Download Aniket Mathematics

MATHEMATICS
CBSE (XII) – 2015
( 7 th Revised Edition )
DATE : 01 ST NOVEMBER 2014
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Page 1 of 27
PREFACE
It is known to every student that 70 – 80 %
questions in CBSE – XII Exam have been asked from NCERT Text Book.
Though the remaining
20 – 30 % HOTS
(High Order Thinking Skills) questions are creating
furor in
As an outcome a lots of students
performed
below the
level , what they
expect from
them.
had
themselves , in
their U.T’s or Terminal Examinations.
So, this ‘Operation Mathematics CBSE XII – 2015’
has been designed for providing relief to such horrified
students.
This package will not only bring confidence,
but help the students in scoring the respectable 70 – 90 % Marks in
coming CBSE XII – 2015 Exam.
Aniket Manohar
Mob : 9013413543
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Page 2 of 27
1. Relation & Functions
(1) Consider f : {1, 2, 3} {a, b, c} and g : {a, b, c} {apple, ball, cat} defined as f (1) = a, f (2) = b,
f (3) = c, g(a) = apple, g(b) = ball and g(c) = cat. Find out f –1, g –1 and ( go f )–1
and show that ( g o f ) –1 = f –1 o g –1.
(2) Show that the function f : R+  [ - 5 ∞) defined by f (x) = 9x2 + 6x – 5 is invertible. Also find f – 1 .
Ans : f –1(x) = {√x + 6 – 1} / 3
(3) Show that the function f : R +  [4
∞ ) defined by f (x) = x2 + 4 is invertible. Also find f – 1 .
Ans : f –1 (x) = √x – 4
(4) Let A = R – {2} and B = R – {3}. Consider the function f : AB defined by f (x) =
.
Show that ‘ f ’ is invertible with f –1(x) =
(5) Let f : N  R be a function defined as f (x) = 4x 2 + 12x + 15. Show that f : N  S, where S is the
range of f , is invertible. Find the inverse of f .
Ans : f –1(x) = √x – 6 – 3
2
(6) Show that the function f : R
(7) Show that the function f(x) =
(–1 1) defined by f (x) =
|
|
is bijective.
is invertible & is inverse of itself .
(8) If R1 and R2 are two equivalence relation , then show that R1  R2 is also an equivalence relation.
(9) Show that the relation R in the set A = { x : x is an integer & 0 ≤ x ≤12 } as :
R = { (a , b) : | a – b | is a multiple of 4 } is an equivalence relation.
(10) Consider the function f : N  N defined by
; if n is odd
f (n) =
Is f injective and surjective?
; if n is even
(11) If the function f : W  W defined by f ( n ) = n – 1 , n is odd
= n + 1, n is even.
Show that f is invertible. Find the inverse of f . Here W is the set of all whole numbers.
(12) Let N br the set of all natural numbers and R be the relation on N  N defined by :
(a,b)
R ( c , d )  ad ( b + c ) = bc ( a + d ). Show that relation R is an equivalence relation.
(13) Let N br the set of all natural numbers and R be the relation on N  N defined by :
(a,b)
R ( c , d )  ad = bc . Show that relation R is an equivalence relation.
(14) Test the commutativity , associativity of the binary operation ‘∗’ defined on N  N as :
( a b ) ∗ ( c d ) = ( ad + bc
bd ). Also find identity & inverse elements if exists.
(15) Let A = N  N and ∗ be the binary operation on A defined by (a , b) ∗ (c , d ) = ( a + c , b + d ).
Show that ‘∗’ is commutative & associative. Find the identity element for ‘∗’ on A if any.
(16) Test the commutativity , associativity of the binary operation ‘∗’ defined on N  N as :
(a
b) ∗ ( c d ) = ( ac bd ). Also find the identity element and inverse element if exist.
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Page 3 of 27
(17) Test commutativity , associativity , also find the identity & inverse elements if exists of the
binary operations defined on Q of rational numbers as :
(i) a ∗ b = ab2
(ii) a ∗ b = (a – b)2.
(iii) a ∗ b = a2 + ab + b2.
(iv) a ∗ b =
(18) Let X be any non empty set . P(X) be its power set. Let ‘∗’ be an operation defined on the
elements of P(X) by setting 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 identity element in P(X) w . r . t . ‘∗’.
(v) Find all invertible elements of P(X).
And , (vi) If ‘o’ be another binary operation in P(X) defined as A o B = A  B
prove that ‘o’ wiil distribute itself over ‘∗’.
(19) Consider the two binary operations ‘o’ and ‘∗’ defined over the set of real numbers R as :
a o b = a and a ∗ b = a – b ;  a , b  R . Then,
(i) show that ‘∗’ is commutative but not associative.
(ii) show that ‘o’ is associative but not commutative.
(iii) does ‘∗’ is distributive over ‘o’.
(iv) does ‘o’ is distributive over ‘∗’.
(20) Define a binary operation ∗ on the set {0,1,2,3,4,5} as: a ∗ b = a + b
;a+b<6
=a+b–6 ;a+b≥6
Show that ‘0’ is the identity of this operation & each element ‘a’ of the set is invertible with
‘6 – a’ being the inverse of a .
******
2. Inverse Trigonometrical Functions
(1) Find ‘x’ if , tan –1( 2x ) + tan –1( 3x ) =
(2) Prove that : sin–1
+ cos–1
.
Ans :
+ tan–1
=π.
(3) Find the value of x if , sin – 1 ( 1 – x ) – 2sin – 1 x =
(4) Prove that : 2 tan –1
(5) If, cos – 1
+ cos – 1
(6) Show that : tan –1
(7) If , tan – 1
+ sec –1
–
√
+ 2tan –1
=  Prove that
+ tan –1
+ tan – 1
–
+ tan –1
=
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.
Ans : x = 0
=
cos  +
+ tan –1
. Find the value of x .
= sin2 
=
.
Ans : x = ±
Page 4 of 27
√
(8) Prove that : tan
cos –1
+
(9) Simplify : tan–1 √ +
+ tan
; x  [0
+√ −
–√ −
√ +
cos –1
–
π]
Ans :
(10) Prove that tan–1 √ +
–√ −
√ +
+√ −
=
=
if x  0
–
cos –1 x ; x  [0
–
(11) Prove that : cos [ tan –1 { sin ( cot –1 x ) } ] =
;
if x 
π
1]
+
+
(12) Prove that tan–1
+
+ √ −
+
–√ −
=
+
cos –1 ( x 2 ) ; |
|≤1
(13) If cos –1 x + cos –1 y + cos –1 z = π , Prove that x2 + y2 + z2 + 2xyz = 1 .
(14) If tan–1x + tan–1y + tan–1z = π , Prove that : x + y + z = xyz .
(15) Prove that : tan–11 + tan–12 + tan–13 = π
–
(16) Solve the equation, tan –1
(17) Prove that, sin – 1
=
+ sin – 1
tan –1 x ; x > 0.
+ sin – 1
Ans : x =
=
(18) Solve for x ; tan–1(x – 1) + tan–1x + tan–1( x + 1 ) = tan–13x .
Ans : 0 , 
(19) If, y = cot –1(√cos x ) – tan–1(√cos x ), prove that sin y = tan2
(20) Evaluate : cos –1 x – cos –1
+
√ −
√
.
; 0 ≤ x ≤
Ans : x =
******
3. Matrices
(1) If, A =
n
, then by using P.M.I, show that, A =
−
–
(2) If A =
(4) If, A =
−
, Show that , ( I + A ) = ( I – A )
n
(3) Let , A =
nN.
−
n
n –1
−
−
nN.
, then by using P.M.I, Show that ( a I + bA ) = a . I + na
−
−
n
, then by using P.M.I, Show that A =
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+
bA ,nN.
Page 5 of 27
−
−
(5) Express the matrix, A = −
, as sum of a symmetric and a skew – symmetric matrices.
