TargeT MaTheMaTics by- agyaT gupTa
Q.1
Q.2
Q.3
Q.4
Q.5
1-Marks
T
If A is a square matrix satisfying A A  I . Write the value of A .
Find the value of a for which the function f ( x) = x 2  2 ax  6 is increasing when x > 0.
If the value of third order determinant is 12, than find the value of the determinant formed by
its cofactors .


Q.6
Q.7
Q.8
Q.9
Q.10
Q.11
Q.12
Q.13
Q.14
Q.15
Q.17
Q.18
Q.19
Q.20
Q.21






sum of 2 i  4 j  5 k and m i  2 j  3 k is equal to unity.
  
Given f(x) = sin x check if function f is one-one for (i) (0,π) (ii)   , 
2 2
How many equivalence relations on the set {1,2,3} containing (1,2) and (2,1) are there in all ?
Justify your answer .
Write the number of all possible matrices of order 2  2 with each entry 0 or 1 and whose
determinant is positive.
Find  when the projection of iˆ  ˆj  kˆ on iˆ  ˆj is 2 units.
x
, find f  f (2) 
If f : R  R is defined by f ( x )  2
x 1
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
If f : R  R be given by f ( x )  ( 3  x 3 ) 1 / 3 , then fof ( x ) .
The operation * define below on Q is binary operation a * b = ab+1. Check the existence
of identity element and the inverse of all elements in Q .
3 
 2 sin x
 is
2 sin x 
 1
In the interval  / 2  x   , find the value of x for which the matrix 
singular.


Let a and b be two vectors such that

Q.16

Find the scalar m, such that the scalar product of i  j k with the unit vector parallel to the

a  3
;

b
=
3

Then what is the angle between a  b ?
2


and a  b is a unit vector.
If * be the binary operation on the set Q of rational numbers defined by a * b =
identity element and inverse .
ab
. Find
4
Write total number of injective functions from set A and set B, where
A = {1, 2, 3}, B = {a, b, c, d}.
If A is a square matrix such that
If

   
 a b    a.b   225 and a  5 ,

 

2
2
A2  I ,
3
3
then find the simplified value of  A  I   A  I  7A .
then write the value of

b
.
Let set A = {3, 5, 6} and set B = {1 ,4 }. A relation R from set A to set B is defined as R =
{(a,b)  A×B: a–b is an even number}. List the elements of relation R.









Let a  2 i  j , b  i  2 j and c  4 i  3 j . Find the values of x and y such that



c  x a y b .
1
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.22
Q.23
TargeT MaTheMaTics by- agyaT gupTa
Prove that if E and F are independent events, then the events E and F are also
independent.

 

 

If a and b are two vectors such that | a  b || a | , then prove that vector 2a  b is
_
perpendicular to vector b .

Q.24
Q.25
Q.26
Q.27
Q.28
Q.29
Q.30
Q.31
Q.32
Q.33
Q.34
Q.35
Let * be binary operation on N defined by a*b=HCF of a and b. Does there exist identity
for this binary operation on N?

a
dx

2

,
(i) If 0
then
find
K
.
(ii) If  3 x dx  8 , write the value of 'a'.
2
16
2  8x
0
k
If
 (3 x
1
2
0
 2 x  k ) dx  0, find the value of k.
 
If f  x   sin2x  cos2x ] find f '  6 .
 
Find the sum of the degree and the order for the following differential equation
d d2y 4
[(
) ]  0.
dx dx 2
In the following matrix equation use elementary operation R2  R2  R1 and write the
equation thus obtained.
Determine the value of the constant ‘k’ so that the function
continuous at x  0 .
If
 3
A  3 and A 1   5


 3
1
2  ,
3 
x
If y  3 , then find
2
f(x)=
Q.3
Q.4
x 0
is
x 0
dy
.
dx
dy
at x  1 .
dx
 d 2 y   dy 
Write the order of the differential equation: log  2      x .
 dx   dx 
3
Find the acute angle which the line with direction cosines
2-Marks
1
3
,
1
6
,n makes with positive direction of z-
Find the value of m and n, where m and n are order and degree of differential equation
 d2 y 
4 2 
3
 dx   d y  x 2  1.
d3 y
dx 3
dx 3
3
Q.2
 kx
 , if
 lxl
 3, if
then write the adj A.
2
2
If f ' ( x)  2 x  1andy  f ( x ), then find
axis.
Q.1
 2 3  1 0   8 3 




 1 4  2 1  9 4 
 cos  sin  
n
 , then for any natural number n, find the value of Det ( A ) .

sin

cos



If A  
If x change from 4 to 4.01, then find the approximate change in logex.
Prove that:

       

a   b  c    a  2 b  3 c
 

 

     
   a b c

  




