CG Board Class 12th Question Bank 2024 Maths

NÙkhlx<+ ek/;fed f'k{kk e.My jk;iqj
}kjk fufeZr ç'u cSad
2023-24

कक्षा 12
गणित (MATHEMATICS)
1

अनक्र
ु मणिका
क्र. अध्याय पृ. क्र.
1 संबध
ं एवं फलन - Relations and Functions 3

2 प्रततलोम तिकोिममतीय फलन – 6

Inverse Trigonometric Functions
3 आव्यूह - Matrix 10

4 सारणिक - Determinant 18

5 सांतत्य तथा अवकलनीयता – 21

Continuity and Differentiability
6 अवकलज के अनुप्रयोग - Applications of Derivatives 30

7 समाकलन - Integration 34

8 समाकलनों के अनुप्रयोग - Applications of Integrals 40

9 अवकल समीकरि - Differential Equations 42

10 सदिश बीजगणित - Vector Algebra 45

11 ति-तवमीय ज्याममती - Three-Dimensional Geometry 48

12 रैखिक प्रोग्रामन - Linear Programming 51

13 प्राययकता - Probability 53

2

(Relation and Function)
(Short Answers Questions : Marks 4)

R N

R={(𝒙, 𝒚): 𝒙 ∈ 𝑵, 𝒚 ∈ 𝑵, 𝟐𝒙 + 𝒚 = 𝟒}

If R is a relation based on the set N of natural numbers such that

R={(𝒙, 𝒚): 𝒙 ∈ 𝑵, 𝒚 ∈ 𝑵, 𝟐𝒙 + 𝒚 = 𝟒}

Then Find its domain and range and prove that it is reflexive, symmetric
and transitive.

A={1,2,3,4,5} R= {(𝒂, 𝒃): |𝒂 − 𝒃|, } |𝒂 − 𝒃|,
R

Suppose A={1,2,3,4,5} and R= {(𝒂, 𝒃): |𝒂 − 𝒃|, } where |𝒂 − 𝒃|, which is
divided by, 2 prove that R is an equivalence relation.

xy- L
L R={(𝑳𝟏 , 𝑳𝟐 ): 𝑳𝟏 ∥ 𝑳𝟐 } R

Let L be the set of all lines in XY plane and R be the relation is in L defined
as R={(𝑳𝟏 , 𝑳𝟐 ): 𝑳𝟏 ∥ 𝑳𝟐 }, show that R is an equivalence relation.

3

𝒏+𝟏
, 𝒏 𝒊𝒔 𝒐𝒅𝒅
n∈ 𝑵 𝒇(𝒙) = { 𝟐𝒏
, 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏
𝟐

f:N→N f

Let 𝒇: 𝑵 → 𝑵 be defined by-

𝒏+𝟏
, 𝒏 𝒊𝒔 𝒐𝒅𝒅
𝒇(𝒙) = {𝒏 𝟐
, 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏
𝟐
State whether the function is bijective justify your answer.

gof fog

Find gof and fog if

(i) f(x)=|𝒙|; g(x) =|𝟓𝒙 − 𝟐|
𝟏
(ii) f(x)= 8x3; g(x)=𝒙𝟑
(iii) f(x)=cosx; g(x)=3x2
𝒙 𝒙
(iv) f(x)= ∀𝒙 ∈ 𝑹; g(x)= ∀𝒙 ∈ 𝑹
𝟏+|𝒙| 𝟏−|𝒙|

f(x)=4x+3 f: R→R
f f

Consider the function f: R→R given by f(x)=4x+3. Prove that f is invertible;
also Find the inverse function of f.

a∗b=a3+b3 N ∗

4

Consider a binary operation * on N defined as a∗b=a3+b3. Check the
commutability and associativity of the operation.