−
−
, find the numbers ‘a’ and ‘b’ such that A2 + aA + bI = O, also find A –1.
(6) For the matrix A =
−
and f (x) = x2 – 5x – 5 , then show that f (A) = O. Hence find A–1 . Ans :
(7) If A =
] then verify ( AB )T = BT . AT .
and B = [
(8) If A =
−
(9) Find ‘D’ if A =
−
,B=
,C=
(10) Find the value of ‘x’ if [
if CD – AB = O.
]
= O.
(11) Find the value of ‘x’, ‘y’& ‘z’if [
]
−
=
A
−
−
−
−
−
Ans : x = –2 , –14
= [
(12) Find the matrix ‘ X ’ so that X .
(13) Find the matrix ‘ A ’ so that
Ans :
] . Ans : x = –1 , y = –1 , z = –2
−
−
Ans :
Ans : −
=
–
Using elementary transformations find inverse of the following :
−
−
−
(14)
Ans :
−
(15) −
−
−
−
−
Ans :
−
−
******
4. Determinants
Using properties of determinant prove that :
(1) (b + c)2
(c + a)2
(a + b)2
(2)
–bc
a + ac
a2 + ab
2
a2
b2
c2
bc
ca = (a – b) (b – c) (c – a) (a + b + c) (a2 + b2 + c2).
ab
b2 + bc
– ac
b2 + ab
(3) If x  y  z , and D =
c2 + bc
c2 + ac = (ab + bc + ac)3.
– ab
+
+
+
= 0 , then show that 1 + xyz = 0.
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Page 6 of 27
−
(4) 1 + a
1
1
1
1+b
1
(5) b + c
c+a
a+b
(6)
1
1
1+ c
q+r
r+p
p+q
= abc + ab + bc + ac
y+z
a p x
z+x = 2 b q y
x+y
c r z
1+a2–b2
2ab
– 2b
2ab
1–a2+b2
2a
2b
–2a
1–a2–b2
=(1+a2+b2)3.
(7) If a , b, c  R, and b + c c + a a + b
Show that either a + b + c = 0 or a = b = c .
c + a a + b b + c = 0,
a+b b+c c+a
(8) x
x2 yz
y
y2
xz = ( x – y )( y – z )( z – x )( xy + yz + zx ) .
z
z2
xy
(9) a2 + 1
ab
ca
ab
b2 + 1
cb
ac
bc
= 1 + a2 + b2 + c2 .
2
c +1
(10) a – b – c
2b
2c
2a
b–c–a
2c
2a
2b
= (a + b + c)3.
c–a–b
(11) ( y + z )2
xy
xz
xy
( x + z )2
yz
zx
yz
( x + y )2
= 2xyz( x + y + z)3.
(12) If a  b  c > 0 then prove that
is always negative.
th
th
th
(13) If x , y & z are all positive and 10 , 15 , and 25 term of a G.P then prove that,
(14) If a  p, b  q, c  r and
(15) If
,
,
= 0, then prove that,
th
, are the p , q
th
+
+
=2
th
and r terms of an A.P then prove that
=0
(16) a + b + 2c
a
b
c
b + c + 2a
b
= 2(a + b + c)3.
c
a
c + a + 2b
(17)
a
a2
b+c
b
b2
c+a
c
c2
= (a – b)(b – c)(c – a)(a + b + c)
a+b
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=0
Page 7 of 27
(18)
a2
a + ab
ab
2
(19) y + z
z+x
x+y
x
z
y
(20) a2 + 2a
2a + 1
3
c2 + ac
ac
c2
bc
b2
b2 + bc
y
x
z
= (x + y + z)(x – z)2
2a + 1
a+2
3
1
1 = (a – 1)3.
1
(21) Solve for ‘x’ , a + x
a–x
a–x
(22) If A =
a–x
a+x
a–x
and B =
−
= 4a2b2c2
a–x
a–x = 0
a+x
Ans : x = 0 , 3a
, then prove that (AB) –1 = B–1. A–1.
−
(23) Let ,
–5
1
3
1
1 2
A = 7
1 –5 B = 3
2 1 Show that AB = BA = 4I3 .
1 –1
1
2
1 3
And , using this relation solve the system of linear equation, x + y + 2z = 0, 3x + 2y + z = 7,
2x + y + 3z = 2
Ans : x = 13 / 4 , y = – 3 / 4 , z = – 5 / 4
(24) If A = 1
2
1
–1
1
1
1 Find A–1. Using A–1 solve the system of linear equations : x + 2y + z = 4 ;
–3
– x + y + z = 0 ; x – 3y + z = 2 .
1
Ans : x = 18 / 10 , y = 4 / 10 , z = 14 / 10
(25) Use the product 1 –1
2
–2 0
1
0
2 –3
9
2 –3
3 –2
4
6
1 –2 ,
to solve the system of linear equations : x – y + 2z = 1 , 2y – 3 z = 1 , 3x – 2y + 4z = 2 ,
by matrix method .
Ans : x = 0 , y = 5, z = 3.
(26) Solve the system of equations : 2x + 3y + 3z = 5 , x – 2y + z = – 4 and , 3x – y – 2z = 3 , by matrix
method .
Ans : x = 1, y = 2 , z = –1
(27) Solve the system of equations : 3x – 2y + 3z = 8 , 2x + y – z = 1 and , 4x – 3y + 2z = 4 , by matrix
method .
Ans : x = 1, y = 2 , z = 3
(28) If A = 2
3
1
–3
5 Find A–1. Using A–1 solve the system of linear equations : 2x – 3y + 5z = 11
2 – 4 ; 3x + 2y – 4z = - 5 ; x + y – 2z = -3 .
1 –2
Ans : x = 1, y = 2, z = 3
(29) Solve the system of linear equations using matrix method ,
–
+
=1,
+
–
=2
+
+
=4
Ans : x = 2, y = 3, z = 5
(30) Test the consistency of the system of linear equation and solve the system by matrix method:
x – y + z = 3, 2x + y – z = 2 , – x – 2y + 2z = 1
Ans : x = 5 / 3, y = (3k – 4) / 3, z = k
*****
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Page 8 of 27
5. Differentiations
(1) Prove that for a = 1/ 2, the function
a sin
(
)
;x ≤ 0,
f (x) =
–
; x > 0 is continuous at x = 0
(2) Test the continuity of the function f ( x ) = (x – a) . sin
1
x–a
=0
; x = a.
1 – sin3x
(3) Prove that for a = 1/ 2, b = 4 the function
; x ≠ a
;x<
Ans : continuous
, is continuous at x =
2
3cos x
f (x) =
a
;x=
b (1 – sin x )
(π – 2x)
;x>
2
(4) Find all points of discontinuities of the function f (x) =
;x<0
= x+1
–
(5) Show that the function
f (x) =
; x < 0 , is continuous at x = 0. for k = 8
k
; x=0
√
;x>0
√
(6) Show that the function
f(x) =
(7) Show that the function
f (x) =
;x 0
Ans : No point of discontinuity
5ax – 2b
11
3ax + b
–
|
|
; x < 1, is continuous at x = 1 if a = 3, b = 2 .
;x=1
;x >1
+a
; x < 4 , is continuous at x = 4. If a = 1, b = –1
a+b
–
;x=4
+b
;x >4
(8) Find all point(s) of discontinuities of the function, f ( x ) = | x | – | x + 1 |
(9) Find the values of ‘k’ so that the function f (x) =
; x ≠
=3
(10) Find the values of ‘k’ so that the function f ( x ) =
Ans : No points of discts.
is continuous at x =
; x=
√
=k
Ans : k = 6
;x ≠
;x=
is continuous at x =
,
Ans : k =
(11) Show that the function, f ( x ) = x – [ x ] is discontinuous at all integral points . Here [ x ] is the
greatest integer function defined by an integer less than or equals to x .