.
2
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.5
Q.6
Q.7
Q.8
Q.9
Q.10
Q.11
Q.12
Q.13
TargeT MaTheMaTics by- agyaT gupTa
1
1 4 
Evaluate: sin  cos
.
5
2
A bag contains 4 green and 6 white balls. Two balls are drawn one by one without
replacement. If the second ball drawn is white, what is the probability that the first ball
drawn is also white?
Find the differential equation for all the straight lines, which are at a unit distance from the origin.
Evaluate

x
1  sin dx.
4
The income I of Dr. Rastogi is given by I ( x)  Rs. ( x 3  3x 2  5x) . Can an insurance
agent ensure him for the growth of his income?
Prove that : tan 1 2  tan 1 3 
Q.15
Q.16
Q.17
Q.18
Q.19
Q.20
Q.21
Q.22
Q.23
Q.24
Q.25
Q.26
Q.27
2
2x
2x

1 1  x
1
3 sin

4
cos

2
tan

1 x2
1 x2
1 x2 3
Solve the following equation :
1
.
Two cards are drawn without replacement from a well shuffled pack of 52 cards. Find the
probability that one is a king and other is a queen of opposite color.
Find
the
integrating
factor
for
the
linear
differential
equation
:
( y 2  1)  2xy
Q.14
3
.
4
dy  2  dy
 .

dx  y 2  1 dx
If â, b̂ and ĉ are mutually perpendicular unit vectors, then find the value of | 2â  b̂  ĉ | .
A pair of fair dice is thrown. Find the probability that the sum is 10 or greater, if 5 appears on
the first die.
Find the differential equation of all the lines in the xy-plane.
4 3 
If A  2 5 , find x and y such that A 2  xA  yI  0 .


Evaluate :






 dx

3
sin x sin( x  a ) 
1
.
A man is walking at the rate of 6.5 km/hr towards the foot of a tower 120 m high. At what rate
is he approaching the top of the tower when he is 50 m away from the tower ?
Solve : cos
1
sin cos
1


x 
.
3
Find the value of c in Rolle’s theorem for the function f  x  x3  3x in 
Let
 2 1
5 2
2 5
A
 ,B  
 ,C  

3 4 
7 4
3 8
If f : R  R be defined as f ( x) 
If
xy  e
xy 
, then show that
3,0  .

, Find a matrix D such that CD-AB=O.
3x  7
, then find f
9
dy y  x  1
.

dx x  y  1
1
( x) .

3 
1
1
1 
3
sin
1

4
tan
(

3
)

5
cos

 cos 1 1.
Find the value of
 2 


Show that f : R  R , given by f ( x)  x  x  , is neither one-one nor onto, where x 
denotes greatest integer less than or equal to x .
If a,b and


c
are mutually perpendicular vectors of equal magnitudes, find the angles which
3
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.28
Q.29
Q.30
Q.31
Q.32
Q.33
Q.34
Q.35
Q.36
Q.37
Q.38
Q.39
Q.40
Q.41
Q.42
Q.43
Q.44
Q.45
Q.46
 
TargeT MaTheMaTics by- agyaT gupTa



the vector 2 a  b  2 c makes with the vector a,b and


c
.
If A  aij is a matrix of order 2  2 , such that A  15 and
C ij
represents the
cofactor of aij then find a 21 c 21  a 22 c 22 .
Let * be a binary operation on the set N, of rational numbers defined by a*b =a+b+10, for
a, b  N. Find the identity elements for * if exists.
Find λ ,  if 2i  26 j  27k   i  j  k   0 .
For what value of k, the matrix
 2k  3 4

0
 4
 5
6



6 
 2k  3 
5
is skew symmetric ?
Let * be a binary operation Q+ , the set of all positive integers, defined by a*b=
 Q . Find the inverse of 0.1.
Write total number of bijective functions from set A and set B, where
A = {1, 2, 3}, B = {a, b, c, d}.
ab
, for a, b
100
Prove that if E and F are independent events, then the events E and F are also
independent.
_
Form the differential equation of the family of circle in the second quadrant and touching
the coordinate axes.
1
Evaluate 1 x x dx.
For what values of x is the rate of increase of x 3  5x 2  5x  8 is twice the rate of increase of x?
a  2b
 1
 1
1 a 
 tan   cos 1  
Prove that : tan   cos
.

b
b
a
4 2
4 2




If the vectors p  aiˆ  ˆj  k,ˆ q  ˆi  bjˆ  kˆ and r  ˆi  ˆj  CK are coplanar, then for
1
1
1


1
1 a 1 b 1 c
a,b,c  1 show
that
A die whose faces are marked 1,2,3 in red and 4,5,6 in green, is tossed. Let A be the event
“number obtained is even” and B be the event “number obtained is red”. Find if A and B
are independent events.
Solve the following equation for x:


3

cos tan1 x  sin  cot 1 
4

1
1
1 x
1    xyz  yz  zx  xy  .
Using properties of determinants show that 1 1  y
1 z
1
1
If y


 sin 1 6 x 1  9 x 2 , 
If a



,b, c
1
3 2
x
1
3 2
, then find


dy
.
dx







are three vectors such that a  b  c  0 , then prove that a b  b c  c a,
  
and hence show that a b c  0 .


