PRACTICE TEST 1
Mathematics
DATE : 01-10-2023
PART – C
SECTION – A
(Single Choice Answer Type)
This section contains 20 multiple choice questions. Each question has four choices (A), (B), (C) and (D)
out of which ONLY ONE is correct.
61.
A triangle is formed by the lines whose combined equation is given by
 x  y  4  xy  2x  y  2  0 The equation of its circumcircle is
(A) x 2  y 2  5 x  3y  8  0
(B) x 2  y 2  3x  5y  8  0
(C) x 2  y 2  3x  5y  8  0
(D) None of these
62.
Two circles, each of radius 5, have a common tangent at (1,1) whose equation is 3x + 4y – 7 = 0.
Then their centres are
(A) (4, -5), (-2, 3)
(B) (4, -3), (-2, 5)
(C) (4, 5), (-2, -3)
(D) None of these
63.
If p and q be the longest distance and the shortest distance respectively of the point (-7, 2) from
any point (, ) on the curve whose equation is x 2  y 2  10 x  14y  51  0, then G.M. of p and
q is equal to
(A) 2 11
(B) 5 5
(C) 13
(D) None of these
64.
Through the point (13, 31) a straight line is drawn to meet the axes of X and Y at Q and S
respectively. If the rectangle OQRS is completed, the co-ordinates of R satisfy the equation.
13 31
31 13
(A)

1
(B)

1
x
y
x
y
(C)
13 31

1
x
y
(D)
31 13

 1.
x
y
65.
The normal chord of parabola, which sub-stends right angle at vertex is inclined at an angle  to
the axis of parabola. Find the value of  ?
 1 
(A) tan1 
(B) tan1  7 

 2

(C)
(D) tan1  2 
3
66.
A ray of light travelling along the line y = k from the positive direction of x-axis strikes a concave
2
mirror, whose intersection with xy plane is parabola y = 4ax. Find the equation of line along
which the reflected ray will travel. Assume ‘a’ and ‘k’ are positive.
(A)  k 2  4a2  y  4ak  x  a 
(B)  k 2  4a2  y  4ak  x  a 
(C) x  a
(D) x  2a
67.
TP and TQ are distinct tangents drawn to a parabola from the point T and P1, P2 and P3 are the
lengths of perpendicular from P, T and Q respectively on any tangent (other than tangent at P and Q)
to the curve. Then,
(A) P1, P2, P3 are in A.P
(B) P1, P2, P3 are in G.P
(C) P1, P2, P3 are in H.P
(D)
1 1
a


(4a is latus rectum)
P1 P2 P2
68.
Any point P on the ellipse
x2 y 2 1


a2 b2 2
x2 y2
(C) 2  2  2
a
b
(A)
69.
70.
x2 y 2

 1 is joined to its centre O. The locus of mid-point of OP is
a2 b2
x2 y 2 1
(B) 2  2 
a
b
4
2
2
a
b
(D) 2  2  1
x
y
If the normal at any point P on an ellipse of eccentricity ‘e’ meets the major axis and minor axis at
T and G respectively, then PT : PG is equal to
(A) e
(B) 1 : e
2
2
(C) (1 – e ) : 1
(D) (1 – e ) : e
C
For a certain value of c, lim  x 5  7 x 4  2   x  is finite & non zero. The value of c and the
x 
value of the limit is
(A) 1/5, 7/5
(C) 1, 7/5
71.
If
 x
3
(B) 0, 1
(D) none of these
 2 x 2  5  e3 x dx  e3 x  Ax 3  Bx 2  Cx  D  then the statement which is incorrect is
(A) C + 3D = 5
(B) A + B + 2/3 = 0
(C) 3C - 2B = 0
2
72.
Let I1 =
e
0
x2
(D) A + B + C = 0
2
sin( x )dx; I2 =
e
 x2
0
2
dx ; I3 =
e
74.
(1  x ) dx
0
and consider the statements
I
I1 < I2 II I2 < I3 III I1 = I3
Which of the following is(are) true?
(A) I only
(C) Neither I nor II nor III
73.
 x2
(B) II only
(D) Both I and II
 