(12) Show that the function f (x) = |
−
| , is continuous but non – differentiable at x = a
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Page 9 of 27
(13) If y =
(14) If
; –1 < x < 1 , show that , ( 1 –
+
= 1, then prove that
–
(15) Differentiate tan–1
(16) If y = cot – 1
(17) If y = x
sin x
sin x
=x
√
√
–√
cos x
+ {sin x}
y
, then prove that
√
√
√ –
√
cos x
=
(
(
–2
=
, then prove that ( 1 +
+
and x =
(25) If √ −
+
−
(26) If f (x) = cot – 1
(27) If y =
;
(28) If y = A e
mx
)
+x
–
) +( −
then prove that
y=0
=
=
.
nx
log x { 1 + log x + (x log x ) –1 }
, show that
– (m + n)
+
y = 0.
(29) If x = a (cos t + t sin t ) & y = a (sin t – t cos t ) ; prove that
(30) If ( −
+y=0
; prove that f 1( 1 ) = – 1 .
=
+Be
+x
then prove that
= a (x – y) ;
.
+ 2y = 0.
(22) If y = A cos (log x) + B sin ( log x ) , prove that ,
(24) If y =
)
)
, then prove that
x
+ √ +
.
. {cos x cot x – sin x log sin x }
=–
(21) If y = e ( sin x + cos x ) , then prove that ;
(23) If y =
=
. Prove that :
, then prove that ;
(20) If y = tan – 1
y=0.
Ans :
;0<x<
+ cos x. log x} + ( sin x )
x–y
–
=
x
y
x
b
(18) If y + x + x = a then prove that
(19) If x = e
–x
with respect to tan –1
√
.{
)
) =
=
for some c > 0,
is a constant independent of ‘a’ and ‘b’.
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Page 10 of 27
(31) If y = sin –1 x , then find
(32) If, y = tan – 1
(33) If y = sin–1
; then prove that
–√
√
.
(34) If y = sin–1
(
; then prove that
(35) If, y = tan – 1
+
(37) If x2 + y2 = t –
n
(38) If x . y = ( x + y )
(39) If y = sin
(40) If, x
–1
+
sin –1
and x4 + y4 = t 2 +
{x√
m+n
−
+y√ +
.
.
(
=
=√
)
(
)
−
; prove that ;
prove that
–√ √ −
=
=
; then prove that
(36) Prove that
m
=
; then prove that
)
Ans : sec2 y tan y
in terms of y alone.
=
.
=
}, prove that
=
–
= 0 ; – 1 < x < 1 , then prove that
= –
(
)
******
6. Applications of dy / dx
3
2
(1) Verify the Mean Value Theorem for the function , f (x) = x – 5x – 3x ; x  [ 1
Find all c  ( 1 3 ) , such that f 1( c ) = 0 .
(2) Verify Rolle’s Theorem for the function f (x) = (x – 1) (x – 2)2 , x  [ 1
(3) Verify Rolle’s Theorem for the function, f (x) = log
(
)
3].
2].
; x [ a , b] where 0 < a < b .
(4) A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower
–1
3
most. Its semivertical angle is tan ( 0.5 ). Water is poured into it at a constant rate of 5 m /min.
Find the rate at which the level of the water is rising at the instant when the depth of water in the
tank is 4m .
Ans :
m/h
(5) A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y - coordinate
is changing 8 times as fast as the x – coordinate.
Ans : (4 , 11); (– 4 , –31/3)
(6) If two equal sides of an isosceles triangle with fixed base ‘b’are decreasing at the rate of 3 cm/s.
How fast is the area of the triangle decreasing ,when the two equal sides are equal to the base.
Ans : √3 b cm2 / s
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Page 11 of 27
(7) The surface area of a spherical balloon is increasing at 2 cm2 / s . At what rate is the volume of the
balloon increasing when the radius of the balloon is 6 cm?
Ans : 6 cm3/s
(8) The volume of a cube is increasing at the rate of 7cm3/s. How fast is its surface area increasing at
the instant when the length of an edge of the cube is 12 cm ?
Ans : 7 / 3 cm2 / s
(9) Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground
in such a way that the height of the cone is always one - sixth of the radius of the base. How
fast is the height of the sand - cone increasing when the height is 4 cm ? Ans : 1 cm / s
48π
(10) A point source of light along a straight road is at a height of a meteres. A boy b meters in height is
walking along the road . How fast is his shadow increasing if he walking away from light at rate
of c meteres per minute ?
Ans : bc m / min
a–b
(11) Water is leaking from a conical funnel at the rate 5 cm3/s. If the radius of the base of the funnel is
5 cm and its altitude is 10 cm, find the rate at which water level is dropping when it is 2.5 cm
from the top.
Ans : 16 cm /sec.
45π
(12) A ladder 5m long is leaning against a wall. The bottom of the ladder is pulled along the ground,
away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the
foot of the ladder is 4m away from the wall ?
Ans : ( 8 / 3 ) cm / sec
(13) The length x of a rectangle is decreasing at the rate of 5cm/minute and the width y is increasing at
the rate of 4cm/minute. When x = 8cm and y = 6 cm, find the rates of change of
(a) the perimeter, and (b) the area of the rectangle.
Ans : – 2 cm / min , 2 cm2 / min
(14) Prove that the function f (x) = log (sin x ) , is strictly increasing on (0 π/2 ) &
strictly decreasing on (π/2 π).
(15) Prove that the function f(x) = cot –1( sin x + cos x ) ; 0 < x < /4 is decreasing in (0 /4).
3
(16) Find the intervals in which the function f (x) = x +
, is (i) increasing & (ii) decreasing .
Ans : (i) ( – –1)  (1
) (ii) (–1
0)  (0
(17) Find the intervals in which the function f (x) = x3 – 12x2 + 36x + 17 , is (i) increasing &
(ii) decreasing .
Ans : (i) ( – 2 ) (6 ) , (ii) (2
1)
3)
(18) Find the intervals in which the function f(x) = (x + 1)3.(x – 3)3 , is (i) increasing & (ii) decreasing
Ans : (i) [1
) (ii) (–  1]
(19) Show that y = log ( 1 + x ) –
; x  –1 ; is an increasing function of x throughout its domain .
(20) Find the intervals in which the function ‘f ’ given by f (x) =
is increasing
& decreasing.
Ans : increasing in (0 π/2)  (3π /2 2π) decreasing in(π/2 3π/ 2)
–x
(21) Find the interval in which the function f (x) = x2 e is (i) increasing & (ii) decreasing
Ans : (i) ( 0 2)
(ii) ( – 0 )  ( 2  )
(22) Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is
(i) perpendicular to the line 5y − 15x = 13. & (ii) parallel to the line 2x – y + 9 = 0.
Ans : (i) 36y + 12x = 227 (ii) y – 2x = 3
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(23) Find the angles between the parabolas x = y2 & x2 = y.
Ans : π / 2 , tan–1( 3 / 4 )
(24) Find the equation of normal to the curve x2 = 4y which passes through the point (1 , 2 ) .
Ans : x + y = 3
2
2
(25) Show that the curves y = 4x & 4xy = k cuts orthogonally if k = 512.
(26) Prove that the curves x = y2 and xy = k cut at right angles if, 8k2 =1.
(27) Prove that the curves x2 + y2 = 8 and xy = 4 touches each other.
(28) Find the equation of tangent to the curve y = cos (x + y) ; x  [–2π 2π ], that are parallel to
the line x + 2y = 0.
Ans : 2x + 4y + 3π = 0, 2x + 4y – π = 0.
(29) Show that the line
+
= 1, touches the curve y = b e
–x/a
, at the point where the curve crosses
the y – axis.
(30) Show that the normal at any point to the curve x a ( cos sin y a(sin  cos , is
at a constant distance from the origin.
(31) Find a point on the parabola y = (x – 3)2, where the tangent is parallel to the chord joining the
points (3 0) and (4 1).
Ans : (7/2
1/4)
(32) Find the point on the curve 9y 2 = x3, where the normal to the curve makes equal intercepts with
the axes.