2
at t  3 when x=10 (t-sin t) and y=12 (1-cos t).
Find the sum of the degree and the order for the following differential equation
Find
dy
dx
4
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
TargeT MaTheMaTics by- agyaT gupTa
d d2y 4
[(
) ]  0.
dx dx2
Q.47
Q.48
Q.49
Q.50
Q.51
Q.52
Q.53
Q.54
Q.55
Q.56
Q.57
Q.58
Q.59
Q.60
Q.61
Q.62
Q.63
Q.64
Q.65
2x 0
1 0
, find the value of x.
and A 1 
x x
1 2
If A 

Evaluate :
2
22
x
2 2 x dx.
2x
2
3
The equation of tangent at (2, 3) on the curve y  ax  b is y  4x  5 Find the value of
a and b.
Solve: tan
1
1 x  1
1

  tan x ; x  0 .
1 x  2
If m and n are the order and degree, respectively of the differential equation
2
3
2
 dy 
3 d y 
y     x  2   xy  sin x , then write the value of m + n.
 dx 
 dx 
 2x
If A  
x
Evaluate: 
0
 1 0
and A 1  

 , find the value of x.
x
 1 2 
sin x  x cos x dx
x ( x  sin x )
3
2
Slope of the normal to the curve x  8a y, a  0 at the point in the first quadrant is
 2
   . Find the point,
3
In a school, there are 1000 students, out of which 430 are girls. It is known that out of
430, 10% of the girls study in class XII. What is the probability that a student chosen
randomly studies in Class XII given that the chosen student is a girl ?
Form the differential equation of all circles which touch the x-axis at the origin.
0 2 
If matrix A  
and f(x)=1+x+x2 + x4 + x8 +x16 , find f(A).

0 0 
Evaluate:

1
x ( x  1) 3 / 5
3
5
dx .





ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
a

2
i

2
j

3
k
,
b


i

2
j

k
c

3
i

j
a


b
If
and
are such that
is
perpendicular to c , then find the value of  .

Find the approximate change in the value of
.
1
, when x changes from x = 2 to x = 2.002
x2
1

 x  1 then cos 1 x  cos 1  x 
Prove that if
2
2

 

.
 3

Let A = Z × Z and * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd).
Find the identity element for * in the set A.
Find:
Find


x
x
4
2
 sin 2 x  sec 2 x
1  x2
 x4
1
x5
3  3x 2
2
dx
dx
If A and B are two independent events, prove that A' and B are independent.
5
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
TargeT MaTheMaTics by- agyaT gupTa
Q.66
Q.67
Q.68
Q.1
Q.2
Q.3
Q.4
Q.5
Q.6
Q.7
Q.8
Q.9
Q.10
Q.11
Q.12
Q.13
4-Marks
 d 2 y  dy  2 
dy
.
Show that Ax  By  1 is a solution of the differential equation x  y 2      y
dx
 dx  
 dx
2
2
x
a
If y  tan 1   log
Evaluate :
 sin
xa
dy
2ax 2
: prove that
.
 4
xa
dx x  a 4
sin x
dx
x  cos 3 x

3

Evaluate: (i) 0 sin 1 x 1  x  x 1  x 2 dx . (ii)
1
Evaluate: (i)  x  x dx . (ii)  sin

2

2
3
1


sin x  cos x
dx
sin
2
x
/6
 /3
x  cos x dx
2
3
Find the equation of the normal’s to the curve y = x  2 x  6 which are parallel
to the line x + 14y + 4 = 0. Why are the fruits good for health?
Find whether the following function is differentiable at x = 1 and x = 2 or not :
x,
x 1


f (x)  
2  x,
1 x  2.
 2  3 x  x 2 ,
x2

The population of a village increases continuously at the rate proportional to the number of
its inhabitants present at any time. If the population of the village was 20000 in 1999 and
25000 in the year 2004, what will be the population of the village in 2009?
For the curve y  4 x  2 x , find all point at which the tangent passes through origin.
3
Evaluate:
(i)


 /2

x2
1  5x
/2
5
dx
(ii)

2
0
dx
1  e sin
x
.
An unbiased coin is tossed 'n' times. Let the random variable X denote the number of times
the head occurs. If P ( X  1), P ( X  2 ) and P ( X  3 ) are in AP, find the value of n.
Water is running into a conical vessel ,15 cm deep and 5 cm in radius , at the rate of 0.1
cm 3 / sec . When the water is 6cm deep, find at what rate is (i) the water level rising ? (ii)
the water –surface area increasing ? (iii) the wetted surface of the vessel increasing ?

