class 12 exampler problem

1. MATHS CLASS XII
CHAPTER – 1
RELATONS AND FUNCTIONS
1. Is the binary operation definedonZ(setof integer) by m  n= m – n + mn  m, n  Z commutative?
2. If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g.
3. Show that the function f: R Rdefinedby f(x) =
𝒙
𝒙 𝟐+ 𝟏
,  x  R, is neitherone – one nor onto.
4. Let  be a binary operationdefinedonQ. findwhich of the followingbinary operationsare associative
a. a  b= a – b for a, b  Q.
b. a  b=
𝒂𝒃
𝟒
for a, b  Q.
c. a  b= a – b + ab for a, b  Q.
d. a  b= ab2
for a, b  Q.
5. the domainof the functionf : R  R definedbyf(x) = √ 𝒙 𝟐 − 𝟑𝒙 + 𝟐 is ………………
6. Let the functionf : R Rdefinedbyf(x) = cos x ,  x  R. Show that f isneitherone – one nor onto.
7. Let n be a fixedpositive integer.Define arelationR in Z as follows:  a,b  Z, aRb if and only ifa – b is
divisible byn. show that R is an equivalence relation.
8. GivenA = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example ofeach of the following:
a. An injectingmappingfrom A to B.
b. A mapping from A to B which isnot injective.
c. A mapping from B to A.
9. Let A = R – {3} , B = R – {1}. Let f : A  B be definedbyf(x) =
𝒙−𝟐
𝒙−𝟑
 x  A. Thenshow that f isbiijective.
10. Let A = [-1, 1]. Then, discusswhetherthe followingfunctionsdefinedonAare one – one,onto or biijective:
a. f(x) =
𝒙
𝟐
b. g(x) = | 𝒙| c. h(x) = x | 𝒙| d. k(x)_ = x2
11. Let A = {1,2,3, …..,9} and R be the relationin A  A definedby{a, b} R (c, d) ifa + d = b + c for (a, b),(c, d) in
AA.Prove that R is an equivalence relationandalso obtain the equivalentclass[(2, 5)].
12. Functionsf, g: R  R are defined,respectively,byf(x) =x2
+ 3x + 1, g(x) = 2x n- 3, find
a. f o g b. g o f c. f o f d. g o g

2. CHAPTER - 2
INVERSE TRIGONOMETRIC FUNCTIONS
1. Evaluate :- 𝐭𝐚𝐧−𝟏 ( 𝒔𝒊𝒏 (
𝝅
𝟐
)) .
2. Findthe value of 𝐭𝐚𝐧−𝟏 (𝒕𝒂𝒏
𝟗 𝝅
𝟖
) .
3. Evaluate :- 𝐬𝐢𝐧−𝟏 [𝒄𝒐𝒔 (𝐬𝐢𝐧−𝟏 √ 𝟑
𝟐
)] .
4. Findthe value of sin [𝟐 𝐜𝐨𝐭−𝟏(
−𝟓
𝟏𝟐
) ] .
5. Evaluate :- cos [𝐬𝐢𝐧−𝟏 𝟏
𝟒
+ 𝐬𝐞𝐜−𝟏 𝟒
𝟑
] .
6. Prove that 2 𝐬𝐢𝐧−𝟏 𝟑
𝟓
− 𝐭𝐚𝐧−𝟏 𝟏𝟕
𝟑𝟏
=
𝝅
𝟒
.
7. Prove that cot-1
+ cot-1
8 + cot-1
18 = cot-1
3.
8. Findthe value of 𝐬𝐢𝐧(𝟐 𝐭𝐚𝐧−𝟏 𝟐
𝟑
) + 𝒄𝒐𝒔 (𝐭𝐚𝐧−𝟏
√ 𝟑 ).
9. Findthe value of x which satisfy the equation sin-1
x + sin-1
(1 – x) = cos-1
x.
10. Solve the equationsin-1
6x + sin-1
6√ 𝟑 x = -
𝝅
𝟐
.
11. Findthe value of 𝐭𝐚𝐧−𝟏 (𝒕𝒂𝒏
𝟓𝝅
𝟔
) + 𝐜𝐨𝐬−𝟏 (𝒄𝒐𝒔
𝟏𝟑𝝅
𝟔
) .
12. Prove that cot (
𝝅
𝟒
− 𝟐 𝐜𝐨𝐭−𝟏 𝟑) = 7.
13. Show that cos (𝟐 𝐭𝐚𝐧−𝟏 𝟏
𝟕
) = sin(𝟒 𝐭𝐚𝐧−𝟏 𝟏
𝟑
) .
14. Solve the followingequationcos(tan-1
x) = sin (cot-1 𝟑
𝟒
).
15. Prove that 𝐭𝐚𝐧−𝟏 (
√ 𝟏+ 𝒙 𝟐 + √ 𝟏− 𝒙 𝟐
√ 𝟏+ 𝒙 𝟐 − √ 𝟏− 𝒙 𝟐
) =
𝝅
𝟒
+
𝟏
𝟐
𝐜𝐨𝐬−𝟏 𝒙 𝟐 .
16. Findthe simplifiedformof 𝐜𝐨𝐬−𝟏(
𝟑
𝟓
𝒄𝒐𝒔 𝒙 +
𝟒
𝟓
𝒔𝒊𝒏 𝒙), where x  [
−𝟑𝝅
𝟒
,
𝝅
𝟒
] .
17. Prove that 𝐬𝐢𝐧−𝟏 𝟖
𝟏𝟕
+ 𝐬𝐢𝐧−𝟏 𝟑
𝟓
= 𝐬𝐢𝐧−𝟏 𝟕𝟕
𝟖𝟓
.
18. Findthe value of 4 tan-1 𝟏
𝟓
− 𝐭𝐚𝐧−𝟏 𝟏
𝟐𝟑𝟗
.
19. Show that 𝐭𝐚𝐧(
𝟏
𝟐
𝐬𝐢𝐧−𝟏 𝟑
𝟒
) =
𝟒− √ 𝟕
𝟑
and justifywhy the othervalue
𝟒 + √ 𝟕
𝟑
is ignored?