Ans : (4 , ± 8 / 3)
(33) Find the equation of tangent to the curve y =
( – )
(
)(
)
.Where it cuts the x – axis.
Ans : x – 20y = 7
(34) The sum of length of hypotenuse and a side of a triangle is given. Show that the area of the triangle is
maximum when the angle between them is π / 3.
(35) A rectangle is inscribed in a semi – circle of radius ‘R’ with one of its side on the diameter of the
semi – circle. Find the dimension of the rectangle so that the rectangle has maximum area.
Ans : √2 R , R / √2 .
(36) Prove that the radius of the right circular cylinder of greatest curved surface which can be
inscribed in a given cone is half of that of the cone , also find the greatest surface of the cylinder
inscribed.
Ans : π r h / 2
(37) Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8 / 27 of
the volume of the sphere, also show that altitude of such cone is 4R / 3.
(38) Show that semi-vertical angle of right circular cone of given surface area and maximum volume is
Sin – 1(1 / 3).
(39) Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of
radius R is 2R / √3. Also find the maximum volume .
Ans : 4πR3 / 3√3
(40) Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone
of height ‘ h’ and having semi-vertical angle ‘’is one - third of that of the cone and the greatest
volume of cylinder is
πh3 tan2 
(41) Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is
tan-1 √2 .
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Page 13 of 27
(42) An apache helicopter of enemy is flying along the curve y = x2 + 7. A soldier, placed at (3 , 7),
wants to hit down the helicopter when it is at the minimum distance . Find the minimum distance.
Ans : √5
(43) The sum of the perimeter of a circle and a square is ‘k’ unit, where k is some constant .Prove
that the sum of their areas is least when the side of square is double the radius of the circle.
(44) An open box with a square base is to be made out of a given quantity of cardboard of area c2
squre units. Show that the maximum volume of the box is c3 / 6√3 cubic units.
(45) Show that the right triangle of maximum area that can be inscribed in a circle is an isosceles triangle.
(46) A given quantity of metal is to be cast into a half cylinder ( a rectangular base and semi – circular
ends). Show that the total surface area will be least when the ratio of the length of the cylinder to the
diameter of the ends is π : (π + 2) .
2
2
2
(47) The length of sides of an isosceles triangle are 9 + x , 9 + x and 18 – 2x units. Calculate the
area of triangle in terms of ‘x’ and find the value of x, which makes the area of triangle
maximum.
Ans : x = √ units
(48) Find the area of the greatest rectangle that can be inscribed in an ellipse
+
= 1.
Ans : 2ab sq.units
(49) Using differentials find the approximate value of √ .
Ans : 0.1925
(50) Using differentials find the approximate value of
Ans : 3.009
*******
7. Indefinite Integrations
(1) Prove that :
∫
(2) Prove that :
∫(
( +
)=
−
√
)
√
log
–
√
( −
√
tan –1
√
2
) = 2 log sin x – 4 sin x + 5
−
+ 7 tan–1 (sin x – 2) + C
(
(3) Prove that :
+
)( +
)
( +
)( +
–1
) = √ + √ tan
(
)
(
)
– 3tan –1 x + C
(4) Prove that :
∫
(5) Prove that :
∫ [log(log x) + {1 / log x} ] dx =
(6) Prove that :
∫ {√tan x + √cot x } dx = √2 . sin
(7) Prove that :
∫ (3sin x – 2 ) cos x dx / {5 – cos x – 4 sin x} = 3 log | 2 – sin x | +
( − ).
( −
)=
(
2
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)
log
x log ( log x ) –
2
–1
+C
x +C
log x
( sin x – cos x ) + C
4
+ C
2 – sin x
Page 14 of 27
∫
(8) Prove that : √1 – √x
(9) Prove that :
(10) Prove that :
∫x
2
/
1 + √x
dx = –2 √1 – x + cos–1 √x + √x – x2 + C
dx / (x2 + 1)(x2 + 4) = –1 tan–1 x + 2 tan–1 x + C
3
3
2
∫ dx / (x
2
+ 1)(x2 + 4) = 1 2 tan–1 x – tan–1 x
6
2
+C
∫
(11) prove that : sin2x cos2x.dx / √9 – cos4(2x) = – 1 . sin–1 1 . cos2 2x + C
4
3
x
x
(12) Prove that : (x2 + 1)e dx / (x + 1)2 = ( x – 1) .e + C
(1 + x)
∫
(13) Prove that :
∫ sin
(14) Prove that :
∫ tan {√1 – x / √1 + x }dx =
–1
{ 2x / ( 1 + x2 )}dx = 2x tan–1 x – log (1 + x2 ) + C
–1
∫{ sin
x . cos–1 x – 1 . √1 – x2
2
2
√x – cos–1√x }dx / ( sin–1√x + cos–1√x )
= 2 . [ (2x – 1) . sin–1 √x + √x – x2 ] – x + C
π
x
x
(16) Prove that : (2 + sin 2x )e dx / (1 + cos 2x) = e tan x + C
(15) Prove that :
–1
∫
∫
(17) Prove that : (sin–1x )2 dx = x . (sin–1 x)2 + 2 √1 – x2 . sin–1 x – 2x + C
(18) Prove that :
∫ x sin
(19) Prove that :
∫√
2
–1
x . dx =
−
√ −
+
√
.(
)
+C
− 2 + sin –1(2x – 3) + C
= – √3 −
x
x
x x
(20) Prove that : dx / (1 + 3e + 2e2 ) = log e ( e + 1 ) + C
∫
x
(2e + 1)2
∫
(21) Prove that : dx / (sin x – sin2x) = – 1 log | 1 – cos x | – 1 log | 1 + cos x | + 2 log | 1 – 2cos x | + C
2
6
3
∫
(22) Prove that : √(x – 1)(2 – x) dx = 2x – 3 √(x – 1)(2 – x) + 1 sin–1 (2x – 3)+ C
4
8
(23) Prove that :
∫ √tan x dx =
(24) Prove that :
∫
(25) Prove that :
∫ sin2x.dx / { sin4x + cos4x} = tan
1 tan–1 tan x – 1 + 1 log tan x – √2 tan x + 1 + C
√2
√2 tan x
2√2
tan x + √2 tan x + 1
.
( +
)
(
= –
√
–1
)
+C
(2sin2 x – 1 ) + C
*******
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Page 15 of 27
8. Definite Integrations
3
(1) Prove that :
∫|x
0
3
– 3x2 + 2x |dx = 11 / 4
π/2
(2) Prove that :
∫ |sin x – cos x | dx = 2√2 – 2.
0
π /2
(3) Prove that :
∫ log (sin x) dx =
0
– { π . log 2 } / 2
1
∫
(4) Prove that : 0 cot – 1(1 – x + x2 ) dx = ( π / 2 ) – log2
π
∫
(5) Prove that : 0 x (tan x) dx / (sec x + tan x) = π (π – 2) / 2
π /2
(6) Prove that : 0
∫
x dx / ( cos x + sin x) = π log ( √2 + 1) / 2√2
π
∫
(7) Prove that : 0 log ( 1 + cos x ) dx = – π . log 2
π
∫
(8) Prove that : 0 x dx / ( a2 cos 2 x + b2 sin 2 x ) = π2 / 2ab.
π/4
∫
(9) Prove that : 0 log ( 1 + tanx ) dx =
log 2
π/2
∫
(10) Prove that : 0 ( 2 log sin x – log sin 2x ) dx = { – π . log 2 } / 2
π
∫
(11) Prove that : 0 x dx / ( 1 + sinx ) = π
π /2
x
∫
(12) Prove that : 0 e ( 1 + sinx )dx / ( 1 + cosx ) = e
π/2
2π
(13) Prove that : 0
∫ dx / { e sin x + 1 } = π
π/4
∫
(14) Prove that : 0 (sin x + cos x) dx / {9 +16 sin (2x)} = { log 9 } / 40
π/3
(15) Prove that : π / 6
∫ dx / ( 1 + √tanx
) = π / 12
1
∫
(16) Prove that : 0 sin–1 {2x / ( 1 + x2 )} dx = ( π / 2 ) – log2
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Page 16 of 27
3/2
(17) Prove that :
–1
∫│x sin ( π x )│dx = ( 3 / π ) + (1 / π
2
)
2
(18) Prove that :
–1
∫│x3 – x │dx = 11 / 4.
