z are non-zero distinct real numbers. then y b y y b is equal
z
to:
(1) y(a–b)
(2) 0
(3) y(b–a)
x 1
y2
z 1
cy
zc
(4) y(a–c)
x2
Q.16 If for some R , the lines L1 :
and L 2 :
2
1
1
the line L2 passes through the point:
(1) (2, –10, –2)
(2) (10, –2, –2)
(3) (10, 2, 2)
y 1 z 1
5
1
are coplanar, then
(4) (–2, 10, 2)
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Q.17
1 i 3
The value of 1 i
(1) 215i
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30
is:
(2) –215
(3) –215i
(4) 65
dy
Q.18 Let y=y(x) be the solution of the differential equation cos x dx 2y sin x sin 2x, x 0, 2 .
If y( / 3) 0 , then y( / 4) is equal to:
(1) 2 2
(2) 2 2
(3)
1
2
1
(4) 2 2
Q.19 If the system of linear equations
x+y+3z=0
x+3y+k2z=0
3x+y+3z=0
y
has a non-zero solution (x,y,z) for some k R , then x z is equal to:
(1) –9
(2) 9
(3) –3
(4) 3
Q.20 Which of the following points lies on the tangent to the curve x 4 e y 2 y 1 3 at the point (1,0)?
(1) (2,6)
(2) (2,2)
(3) (–2,6)
(4) (–2,4)
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 Let A={a,b,c} and B={1,2,3,4}. Then the number of elements in the set C {f : A B | 2 f (A) and
f is not one–one} is_______
Q.22 The coefficient of x4 in the expansion of (1+x+x2+x3)6 in powers of x, is ______
Q.23 Let the vectors a, b, c be such that | a | 2,| b | 4 and | c | 4 . If the projection of b on a is equal to
the projection of c on a and b is perpendicular to c , then the value of | a b c | is_______
Q.24 If the lines x+y=a and x-y=b touch the curve y=x2–3x+2 at the points where the curve intersects the x–
axis, then
a
b
is equal to_______
Q.25 In a bombing attack, there is 50% chance that a bomb will hit the target. At least two independent hits
are required to destroy the target completely. Then the minimum number of bombs, that must be dropped
to ensure that there is at least 99% chance of completely destroying the target, is________
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (6/09/2020) SHIFT-1
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 The region represented by {z = x + iy C : |z| – Re(z) 1} is also given by the inequality:
{z = x + iy C : |z| – Re(z) 1}
1
(1) y2 2 x 2
(2) y2 x +
1
2
(3) y2 2(x + 1)
(4) y2 x + 1
Q.2
The negation of the Boolean expression p (~p q) is equivalent to:
(1) p ~q
(2) ~p ~q
(3) ~p q
(4) ~p ~q
Q.3
The general solution of the differential equation 1 x2 y2 x2y2 + xy
= 0 is:
dx
dy
(where C is a constant of integration)
2
(1) 1 y + 1 x
2
1 x2 – 1
1
= loge
+C
2
2
1 x 1
1
1 x2 – 1
1
1 x2 1
(2) 1 y2 – 1 x2 = loge
+C
2
2
1 x 1
(3) 1 y2 + 1 x2 = loge
+C
2
2
1 x –1
2
(4) 1 y – 1 x2
1 x2 1
1
= loge
+C
2
2
1 x –1
Q.4
Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola
y2 = 8(x + 2) such that L1 and L2 intersect at right angles. Then L1 and L2 meet on the straight line:
(1) x + 2y = 0
(2) x + 2 = 0
(3) 2x + 1 = 0
(4) x + 3 = 0
Q.5
The area (in sq. units) of the region A = {(x, y): |x| + |y| 1, 2y2 |x|}
(1)
Q.6
1
6
(2)
5
6
(3)
The shortest distance between the lines
1
3
(4)
7
6
x –1
y 1
z
=
=
and x + y + z + 1 = 0,
0
–1
1
2x – y + z + 3 = 0 is:
(1) 1
(2)
1
2
(3)
1
3
(4)
1
2
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Q.7
Let a, b, c, d and p be any non zero distinct real numbers such that
(a2 + b2 + c2)p2 – 2(ab + bc + cd)p + (b2 + c2 + d2) = 0. Then:
(1) a, c, p are in G.P.