3. CHAPTER – 3
MATRICES
1. If [ 𝟐𝒙 𝟑] [
𝟏 𝟐
−𝟑 𝟎
] [
𝒙
𝟖
] = 0, findthe value of x.
2. If A = [
𝟏 𝟑 𝟐
𝟐 𝟎 −𝟏
𝟏 𝟐 𝟑
] , then show that A satisfiesthe equationA3
- 4A2
- 3A + 11I = 0.
3. Let A = [
𝟐 𝟑
−𝟏 𝟐
] . then show that A2
- 4A + 7I = 0. Usingthis resultcalculate A5
also.
4. If a matrix has 28 elements,whatare the possible ordersit can have?What if it has 13 elements?
5. If X = [
𝟑 𝟏 −𝟏
𝟓 −𝟐 −𝟑
] 𝒂𝒏𝒅 𝒀 = [
𝟐 𝟏 −𝟏
𝟕 𝟐 𝟒
] , find (i) X + Y (ii) 2X – 3Y (iii) A matrix Z such that X + Y + Z is
a zeromatrix.
6. If possible,findBA and AB, where A = [
𝟐 𝟏 𝟐
𝟏 𝟐 𝟒
] , 𝑩 = [
𝟒 𝟏
𝟐 𝟑
𝟏 𝟐
] .
7. Show that AAand AA are both symmetricmatrices for any matrix A.
8. Let A = [
𝟏 𝟐
−𝟏 𝟑
] , B = [
𝟒 𝟎
𝟏 𝟓
] , C = [
𝟐 𝟎
𝟏 −𝟐
] and a = 4, b = -2. Showthat :
a. A + (B + C) = (A + B) + C
b. A(BC) = (AB) C
c. (a + b)B = aB + bB
d. A(C – A) = aC – aA
e. (AT
)T
= A
f. (bA)T
= b AT
g. (AB)T
= BT
AT
h. (A – B)C = AC – BC
i. (A – B)T
= AT
- BT
9. If A = [
𝟎 −𝒙
𝒙 𝟎
] , B = [
𝟎 𝟏
𝟏 𝟎
] and x2
= -1, then showthat (A + B)2
= A2
+ B2
.
10. If A = [
𝒄𝒐𝒔  𝒔𝒊𝒏 
− 𝒔𝒊𝒏  𝒄𝒐𝒔 
], and A-1
= A , findthe value of  .

4. CHAPTER – 4
DETERMINANTS
1. If  = |
𝟏 𝒙 𝒙 𝟐
𝟏 𝒚 𝒙 𝟐
𝟏 𝒛 𝒙 𝟐
|, ∆ 𝟏= |
𝟏 𝟏 𝟏
𝒚𝒛 𝒛𝒙 𝒙𝒚
𝒙 𝒚 𝒛
| , thenprove that  + ∆ 𝟏 = 0.
2. Withoutexpanding,showthat  = |
𝒄𝒐𝒔𝒆𝒄 𝟐 𝒄𝒐𝒕 𝟐 𝟏
𝒄𝒐𝒕 𝟐 𝒄𝒐𝒔𝒆𝒄 𝟐 −𝟏
𝟒𝟐 𝟒𝟎 𝟐
| = 0.
3. If x = -4 is a root of  = |
𝒙 𝟐 𝟑
𝟏 𝒙 𝟏
𝟑 𝟐 𝒙
| = 0, thenfind the other two roots.
4. Evaluate : |
𝟑𝒙 −𝒙 + 𝒚 −𝒙 + 𝒛
𝒙 − 𝒚 𝟑𝒚 𝒛 − 𝒚
𝒙 − 𝒛 𝒚 − 𝒛 𝟑𝒛
| .
5. Evaluate: |
𝒂 − 𝒃 − 𝒄 𝟐𝒂 𝟐𝒂
𝟐𝒃 𝒃 − 𝒄 − 𝒂 𝟐𝒃
𝟐𝒄 𝟐𝒄 𝒄 − 𝒂 − 𝒃
|
6. Evaluate: |
𝒂 𝟐 + 𝟐𝒂 𝟐𝒂 + 𝟏 𝟏
𝟐𝒂 + 𝟏 𝒂 + 𝟐 𝟏
𝟑 𝟑 𝟏
| = (a – 1)3
.
7. If [
𝟒 − 𝒙 𝟒 + 𝒙 𝟒 + 𝒙
𝟒 + 𝒙 𝟒 − 𝒙 𝟒 + 𝒙
𝟒 + 𝒙 𝟒 + 𝒙 𝟒 − 𝒙
] = 0. Then findvaluesof x.
8. If A = [
𝟏 𝟐 𝟎
−𝟐 −𝟏 −𝟐
𝟎 −𝟏 𝟏
] , find A1
.
9. Usingmatrix method,solve the system ofequations3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.
10. GivenA = [
𝟐 𝟐 −𝟒
−𝟒 𝟐 −𝟒
𝟐 −𝟏 𝟓
] , 𝑩 = [
𝟏 −𝟏 𝟎
𝟐 𝟑 𝟒
𝟎 𝟏 𝟐
] , findBA and use thisto solve the system ofequations
y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.
11. If A = [
𝒙 𝟓 𝟐
𝟐 𝒚 𝟑
𝟏 𝟏 𝒛
], xyz = 80, 3x + 2y + 10z = 20, then A adj. A = [
𝟖𝟏 𝟎 𝟎
𝟎 𝟖𝟏 𝟎
𝟎 𝟎 𝟖𝟏
] .
12. If A, B, C are the anglesof a triangle,then ∆ = |
𝒔𝒊𝒏 𝟐 𝑨 𝒄𝒐𝒕 𝑨 𝟏
𝒔𝒊𝒏 𝟐 𝑩 𝒄𝒐𝒕 𝑩 𝟏
𝒔𝒊𝒏 𝟐 𝑪 𝒄𝒐𝒕 𝑪 𝟏
| = ……..
13. The determinant ∆ = |
√ 𝟐𝟑 + √ 𝟑 √ 𝟓 √ 𝟓
√ 𝟏𝟓 + √ 𝟒𝟔 𝟓 √ 𝟏𝟎
𝟑 + √ 𝟏𝟏𝟓 √ 𝟏𝟓 𝟓
| isequal to ………………………