2
(19) Prove that :
0
∫ dx / {x + 4 – x
2
}=
1 . log 21 + 5 √17
√17
4
π/4
(20) Prove that :
0
∫ sin x cos x dx / (cos x + sin
4
4
x)=π/8
Prove the following integrals by limits of a sum
4
4
∫
2
(21) 1 (x – x ) dx = 27 / 2
(22) 0
∫
2
2x
8
( x + e ) dx = ( e + 15) / 2
3
(24)
1
(23)
∫( x + e–x ) dx = 3 – e–2
0
4
∫ {2x
2
∫
– 3x + 5}dx = 46 / 3
2
(25) 0 ( x – x – 1 ) dx = 28 / 3
******
9. Area Bounding
(1) Using the method of integration find the area of the region bounded by the lines: 2x + y = 4,
3x – 2y = 6 & x – 3y + 5 = 0.
Ans : 7 / 2
(2) Find the area of the region bounded by the curves y = x2 + 2 , y = x , x = 0 & x = 3 .
Ans : 21 / 2
(3) Find the area of the region bounded by the parabola y = x2 and y = | x |.
Ans : 1 / 3
(4) Find the area bounded by the curve x2 = 4y and the line x = 4y – 2.
Ans : 9 / 8
(5) Find the area of the region enclosed by the parabola x2 = y, the line y = x + 2 & the x-axis.
Ans : 5 / 6
(6) Find the area of the region { (x, y) : x2 + y2 ≤ 16 ; y2 ≥ 6x }
Ans : 4(8π – √3 ) / 3
(7) Find the area of the region enclosed between the two circles x2 + y2 = 1 and (x – 1)2 + y2 = 1.
Ans : (4π – 3√3) / 6
(8) Find the area of region { ( x , y ) : 0 ≤ y ≤ x2 + 1 , 0 ≤ y ≤ x + 1 , 0 ≤ x ≤ 2 } .
Ans : 23 / 6
(9) Using the method of integration find the area of the triangle ABC, coordinates of whose
vertices are A( 2, 0 ) ; B( 4, 5 ) & C( 6 , 3 ) .
Ans : 7
(10) Find the area lying above x-axis & included between the circle x2 + y2 =8x and parabola y2 = 4x.
Ans : 4 ( 3π – 8 ) / 3
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(11) Find the area of the region (x , y) :
+
≤ 1 ≤
+
(12) Find the area of the region { ( x, y) : y2 ≤ 4x , 4x2 + 4y2 ≤ 9 } .
Ans :
( π – 2)
Ans : 9 sin–1 2√2 + 1
4
3
3√2
(13) Prove that the area of the region enclosed by x-axis, line x = √3 y and the circle x2 + y2 = 4 is 2π / 3
(14) Prove that the area of the region bounded by the two parabolas y2 = 4x & x2 = 4y is 16 / 3.
(15) Find the area of region { ( x , y ) : | x | ≤ y ≤ √4 – x2 } .
Ans : π
(16) Prove that the area enclosed between the curves y = | x – 1 | and y = 3 – | x | is 4 sq. units.
(17) Prove that, the area of the region between the curves y2 = x and x + y = 2 is 9 / 2 sq. units
(18) Prove that the area enclosed by the curve y = x | x | , x = 1, x = –1 and x - axis is 2 / 3 sq units.
(19) Find area of the region {(x, y) : x2 + y2 ≤ 4 ; x + y ≥ 2 }
Ans : π – 2
(20) Prove that the area enclosed between the curves y2 = 4x , x = y is 8 / 3 sq units.
******
10. Differential Equation
(1) Solve the initial value problem : 2y e
x/y
x/y
dx + ( y – 2x e
) dy = 0 ; x = 0 , y = 1 .
x/y
Ans : 2 e
+ log | y | = 2
(2) Solve the initial value problem : ( x 3 + x 2 + x + 1)dy = ( 2x 2 + x ) dx ; y(0) = 1.
Ans : 4y = log{(x + 1)2. (x2 + 1)3} – 2 tan–1 x + 4
(3) Solve the initial value problem : (x – y)(dx + dy) = dx – dy ; y(0) = –1
Ans : log | x – y | = x + y + 1
x
x
(4) Solve the initial value problem : (1 + e 2 )dy + (1 + y2 ) e dx = 0 ; y(0) = 1
x
Ans : tan–1 y + tan – 1( e ) = π / 2
(5) Solve the initial value problem : y1 + y cot x = 2x + x2cot x
;
y(π/2) = 0
Ans : y = x2 – π2 / 4 sin x
(6) Solve the initial value problem : dy = y2 – 2xy – x2
dx y2 + 2xy – x2
;
y(1) = –1
Ans : x + y = 0.
Ans : log ( 1 + y ) = x + x2
2
(8) Solve the initial value problem : (x3 – 3xy2)dx = (y3 – 3x2y)dy ; y(1) = 0
Ans : ( x2 – y2 ) = ( x2 + y2 )2
(7) Solve the initial value problem : dy = (1 + x + y + xy)dx ; y(0) = 0
(9) Solve the initial value problem : x2dy = (x2 – 2y2 + xy)dx ; y(1) = 0
Ans : log (x + √2y) – log (x – √2y) = 2√2log x
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(10) Solve the initial value problem : x2dy + (xy + y2)dx = 0 ; y(1) = 1.
Ans : y + 2x = 3x2 y
(11) Show that the general solution of the differential equation (x2 + x + 1) dy + (y2 + y + 1) dx = 0 is
given by (x + y + 1) = A( 1 – x – y – 2xy)
(12) Solve the initial value problem :
{ x cos ( y / x ) + y sin ( y / x ) } y dx = { y sin ( y / x) – x cos ( y / x) }x dy , y(1) = π / 4
Ans : xy cos ( y / x ) = π / 4√2
(13) Solve the initial value problem : (tan–1y – x) dy = ( 1 + y 2 ) dx ; x = –1 , y = 0
Ans : y = tan (x + 1)
(14) Solve the initial value problem : x . cos ( y / x ) dy = [ x + y cos( y / x )] dx ; y (1) = 0
Ans : sin ( y / x) = log | x |
(15) Solve the initial value problem : x dy + y dx – x dx + xy cot x dx = 0 ; x = 0 , y = 1
Ans : xy sin x = sin x – x cos x
(16) Solve the initial value problem : ( 1 + x 2 )dy + 2xy dx = cot x dx ; x = π / 2 , y = 0
Ans : y(1 + x)2 = log | sin x |
(17) Solve the initial value problem : sin x dy = (x sin x – y cos x )dx ; y (0) = 1
Ans : y = – x cot x + 1
(18) Show that the family of curves for which the slope of the tangent at any point (x , y) on it is
x 2 + y 2 , is given by x2 – y2 = cx .
2xy
(19) The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is
3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
Ans : (63 + 27)
/
(20) Find the equation of the curve passing through the point (–2 , 3), given that the slope of the
tangent to the curve at any point (x , y) is 2x / y 2 .
Ans : y = ( 3x2 + 15 ) 1 / 3
(21) Find the equation of the curve passing through the point (0 , π / 4) whose differential
equation is sin x .cos y dx + cos x sin y dy = 0 .
Ans : √2 cos y = sec x
ax
ax
(22) Form the differential equation representing the family of curves : y = c1e cos bx + c2e sin bx.
If a and b are known constants and , are unknown constants.
Ans : y11 – 2ay1 +( a2 + b2 ) y = 0
x
(23) Form the differential equation representing the family of curves : y = e ( a cosx + b sinx ) .