(2) a, b, c, d are in G.P.
(3) a, b, c, d are in A.P.
(4) a, c, p are in A.P.
Q.8
Two families with three members each and one family with four members are to be seated in a row. In
how many ways can they be seated so that the same family members are not separated?
(1) 2! 3! 4!
(2) (3!)3(4!)
(3) 3! (4!)3
(4) (3!)2(4!)
Q.9
The values of and µ for which the system of linear equations
x+ y+ z= 2
x + 2y + 3z = 5
x + 3y + z = µ
has infinitely many solutions are, respectively:
(1) 6 and 8
(2) 5 and 8
(3) 5 and 7
(4) 4 and 9
Q.10 Let m and M be respectively the minimum and maximum values of
cos2 x
2
1 cos x
cos2 x
1 sin2 x
2
sin x
sin2 x
sin 2x
sin 2x
1 sin 2x
Then the ordered pair (m, M) is equal to:
(1) (–3, –1)
(2) (–4, –1)
Q.11
(3) (1, 3)
(4) (–3, 3)
A ray of light coming from the point (2, 2 3 ) is incident at an angle 30º on the line x = 1 at the point A.
The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB passes through
the point:
(1) (4, – 3 )
1
(2) 3, –
3
(3) (3, – 3 )
3
(4) 4, – 2
Q.12 Out of 11 consecutive natural numbers if three numbers are selected at random
(without repetition), then the probability that they are in A.P. with positive common difference, is:
(1)
10
99
(2)
5
33
(3)
15
101
(4)
5
101
Q.13 If f(x + y) = f(x) f(y) and f(x) 2 , x, yN, where N is the set of all natural number, then the value of
x 1
f(4)
is :
f(2)
(1)
2
3
(2)
1
9
(3)
1
3
(4)
4
9
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3200
Q.14 If {p} denotes the fractional part of the number p, then 8 , is equal to :
(1)
5
8
(2)
1
8
(3)
7
8
(4)
3
8
Q.15 Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent to the
x2
4
ellipse,
+
y2
2
= 1 from any of its foci ?
(1) (–1, 3 )
(2) (–2, 3 )
(3) (–1, 2 )
(4) (1, 2)
(3) does not xist
(4) is equal to –
2
Q.16
(x 1)
t cos(t2 )dt
lim 0
x 1
(x 1)sin(x 1)
(1) is equal to 1
1
2
1
2
n
n
Q.17 If
(2) is equal to
(x
i
i1
2
a) n and (xi a) na , (n, a > 1) then t he standard deviation of n
i1
observations x1, x2, ..., xn is :
(1) n a 1
(2) na 1
(3) a – 1
(4) a 1
Q.18 If and be two roots of t he equat ion x 2 – 64x + 256 = 0. Then the value of
1/8
3
5
1/8
3
+ 5
(1) 1
is :
(2) 3
(3) 2
(4) 4
Q.19 The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real
numbers greater than 1. Then the average speed of the car over the time interval [t1, t2] is attained at the
point :
(1) (t1 + t2)/ 2
(2) 2a(t1 + t2) + b
(3) (t2– t1)/2
(4) a(t2– t1) + b
1
Q.20 If I1 = 0 (1 x50 )100 dx and I2 =
(1)
5050
5049
(2)
1
0
5050
5051
(1 x50 )101 dx such that I2 = I1 then equals to :
(3)
5051
5050
(4)
5049
5050
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SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 If a and b are unit vectors, then the greatest value of 3 a b a b is_____.
Q.22 Let AD and BC be two vertical poles at A and B respectively on a horizontal ground.
If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point
A such that MD2 +MC2 is minimum is ______.
Q.23 Let f : R R be defined as
5
1
2
x sin 5x , x 0
x
0,
x0
f(x) =
x5 cos 1 x2 , x 0
x
The value of for which f(0) exists, is _______.
Q.24 The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of
the hill is found to be 45°. After walking a distance of 80 meters towards the top, up a slope inclined at
an angle of 30° to the horizontal plane, the angle of elevation of the top of the hill becomes 75°. Then the
height of the hill (in meters) is ______.
Q.25 Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the
total number of subsets of B, then the value of m.n is ______.