6
For what value of K is the following function continuous at x   ?
6
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.14
Q.15
Q.16



f ( x)  



Q.18
3 sin x  cos x
x
k

6
Q.20
Q.21
Q.22
Q.23
x 

6

6
.
At what point on the curve x  y  2x  3  0 , is the tangent parallel to X-axis.
2
y/x
(i)From the differential equation of the family of curves given by (a  bx)e  x .
(ii)Solve the differential equation, (1  y  x 2 y ) dx  ( x  x 3 ) dy  0 where y = o when x = 1
If
x
cos(a+y)=
d
2
dx
y
2
cos
y
then
 sin 2 ( a  y )
sec x
dx
Find : 
1  cos ecx
prove
dy
 0 .
dx
that
2
dy cos  a  y  .

dx
sina
Hence
show
that
 3 2 
Find the inverse of the matrix 
 . Hence, find the matrix P satisfying the
 5  3
 3
Q.19
x 
2
sin a
Q.17
TargeT MaTheMaTics by- agyaT gupTa
matrix equation P 
5
2  1 2 

.
 3  2  1
Find the value of p for when the curves x 2  9p(9  y ) and x 2  p( y  1) cut each other at right
angles.
by c z
a
If a  x
c  z  0 , then using properties of determinants, find the value of
ax by
c
b
a b c
 
x, y,z  0
x y z , where

3

1  cosx  1  cosx
 x
1
Prove that : tan  1  cosx  1  cosx   4  2 , where   x  2


Find:  e
x
1  sin 2 x
dx .
1  cos 2 x

x  a 2,
0  x   /4

f ( x )   2 x cot x  b,
 /4  x   /2
Let
a cos 2 x  b sin x,  / 2  x   is continuous function on 0  x   . Then

determine the values of ‘a’ and ‘b’ . What are your views about ‘’learning ‘ ? Is ‘learning’ a
continuous process?
Q.24
Q.25
Q.26
Form the differential equation of the family of curve y  ae  be  ce ; where a, b, c are some
arbitrary constants.
dy
1x
Find the particular solution of the differential equation (1 x 2 )
 (e m tan  y ) , given that y = 1 when x
dx
= 0.
x
2x
3x
Find the point on the curve 9y 2  x 3 , where the normal to the curve makes equal intercepts on the axes.
7
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.27
Q.28
Q.29
Q.30
Q.31
TargeT MaTheMaTics by- agyaT gupTa
2
 dy

x 
3 d y
x
 x
 y .
If y  x log 
then,
prove
that

2
dx
 dx

 a  bx 
2
Vectors a, b, c are of the same magnitude and taken pairwise in order form equal angles. If a  ˆi  ˆj and
  



b  ˆj  kˆ find c .
Show that the equation of a plane, which meets the axes in A, B and C and the given controid of the
triangle ABC is the point   ,  ,   , is x  y  z  3. If 3p is distance of plane from origin, show that

     p .
2
2
2
2
Find the image of the line
Evaluate :

/4
0
sec x
dx
1  2 sin 2 x


x 1 y  3 z  4


0
1
7
in the plane 2 x  y  z  3  0.
.
Q.32
Solve the differential equation
Q.33
2
1
1
If y  cot ( cos x )  tan ( cos x ) Prove that sin y  tan
Q.34
Q.35
Q.36
Q.37
Q.38
Q.39
Q.40
Q.41
Q.42
Q.43
Q.44


 y
 y 
 y
 y 
 x cos   y sin   ydx   y sin   x cos   xdy .
x
x
x
 
 
 
 x 


If
yx ,
x
d 2 y 1  dy 
y


 0.


prove that
dx 2 y  dx 
x
2
x
.
2
2
Slope of the normal to the curve x3  8a2 y,a  0 at the point in the first quadrant is    .
 3
Find the point.
Is f (x) = x  1  x  2 continuous and differentiable at x = 1,2 .
Bag I contains 5 red and 4 white balls and bag II contains 3 red and 3 white balls. Two
balls are transferred from bag I to bag II and then one ball is drawn from bag II. The balls
so drawn are both found to be red. Find the probability that the transferred ball is 1 red and
1 white.
Find the equation of tangent to the curve y  cos x  y , 2  x  0 , that is parallel to the
line x  2y  0 .
Solve the differential equation : x
dy
 y  x  xy cot x  0;x  0.
dx
Determine the values of ‘a’ and ‘b’ such that the following function is continuous at x= 0:
 x  sin x
 sin  a  1 x ,if    x  0


2, if x  0

.

sin b x
e

1

2
,ifx  0

bx
Find the intervals in which the function f ( x)  3log(1  x)  4 log(2  x) 
4
2 x
is strictly increasing or strictly decreasing.
Show that the right circular cone of least curved surface and given volume has an
altitude equal to 2 times the radius of the base.
 1  sin x  1  sin x 