Ans : y11 – 2y1 + 2y = 0
(24) Form the differential equation representing the family of ellipse having foci on x – axis and centre
at origin .
Ans : xy + x
–y =0
x
x
(25) Form the differential equation representing the family of curves : y = a e 3 + b e– 2 .
Ans : y11 – y1 – 6y = 0
******
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11. Vectors
(1) Find the magnitude and direction of resultant of the vectors a = i – 2j + k , b = –2i + 4j + 5k
&c =i–6j–7k.
Ans : – 4j – k ; d.c’s ( 0 , – 4 / √17 , – 1 / √17 )
(2) If a , b , c are unit vectors such that a + b + c = 0 , find the value of
a .b + b .c + c . a .
Ans : – 3 / 2
(3) Three vectors a , b , c satisfy the condition a + b + c = 0 . Evaluate the quantity
a . b + b . c + c . a , if | a | = 1 , | b | = 4 , | c | = 2 .
Ans : – 21 / 2
(4) Let a , b and c be three vectors such that | a | = 3 ,| b | = 4 , | c | = 5 and each one of
them being perpendicular to the sum of the two, find | a + b + c | .
Ans : 5√2
(5) If a unit vector a makes angles π/ 3 with i , π / 4 with j and an acute angle θ with k , then find θ
and hence the vector a .
Ans : π/ 3 , ( i + √2j + k ) / 2
(6) Find a unit vector perpendicular to each of the vector a + b & a – b,
where a = i + j + k , b = i + 2 j + 3k.
Ans : ( – i + 2j – k ) / √6
(7) If the vertices of a triangle ABC are A( 1, 2, 3) ; B(–1, 0, 0) & C(0, 1, 2), then find ABC.
Ans : cos –1( 10 / √102 )
(8) If a , b , c are mutually perpendicular vectors of equal magnitude , show that the vector
a + b + c is equally inclined to a , b , c . Also find the respective angles. Ans : cos–1(1 / √3)
(9) Let a = i + 4j + 2k , b = 3i – 2j + 7k , c = 2i – j + 4k . Find a vector d which is
perpendicular to both a and b , & c . d = 15 .
Ans : (160i – 5j + 70k) / 3
(10) Prove that for any two vectors a and b, | a × b |2 = a . a
a .b
a .b
b.b
(11) If a , b and c are three vectors, such that a + b + c = 0, and | a | = 3, | b | = 5, | c | = 7,
°
then prove that angle between a and b is 60 .
(12) If with reference to the right handed system of mutually perpendicular unit vectors i, j, k ;
a = 3i – j , b = 2i + j – 3k , then express b in the form b = b1 + b2 , where b1 is parallel to a and
b2 is perpendicular to a .
Ans : b1 = ( 3i – j ) / 2, b2 = ( i + 3j – 6k) / 2
(13) Decompose the vector 5i – 2j + 5k into vectors which are parallel & perpendicular to 3i + k .
Ans : 6i + 2k , – i – 2j + 3k
(14) If a = i + j + k and b = j – k, find a vector c such that a × c = b and a . c = 3
Ans : (5i + 2j + 2k) / 3
(15) Let and a and b are two non – zero vectors. Then prove that a and b are perpendicular if and
only if | a + b | = | a – b | .
(16) If a  b = c  d and a  c = b  d. Show that a – d is parallel to b – c.
(17) If a  c = b  c and a . c = b . c ; | c | ≠ 0. Show that a = b
(18) If A, B, C, D are four points in space, prove that :
| AB × CD + BC × AD + CA × BD | = 4 (area  ABC ).
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(19) A vector a of magnitude 4 makes angles π/ 4 with i , π / 3 with j and an obtuse angle θ with k ,
then find θ and hence the vector a .
Ans : 2π/ 3 , 2( i + √2j – √2k )
(20) Find area of the parallelogram
(i) whose diagonals are given by, a = 3i + j – 2k , b = i – 3j + 4k
(ii) whose adjacent sides are given, a = 3i + j + 4k ,b = i – j + k.
Ans : 5√3
Ans : √42
(21) Prove that if three vectors a , b , c satisfies a + b + c = 0 then b × c = c × a = a × b .
(22) (i) Show that the four points with position vectors 4i + 8j + 12k, 2i + 4j + 6k, 3i + 5j + 4k and
5i + 8j + 5k are coplanar.
(ii) Find x such that the four points A(3, 2, 1), B(4, x, 5), C(4, 2, – 2) and D(6, 5, –1) are coplanar
(23) If the vectors ⃗ = xi + j + k , ⃗= i + yj + k and ⃗ = i + j + z k are coplanar, then prove that
+
+
(24) Show that the vectors
= 1, where x, y, z ≠ 1
⃗ , ⃗ and ⃗ coplanar if
⃗ + ⃗ , ⃗ + ⃗ and ⃗ + ⃗ coplanar.
(25) Find the volume of the parallelepipeds whose co – terminus edges are
⃗ = 2i – 3j + 4k , ⃗= 3i – j + 2k and ⃗ = i + 2j – k
*****
12. Three – Dimensions
(1) Find the equation of the plane passing through the points (1 , 2 , 3) and (0 , –1 , 0) and parallel to the
line r = i – 2j + λ( 2i + 3j – 3k). Ans : 6x – 3y + z + 3 = 0 ; r . (6i – 3j + k) = 3
(2) Find the value of ‘p’ so that the lines: 1 – x = 7y – 14 = z – 3 ; 7 – 7x = y – 5 = 6 – z ,
are perpendicular.
3
2p
2
3p
1
5
Ans : p = 70 / 11
(3) If and are the angles made by any line with the four diagonals of any cube, then prove
that : cos2 cos2 cos2 cos2 4 / 3 .
(4) Find the shortest distance between the lines : x + 1 = y + 1 = z + 1 and x – 3 = y – 5 = z – 7
Ans : 2√29
7
–6
1
1
–2
1
(5) Find the distance between the lines : r = i + 2j – 4k + λ( 2i + 3j + 6k ) ;
r = 3i + 3j – 5k + µ( 2i + 3j + 6k ).
Ans : √293 / 7
(6) Find the coordinates of the point where the line through ( 3, – 4, – 5) and ( 2, – 3, 1 ) crosses the
plane 2x + y + z = 7 .
Ans : ( 1 , – 2 , 7 )
(7) Find the equation of the plane passing trough (–1, 3, 2) and perpendicular to each of the planes
x + 2y + 3z = 5 , 3x + 3y + z = 0 .
Ans : 7x – 8y + 3z + 25 = 0
(8) Find the distance of the point P( 6, 5, 9 ) and the plane determined by the points A( 3, –1, 2 ),
B( 5, 2, 4 ) and C( – 1 , – 1 , 6 ).
Ans : 6 / √34
(9) Find the equation of the plane passing through the line of intersection of the planes
r . ( i + 2j + 3k ) = 4, and r . ( 2i + j – k ) + 5 = 0 , and which is perpendicular to the plane
r . ( 5i + 3j – 6k ) + 8 = 0 .
Ans : 33x + 45y + 50z = 41
(10) Show that the lines : x – a + d = y – a = z – a – d & x – b + c = y – b = z – b – c

are coplanar. Also find the equation of plane .
Ans : x – 2y + z = 0
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(11) Show that the lines : x + 3 = y – 1 = z – 5 ; x + 1 = y – 2 = z – 5 are coplanar.
–3
1
5
–1
2
5
Also find the equation of plane.
Ans : x – 2y + z = 0
(12) Find the equation of the plane passing through the line of intersection of the planes
r . ( i + j + k ) = 1 , r . ( 2i + 3j – k ) + 4 = 0 , and parallel to the x – axis .
Ans : y – 3z + 6 = 0
(13) Find the distance of the point (–1, –5, –10) from the point of intersection of the line
⃗ = 2i – j + 2k + λ (3i + 4j + 2k) and the plane ⃗. (i – j + k) = 5 .