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MATHEMATICS BY MANOJ CHAUHAN SIR
JEE MAIN-2020 (6/09/2020) SHIFT-2
SECTION – 1 : (Maximum Marks : 80)
Straight Objective Type
This section contains 20 multiple choice questions. Each question has 4 choices (1), (2), (3) and
(4) for its answer, out of which Only One is correct.
Single Correct Questions
Q.1 If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then
the eccentricity e of the ellipse satisfies:
(1) e4+2e2–1=0
(2) e2+2e–1=0
(3) e4+e2–1=0
(4) e2+e–1=0
Q.2
The set of all real values of for which the function f (x) (1 cos2 x) . ( sin x) , x 2 , 2 , has
exactly one maxima and exactly one minima, is:
3 3
(1) 2 , 2 {0}
1 1
(2) 2 , 2 {0}
3 3
(3) 2 , 2
1 1
(4) 2 , 2
Q.3
The probabilities of three events A, B and C are given by P(A)=0.6, P(B)=0.4 and P(C)=0.5.
If P (A B) =0.8, P (A C) =0.3, P (A B C) =0.2, P (B C) and P (A B C) , where
0.85 0.95 , then lies in the interval:
(1) [0.36,0.40]
(2) [0.25,0.35]
(3) [0.35,0.36]
(4) [0.20,0.25]
Q.4
The common difference of the A.P. b1, b2,..... bm is 2 more than the common difference of A.P. a1, a2,
...an. If a40 =–159, a100=–399 and b100=a70, then b1 is equal to:
(1) –127
(2) 81
(3) 127
(4) –81
2
Q.5
x
x
The integral e .x (2 loge x)dx equal :
1
(1) e(4e–1)
Q.6
(3) 4e2–1
(4) e(2e–1)
If the tangent to the curve, y=f(x)=xlogex, (x>0) at a point (c,f(c)) is parallel to the line-segment joining
the points (1,0) and (e,e), then c is equal to:
(1)
Q.7
(2) e(4e+1)
e
1
1 e
(2)
e 1
e
1
(3) e 1
(4)
e
1
e 1
2
dy
2
If y x 1 cosecx is the solution of the differential equation, dx p(x)y cosecx, 0 x 2 ,
then the function p(x) is equal to:
(1) cosec x
(2) cot x
(3) tan x
(4) sec x
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Q.8
If and are the roots of the equation 2x(2x+1)=1, then is equal to:
(1) 2( 1)
(2) 2( 1)
(3) 2 2
(4) 2( 1)
Q.9
For all twice differentiable functions f: R R, with f(0)=f(1)=f’(0)=0,
(1) f”(x)=0, at every point x (0,1)
(2) f”(x) 0, at every point x (0,1)
(3) f”(x)=0, for some x (0,1)
(4) f”(0)=0
Q.10 The area (in sq.units) of the region enclosed by the curves y=x2–1 and y=1–x2 is equal to :
(1)
Q.11
4
3
(2)
7
2
(3)
16
3
(4)
8
3
For a suitably chosen real constant a, let a function, f:R–{–a} R be defined by f (x)
ax
ax
. Further
1
suppose that for any real number x a and f(x) –a, (fof)(x)=x.Then f 2 is equal to:
(1) –3
1
(2) 3
cos
1
(4) 3
(3) 3
sin
Q.12 Let 5 and A sin cos . If B=A+A4, then det (B):
(1) is one
(2) lies in (1,2)
(3) lies in (2,3)
(4) is zero
Q.13 The centre of the circle passing through the point (0,1) and touching the parabola y=x2 at the point (2,4)
is :
3 16
(1) 10 , 5
6 53
(2) 5 , 10
16 53
(3) 5 , 10
53 16
(4) 10 , 5
Q.14 A plane P meets the coordinate axes at A, B and C respectively. The centroid of ABC is given to be
(1,1,2). Then the equation of the line through this centroid and perpendicular to the plane P is:
(1)
x 1 y 1 z 2
2
1
1
(2)
x 1 y 1 z 2
2
2
1
(3)
x 1 y 1 z 2
1
2
2
(4)
x 1 y 1 z 2
1
1
2
Q.15 Let f : R R be a function defined by f(x)=max {x,x2}. Let S denote the set of all points in R, where
f is not differentiable. Then