1
   ; x   ,  
Prove that: cot 

2 
 1  sin x  1  sin x  2 2
x



If the vectors a  2iˆ  ˆj  kˆ, b  iˆ  2 ˆj  3kˆ and c  3iˆ  ˆj  5kˆ are coplanar, find the value of .
8
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.45
TargeT MaTheMaTics by- agyaT gupTa
  x  1
x2  x  1
Evaluate:
3
dx
Q.46
Find the value (s) of  , such that the volume of a parallelepiped whose adjacent edges are
Q.47
Evaluate :eku Kkr dhft, % 
Q.48
Q.49
Q.50
Q.51
Q.52
ˆ 3jˆ  k,
ˆ 4iˆ  4kˆ is 40 cu units.
represented by vectors ˆi  2j,
 2x 

2
2 
.
Differentiate tan  1  1  x  1  x  with respect to sin  1 

2
 1  x2  1  x2 
 1 x 


If a

Q.54
Q.55
Q.56
Q.57
Q.58
Q.59
Q.60










are three vectors such that a  b  c  0 , then prove that a b  b c  c a,

,b , c
  
and hence show that a b c  0 .



2
 , then shown that x  x  1  y 2  ( x  1) 2 y 1  2 .

x 1 y 1 z

 and
Find the equation of the plane containing two parallel lines
2
1
3
x y  2 z 1


4
2
6 . Also, find if the plane thus obtained contains the line
x  2 y 1 z  2


3
1
5 or not.
Check whether the following differential equation is homogeneous or not

If y  log 

x2
Q.53
x3  x
dx .
x4  9
1
x
x
2
x
dy
 xy  1  cos   ,x  0
dx
y
Find the general solution of the differential equation using substitution y=vx.
Evaluate
Evaluate :

2

0
(3
x
 5x
2
 7 x  2 ) dx
as limit of sums.
dx
.
x  5 x 2  16
4
Check whether the following differential equation is homogeneous or not x
cos
, x ≠ 0 . Find the general solution of the differential equation .
Find
, if y = e
− xy = 1 +
Obtain the differential equation of all circles of radius r.
If the curves
2tan
.
y  aex and y= be  x cut orthogonally, find the relation between a and b.
 x 2  ax  b ,

The function f(x) is defined as f(x)   3 x  2 ,
 2 ax  5 b ,

0  x  2
2  x  4 . If f(x) is continuous on
4  x  8
[0, 8], find the value of a and b.
How many times must a man toss a fair coin so that the probability of having at least one head
is more than 90%?
9
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Q.61
Q.62
Q.63
Q.64
TargeT MaTheMaTics by- agyaT gupTa
( + + )
−
If A+B+C = , then find the value of
.
( + )
−
Find the equation of the line drawn through point (1, 0, 2) to meet at right angles
the line
Q.66
Q.67
Q.68
Q.69
Q.70
Q.71
Q.72
Fine the equation of the plane through the point (4,-3,2) and perpendicular to the line of
intersection of the planes x-y+2z-3=0 and 2x-y-3z = 0. Find the point of intersection of the
line r  i  2 j  k  ( i  3 j  9k) and the plane obtained above.
 

Evaluate :

 /4
0
 


sin x cos x
dx
1  sin 4 x .
Find the foot of the perpendicular from P(1, 2, 3) on the line


Given that vectors
triangle is 5



a, b, c

from a triangle such that



Evaluate  (e3 x  7x  5)dx as a limit of sums.
2



a  b c
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
6 where a  p i  q j  rk, b  s i  3 j  4k
. Find p, q, r, s such that area of
ˆ ˆ ˆ
and c  3i  j  2k.

1
Suppose every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates
and the corresponding values for rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs
5 per kg and rice Rs 20. The minimum daily requirements of proteins and carbohydrates
for an average child are 50 gm and 200 gm respectively. In what quantities should wheat
and rice be mixed in the daily diet to provide the minimum daily requirements of proteins
and carbohydrates, if both the items are to be taken up in each diet ?
A bag contains 4 balls. Two balls are drawn at random (without replacement) and are
found to be white. What is the probability that all balls in the bag are white?
Using properties of determinates, show that ABC is isosceles if:
1
1
1
1  cosB
1  cosC
cos2 A  cos A cos2 B  cosB cos2 C  cosC



0



Given that vectors a , b , c from a triangle such that a  b  c . Find p, q, r, s such that

Q.74
x 6 y 7 z 7


.Also
3
2
2
obtain the equation of the plane containing the line and the point (1, 2, 3).
2
2 
2     2
 
1

a
1

b

1

a
.b  a  b  (a  b ) .


For any two vectors a & b , show that


1  cos A
Q.73
.
A man is known to speak truth 5 out of 6 times. He draws a ball from the bag containing 4
white and 6 black balls and reports that it is white. Find the probability that it is actually
white? Do you think that speaking truth is always good?