Ans : 13
(14) Find the point of intersection of the lines : x – 5 = y – 7 = z + 3
If the lines are intersecting.
4
4
–5
; x–8 = y–4 = z–5 .
7
1
3
Ans : (1, 3, 2)
(15) Find the image of the point ( 1 , 2 , 3 ) in the line , x – 6 = y – 7 = z – 7 .
Ans : (5, 8, 15)
3
2
–2
(16) Find the length and foot of perpendicular from the point (7, 14, 5) on the plane 2x + 4y – z = 2.
Ans : 3√21 ; (1, 2, 8)
(17) A variable plane which always remains at a constant distance ‘ p’ from origin, cuts the coordinate
axes at A, B, C respectively. Prove that the locus of the centroid of the triangle ABC is
x – 2 + y – 2 + z – 2 = 9p– 2.
(18) Find the foot and length of perpendicular drawn from the point (2, 3, 4) to the line
4–x = y = 1–z.
Ans : 170 , 78 , 10 ; 3√101 units
2
6
3
49 49 49
7
(19) Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes
⃗. (i – j + 2k) = 5 and ⃗. (3i + j + k) = 6 .
Ans : ⃗ = (i + 2j + 3k) + λ (–3i + 5j + 4k)
(20) Find the cartesian as well as the vector equation of the plane through the intersection of the planes
r . ( 2i + 6j ) + 12 = 0 and r . ( 3i – j + 4k ) = 0, and at a unit distance from origin .
Ans : r . (2i + j + 2k ) + 3 = 0 : 2x + y + 2z + 3 = 0 & r . (–i + 2j – 2k ) + 3 = 0 : x – 2y + 2z = 3
(21) Show that the line r = 2i – 2j + 3k + λ( i – j + 4k ) is parallel to the plane r . (i + 5j + k) = 5.
Also find the distance between them.
Ans : 10 / 3√3
(22) Find the image of the point (1, 3, 4) in the plane 2x – y + z + 3= 0.
Ans : (–3, 5, 2)
(23) Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5 = 0 ; measured parallel to
the line x + 3 = y – 2 = z .
Ans : 7 units
3
6
2
(24) A variable plane which always remains at a constant distance ‘ p’ from origin, cuts the coordinate
axes at A, B, C respectively. Prove that the locus of the point of intersection of the planes drawn
parallel to the coordinate planes through the points A , B and C respectively is 1 + 1 + 1 = 1 .
x2 y2 z 2
p2
(25) Find the distance of the point ( – 2, 3 , – 4 ) from the line
=
=
measured
parallel to the plane 4x + 12y – 3z + 1 = 0
Ans :
******
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13. Linear Programming
(1) There are two type of fertilizers F1 & F2. F1 consists of 10 % nitrogen and 6 % phosphoric acid and
F2 consists of 5% nitrogen & 10% phosphoric acid. After testing the soil condition, a farmer finds
that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop. If F1 costs Rs 6 / kg
and F2 costs Rs 5 / kg, determine how much of each type of fertilizers should be used so that the
nutrient requirement are met at a minimum cost. What is the minimum cost.
Ans : F1 = 100 kg, F2 = 80 kg ; Min Cost = Rs 1000
(2) To maintain one’s health, a person must fulfil minimum daily requirements for the following three
nutrients – calcium, protein and calories. His diet consists of only food items I and II whose prices
and nutrient contents are shown below :
Price
Food I
Rs 0. 60 per unit
Food II
Rs 1 per unit
Minimum
requirements
Calcium
Protein
Calories
10
5
2
4
5
6
20
20
12
Find the combination of food items so that the cost may be minimum.
Ans : Min Cost = Rs 2.80, Food I = 3 units, Food II = 1 unit.
(3) A company produces two types of belts, A and B. Profits on these types are Rs 2 and Rs 1.5 on each
belt, respectively. A belt of type A requires twice as much time as a belt of type B. The company can
produce at the most 1000 belts of type B per day. Material for 800 belts per day is available. At the
most 400 buckles for belts of type A and 700 for those of type B are available per day. How many
belts of each type should the company produce so as to maximize the profit ?
Ans : A = 200, B = 600
(4) A manufacturer produces nuts and bolts for industrial machinery. It takes 1hr of work on machine A
and 3hrs on machine B to produce a package of nuts, while it takes 3hrs on machine A and 1hr on
machine B to produce a package of bolts. He earns a profit of Rs 2.50 per package on nuts and Re1 per
package on bolts. Form a linear programming problem to maximize his profit, if he operates each
machine for at the most 12 hrs a day. Find the maximum profit also .
Ans : A = 3, B = 3, Max Profit = Rs 10.50
(5) A young man rides his motorcycle at 25 km / hr, he has to spend Rs 2 / km on petrol ; if he rides it at
a faster speed of 40 km / hr, the petrol cost increases to Rs 5 / km. He has Rs 100 to spend on petrol
and wishes to find the maximum distance he can travel within 1 hr. Express this as an L.P.P. and
solve it.
Ans : 50 / 3 km at 25 km / hr & 40 / 3 km at 40 km / hr.
(6) A house wife wishes to mix together two kinds of food, I and II, in such a way that the mixture
contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin
contents of one kg of food is given below :
Vitamin A
Vitamin B
Vitamin C
Food I
1
2
3
Food II
2
2
1
One kg of food I costs Rs 6 and one kg of food II costs Rs 10. Formulate and solve the above
problem to find the least cost of the mixture which will produce the diet.
Ans : Food I = 2 kgs , Food II = 4 kgs .
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(7) A toy company manufactures two types of dolls, A and B. Each doll of type B takes twice as long as
to produce as one of type A. If the company produces only type A, it can make a maximum of 2000
dolls per day. The supply of plastic is sufficient to produce 1500 dolls per day. Type B requires a
fancy dress which cannot be available for more than 600 per day. If the company makes profits of Rs
3 and Rs 5 per doll respectively on dolls A and B, how many of each should be produced per day in
order to maximize the profit?
Ans : Max Profit = Rs 5500, A = 1000, B = 500.
(8) A manufacturing company makes two models A and B of a product. Each piece of model A require 9
labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 labour
hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum
labour hours available are 180 and 30 respectively. The company makes a profit of Rs 8000 on each
piece of model A and Rs 12000 on each piece of model B. How many piece of model A and model B
should be manufactured per week to realize a maximum profit? What is the maximum profit ?
Ans : A = 12, B = 6 , Max Profit = Rs 168000.
(9) 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 ?
Ans : Executive Class = 40, Economy Class = 160 Max Profit = Rs 136000.
(10) A toy company manufactures two type of dolls, A and B. Market tests and available resources have
indicated that the combine production level should not exceed 1200 dolls / week and the demand for
the dolls of type B is at most half of that for the 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 the profit of Rs12 and Rs16 per doll respectively on dolls A and B, how many of
each should be produced weekly in order to maximize the profit .
Ans : A = 800, B = 400
(11) A factory manufactures two types of screws, A and B, each type requiring the use of two machines,
an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand
operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and
3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is
available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a
profit of Rs7 and screws B at a profit of Rs10. Assuming that he can sell all the screws he can
manufacture, how many packages of each type should the factory owner produce in a day in order to
maximize his profit? Determine the maximum profit.
Ans : A = 30, B = 20 Max Profit = Rs 410
(12) A furniture dealer deals in only two items : tables and chairs. He has Rs 5000 to invest and a space to
store at most 60 pieces. A table costs him Rs 250 and a chair, Rs 50. He can sell a table at a profit of
Rs 50 and a chair at a profit of Rs 15. Assuming that he can sell all the items that he buys, how should
he invest his money in order that he may maximize his profit?
Ans : 10 table, 50 chair.
(14) A retired person wants to invest an amount of up to Rs 20000. His broker recommends investing in
two types of bonds A and B, bond A yields 10% return on the amount invested and bond B yields
15% return on the amount invested. After some consideration, he decides to invest at least Rs 5000
in bond A and no more than Rs 8000 in bond B. He also wants to invest at least as much in bond A as
in bond B. What should his broker suggest if he wants to maximize his return on investments?