(1) {0,1}
(2) (an empty set) (3) {1}
(4) {0}
Q.16 The angle of elevation of the summit of a mountain from a point on the ground is 45°. After climbing up
one km towards the summit at an inclination of 30° from the ground, the angle of elevation of the summit
is found to be 60°. Then the height (in km) of the summit from the ground is:
(1)
1
3 1
(2)
3 1
3 1
(3)
3 1
3 1
(4)
1
3 1
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Q.17
k
If the constant term in the binomial expansion of x x 2 is 405, then |k| equals:
(1) 1
(2) 9
(3) 2
(4) 3
2
2
Q.18 Let z=x+iy be a non-zero complex number such that z = i|z| , where i 1 , then z lies on the
(1) line, y=x
(2) real axis
(3) imaginary axis
(4) line, y=–x
Q.19 Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the
point (–1, –4) in this line is:
11 28
(1) 5 , 5
8 29
(2) 5 , 5
29 11
(3) 5 , 5
29 8
(4) 5 , 5
Q.20 Consider the statement : “For an integer n, if n3–1 is even, then n is odd.” The contrapositive statement
of this statement is:
(1) For an integer n, if n is even, then n3–1 is even
(2) For an integer n, if n is odd, then n3–1 is even
(3) For an integer n, if n3–1 is not even, then n is not odd.
(4) For an integer n, if n is even, then n3–1 is odd
SECTION – 2 : (Maximum Marks : 20)
This section contains FIVE (05) questions. The answer to each question is NUMERICAL
VALUE with two digit integer and decimal upto one digit.
If the numerical value has more than two decimal places truncate/round-off the value upto
TWO decimal places.
Full Marks : +4 If ONLY the correct option is chosen.
Zero Marks : 0 In all other cases
Integer Type Questions
Q.21 The number of words (with or without meaning) that can be formed from all the letters of the word
“LETTER” in which vowels never come together is______
Q.22 If x and y be two non-zero vectors such that x y x and 2x y is perpendicular to y , then the
value of is ______
Q.23 Consider the data on x taking the values 0, 2, 4, 8, .....,2n with frequencies n C0 ,n C1,n C2 ,....n Cn ,
respectively. If the mean of this data is
728
, then n is equal to ______
2n
Q.24 Suppose that function f : R R satisfies f(x+y)=f(x)f(y) for all x, y R and f(1)=3.
n
If f (i) 363 , then n is equal to .......
i 1
Q.25 The sum of distinct values of for which the system of equations
( 1)x (3 1)y 2z 0
( 1)x (4 2)y ( 3)z 0
2x (3 1)y 3( 1)z 0 ,
has non-zero solutions,is ______
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ANSWER KEY
1.
6.
11.
16.
21.
26.
(4)
(4)
(3)
(1)
(2)
(2)
2.
7.
12.
17.
22.
27.
(1)
(3)
(2)
(1)
(3)
(4)
AIEEE 2011 PAPER
3.
(4)
8.
(3)
13.
(4)
18.
(1, 2)
23.
(3)
28.
(2)
4.
9.
14.
19.
24.
29.
(2)
(3)
(2)
(3)
(2)
(1)
5.
10.
15.
20.
25.
30.
(3)
(1)
(2)
(1)
(2)
(2)
1.
6.
11.
16.
21.
26.
(2)
(4)
(2)
(2)
(2)
(2)
2.
7.
12.
17.
22.
27.
(3)
(2)
(1)
(3)
(1)
(4)
AIEEE 2012 PAPER
3.
(3)
8.
(4)
13.
(3)
18.
(4)
23.
(3)
28.
(2)
4.
9.
14.
19.
24.
29.
(2)
(1)
(4)
(1)
(2 or 4)
(3)
5.
10.
15.
20.
25.
30.
(1)
(4)
(1)
(3)
(1)
(2)
(3)
(3)
(1)
(1)
(2)
(2)
JEE MAIN 2013 PAPER
3.
(3)
8.
(2)
13.
(2)
18.
(3)
23.
(3)
28.
(1)
4.
9.
14.
19.
24.
29.
(3)
(2)
(3)
(2)
(1)
(2)
5.
10.
15.
20.
25.
30.