If the function f(x) = 2 tan x + (2a + 1)ln |sec x| + (a –2)x is increasing in  0,  then range of ‘a’
 2
equal to
(A) (–, 0]
(B) [0, 1]
(C) [0, 3]
(D) [0, )
 7  5  x2  3 
The maximum value of sec-1 
 is:
2
 2  x  2 
5
6
7
(C)
12
(A)
5
12
2
(D)
3
(B)
75.
Suppose p, q, r  0 and system of equations
 p  a  x  by  cz  0
ax   q  b  y  cz  0
ax  by  r  c  z  0
has a non-trivial solution, then value of
(A) -1
(C) 1
76.
77.
78.
79.
2
2
If the system of equations ax + hy + g = 0, hx + by + f = 0 and ax + 2hxy + by + 2gx + 2fy + c + t
= 0 has a unique solution, then
(A) t = 1
(B) t = 0
2
2
2
abc  2fgh  af  bg  ch
1
(C) t 
(D) t  2
2
h  ab
h  ab


Let p and q be the position vectors of P and Q respectively, with respect to O and


p  p, q  q. The points R and S divide PQ internally and externally in the ratio 2 : 3


respectively. If OR and OS are perpendicular, then
(A) 9p2 = 4q2
(B) 4p2 = 9q2
(C) 9p = 4q
(D) 4p = 9q

 
 
 

If d   (a  b)   (b  c)   (c  a) and [a,b,c]  1
Then      is equal to
   
  
(A) d.( a  b  c )
(B) a  b  c
(C) 1
(D) none of these
Area of the region bounded by the curve y 
the greatest integer function) is
1
3/ 2
(A)  4   
3
8
3/ 2
(C)  4   
3
80.
a b c
  is
p q r
(B) 0
(D) 2
16  x2
and y = sec-1 [– sin2 x] (where [.] denotes
4
(B) 8  4   
(D)
3/2
8
 4   1/ 2
3
A contest consists of predicting the results (win, draw or defeat) of 10 football matches. The
probability that one entry contains at least 5 correct answers is
12585
12385
(A)
(B)
310
310
9385
(C)
(D) none of these
310
SECTION – B
(Numerical Answer Type)
This section contains 10 Numerical based questions. The answer to each question is rounded off to the
nearest integer value.
81.
If sinx + cosx + tanx + cot x + secx + cosecx = 7 and sin2x = a – b 7 , where a, b are rational ,
ab
then
is
10
82.
If tanx – tan2x = 1, then the value of tan4x – 2tan3x – tan2x + 2tanx + 1 is
83.
The hyperbola
84.
Co-ordinates of points common to hyperbola 25x2 – 9y2 = 225 and 25x + 12y = 45 is (x1, y1) then
3  x1  y1  is
85.
For x satisfying log1/ (6x+1 – 36x)  – 2, x  (, a]  [logb c,1 ) then a + b + c is
86.
S.D of a data is 6. When each observation is increased by 1, then the S.D of new data is
87.
Let u =
x2 y2
4

 1 passes through (4, 2) and length of latus rectum is , the angle
a2 b2
3
6
between asymptotes is  then  is

1

0
ln( x  1)
x2  1
2
dx and v =
 ln(sin2x ) dx
then 4u + v =
0
88.
Let f(x) = Max. (cos x, x, 2x –1) where x  0. Then number of points of non-differentiability of f(x),
is equal to
89.
If a1 is the value of a for which function f(x) = x 2 
90.
If n positive integers are taken at random and multiplied together, then the probability that the last
an  bn
digit of the product is 2, 4, 6 or 8 is
then a + b + c is equal to (a,b,c are in lowest form)
cn
a
has a local minimum at x = 2 and a2 is the
x
a a 
value of a for which f(x) has a point of inflection at x = 1, then  1 2  is equal to
 3 