5. CHAPTER – 5
CONTINUITY AND DIFFERENTIABILITY
1. If f(x) = {
𝒙 𝟑+ 𝒙 𝟐− 𝟏𝟔𝒙+𝟐𝟎
(𝒙−𝟐) 𝟐
, 𝒙 ≠ 𝟐
𝒌 𝒙 = 𝟐
is continuouse at x = 2, findthe value of k.
2. Differentiate √ 𝒕𝒂𝒏 √ 𝒙 w.r.ta.
3. Find
𝒅𝒚
𝒅𝒙
, ify = tan-1
(
𝟑𝒙− 𝒙 𝟑
𝟏−𝟑𝒙 𝟐
) , −
𝟏
√ 𝟑
< 𝑥 <
𝟏
√ 𝟑
.
4. If y = sin-1 { 𝒙 √ 𝟏− 𝒙 − √ 𝒙 √𝟏− 𝒙 𝟐} and 0 < x < 1, thenfind
𝒅𝒚
𝒅𝒙
.
5. If x = a sec3
 and y = a tan3
 , find
𝒅𝒚
𝒅𝒙
at  =

𝟑
.
6. If xy
= ex-y
, prove that
𝒅𝒚
𝒅𝒙
=
𝒍𝒐𝒈 𝒙
(𝟏+𝒍𝒐𝒈 𝒙) 𝟐
.
7. If y = tan x + sec x, prove that
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
=
𝒄𝒐𝒔 𝒙
(𝟏−𝒔𝒊𝒏 𝒙) 𝟐
.
8. VerifyRolle’stheoremfor the function,f(x) = sin 2x in [ 0,
𝝅
𝟐
].
9. Let f(x) =
{
𝟏−𝒄𝒐𝒔 𝟒𝒙
𝒙 𝟐
, 𝒊𝒇 𝒙 < 0
𝒂, 𝒊𝒇 𝒙 = 𝟎
√ 𝒙
√ 𝟏𝟔+ √ 𝒙− 𝟒
, 𝒊𝒇 𝒙 > 0
, for what value of a, f iscontinuous at x = 0?
10. Examine the differentiabilityofthe functionf definedby
2x + 3, if -3  x < -2
f (x) = x + 1, if -2  x < 0
x + 2, if 0  x ≤ 1
11. Differentiate tan-1
√ 𝟏− 𝒙 𝟐
𝒙
with respectto cos-1
(2x √𝟏 − 𝒙 𝟐 ),where x 
𝟏
√ 𝟐
, 1.
12. Findwhich of the functionsin Exercise 2 to 10 iscontinuous or discontinuousat the indicatedpoints:
a. f (x) = {
𝟏−𝒄𝒐𝒔 𝟐𝒙
𝒙 𝟐
, 𝒊𝒇 𝒙 ≠ 𝟎
𝟓, 𝒊𝒇 𝒙 = 𝟎
. at x = 0
b. f (x) = {
| 𝒙| 𝒄𝒐𝒔
𝟏
𝒙
, 𝒊𝒇 𝒙 ≠ 𝟎
𝟎, 𝒊𝒇 𝒙 = 𝟎
. at x = 0
c. f (x) = {
𝒙 𝟐
𝟐
, 𝒊𝒇 𝟎 ≤ 𝒙 ≤ 𝟏
𝟐𝒙 𝟐 − 𝟑𝒙 +
𝟑
𝟐
, 𝒊𝒇 𝟏 < 𝑥 ≤ 2
, at x = 1
d. f (x) = | 𝒙| + | 𝒙 − 𝟏|at x = 1
13. findthe value of k ineach of the Exercise 11 to 14 so that the functionf is continuousat the indicatedpoint:
a. (x) = {
√ 𝟏+𝒌𝒙− √ 𝟏−𝒌𝒙
𝒙
, 𝒊𝒇 − 𝟏 ≤ 𝒙 < 0
𝟐𝒙+𝟏
𝒙−𝟏
, 𝒊𝒇 𝟎 ≤ 𝒙 ≤ 𝟏
at x = 0

6. b. f(x) = {
𝟏−𝒄𝒐𝒔 𝒌𝒙
𝒙 𝒔𝒊𝒏 𝒙
, 𝒊𝒇 𝒙 ≠ 𝟎
𝟏
𝟐
, 𝒊𝒇𝒙 = 𝟎
at x = 0
14. findthe valuesof a and b such that the function f definedby f(x) =
{
𝒙−𝟒
| 𝒙−𝟒|
+ 𝒂, 𝒊𝒇𝒙 < 4
𝒂 + 𝒃, 𝒊𝒇 𝒙 = 𝟒
𝒙−𝟒
| 𝒙−𝟒|
+ 𝒃, 𝒊𝒇 𝒙 > 4
is a continuous
functionat x = 4.
15. Find
𝒅𝒚
𝒅𝒙
,if y = xtanx
+ √ 𝒙 𝟐+𝟏
𝟐
.
16. If x = sintand y = sin pt, prove that (1 - 𝒙 𝟐)
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
- x
𝒅𝒚
𝒅𝒙
+ p2
y = 0.
17. If xm
.yn
= (x+ y)m+n
, prove that:- (i)
𝒅𝒚
𝒅𝒙
=
𝒚
𝒙
and (ii)
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
= 0.
18. Findthe valuesof p and q so that f(x) = {
𝒙 𝟐 + 𝟑𝒙 + 𝒑, 𝒊𝒇 𝒙 ≤ 𝟏
𝒒𝒙 + 𝟐, 𝒊𝒇 𝒙 > 1
, isdifferentiable atx = 1.
19. Finda point on the curve y = (x – 3)2
, where the tangent is parallel to the chord joiningthe points(3,0) and
(4,1).
20. Find
𝒅𝒚
𝒅𝒙
whenx and y are connectedby the relationgivenin each of the Exercise 54 to 57.
a. tan-1
(x2
+ y2
)2
= a
b. if ax2
+ 2hxy + by2
+ 2gx + 2fy + c = 0, then showthat
𝒅𝒚
𝒅𝒙
.
𝒅𝒙
𝒅𝒚
= 1.
c. If x = 𝒆
𝒙
𝒚 , prove that
𝒅𝒚
𝒅𝒙
=
𝒙−𝒚
𝒙 𝒍𝒐𝒈 𝒙
.
d. If yx
= ey-x
, prove that
𝒅𝒚
𝒅𝒙
=
(𝟏+𝒍𝒐𝒈 𝒚) 𝟐
𝒍𝒐𝒈 𝒚
.
e. If x sin (a + y) + sin cos (a + y) = 0, prove that
𝒅𝒚
𝒅𝒙
= √ 𝟏− 𝒚 𝟐
𝟏− 𝒙 𝟐
.