(3)
(1)
(4)
(3)
(2)
(4)
4
4
4
4
2
3
JEE MAIN 2014 PAPER
3.
1
8.
1
13.
3
18.
4
23.
4
28.
1
4.
9.
14.
19.
24.
29.
4
2
4
4
4
1
5.
10.
15.
20.
25.
30.
2
2
4
1
3
3
JEE MAIN 2015 PAPER
3.
(1)
8.
(4)
13.
(4)
18.
(3)
23.
(1)
28.
(1)
4.
9.
14.
19.
24.
29.
(3)
(*)
(1)
(2)
(2)
(2)
5.
10.
15.
20.
25.
30.
(1)
(2)
(2)
(4)
(4)
(4)
3
1
3
1
4
1
5.
10.
15.
20.
25.
30.
2
1
2
1
3
3
1.
6.
11.
16.
21.
26.
1.
6.
11.
16.
21.
26.
(3)
(4)
(3)
(4)
(1)
(1)
1
4
2
3
3
4
2.
7.
12.
17.
22.
27.
2.
7.
12.
17.
22.
27.
1.
6.
11.
16.
21.
26.
(3)
(1)
(2)
(1)
(3)
(3)
2.
7.
12.
17.
22.
27.
(3)
(2)
(2)
(4)
(2)
(4)
1.
6.
11.
16.
21.
26.
2
2
3
1
3
3
2.
7.
12.
17.
22.
27.
1
3
2
3
1
1
JEE MAIN 2016 PAPER (OFFLINE)
3.
4
4.
8.
2
9.
13.
1
14.
18.
3
19.
23.
2
24.
28.
1
29.
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1.
6.
11.
16.
21.
26.
3
2
2
4
3
1
2.
7.
12.
17.
22.
27.
JEE-MAIN (09-04-2016) ONLINE PAPER
3
3.
4
4.
1
8.
4
9.
2
13.
3
14.
1
18.
1
19.
1
23.
1
24.
3
28.
4
29.
3
3
3
3
1
3
5.
10.
15.
20.
25.
30.
1
3
1
3
1
4
1.
6.
11.
16.
21.
26.
2
3
3
1
3
4
2.
7.
12.
17.
22.
27.
JEE-MAIN (10-04-2016) ONLINE PAPER
1
3.
2
4.
2
8.
1
9.
4
13.
Bonus
14.
1
18.
3
19.
2
23.
4
24.
4
28.
2
29.
4
4
1
2
1
2
5.
10.
15.
20.
25.
30.
1
1
1
4
3
2
2.
7.
12.
17.
22.
27.
JEE MAIN 2017 (ONLINE PAPER)
(4)
3.
(1)
4.
(4)
8.
(4)
9.
(1)
13.
(1)
14.
(4)
18.
(2)
19.
(4)
23.
(3)
24.
(1)
28.
(1)
29.
(4)
(3)
(1)
(3)
(2)
(1)
5.
10.
15.
20.
25.
30.
(2)
(3)
(1)
(3)
(2)
(1)
2.
7.
12.
17.
22.
27.
JEE MAIN 2018 (ONLINE PAPER)
(3)
3.
(3)
4.
(4)
8.
(3)
9.
(4)
13.
(4)
14.
(3)
18.
(3)
19.
(1)
23.
(2)
24.
(3)
28.
(2)
29.
(3)
(2)
(2)
(3)
(4)
(1)
5.
10.
15.
20.
25.
30.
(2)
(3)
(4)
(4)
(1)
(1)
4.
9.
14.
19.
24.
29.
C
D
C
C
A
A
5.
10.
15.
20.
25.
30.
B
A
A
A
D
D
4.
9.
14.
19.
24.
29.
A
A
B
C
D
B
5.
10.
15.
20.
25.
30.
A
D
D
A
A
B
1.
6.
11.
16.
21.
26.
1.
6.
11.
16.
21.
26.
(4)
(1)
(1)
(4)
(4)
(2)
(2)
(1)
(1)
(1)
(2)
(2)
JEE MAIN 2019 PAPER
(9/01/2019) FIRST SHIFT
1.
6.
11.
16.
21.
26.
A
A
B
A
C
D
2.
7.
12.
17.
22.
27.
D
A
D
D
B
C
3.
8.
13.
18.
23.
28.