Q.65
x 1 y  2 z 1


3
2
1
area of triangle is 5 6 where a  p i  q j  r k

b  si  3 j  4 k

& c  3i  j  2 k .
If the following function is differentiable at x=2, then find the values of a and b.
 x 2 ,ifx  2
f x 
.
 ax  b,ifx  2
Q.75
Determine the intervals in which the function f ( x )  x 4  8 x 3  2 2 x 2  2 4 x  2 1 is
Q.76
Show that the equation of normal at any point t on the curve x  3 co s t  cos 3 t
strictly increasing or strictly decreasing.
10
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
and
TargeT MaTheMaTics by- agyaT gupTa
Q.77
Q.78
Q.79
Q.80
Q.81
Q.82
Q.83
Q.84
Q.85
Q.86
Q.87
Q.88
Q.89
Q.90
Q.91
Q.92
Q.93
y  3 s in t  sin 3 t is 4  y co s 3 t  x sin 3 t   3 sin 4 t .



Evaluate :

2
 /4
0


d2 y
dy
 2x 3  x
2
2
dx
dx
dx
cos 3 x 2 sin 2 x
x cos x
. Then find
Let y   log X   X
x
dy
dx
.


3
Find the equation (s) of the tangent (s) to the curve y  x  1  x  2  at the
points where the curve interests the x-axis.
Find the equation of tangent to the curve
x  acos   a sin , y  a sin   a cos  at
any point

of
the curve. Also show that at any point  of the curve the normal is at a constant distance
from origin.


2
Find the area of the region  x , y  : x  y  x .
d2y
dy

cot
x
 y sin 2 x  0 .
If y  cos(cos x) ;Prove that
2
dx
dx
Prove that the function f :0,    R Given by f (x) = 9x2 + 6x – 5 is not invertible. Modify the codomain
of the function f to make it invertible, and hence find f 1 .


1
Find the value of : sin  2 tan 1   cos tan 1 2 2 .
Find:
x
x4 1
x
2
 1

2
4
dx .
Check whether the relation R in the set R of real numbers, defined by R= {(a, b) : 1 + ab > 0}, is
reflexive, symmetric or transitive.
If y  log 1  2t 2  t 4  , x  tan 1 t, find
Evaluate:
x
1
1
x  | x | 1
dx .
 2 | x | 1
d2 y
dx 2
2
Find the particular solution of the following differential equation cos y dx  1  2e  x  sin y dy  0; y  0  
Find the general solution of the differential equation:
dx y tan y  x tan y  xy

dy
y tan y
.

.
4
ˆ find a vector of magnitude 5 3 units perpendicular to the vector q and
If P  ˆi  ˆj  kˆ and q  ˆi  2jˆ  k,


coplanar with vectors p and q .

Q.94

2
2
1
If y  sec x , then shown that x x  1

11
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89

TargeT MaTheMaTics by- agyaT gupTa
Q.95
Q.96
Q.97
Q.1
Q.2
Q.3
Q.4
Q.5
Q.6
Q.7
6-Marks
Find the vector equation of the line parallel to the line
x 1
2  y
z  3


2
 3
4
and passing through
the point ( 2 , 4, 5 ) . Also find the distance between two lines
Find the equation of the plane passing through the line of intersection of the planes x - 2y +z = 1
and 2x + y + z = 8 and parallel to the line with direction ratio 1,2,1. Also find the distance of P(1,2,-2) from this plane measured along a line parallel to r = t (i – 2j – 5k ) .
Find the equation of the line passing through the point (-4, 3, 1), parallel to the plane
y  3
x  1
z  2
x  2 y  z  0 and intersecting the line


.
3
 2
1
Using integration, find the area bounded by the curves y = |x – 1| & y  3  x .
Use product
 1  1 2   2 0 1 



0 2 3   9 2 3  To
3 2 4   6 1 2 
solve the system of equations x+3z=9, -x+2y-2z = 4, 2 - 3
y+4z =-3.
A toy company manufactures two types of dolls, A and B. Market tests and available
resources have indicated that the combined production level should not exceed 1200 dolls
per week and the demand for dolls of type B is at most half of that for dolls of type A.
Further, the production level of dolls of type A can exceed three times the production of
dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16
per doll respectively on dolls A and B, how many of each should be produced weekly in
order to maximize the profit?
A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in
kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each rand are given in the table.
Tests indicate that the garden needs at least 240 kg of phosphoric acid at least 270 kg of potash
and at most 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to
the garden, how many ags of each brand should be used? What is the minimum amount of
nitrogen added in the garden?
Kg per bag
Brand P
Brand Q
12
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89
Nitrogen
TargeT MaTheMaTics by- agyaT gupTa
Phosphoric
Chlorine
Q.9
Q.10
Q.11
3.5
3
1.2
1
acid Potash
Q.8
3
2
1.5
2
A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where
the proportions are 4:4:2 respectively. The germination rates of three types of seeds are 45%, 60%
and 35%. Calculate the probability (a) of a randomly chosen seed to germinate.(b) that it is of the
type A2, given that a randomly chosen seed does not germinate.
A toy company manufactures two types of dolls, A and B. Market tests and available
resources have indicated that the combined production level should not exceed 1200 dolls
per week and the demand for dolls of type B is at most half of that for dolls of type A.
Further, the production level of dolls of type A can exceed three times the production of
dolls of other type by at most 600 units. If the company makes profit of Rs 12 and Rs 16
per doll respectively on dolls A and B, how many of each should be produced weekly in
order to maximize the profit?
6 
2 4
3 6
9  , find A-1 and hence solve the system
For A = 
10 5  20
2
3
10