7. CHAPTER – 6
APPLICATION OF DERIVATIVES
1. For the curve y = 5x – 2x3
, if x increase at the rate of 2 units/sec,thenhow fast isthe slope of curve changing
whenx = 3?
2. Water isdripping out from a conical funnel ofsemi vertical angle
𝝅
𝟒
at the uniformrate of 2cm2
/sec in the
surface area, through a tinyhole at the vertexof the bottom. Whenthe slant heightof the cone is 4 cm, find
the rate of decrease ofthe slant heightof water.
3. Determine for whichvaluesof x, the functiony = x4
-
𝟒𝒙 𝟑
𝟑
isincreasingand for which values,it isdecreasing.
4. Show that the function f(x) = 4x3
- 18x2
+ 27x – 7 has neithermaximanor minima.
5. Usingdifferentials,findthe approximate value of √ 𝟎. 𝟎𝟖𝟐 .
6. Findall the pointsof local maxima and local minima ofthe function f(x) = -
𝟑
𝟒
𝒙 𝟒 − 𝟖𝒙 𝟑 −
𝟒𝟓
𝟐
𝒙 𝟐 + 𝟏𝟎𝟓.
7. Findthe equationof all the tangentsto the curve y = cos (x + y), -2  x  2,that are parallel to the line x+ 2y
= 0.
8. Show that the equationof normal at any point on the curve x = 3cos - cos3
 , y = 3sin - sin3
 is 4
(y cos3
 -xsin3
 ) = 3 sin4 .
9. Findthe area of greatest rectangle that can be increasedin an ellipse
𝒙 𝟐
𝒂 𝟐
+
𝒚 𝟐
𝒃 𝟐
= 1.
10. Findthe difference betweenthe greatestandleast valuesof the functionf(x) = sin 2x – x, on [−
𝝅
𝟐
,
𝝅
𝟐
] .
11. An isoscelestriangle ofvertical angle 2 is inscribedina circle of radius a. show that the area of triangle is
maximum when =
𝝅
𝟔
.
12. If the area of a circle increasesat a uniformrate, then prove that perimetervariesinverselyas the radius.
13. Findthe approximate value of (1.999)5
.
14. The volume of a cube increasesat a constant rate. Prove that the increase in its surface area varies inversely
as the lengthof the side.
15. Findthe conditionthat the curves 2x = y2
and 2xy = k intersectorthogonally.
16. Findthe co-ordinatesof the point on the curve √ 𝒙 + √ 𝒚 = 4 at which tangent is equallyinclinedtothe axes.
17. At what pointson the curve x2
+ y2
- 2x – 4y + 1 0 0, the tangents are parallel to the y – axis?
18. Show that the line
𝒙
𝒂
+
𝒚
𝒃
= 𝟏 , touches the curve y = b. 𝒆
−𝒙
𝒂 at the point where the curve intersectsthe axis of
y.
19. Show that f(x) = tan-1
(sinx + cos x) is an increasingfunctionin (𝟎,
𝝅
𝟒
) .
20. If the sum ofthe lengthsof the hypotenuse anda side of a right angled triangle is given,showthat the area of
the triangle is maximumwhen the angle betweenthemis (
𝝅
𝟑
)

8. 21. A telephone companyin a town has 500 subscribers on itslist and collectsfixedcharges of Rs. 300/- per
subscriberper year. The company proposesto increase the annual subscriptionand it isbelivedthat for every
increase of Re. 1/- one subscriberwill discontinue the service.Finswhat increase will bring maximumprofit?
22. An openbox with square base isto be made of a givenquantity of card board of area c2
. show that the
maximum volume ofthe box is
𝒄 𝟑
𝟔√ 𝟑
cubic units.
23. If the sum ofthe surface areas of cube and a sphere is constant, what isthe ratio of an edge ofthe cube of the
diameterof the sphere,whenthe sum of their volumesisminimum?
24. The sum of the surface areas of a rectangular parallelepipedwithsidesx,2x and
𝒙
𝟑
and a sphere is givento be
constant. Prove that the sum of theirvolumesis minimum,ifx isequal to three timesthe radius of the
sphere.Alsofind the minimumvalue of the sum of their volumes.