f:X→Y
f-1 f (𝒇−𝟏 )−𝟏 = 𝒇

Let f:X→Y be an investible function then show that the inverse of 𝒇−𝟏 is f

i.e. (𝒇−𝟏 )−𝟏 = 𝒇

𝒇(𝒙) = 𝒙𝟐 + 𝟒 𝒇: 𝑹+ → [𝟒, ∞)
f f 𝒇−𝟏 , 𝒇−𝟏 =

√𝒚 − 𝟒 𝑹+

Consider 𝒇: 𝑹+ → [𝟒, ∞) given by (𝒙) = 𝒙𝟐 + 𝟒 . show that f is
invertible with the inverse 𝒇−𝟏 of 𝒇 given by 𝒇−𝟏 (𝒚) = √𝒚 − 𝟒
where 𝑹+ is the set of all non-negative real numbers.
𝟒𝒙+𝟑 𝟐 𝟐
𝒇(𝒙) = ,𝒙 ≠ 𝒙≠
𝟔𝒙−𝟒 𝟑 𝟑

𝒇𝒐𝒇(𝒙) = 𝒙 f ?
𝟒𝒙+𝟑 𝟐 𝟐
if 𝒇(𝒙) = , 𝒙 ≠ , show that 𝒇𝒐𝒇(𝒙) = 𝒙 for all 𝒙 ≠ , what is
𝟔𝒙−𝟒 𝟑 𝟑

the inverse of f ?

5

(Inverse Trigonometric Function)
(Very Short Answers Questions : Marks 2)

tan2(sec-12)+cot2(cosec-13)

Find the value of tan2(sec-12)+cot2(cosec-13)

𝟏
𝐬𝐢𝐧 (𝐬𝐢𝐧−𝟏 + 𝐜𝐨𝐬 −𝟏 𝒙) = 𝟏 𝒙
𝟓

𝟏
If 𝐬𝐢𝐧 (𝐬𝐢𝐧−𝟏 + 𝐜𝐨𝐬 −𝟏 𝒙) = 𝟏then Find the value of 𝒙.
𝟓

𝟏 𝟏 𝟏
𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 + 𝐭𝐚𝐧−𝟏 𝒛 = 𝝅 + +
𝒙𝒚 𝒚𝒛 𝒛𝒙

If 𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 + 𝐭𝐚𝐧−𝟏 𝒛 = 𝝅 then Find the value of

𝟏 𝟏 𝟏
+ + .
𝒙𝒚 𝒚𝒛 𝒛𝒙

−𝟏
𝐜𝐨𝐭 −𝟏 ( )
√𝟑

−𝟏
Find the principal value of 𝐜𝐨𝐭 −𝟏 ( ).
√𝟑

cot(𝐬𝐢𝐧−𝟏 𝒙)

Find the value of cot(𝐬𝐢𝐧−𝟏 𝒙)