 4
x
y
z
1
If   a
a2
a
6
9 20
4
6
5
 2


 1&  
x
y
z
x
y
z
;
a2
.
1  4 , then using properties of determinants, find the value of
a
a2
1
a3 1
0
a  a4
0
a  a4 a3 1 .
a  a4 a3 1
0
Q.12
Q.13
Q.14
Find the foot of the perpendicular from P(1, 2, 3) on the line
x6 y7 z 7


.Also
3
2
2
obtain the equation of the plane containing the line and the point (1, 2, 3).
If
1  1 0
A  2 5 3 ,
0 2 1
(i)Using
1
2x
find A 1 using elementary row transformation.
the
properties
x 1
x(x  1)
x
x(x  1)
3x(1  x) x(x  1)(x  2) x(x  1)(x  1)
(ii)Prove
,
using
 a(b2  c2  a2)
2b3
2a3
 b(c2  a2  b2)
Q.15
of equations .
2a3
2b3
2c3
2c3
of
 6x2 (1  x2 )
properties
 c(a2  b2  c2)
determinants:
.
of
 abc (a2  b2  c2)3
determinants:
.
Three shopkeepers A, B, C are using polythene, handmade bags (prepared by prisoners), and
newspaper’s envelope as carry bags. It is found that the shopkeepers A, B, C are using (20,
13
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TargeT MaTheMaTics by- agyaT gupTa
30, 40), (30, 40, 20), (40, 20, 30) polythene, handmade bags and newspapers envelopes
respectively. The shopkeepers A, B, C spent Rs. 250, Rs. 270& Rs. 200 on these carry bags
respectively. Find the cost of each carry bags using matrices. Keeping in the mind the social
& environmental conditions, which shopkeeper is better? Why?
Q.17
Of the students in a school, it is known that 30% have 100% attendance and 70% students are
irregular. Previous year results that 70% of all students who have 100% attendance attain A
grade and 10% irregular students attain A grade in their annual examination. At the end of the
year, one students is chosen at random from the school and he was found to have an A grade.
What is the probability that the students has 100% attendance? Is regularity required only in
school?
Using integration, find the area of the triangle bounded by the lines x + 2y = 2, y – x = 1 and
Q.18
Using the method of integration find the area of the region enclosed between the circles
Q.16
Q.19
2x + y = 7.
+
= 1 and
−
+
= 1.
p
q
p  q
p  0,q  0and q
r
q  r  0
If
, then, using properties of determinants,
p  q q  r
0
prove that at least one of the following statements is true (a) p,q,r, are in G.P.,(b)  is a root of
Q.20
2
the equation px  2qx  r  0 .
An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively.
The company is to supply oil to three petrol pumps, D, E and F whose requirements are
4500L, 3000L and 3500L respectively. The distance (in km) between the depots and the
petrol pumps is given in the following table:
Distance in (km)
From/To
A
B
E
6
4
D
7
F
Q.21
Q.22
Q.23
3
Q.25
2
Assuming that the transportation cost of 10 litres of oil is Re 1 per km, how should the
delivery be scheduled in order that the transportation cost is minimum? What is the minimum
cost?
Using integration, find the area of the region
{( x , y ) : x 2  y 2  1  x 
y
; x, y  R }
2
An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on
each executive class ticket and a profit of Rs 600 is made on each economy class ticket.
The airline reserves at least 20 seats for executive class. However, at least 4 times as many
passengers prefer to travel by economy class than by the executive class. Determine how
many tickets of each type must be sold in order to maximize the profit for the airline. What
is the maximum profit?
Using integration find the area of the region bounded by the parabola y  4 x and the
2
circle 4x  4 y  9 .
2
Q.24
3
2
Find the area cut off the parabola 4 y  3x 2 by the straight line 2 y  3x  12 .
There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6%
phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the
soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of
14
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Q.26
TargeT MaTheMaTics by- agyaT gupTa
phosphoric acid for her crop. If F1 cost Rs 6/kg and F2 costs Rs 5/ kg, determine how
much of each type of fertilizer should be used so that nutrient requirements are met at a
minimum cost. What is the minimum cost?
Sketch the graph y  x  3 . Evaluate 
0
6
Q.27
Q.28
Q.29
Q.30
Q.31
Q.32
Q.33
x  3 dx . what
dose this integral represent on the
graph?
Let A = Q × Q, where Q is the set of all rational numbers, and * be a binary operation on a
defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d)  A. Then find
(i) The identity element of * in A.
(ii) Invertible elements of A, and hence write the inverse of elements (5, 3) and
1 .
 ,4 
2 
Two trusts A & B receive Rs. 70000 and 55000 respectively from central government to
award prize to persons of a district in 3 fields agriculture, education and social services.
Trust a awarded 10, 5 and 15 persons in the field of agriculture, education and social
services respectively while trust B awarded 15, 10 and 5 persons in the field of agriculture,
education and social services respectively. If all three prizes together amount to Rs. 6000,
them find amount of each prize by matrix method. What field you prefer most for award
for development of society? Give answer with justifications.
A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The
profit from crops X and Y per hectare are estimated as Rs 10,500 and Rs 9,000
respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates
of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should
be used in order to protect fish and wild life using a pond which collects drainage from this
land. How much land should be allocated to each crop so as to maximise the total profit of
the society?
A company produces two different products. One of them needs 1/4 of an hour of assembly
work per unit, 1/8 of an hour in quality control work and Rs1.2 in raw materials. The other
product requires 1/3 of an hour of assembly work per unit, 1/3 of an hour in quality control
work and Rs 0.9 in raw materials. Given the current availability of staff in the company, each
day there is at most a total of 90 hours available for assembly and 80 hours for quality control.
The first product described has a market value (sale price) of Rs 9 per unit and the second
product described has a market value (sale price) of Rs 8 per unit. In addition, the maximum
amount of daily sales for the first product is estimated to be 200 units, without there being a
maximum limit of daily sales for the second product. Formulate and solve graphically the
LPP and find the maximum profit.
Find the direction ratios of the normal to the plane, which passes through the points (1, 0,
0) and (0, 1, 0) and makes angle