D
A
B
B
A
A
(9/01/2019) SECOND SHIFT
1.
6.
11.
16.
21.
26.
D
C
A
A
A
B
2.
7.
12.
17.
22.
27.
A
C
B
A
D
B
3.
8.
13.
18.
23.
28.
B
A
A
B
B
B
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(10/01/2019) FIRST SHIFT
1.
6.
11.
16.
21.
26.
C
D
A
*
C
A
2.
7.
12.
17.
22.
27.
A
A
B
D
B
B
1.
6.
11.
16.
21.
26.
D
B
A
D
D
C
2.
7.
12.
17.
22.
27.
C
A
D
D
A
C
3.
8.
13.
18.
23.
28.
B
A
B
C
A
D
4.
9.
14.
19.
24.
29.
B
A
B
B
A
B
5.
10.
15.
20.
25.
30.
D
D
C
C
B
D
4.
9.
14.
19.
24.
29.
C
A
B
B
B
A
5.
10.
15.
20.
25.
30.
A
D
B
B
B
C
4.
9.
14.
19.
24.
29.
B
C
B
B
A
D
5.
10.
15.
20.
25.
30.
B
B
A
A
C
A
4.
9.
14.
19.
24.
29.
B
B
B
A
B
A
5.
10.
15.
20.
25.
30.
A
D
D
B
B
C
4.
9.
14.
19.
24.
29.
A
B
D
A
D
B
5.
10.
15.
20.
25.
30.
B
A
A
C
B
D
4.
9.
14.
19.
24.
29.
A
D
B
A
C
B
5.
10.
15.
20.
25.
30.
B
B
A
D
B
C
(10/01/2019) SECOND SHIFT
3.
8.
13.
18.
23.
28.
A
B
C
A
D
A
(11/01/2019) FIRST SHIFT
1.
6.
11.
16.
21.
26.
B
A
B
A
B
C
2.
7.
12.
17.
22.
27.
D
D
C
C
C
B/C
1.
6.
11.
16.
21.
26.
D
C
A
C
C
C
2.
7.
12.
17.
22.
27.
B
A
B
D
B
C
1.
6.
11.
16.
21.
26.
A
C
C
C
C
C
2.
7.
12.
17.
22.
27.
D
D
D
B
C
D
3.
8.
13.
18.
23.
28.
C
D
D
A
A
D
(11/01/2019) SECOND SHIFT
3.
8.
13.
18.
23.
28.
A
B
B
C
B
B
(12/01/2019) FIRST SHIFT
3.
8.
13.
18.
23.
28.
C
A
D
A
B
A
(12/01/2019) SECOND SHIFT
1.
6.
11.
16.
21.
26.
B
B
B
D
C
A
2.
7.
12.
17.
22.
27.
B
B
A
C
B
B
3.
8.
13.
18.
23.
28.
D
A
B
B
C
A
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(8/04/2019) FIRST SHIFT
1.
6.
11.
16.
21.
26.
A
B
B
A
D
D
2.
7.
12.
17.
22.
27.
B
B
A
B
D
C
1.
6.
11.
16.
21.
26.
D
A
A
D
A
C
2.
7.
12.
17.
22.
27.
A
C
C
C
D
D
1.
6.
11.
16.
21.
26.
C
A
A
A
B
C
2.
7.
12.
17.
22.
27.
D
C
B
B
C
A
3.
8.
13.
18.
23.
28.
A
B
A
B
B
D
4.
9.
14.
19.
24.
29.
C
C
B
B
B
A
5.
10.
15.
20.
25.
30.
B
D
D
C
D
D
4.
9.
14.
19.
24.
29.
D
D
D
A
B
D
5.
10.
15.
20.
25.
30.
C
B
C
A
D
A
4.
9.
14.
19.
24.
29.
B
B
B
B
C
A
5.
10.
15.
20.
25.
30.
B
C
D
A
B
A
4.
9.
14.
19.
24.
29.
C
C
C
D
B
C
5.
10.
15.
20.
25.
30.
B
D
B
D
C
B
4.
9.
14.
19.
24.
29.
B
A
D
C
D
A
5.
10.
15.
20.
25.
30.
C
B
C
C
C
C
4.
9.
14.
19.
24.
29.
C
D
A
A
C
A
5.