𝟏 𝟏
𝐜𝐨𝐬 −𝟏 + 𝟐 𝐬𝐢𝐧−𝟏
𝟐 𝟐

𝟏 𝟏
Find the value of 𝐜𝐨𝐬 −𝟏 + 𝟐 𝐬𝐢𝐧−𝟏
𝟐 𝟐

6

√𝟏−𝒄𝒐𝒔𝒙
𝐭𝐚𝐧−𝟏 ( ) ; (𝟎 < 𝒙 < 𝝅)
√𝟏+𝒄𝒐𝒔𝒙

Write the following function in the simplest form

√𝟏 − 𝒄𝒐𝒔𝒙
𝐭𝐚𝐧−𝟏 ( ) ;𝟎 < 𝒙 < 𝝅
√𝟏 + 𝒄𝒐𝒔𝒙

𝟑𝝅
𝐭𝐚𝐧−𝟏 (𝒕𝒂𝒏 )
𝟒

𝟑𝝅
Find the value of 𝐭𝐚𝐧−𝟏 (𝒕𝒂𝒏 )
𝟒

−𝟏 𝟏
𝟑 𝐬𝐢𝐧−𝟏 𝒙 = 𝐬𝐢𝐧−𝟏 (𝟑𝒙 − 𝟒𝒙𝟑 ) , 𝒙 ∈ [ , ]
𝟐 𝟐

−𝟏 𝟏
Prove that 𝟑 𝐬𝐢𝐧−𝟏 𝒙 = 𝐬𝐢𝐧−𝟏 (𝟑𝒙 − 𝟒𝒙𝟑 ) , 𝒙 ∈ [ , ]
𝟐 𝟐

𝟏
𝐭𝐚𝐧−𝟏 [𝟐𝒄𝒐𝒔 (𝟐 𝐬𝐢𝐧−𝟏 )]
𝟐

𝟏
Find the value of 𝐭𝐚𝐧−𝟏 [𝟐𝒄𝒐𝒔 (𝟐 𝐬𝐢𝐧−𝟏 )]
𝟐

𝝅
𝐬𝐢𝐧−𝟏 𝒙 + 𝐜𝐨𝐬 −𝟏 𝒙 = , 𝒙 ∈ [−𝟏, 𝟏]
𝟐

𝝅
Prove that 𝐬𝐢𝐧−𝟏 𝒙 + 𝐜𝐨𝐬 −𝟏 𝒙 = , 𝒙 ∈ [−𝟏, 𝟏]
𝟐

𝒙+𝒚
𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 = 𝐭𝐚𝐧−𝟏 , 𝒙𝒚 < 𝟏
𝟏−𝒙𝒚

𝒙+𝒚
Prove that 𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 = 𝐭𝐚𝐧−𝟏 , 𝒙𝒚 < 𝟏
𝟏−𝒙𝒚

𝟐 𝟕 𝟏
𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 = 𝐭𝐚𝐧−𝟏
𝟏𝟏 𝟐𝟒 𝟐

𝟐 𝟕 𝟏
Prove that 𝐭𝐚𝐧−𝟏 𝒙 + 𝐭𝐚𝐧−𝟏 𝒚 = 𝐭𝐚𝐧−𝟏
𝟏𝟏 𝟐𝟒 𝟐

7

(Short Answers Questions : Marks 4)

𝟐𝒂 𝟏−𝒃𝟐 𝟐𝒙
𝐬𝐢𝐧−𝟏 𝟐 − 𝐜𝐨𝐬
−𝟏
𝟐 = 𝐭𝐚𝐧
−𝟏
𝟏+𝒂 𝟏+𝒃 𝟏−𝒙𝟐

𝒂−𝒃
𝒙=
𝟏 + 𝒂𝒃
𝟐𝒂 𝟏−𝒃𝟐 𝟐𝒙
If 𝐬𝐢𝐧−𝟏 𝟐 − 𝐜𝐨𝐬
−𝟏
= 𝐭𝐚𝐧−𝟏 Then prove that
𝟏+𝒂 𝟏+𝒃𝟐 𝟏−𝒙𝟐