9. CHAPTER – 7
INTEGRALS
1. Evaluate :- ∫√
𝟏+𝒙
𝟏−𝒙
dx , x  1.
2. Evaluate :- ∫ 𝒕𝒂𝒏 𝟖 𝒙 𝒔𝒆𝒄 𝟒 xdx.
3. Find ∫
𝒙 𝟑
𝒙 𝟒+ 𝟑𝒙 𝟐+ 𝟐
dx.
4. Find ∫
𝒅𝒙
𝟐𝒔𝒊𝒏 𝟐 𝒙+ 𝟓𝒄𝒐𝒔 𝟐 𝒙
5. Evaluate ∫ ( 𝟕𝒙 − 𝟓) 𝒅𝒙
𝟐
−𝟏 as a limitof sums.
6. Evaluate ∫
𝒕𝒂𝒏 𝟕 𝒙
𝒄𝒐𝒕 𝟕 𝒙+ 𝒕𝒂𝒏 𝟕 𝒙
𝝅
𝟐
𝟎 dx
7. Find∫
√ 𝟏𝟎−𝒙
√ 𝒙+ √ 𝟏𝟎−𝒙
𝟖
𝟐 dx
8. Find∫ √ 𝟏 + 𝒔𝒊𝒏 𝟐𝒙
𝝅
𝟒
𝟎 dx
9. Find∫ 𝒙 𝟐 𝐭𝐚𝐧−𝟏 𝒙 dx
10. Find∫√ 𝟏𝟎 − 𝟒𝒙 + 𝟒𝒙 𝟐 dx
11. Evaluate ∫
𝒙 𝟐 𝒅𝒙
𝒙 𝟒+ 𝒙 𝟐− 𝟐
12. Evaluate ∫
𝒙 𝟑+ 𝒙
𝒙 𝟒− 𝟗
dx
13. Show that ∫
𝒔𝒊𝒏𝟐 𝒙
𝒔𝒊𝒏𝒙+𝒄𝒐𝒔 𝒙
=
𝟏
√ 𝟐
𝒍𝒐𝒈(√ 𝟐
𝝅
𝟐
𝟎 + 𝟏)
14. Find∫ 𝒙 ( 𝐭𝐚𝐧−𝟏 𝒙)
𝟐𝟏
𝟎 dx
15. ∫ 𝒕𝒂𝒏 𝟐 𝒙 𝒔𝒆𝒄 𝟒 𝒙 𝒅𝒙
16. ∫√ 𝟏 + 𝒔𝒊𝒏𝒙 𝒅𝒙
17. ∫
𝒙
√ 𝒙+ 𝟏
𝒅𝒙
18. ∫
𝒙
𝒙 𝟒− 𝟏
dx
19. ∫
𝒙
𝟏
𝟐
𝟏+ 𝒙
𝟑
𝟒
20. ∫√ 𝟓 − 𝟐𝒙 + 𝒙 𝟐 dx
21. ∫
𝒙 𝟐
𝟏− 𝒙 𝟒
dx put x2
= t
22. ∫
(𝒄𝒐𝒔 𝟓𝒙+𝒄𝒐𝒔 𝟒𝒙)
𝟏−𝟐 𝒄𝒐𝒔 𝟑𝒙
dx
23. ∫
𝒔𝒊𝒏 𝟔 𝒙+ 𝒄𝒐𝒔 𝟔 𝒙
𝒔𝒊𝒏 𝟐 𝒙 𝒄𝒐𝒔 𝟐 𝒙
dx
24. ∫
√ 𝒙
√ 𝒂 𝟑− 𝒙 𝟑
dx
25. ∫
𝒄𝒐𝒔 𝒙−𝒄𝒐𝒔 𝟐𝒙
𝟏−𝒄𝒐𝒔 𝒙
dx

10. 26. ∫
𝒅𝒙
𝒆 𝒙+ 𝒆−𝒙
𝟏
𝟎
27. ∫
𝒙 𝒅𝒙
√ 𝟏+ 𝒙 𝟐
𝟏
𝟎
28. ∫
𝒙
𝟏+𝒔𝒊𝒏 𝒙
𝝅
𝟎
29. ∫
𝒙 𝟐
( 𝒙 𝟐+ 𝒂 𝟐)( 𝒙 𝟐+ 𝒃 𝟐)
30. ∫ 𝐬𝐢𝐧−𝟏 √
𝒙
𝒂+𝒙
dx
31. ∫√ 𝒕𝒂𝒏 𝒙 dx
32. ∫
𝒅𝒙
( 𝒂 𝟐 𝒄𝒐𝒔 𝟐 𝒙+ 𝒃 𝟐 𝒔𝒊𝒏 𝟐 𝒙) 𝟐
𝝅
𝟐
𝟎
33. ∫ 𝒍𝒐𝒈 (𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔 𝒙)
𝝅
𝟒
−
𝝅
𝟒
dx

11. CHAPTER – 8
APPLICATION OF INTEGRALS
1. Findthe area of the regionboundedby the parabola y2
= 2x and the straight line x – y = 4.
2. Findthe area enclosedby the curve x = 3 cost, y = 2 sint.
3. Findthe area of the regionincludedbetweenthe parabola y =
𝟑 𝒙 𝟐
𝟒
and the line 3x – 2y + 12 = 0.
4. Findthe area of the regionabove the x – axis, includedbetweenthe parabola y2
= ax and the circle x2
+ y2
=
2ax.
5. Findthe area of the regionboundedby the curves y2
= 9x, y = 3x.
6. Findthe area of the regionboundedby the parabola y2
= 2px, x2
= 2py.
7. Findthe area of the regionincludedbetweeny2
= 9x and y = x.
8. Sketch the region {(x,0) : y = √𝟒− 𝒙 𝟐 and x – axis. Findthe area of the regionusingintegration.
9. Usingintegration,findthe area of the regionboundedby the line 2y = 5x + 7, x – axis and the linesx = 2 and x
= 8.
10. Findthe area of the regionboundedby the curve y2
= 2x and x2
+ y2
= 4x.
11. Findthe area boundedby the curve y = sin x betweenx= 0 and x = 2.
12. Findthe area of region boundedby the triangle whose verticesare (-1, 1), (0, 5) and (3, 2), usingintegration.
13. Draw a rough sketchof the region{(x,y):y2
≤ 16a2
}. Also findthe area of the regionsketchedusingmethod
of integration.

12. CHAPTER – 9
DIFFERENTIAL EQUATIONS
1. Solve the differential equation
𝒅𝒚
𝒅𝒙
+
𝒚
𝒙
= x2
.
2. Findthe general solutionof the differential equation
𝒅𝒚
𝒅𝒙
=
𝒚
𝒙
.
3. Findthe equationof a curve whose tangent at any pointon it, differentfromorigin,has slope y +
𝒚
𝒙
.
4. Solve x2 𝒅𝒚
𝒅𝒙
- xy = 1 + cos (
𝒚
𝒙
),x  0 and x = 1, y =
𝝅
𝟐
.
5. State the type of the differential equationforthe equation.xdy – ydx = √ 𝐱𝟐 + 𝐲𝟐 dx and solve it.
6. Solve the differential equation(x2
–1)
𝒅𝒚
𝒅𝒙
+ 2xy =
𝟏
𝒙 𝟐− 𝟏
7. Solve the differential equation
𝒅𝒚
𝒅𝒙
+1 = ex + y
8. Solve : ydx – xdy = x2
ydx.
9. Solve the differential equation
𝒅𝒚
𝒅𝒙
=1 + x + y2
+ xy2
, wheny = 0, x = 0.
10. Findthe general solutionof (x + 2y3
)
𝒅𝒚
𝒅𝒙
= y.
11. Findthe equationof a curve passingthrough origin and satisfyingthe differential equation
(1 + x2
)
𝒅𝒚
𝒅𝒙
+ 2xy = 4x2
.
12. Solve : x2 𝒅𝒚
𝒅𝒙
= x2
+ xy + y2
.
13. Solve : 2(y + 3) – xy
𝒅𝒚
𝒅𝒙
= 0, giventhat y(1) = -2.
14. Solve :
𝒅𝒚
𝒅𝒙
= cos (x + y0 + sin (x + y).
15. Findthe general solutionof
𝒅𝒚
𝒅𝒙
- 3y = sin 2x.
16. Findthe equationof a curve passingthrough (2, 1) if the slope of the tangent to the curve at any point (x,y) is
𝒙 𝟐+ 𝒚 𝟐
𝟐𝒙𝒚