10.
15.
20.
25.
30.
B
C
A
B
C
B
(8/04/2019) SECOND SHIFT
3.
8.
13.
18.
23.
28.
B
A
D
A
C
C
(9/04/2019) FIRST SHIFT
3.
8.
13.
18.
23.
28.
C
C
D
A
C
A
(9/04/2019) SECOND SHIFT
1.
6.
11.
16.
21.
26.
C
A
C
D
D
B
2.
7.
12.
17.
22.
27.
A
A
D
B
C
A
1.
6.
11.
16.
21.
26.
A
C
A
A
A
C
2.
7.
12.
17.
22.
27.
B
B
A
C
D
C
1.
6.
11.
16.
21.
26.
A
B
C
C
B
D
2.
7.
12.
17.
22.
27.
D
C
A
B
C
A
3.
8.
13.
18.
23.
28.
C
D
B
A
D
B
(10/04/2019) FIRST SHIFT
3.
8.
13.
18.
23.
28.
D
D
A
A
A
D
(10/04/2019) SECOND SHIFT
3.
8.
13.
18.
23.
28.
A
C
D
B
C
D
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(12/04/2019) FIRST SHIFT
1.
6.
11.
16.
21.
26.
B
A
A
D
B
C
2.
7.
12.
17.
22.
27.
D
A
B
C
B
C
3.
8.
13.
18.
23.
28.
C
C
A
D
A
A
4.
9.
14.
19.
24.
29.
B
C
A
D
C
B
5.
10.
15.
20.
25.
30.
D
A
C
B
C
C
4.
9.
14.
19.
24.
29.
C
A
C
C
B
A
5.
10.
15.
20.
25.
30.
B
D
A
A
A
D
4.
7.
12.
17.
22.
3
2
3
2
18
8.
13.
18.
23.
4
2
4
30
4.
9.
14.
18.
23.
4
1
3
1
5
5.
10.
15.
19.
24.
3
4
2
1
54
4.
9.
14.
19.
24.
1
2
2
4
1540
5.
10.
15.
20.
25.
3
4
3
2
672
4.
9.
14.
18.
23.
3
4
3
2
1
5.
10.
15.
19.
24.
4
3
2
1
2454
(12/04/2019) SECOND SHIFT
1.
6.
11.
16.
21.
26.
1.
5.
9.
14.
19.
24.
A
B
D
B
D
B
2.
7.
12.
17.
22.
27.
2
2.
1, 3 (NTA given 1)
4
10.
3
15.
1
20.
3
25.
A
A
D
D
C
D
4
3
2
1
5
3.
8.
13.
18.
23.
28.
A
A
B
A
A
B
JEE MAIN 2020 PAPER
(07/01/2020)_FIRST SHIFT
3.
1
6.
4
11.
3
16.
3
21.
36
(07/01/2020)_SECOND SHIFT
1.
6.
11.
16.
20.
25.
2
1
1
3
2
29
2.
7.
12.
17.
21.
4
3.
4
8.
3
13.
Bonus (NTA given 2)
13
22.
1.
6.
11.
16.
21.
4
1
3
4
490
2.
7.
12.
17.
22.
3
1
1
4
4
2
1
3
4
(08/01/2020) FIRST SHIFT)
3.
8.
13.
18.
23.
1
2
4
4
8
(08/01/2020)_SECOND SHIFT
1.
6.
11.
16.
20.
25.
1
2
1
2
4
504
2.
7.
12.
17.
21.
1
3.
3
8.
2
13.
2 (NTA given 2)
0.5
22.
2
1
4
3
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(09/01/2020)_FIRST SHIFT
1.
6.
11.
16.
21.
2
3
3
4
1
2.
7.
12.
17.
22.
3
4
3
1
8
1.
4.
9.
14.
19.
24.
2
4
3
1
2
14
2.
5.
10.
15.
20.
25.
2
1
1
3
2
3
1.
6.
11.
16.
21.
1
3
4
2
9
2.
7.
12.
17.
22.
3
4
4
4
2
3.
8.
13.
18.
23.
4
2
2
3
8
3
4
1
3
3
5.
10.
15.
20.
25.
2
3
4
4
615
3
4
3
51.
8.
13.
18.
23.
1
2
4
36
4.