𝒂−𝒃
𝒙=
𝟏 + 𝒂𝒃
√𝟏+𝒙−√𝟏−𝒙 𝝅 𝟏 −𝟏
𝐭𝐚𝐧−𝟏 = − 𝐜𝐨𝐬−𝟏 𝒙 , ≤𝒙≤𝟏
√𝟏+𝒙+√𝟏−𝒙 𝟒 𝟐 √𝟐

√𝟏+𝒙−√𝟏−𝒙 𝝅 𝟏
Prove that 𝐭𝐚𝐧−𝟏 = − 𝐜𝐨𝐬 −𝟏 𝒙
√𝟏+𝒙+√𝟏−𝒙 𝟒 𝟐

𝟏 𝟏 𝟏 𝟏 𝝅
𝐭𝐚𝐧−𝟏 + 𝐭𝐚𝐧−𝟏 + 𝐭𝐚𝐧−𝟏 + 𝐭𝐚𝐧−𝟏 =
𝟓 𝟕 𝟑 𝟖 𝟒

𝟏 𝟏 𝟏 𝟏 𝝅
Prove that 𝐭𝐚𝐧−𝟏 + 𝐭𝐚𝐧−𝟏 + 𝐭𝐚𝐧−𝟏 + 𝐭𝐚𝐧−𝟏 =
𝟓 𝟕 𝟑 𝟖 𝟒

𝟒 𝟏𝟐 𝟑𝟑
𝐜𝐨𝐬 −𝟏 + 𝐜𝐨𝐬 −𝟏 = 𝐜𝐨𝐬 −𝟏
𝟓 𝟏𝟑 𝟔𝟓

𝟒 𝟏𝟐 𝟑𝟑
Prove that 𝐜𝐨𝐬 −𝟏 + 𝐜𝐨𝐬 −𝟏 = 𝐜𝐨𝐬 −𝟏
𝟓 𝟏𝟑 𝟔𝟓

𝟏 𝟏 𝝅
𝐭𝐚𝐧−𝟏 ( ) + 𝐭𝐚𝐧−𝟏 ( ) = k
𝟐 𝒌 𝟒

𝟏 𝟏 𝝅
If 𝐭𝐚𝐧−𝟏 ( ) + 𝐭𝐚𝐧−𝟏 ( ) = then Find the value of K.
𝟐 𝒌 𝟒

𝝅
𝐬𝐢𝐧−𝟏 (𝟏 − 𝒙) − 𝟐 𝐬𝐢𝐧−𝟏 𝒙 = x
𝟐

𝝅
If 𝐬𝐢𝐧−𝟏 (𝟏 − 𝒙) − 𝟐 𝐬𝐢𝐧−𝟏 𝒙 = then Find the value of X.
𝟐

8

𝒙
𝐭𝐚𝐧−𝟏 , |𝒙| < 𝒂
√𝒂𝟐 −𝒙𝟐

𝒙
Write 𝐭𝐚𝐧−𝟏 , |𝒙| < 𝒂 in its simplest form.
√𝒂𝟐 −𝒙𝟐

√𝟏+𝒔𝒊𝒏𝒙+√𝟏−𝒔𝒊𝒏𝒙 𝝅 𝝅
𝐜𝐨𝐭 −𝟏 [ ] = , 𝒙 ∈ (𝟎, )
√𝟏+𝒔𝒊𝒏𝒙−√𝟏−𝒔𝒊𝒏𝒙 𝟐 𝟒

√𝟏+𝒔𝒊𝒏𝒙+√𝟏−𝒔𝒊𝒏𝒙 𝝅 𝝅
Prove that 𝐜𝐨𝐭 −𝟏 [ ] = , 𝒙 ∈ (𝟎, )
√𝟏+𝒔𝒊𝒏𝒙−√𝟏−𝒔𝒊𝒏𝒙 𝟐 𝟒

𝒄𝒐𝒔𝒙 −𝟑𝝅 𝝅
𝐭𝐚𝐧−𝟏 ( ), 𝟐 < 𝒙 < 𝟐
𝟏−𝒔𝒊𝒏𝒙

Write the following in the simplest form –

𝒄𝒐𝒔𝒙 −𝟑𝝅 𝝅
𝐭𝐚𝐧−𝟏 ( ), 𝟐 < 𝒙 < 𝟐
𝟏−𝒔𝒊𝒏𝒙

𝟑 −𝟖 𝟖𝟒
𝐬𝐢𝐧−𝟏 − 𝐬𝐢𝐧−𝟏 = 𝐜𝐨𝐬 −𝟏
𝟓 𝟏𝟕 𝟖𝟓

𝟑 −𝟖 𝟖𝟒
show that: 𝐬𝐢𝐧−𝟏 − 𝐬𝐢𝐧−𝟏 = 𝐜𝐨𝐬 −𝟏
𝟓 𝟏𝟕 𝟖𝟓

9

(Unit- 2)

(Matrices)
(Very Short Answers Questions : Marks 1)

3x4
𝒊
(i) aij=2i-j (ii) aij= (iii) aij=|−𝟐𝒊 + 𝟑𝒋|
𝒋

Construct a 3x4 matrix whose elements are obtained in the following
𝒊
manner: (i) aij=2i-j (ii) aij= (iii) aij=|−𝟐𝒊 + 𝟑𝒋|
𝒋