13. CHAPTER – 10
VECTOR ALGEBRA
1. Findthe unit vector in the directionof the sum of the vectors 𝒂⃗⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝟐𝒌̂ and 𝒃⃗⃗ = −𝒊̂ + 𝒋̂ + 𝟑𝒌̂
2. Findthe positionvector of a point R whichdividesthe line joiningthe two points P and Q with position
vectors 𝑶𝑷⃗⃗⃗⃗⃗⃗ = 𝟐 𝒂⃗⃗ + 𝒃⃗⃗ 𝒂𝒏𝒅 𝑶𝑸⃗⃗⃗⃗⃗⃗ = 𝒂⃗⃗ − 𝟐𝒃⃗⃗ , respectively,inthe ratio 1 : 2, (i) internally and (ii) externally
3. If 𝒂⃗⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝒌̂ , 𝒃⃗⃗ = 𝒊̂ + 𝒋̂ − 𝟐𝒌̂ and 𝒄⃗ = 𝒊̂ + 𝟑𝒋̂ − 𝒌̂ , find such that isperpendicularto  𝒃⃗⃗ + 𝒄⃗
4. If the points(-1, -1, 2), (2, m, 5) and (3, 11, 6) are collinear,findthe value of m.
5. Findthe vector 𝒓⃗ of magnitude 3√ 𝟐 units which makes an angle of
𝝅
𝟒
and
𝝅
𝟐
with y and z – axes,
respectively.
6. If 𝒂⃗⃗ = 𝒊̂ + 𝒋̂ + 𝟐𝒌̂ and 𝒃⃗⃗ = 𝟐𝒊̂ + 𝒋̂ − 𝟐𝒌̂ , findthe unit vector inthe directionof (i) 6𝒃⃗⃗ (ii) 2𝒂⃗⃗ − 𝒃⃗⃗
7. Usingvectors, findthe value of k such that the points(k, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.
8. Finda vector of magnitude 6, which is perpendicularto both the vectors 2𝒊̂ - 𝒋̂ + 2𝒌̂ and 4𝒊̂ - 𝒋̂ + 3𝒌̂ .
9. Findthe angle betweenthe vectors 2𝒊̂ - 𝒋̂ + 𝒌̂ and 3𝒊̂ + 4𝒋̂ - 𝒌̂ .
10. If 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 0, showthat 𝒂⃗⃗  𝒃⃗⃗ = 𝒃⃗⃗  𝒄⃗ = 𝒄⃗  𝒂⃗⃗ . Interpretthe resultgeometrically?
11. Findthe sine of the angle betweenthe vectors 𝒂⃗⃗ = 𝟑𝒊̂ + 𝒋̂ + 𝟐𝒌̂ and 𝒃⃗⃗ = 𝟐𝒊̂ − 𝟐𝒋̂ + 𝟒𝒌̂
12. If A, B, C, D are the points with positionvectors 𝒊̂ + 𝒋̂ − 𝒌̂, 𝟐𝒊̂ − 𝒋̂ + 𝟑𝒌̂ , 𝟐𝒊̂ − 𝟑𝒌̂, 𝟑𝒊̂ − 𝟐𝒋̂ + 𝒌̂ , respectively,
findthe projectionof 𝑨𝑩⃗⃗⃗⃗⃗⃗ along 𝑪𝑫⃗⃗⃗⃗⃗⃗ .
13. Usingvectors, findthe area of the triangle ABC with verticesA(1, 2, 3), B(2, -1, 4) and C(4,5, -1).
14. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ determine the verticesof a triangle,show that ½ [𝒃⃗⃗  𝒄⃗ = 𝒄⃗  𝒂⃗⃗ = 𝒂⃗⃗  𝒃⃗⃗ ] givesthe vector area of
the triangle.Hence deduce the conditionthat the three points are collinear.Alsofind the unit vector
normal to the plane of the triangle.
15. If 𝒂⃗⃗ = 𝒊̂ + 𝒋̂ + 𝒌̂ and 𝒃⃗⃗ = 𝒋̂ − 𝒌̂ , finda vector 𝒄⃗ such that 𝒂⃗⃗  𝒄⃗ = 𝒃⃗⃗ and 𝒂⃗⃗ . 𝒄⃗ = 3