which the plane x + y = 3. Also find the equation of the
4
plane.
A dietician wants to develop a special diet using two foods X and Y. Each packet (contains
30g) of food X contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6
units of vitamin A. Each packet of the same quantity of food Y contains 3 units of calcium,
20 units of iron, 4 units of cholesterol and 3 units of vitamin A. the diet requires at least
240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. Make
an LPP to find how many packets of each food should be used to minimize the amount
vitamin A in the diet, and solve it graphically.
2

Let A   1
 0
1
0
2
1 
1 
 1  ,find the inverse of A using elementary row transformations and
hence solve the following matrix equation
XA  1
0
1
15
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TMC/D/79/89
Q.34
Q.35
Q.36
Q.37
Q.38
Q.39
TargeT MaTheMaTics by- agyaT gupTa
Find 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. Hence find
whether the plane thus obtained contains the line
x2 y3 z

 5
4
5
Let X be a non – empty set. P(x) be its power set. Let ‘* be an operation defined on
element of P(x) by A*B = A  B  A, B  P(X) Then,
(i) Prove that * is a binary operation in P(X).
(ii) Is* commutative ?
(iii) Is* associative?
(iv)find the identity element in P(X) w.r.t * .
(v) find the all the invertible element of P(X)
(vi) if O is another binary operation defined on P(X) as A O B = A  B then verify that O
distribution itself over *.
Consider f : R   4   R   4  , given by f  x  
 3
3 
inverse of f and hence find
f 1  0  and
4x  3
. Show that f is bijective. Find the
3x  4
x such that f 1  x   2.
A cuboidal shaped godown with square base is to be constructed. Three times as much cost per square
meter is incurred for constructing the roof as compared to the walls. Find the dimensions of the godown if
it is to enclose a given volume and minimize the cost of constructing the roof and the walls.
Find the area bounded by the curves y  x , 2y  3  x and x – axis.
Find the equation of the plane through the line
x 1 y  4 z  4


3
2
2
x 1 1 y z  2


. Hence, find the shortest distance between the lines.
2
4
1
and parallel to the line
Q.40
Show that the line of intersection of the planes x + 2y + 3z = 8 and 2x + 3y + 4z = 11 is coplanar with the
Q.41
The members of a consulting firm rent cars from three rental agencies: 50% from agency X, 30% from
agency Y and 20% from agency Z.
Form past experience it is known that 9% of the cars from agency X need a service and tuning before
renting, 12% of the cars from agency Y need a service and tuning before renting and 10% of the cars from
agency Z need a service and turning before renting. If the rental car delivered to the firm needs service
and tuning, find the probability that agency Z is not to blamed.
Q.42
Q.43
line
x 1 y 1 z 1
. Also find the equation of the plane containing them.


1
2
3
3 1 2 
If A   3 2 3 , find A-1.
 2 0 1
Hence, solve the system of equations:
3x + 3y + 2z = 1
X + 2y = 4
2x – 3y – z = 5
 2 1 3
Find the inverse of the following matrix using elementary transformations.  5 3 1
 3 2 3
****************
16
P.T.O.
Resi.: D-79 Vasant Vihar ; Office : 89-Laxmi bai colony
Ph. :2337615; 4010685®, 2630601(O) Mobile : 9425109601 (P); 9425110860; 425772164; www:agyatgupta.com.
TMC/D/79/89