9.
14.
19.
24.
4
2
2
2
40
5.
10.
15.
20.
25.
1
2
2
3
1.5
4.
9.
14.
19.
24.
(2)
(3)
(2)
(2)
3
5.
10.
15.
20.
25.
(2)
(2)
(2)
(4)
118
4.
9.
14.
19.
24.
3
4
3
1
10
5.
10.
15.
20.
25.
4
3
1
2
4
4.
9.
14.
19.
24.
4
1
2
1
54
5.
10.
15.
20.
25.
4
3
3
4
4
4.
9.
14.
19.
24.
(3)
(3)
(3)
(4)
8
5.
10.
15.
20.
25.
(3)
(1)
(4)
(1)
3
4.
9.
14.
19.
24.
(09-01-2020)_SECOND SHIFT
3.
6.
11.
16.
21.
Bonus ( NTA given 1)
2
7.
2
12.
1
17.
30
22.
(02/09/2020)_FIRST SHIFT
3.
8.
13.
18.
23.
4
4
3
3
309
(02/09/2020)_SECOND SHIFT
1.
6.
11.
16.
21.
(2)
(2)
(4)
(1)
0.8
2.
7.
12.
17.
22.
(2)
(3)
(2)
(3)
1
1.
6.
11.
16.
21.
2
2
1
4
8
2.
7.
12.
17.
22.
4
1
1
4
2
3.
8.
13.
18.
23.
(3)
(3)
(3)
(1)
91
(03/09/2020)_FIRST SHIFT
3.
8.
13.
18.
23.
3
3
4
3
4
(03/09/2020)_SECOND SHIFT
1.
6.
11.
16.
21.
2
3
1
1
39
2.
7.
12.
17.
22.
3
1
3
3
5
1.
6.
11.
16.
21.
(2)
(4)
(2)
(3)
10
2.
7.
12.
17.
22.
(1)
(2)
(1)
(4)
3
3.
8.
13.
18.
23.
1
4
3
3
8
(04/09/2020)_FIRST SHIFT
3.
8.
13.
18.
23.
(2)
(1)
(2)
(4)
5
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Maths IIT-JEE ‘Best Approach’
MCSIR
JEE Main Papers
(04/09/2020)_SECOND SHIFT
1.
6.
11.
16.
21.
(1)
(2)
(2)
(3)
(21)
2.
7.
12.
17.
22.
(3)
(4)
(3)
(2)
(135)
1.
6.
11.
16.
21.
2
3
4
4
13
2.
7.
12.
17.
22.
4
2
2
1
11
1.
6.
11.
16.
21.
1
1
1
19
2.
7.
12.
17.
22.
2
2
4
3
120
1.
6.
11.
16.
21.
1
3
3
Bouns
4
2.
7.
12.
17.
22.
4
2
2
4
5
3.
8.
13.
18.
23.
(4)
(4)
(3)
(2)
(18)
4.
9.
14.
19.
24.
(3)
(2)
(2)
(4)
(7)
5.
10.
15.
20.
25.
(1)
(2)
(3)
(3)
(4)
4.
9.
14.
19.
24.
1
4
1
2
240
5.
10.
15.
20.
25.
4
1
4
4
30
4.
9.
14.
19.
24.
1
1
2
3
0.5
5.
10.
15.
20.
25.
2
4
1
3
11
4.
9.
14.
19.
24.
4
2
2
1
80
5.
10.
15.
20.
25.
2
1
4
2
28
4.
9.
14.
19.
24.
(4)
(3)
(2)
(1)
5
5.
10.
15.
20.
25.
(1)
(4)
(1)
(4)
3
(05/09/2020)_FIRST SHIFT
3.
8.
13.
18.
23.
4
2
1
2
8
(05/09/2020)_SECOND SHIFT
3.
8.
13.
18.
23.
4
4
1
2
6
(06/09/2020)_FIRST SHIFT
3.
8.
13.
18.
23.
3
2
4
3
5
(06/09/2020)_SECOND SHIFT
1.
6.
11.
16.
21.
(3)
(4)
(2)
(4)
120
2.
7.
12.
17.
22.
(1)
(2)
(2)
(4)
1
3.
8.
13.
18.
23.
(2)
(2)
(3)
(1)
6
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