14. CHAPTER – 11
THREE DIMENSIONAL GEOMETRY
1. The angle betweenthe line 𝒓⃗ = (5𝒊̂ − 𝒋̂ − 𝟒𝒌̂) + (𝟐𝒊̂ − 𝒋̂ + 𝒌̂ ) and the plane 𝒓⃗ .(3𝒊̂ − 𝟒𝒋̂ − 𝒌̂) +5 = 0 is
𝐬𝐢𝐧−𝟏 (
𝟓
𝟐√ 𝟗𝟏
) .
2. The line 𝒓⃗ = 2𝒊̂ − 𝟑𝒋̂ − 𝒌̂ + (𝒊̂ − 𝒋̂ + 𝟐𝒌̂ ) liesinthe plane 𝒓⃗ .(3𝒊̂ + 𝒋̂ − 𝒌̂) − 𝟐 = 0.
3. The vector equationof the line is
𝒙−𝟓
𝟑
=
𝒚+𝟒
𝟕
=
𝒛−𝟔
𝟐
is 𝒓⃗ = 5𝒊̂ − 𝟒𝒋̂+ 𝟔𝒌̂ + (𝟑𝒊̂+ 𝟕𝒋̂+ 𝟐𝒌̂ ).
4. Findthe equationof the plane through the points(2, 1, -1) and (-1, 3, 4), and perpendiculartothe plane x – 2y
+ 4z = 10.
5. Findthe equationof the plane through the intersectionofthe planes 𝒓⃗ .( 𝒊̂ + 𝒋̂) − 𝟔 = 0 and 𝒓⃗ .(3𝒊̂ − 𝒋̂ − 𝟒𝒌̂)=0
, whose perpendiculardistance from origin isunity.
6. Findthe foot of perpendicularfromthe point (2, 3, -8) to the line
𝟒−𝒙
𝟐
=
𝒚
𝟔
=
𝟏−𝒛
𝟑
. Also,find the
perpendiculardistance from the givenpointto the line.
7. Findthe distance of a point (2, 4, -1) from the line
𝒙+𝟓
𝟏
=
𝒚+𝟑
𝟒
=
𝒛−𝟔
𝟗
8. Findthe lengthand the foot of perpendicularfromthe point (1,
𝟑
𝟐
, 2) to the plane 2x – 2y + 4z + 5 = 0.
9. Findthe equationsof the line passingthrough the point(3, 0, 1) and parallel to the plane x + 2y = 0 and 3y – z
= 0
10. Show that the lines
𝒙−𝟏
𝟐
=
𝒚−𝟐
𝟑
=
𝒛−𝟑
𝟒
and
𝒙−𝟒
𝟓
=
𝒚−𝟏
𝟐
= z intersect.Also,findtheir pointof intersection.
11. Findthe angle betweenthe lines 𝒓⃗ = 3𝒊̂ − 𝟐𝒋̂ + 𝟔𝒌̂ + (𝟐𝒊̂ + 𝒋̂ + 𝟐𝒌̂) and 𝒓⃗ =(𝟐𝒊̂ − 𝟓𝒌̂) + (𝟔𝒊̂ + 𝟑𝒋̂ + 𝟐𝒌̂).
12. Findthe equationof a plane which isat a distance 3√ 𝟑 units from originand the normal to whichis equally
inclinedto coordinate axis.
13. Findthe equationof the plane through the points(2, 1, 0), (3, -2, -2) and (3, 1, 7).
14. The vector equationof the line passing through the points(3, 5, 4) and (5, 8, 11) is
𝒓⃗ = 3𝒊̂ + 𝟓𝒋̂+ 𝟒𝒌̂ + (𝟐𝒊̂+ 𝟑𝒋̂+ 𝟕𝒌̂ ).
15. Findthe image ofthe point havingpositionvector 𝒊̂ + 𝟑𝒋̂ + 𝟒𝒌̂ in the plane 𝒓⃗ .( 𝟐𝒊̂ − 𝒋̂ + 𝒌̂ ) + 3 = 0
16. Findthe image ofthe point (1, 6, 3) inthe line
𝒙
𝟏
=
𝒚−𝟏
𝟐
=
𝒛−𝟐
𝟑
17. Findthe co-ordinatesof the foot of perpendiculardrawn from the pointA(1, 8, 4) to the joiningthe pointsB(0,
-1, 3) and C(2, -3, -1).
18. Findthe directioncosinesof the line passing through the pointsP(2, 3, 5) and Q(-1, 2, 4)
19. If a line makes an angle of 30o
, 60o
, 90o
with the positive directionof x, y, z- axes, respectively,thenfindits
directioncosines.
20. The x – coordinate of a point on the line joiningthe points Q(2, 2, 1) and R(5, 1, -2) is 4. Find itsz – coordinate.
21. Findthe distance of the point (-2,4, -5) from the line
𝒙+𝟑
𝟑
=
𝒚−𝟒
𝟓
=
𝒛+𝟖
𝟔

15. 22. Findthe coordinatesof the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane passing
through three points (2, 2, 1), (3, 0, 1) and (4, -1, 0).
23. Findthe distance of the point (-1,-5, -10) from the point of intersectionofthe line
𝒓⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝟐𝒌̂ + ( 3𝒊̂ + 𝟒𝒋̂ + 𝟐𝒌̂ ) and the plane 𝒓⃗ .( 𝒊̂ − 𝒋̂ + 𝒌̂ ) = 5
24. A plane meetsthe co-ordinatesaxis in A, B, C such that the centroid of the ABCis the point(,,).Show
that the equationof the plane is
𝒙
𝜶
+
𝒚
𝜷
+
𝒛
𝜸
= 𝟑

16. CHAPTER – 12
LINEAR PROGRAMMING
1. Solve the followingLPPgraphically: Maximise Z = 2x + 3y, subjectto x + y  4, x  0, y  0.
2. A manufacturing company makes two types of televisionsets;one isblack and white and the otheris colour.
The company has resourcesto make at most 300 sets a week.It takes Rs. 1800 to make a black and white set
and Rs. 2700 to make a colouredset.The company can spendnot more than Rs. 648000 a weekto make
televisionsets.Ifit makes a profit of Rs. 510 perblack and white set and rs. 675 per colouredset, how many
setsof each type shouldbe producedso that the company has maximum profit?Formulate this problemas a
LPP giventhat the objective isto maximize the profit.
3. Referto Example 4. Solve the LPP.
4. Minimise Z= 3x + 5y subjectto the constraints:
x + 2y  10
x + y  6
3x + y  8
x, y  0
5. Determine the maximum value of Z = 11x + 7y subjectto the constraints: 2x + y  6, x  2, x  0, y  0.
6. Maximise Z = 3x + 4y, subjectto the constraints: x + y  7, 2x – 3y + 6  0, x  0, y  0.
7. A firm has to transport 1200 packages using large vans whichcan carry 200 packages each and small vans
which can take 80 packages each. The cost for engagingeach large van is Rs. 400 and each small van is Rs. 200.
not more than Rs. 3000 is to be spent on the joband the number of large vans can not exceedthe number of
small vans. Formulate this problemas a LPP giventhat the objective isto minimize cost.
8. A company manufactures two types of screwsA and B. All the screws have to pass through a threading
machine and a slotting machine.A box of Type A screwsrequires2 minuteson the threadingmachine and 3
minuteson the slottingmachine.A box of type B screws requires8 minutesof threading on the threading
machine and 2 minuteson the slottingmachine. In a week,each machine isavailable for 60 hours.
9. A company manufactures two types of sweaters: type A and type B. It costs Rs. 360 to make a type A sweater
and Rs. 120 to make a type B sweater. The company can make at most 300 sweatersand spendat most Rs.
72000 a day. The number of sweatersof type B cannot exceedthe number ofsweaters of type A by more than
100. The company makesa profit ofRs. 200 for each sweater of type A and Rs. 120 for everysweater oftype
B. Formulate this problemas a LPP to maximize the profitto the company.
10. A man rideshismotorcycle at the speedof 50 km/hour. He has to spendrs. 2 per km on petrol. If he ridesit at
a fasterspeedof 80 km/hour, the petrol cost increasesto Rs. 3 per km. He has atmost Rs. 120 to spendon
petrol and one hour’s time.He wishesto find the maximum distance that he can travel. Expressthis problem
as a linearprogramming problem.