Formulate and solve the LPP.
Ans : A = Rs 12000, B = Rs 8000.
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(13) 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 requirement are 4500 L, 3000 L and
3500 L, respectively. The distances (in km) between the depots and the petrol pumps are given in the
following table:
From / To
D
E
F
Distance (in km)
A
7
6
3
B
3
4
2
Assuming that the transportation cost per km is Re 1 per l0L, how should the delivery be scheduled in
order that the transportation cost is minimum?
Ans : From A : 500L, 3000L & 3500 L and from B : 4000L, 0 , 0 to D, E and F
respectively Min cost : Rs 510
(15) A dietician mixes together two kinds of food, say X and Y in such a way that the mixture contains at
least 6 units of vitamin A, 7 units of vitamin B, 12 units of vitamin C and 9 units of vitamin D. The
vitamin contents of 1 kg of food X and 1 kg of food Y are given below :
Food X
Food Y
Vitamin A
1
2
Vitamin B
1
1
Vitamin C
1
3
Vitamin D
2
1
One kg of food X costs Rs 5, whereas one kg of food Y costs Rs 8. Formulate the L.P.P and find the
least cost of mixture which will produce the desired diet.
Ans : X = 4.5 kgs , Y = 2.5 kgs, Cost = Rs 42.5
******
14. Probability
(1) If E and F are two independent events , prove that the events (i) E and F1 are also independent .
(ii) E1 and F are also independent .
(iii) E1 and F1 are also independent.
(2) Given that the events A and B are such that P(A) = 1 / 2 , P(AB) = 3 / 5 and P(B) = p.
Find ‘p’ if A and B are (i) mutually exclusive
Ans : 1 / 10
(ii) independent .
Ans : 1 / 5
(3) Suppose a girl throws a die. If she gets 5 or 6 she tosses a coin three times and notes the number of
heads. If she gets 1, 2, 3 or 4, she tosses the coin once and notes whether a head or tail is obtained. If
she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die ?
Ans : 8 / 11
(4) A man is to known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find
the probability that it is actually a six .
Ans : 3 / 8
(5) A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn
and are found to be both diamonds. Find the probability of the lost card being a diamond.
Ans : 11 / 50
(6) An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The
probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets an
accident. What is the probability that he is a scooter driver ?
Ans : 1 / 52
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(7) There are three coins. One is a two headed coin ( having head on both face ), another is a biased coin
that comes up heads 75 % of the time and third is an unbiased coin. One of the three coin is chosen
at random and tossed, it shows heads, what is the probability that it was a two headed coin ?
Ans : 4 / 9
(8) Suppose that the reliability of a HIV test is specified as follows:
Of people having HIV, 90% of the test detect the disease but 10% go undetected. Of people free of
HIV, 99% of the test are judged HIV(– ve) but 1% are diagnosed as showing HIV(+ ve). From a
large population of which only 0.1% have HIV, one person is selected at random, given the HIVtest,
and the pathologist reports him/her as HIV(+ve). What is the probability that the person actually has
HIV?
Ans : 90 / 1089
(9) A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present.
However, the test also yields a false positive result for 0.5% of the healthy person tested ( i.e. if a
healthy person is tested, then, with probability 0.005, the test will imply he has the disease ). If 0.1
percent of the population actually has the disease, what is the probability that a person has the
disease given that his test result is positive ?
Ans : 198 / 1197
(10) A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will
come by train, bus, scooter or by other means of transport are respectively 3 / 10, 1 / 5, 1 / 10 and
2 / 5. The probabilities that he will be late are 1 / 4, 1/ 3 and 1 / 12 if he comes by train, bus and
scooter respectively, but if he comes by other means of transport, then he will not be late. When he
arrives, he is late. What is the probability that he comes by train?
Ans : 1 / 2
(11) If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it
produces only 40% acceptable items. Past experience shows that 80% of the set ups are correctly
done. If the machine produces 2 acceptable items, find the probability that the machine is correctly
set up.
Ans : 0.95
(12) Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. Find the
probability distribution of the number of aces . Also show that mean, variance and standard
deviation for above distribution are 2 / 13, 0.14 & 0.38 respectively.
Ans :
X
0
1
2
P(X)
144 / 169
24 / 169
1 / 169
(13) Two cards are drawn without replacement from a well-shuffled deck of cards. Determine the
probability distribution of the number of ‘face cards’.
Ans :
X
0
1
2
P(X)
130 / 221
80 / 221
11 / 221
(14) Find the mean and variance of heads in three tosses of a fair coin.
Ans : mean = 3 / 2 , var = 3 / 4
(15) A and B throws a die alternately till one of them gets a ‘6’ and wins the game. Find their respective
probabilities of winning, if A starts the game.
Ans : A’s win 6 / 11, B’s win 5 /11
(16) In a hurdle race, a player has to cross 10 hurdles. The probability that he clear each hurdle is
5 / 6 .What is the probability that he will knock down fewer than 2 hurdles?
Ans : 5 10 / 2 . 6 9
(17) 3 defective bulbs are accidently mixed with 7 good ones. If three bulbs are drawn, what is the
average number of defective bulbs drawn ?
Ans : 0.9
(18) A coin is biased so that the head is three times as likely to occur as tail. If the coin is tossed twice,
find the probability distribution of number of tails.
Ans :
X
0
1
2
P(X)
9 / 16
6 / 16
1 / 16
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(19) Two cards are drawn at random without replacement from a well shuffled deck of 52 cards. Find the
probability distribution of the number of aces and the mean, variance and standard deviation of
the number of aces in the distribution.
Ans :
X
0
1
2
P(X)
188/ 221
32 / 221
1 / 221
Mean = 34 / 221, Variance = 6800 / (221)2, Standard deviation = 0.37
(20) Probability of solving specific problem independently by A, B and C are 1 / 2, 1 / 3 and 1 / 6
respectively. If all try to solve the problem independently, find the probability that
(i) the problem is solved .
Ans : 13 / 18
(ii) problem is solved by exactly one.
Ans : 17 / 36
(iii) problem is solved by exactly two.
Ans : 2 / 9
(21) A letter is known to have come from ‘LONDON’ or ‘CLIFTON’. On the envelope just two
consecutive letters ‘ON’ are visible. What is the probability that the letter has come from CLIFTON.
Ans : 5 / 17
(22) A letter is known to have come from ‘TATANAGAR’ or ‘KOLKATA’. On the envelope just two
regular letters ‘TA’ are visible. What is the probability that the letter has come from ‘KOLKATA’.
Ans : 2 / 5
(23) In a test, an examinee either guesses or copies or knows the answer to a multiple-choice question
with four choices. The probability that he makes a guess is 1 / 3and the probability that he copies the
answer is 1 / 6. The probability that his answer is correct, given that he copied it, is 1 / 8. Find the
probability that he knew the answer to the question, given that he correctly answered it.
Ans : 24 / 29
(24) Three bags contains balls as shown in the table below:
Bag
I
II
III
No. of White balls
1
2
4
No. of Black balls
2
1
3
No. of Red balls
3
1
2
A bag is chosen at random and two balls are drawn from it. They happen to be white and red. What
is the probability that they came from the bag III ?
Ans : 5 / 17
(25) Bag I and Bag II contains 3 red, 4 black balls and 4 red, 5 black balls respectively. Two balls are
transferred from the Bag I to the Bag II without observing its colour. Then a ball is drawn from the
Bag II. The ball thus drawn is found to red in colour. Find the probability that the transferred balls
are black in colour.
Ans : 4 / 17
(26) The probability of a shooter hitting a target is 0.75. How many minimum number of times must
he/she fire so that the probability of hitting the target at least once is more than 0.99?
Ans : 4
(27) A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the
third six in the sixth throw of the die.
Ans : 625 / 23328
******
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