17. CHAPTER – 13
PROBABILITY
1. The probabilityof simultaneousoccurrence of at least one of two eventsA and B is p. If the probabilitythat
exactlyone of A, B occurs is q, thenprove that P(A) + P(B) =2 – 2p + q.
2. 10% of the bulbs produced in a factory are of red colour and 2% are red and defective.Ifone bulbis pickedup
at random, determine the probabilityof itsbeing defective ifit is red.
3. Two dice are thrown together.Let A be the event‘getting6 on the first die’and B be the event‘ getting on
the seconddie’.Are the eventsA and B independent?
4. Three machines E1 , E2 , E3 in a certain factory produce 50%, 25% and 25%, respectively,ofthe total daily
output of electrictubes.It is known that 4% of the tubesproducedone each of machinesE1 and E2 are
defective,andthat 5% of those produced on E3 are defective.Ifone tube is pickedup at random from a day’s
production,calculate the probabilitythat it is defective.
5. Findthe probabilitythat in 10 throws of a fair die a score which isa multiple of3 will be obtainedin at least 8
of the throws.
6. A discrete random variable X has the followingprobabilitydistribution:
X 1 2 3 4 5 6 7
P(X) C 2C 2C 3C C2
2C2
7C2
+ C
Findthe value of C. also find the mean of the distribution.
7. Determine variance and standard deviationof the numberof heads inthree tossesof a coin.
8. Four balls are to be drawn without replacementfroma box containing red and white balls.If X denotesthe
numberof red ball drawn, findthe probabilitydistributionof X.
9. If A and B are independenteventssuchthat P(A) = p, P(B) = 2p and P( Exactly one of A, B) =
𝟓
𝟗
, thenp =
………………….
10. A and B are two eventssuch that P(A) = ½, P(B) =
𝟏
𝟑
and P(A∩ B) =
𝟏
𝟒
. find
(i) P(A I B) (ii) P(B I A) (iii) P(A IB) (iv) P(A I B)
11. Three eventsA, B and C have probabilities
𝟐
𝟓
,
𝟏
𝟑
,
𝟏
𝟐
, respectively.GiventhatP(A C) =
𝟏
𝟓
and P(B C) =
𝟏
𝟒
, find
the valuesof P(CIB) and P(A  C).
12. A discrete random variable X has the probabilitydistribution given as below:
X 0.5 1 1.5 2
P(X) K k2
2k2
k
(i) Findthe value of k.
(ii) Determine the mean of the distribution.

18. 13. Bag I contains 3 black and 2 white balls,Bag II contains 2 black and 4 white balls. A bag and a ball is selectedat
random. Determine the probability ofselectinga black ball.
14. A box has 5 blue and 4 red balls.One ball isdrawn at random and not replaced.Its colour is also not noted.
Then another ball is drawn at random. What is the probabilityof secondball beingblue?
15. A die is thrown 5 times.Find the probabilitythat an odd number will come up exactlythree times.
16. The probabilityof a man hittinga target is 0.25. he shoots 7 times.What is the probabilityof hishitting at
leasttwice?
17. The probabilitydistributionof a random variable X is givenbelow:
X 0 1 2 3
P(X) K 𝒌
𝟐
𝒌
𝟒
𝒌
𝟖
(i) Determine the value of k.
(ii) Determine P(X 2) and P(X > 2)
(iii) FindP(X  2) + P(X> 2)
18. A die is thrown three times.Let X be ‘the number oftwos seen’.Findthe expectationof X.
19. A factory producesbulbs.The probabilitythat any one bulbis defective is
𝟏
𝟓𝟎
and they are packed in boxesof
10. From a single box,findthe probabilitythat
a. None of the bulbsis defective
b. Exactly two bulbs are defective
c. More than 8 bulbswork properly
20. The random variable X can be take only the values0, 1, 2. Giventhat P(X= 0) = P(X = 1) = p and that E(X2
) =
E[X], findthe value of p.
21. Findthe variance of the distribution:
X 0 1 2 3 4 5
P(X) 𝟏
𝟔
𝟓
𝟏𝟖
𝟐
𝟗
𝟏
𝟔
𝟏
𝟗
𝟏
𝟏𝟖
22. A and B throw a pair of dice alternately.A wins the game ifhe getsa total of 6 and B winsif she gets a total of
7. It A starts the game,findthe probabilityof winningthe game by A in third throw of the pair of dice.
23. A letteris known to have come eitherfrom TATA NAGAR or from CALCUTTA. On the envelope,justtwo
consecutive letterTA are visible. Whatis the probabilitythat the lettercame from TATA NAGAR.
24. There are three urns containing2 white and 3 black balls,3 white and 2 black balls, and 4 white and 1 black
balls,respectively.There isan equal probabilityof each urn beingchosen.A ball is drawn at random from the
chosenurn and it is foundto be white.Find the probabilitythat the ball drawn was from the secondurn.
25. By examiningthe chestX ray, the probabilitythat TB is detectedwhena personis actually sufferingis0.99.
the probabilityof an healthyperson diagnosedto have TB is0.001. in a certaincity, 1 in 1000 people suffers

19. from TB. A personis selectedat random and is diagnosedto have TB. What is the probabilitythat he actually
has TB?
26. The probabilitydistributionof a random variable x is givenas under:
P(X = x) = {
𝒌𝒙 𝟐 𝒇𝒐𝒓 𝒙 = 𝟏, 𝟐, 𝟑
𝟐𝒌𝒙 𝒇𝒐𝒓 𝒙 = 𝟒, 𝟓, 𝟔
𝟎 𝒐𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
Where k is a constant. Calculate
(i) E(X) (ii) E(3X2
) (iii) P(X 4)