x,y z

𝒙+𝒚 𝟐 𝟔 𝟐
(i)[ ]=[ ]
𝟓+𝒛 𝒙𝒚 𝟓 𝟖

𝒙+𝒚+𝒛 𝟗
(ii)[ 𝒙 + 𝒛 ] = [𝟓]
𝒚+𝒛 𝟕

Find the values of 𝒙, 𝒚 and 𝒛 in the following equation-

𝒙+𝒚 𝟐 𝟔 𝟐
(i)[ ]=[ ]
𝟓+𝒛 𝒙𝒚 𝟓 𝟖

𝒙+𝒚+𝒛 𝟗
(ii) [ 𝒙 + 𝒛 ] = [𝟓]
𝒚+𝒛 𝟕

10

𝟐 𝟒 𝟏 𝟑 −𝟐 𝟓
A =[ ] ,B=[ ],𝑪 = [ ]
𝟑 𝟐 −𝟐 𝟓 𝟑 𝟒

(i) A-B (ii) A+B (iii) AB (iv) BA

𝟐 𝟒 𝟏 𝟑 −𝟐 𝟓
If A =[ ] ,B=[ ],𝑪 = [ ] then Find the value of the
𝟑 𝟐 −𝟐 𝟓 𝟑 𝟒

following (i) A-B (ii) A+B (iii) AB (iv) BA

𝟒 𝟎
𝟎 −𝟏 𝟐
A=[ ] B=[𝟏 𝟑]
𝟒 𝟑 −𝟒
𝟐 𝟔

(i) (𝑨′ )′ = 𝑨 (ii) (𝑨𝑩)′ = 𝑩′ 𝑨′ (iii) (𝑲𝑨)′ =k𝑨′

𝟒 𝟎
𝟎 −𝟏 𝟐
If A=[ ] and B=[𝟏 𝟑]show that
𝟒 𝟑 −𝟒
𝟐 𝟔

(i) (𝑨′ )′ = 𝑨 (ii) (𝑨𝑩)′ = 𝑩′ 𝑨′ (iii) (𝑲𝑨)′ =k𝑨′

𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽
A=[ ]
−𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝜽

𝒄𝒐𝒔𝟐𝜽 𝒔𝒊𝒏𝟐𝜽
A2=[ ]
−𝒔𝒊𝒏𝟐𝜽 𝒄𝒐𝒔𝟐𝜽

𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝟐𝜽 𝒔𝒊𝒏𝟐𝜽
If A=[ ] is, show that A2=[ ]
−𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝜽 −𝒔𝒊𝒏𝟐𝜽 𝒄𝒐𝒔𝟐𝜽

𝟎 −𝒙
A=[ ], x2=-1 A2
𝒙 𝟎

𝟎 −𝒙
If A=[ ], and x2=-1 then Find the value of A2.
𝒙 𝟎

11

𝟑 𝟒
−𝟏 𝟐 𝟏
A’=[−𝟏 𝟐] B=[ ]
𝟏 𝟐 𝟑
𝟎 𝟏

(i) (A+B)’=A’+B’ (ii) (A- B)’=A’-B’

𝟑 𝟒
−𝟏 𝟐 𝟏
If A’=[−𝟏 𝟐] and B=[ ] are, verify that
𝟏 𝟐 𝟑
𝟎 𝟏

(i) (A+B)’=A’+B’ (ii) (A- B)’=A’-B’

𝟏 −𝟏 𝟓
(i) A= [−𝟏 𝟐 𝟏]
𝟓 𝟏 𝟑

𝟎 𝟏 −𝟏
(ii) A=[−𝟏 𝟎 𝟏]
𝟏 −𝟏 𝟎

𝟏 −𝟏 𝟓
(i) Prove that the matrix A= [−𝟏 𝟐 𝟏] is a symmetric matrix.
𝟓 𝟏 𝟑

𝟎 𝟏 −𝟏
(ii) Prove that the matrix A=[−𝟏 𝟎 𝟏 ] is skew symmetric matrix.
𝟏 −𝟏 𝟎

𝟏 𝟓
A= [ ]
𝟔 𝟕

(A+A’)

(A-A’)

12

𝟏 𝟓
For the matrix A= [ ] verify that
𝟔 𝟕

(A+A’) is a symmetric matrix.

(A-A’) is a skew symmetric matrix.

𝒄𝒐𝒔∅ −𝒔𝒊𝒏∅
A=[ ] A+A’=I ∅
𝒔𝒊𝒏∅ −𝒄𝒐𝒔∅

𝒄𝒐𝒔∅ −𝒔𝒊𝒏∅
If A=[ ]and A+A’=I, then Find the value of ∅.
𝒔𝒊𝒏∅ −𝒄𝒐𝒔∅

𝟐 −𝟏 𝟏𝟎
x[ ] + 𝒚 [ ] = [ ] x y
𝟑 𝟏 𝟓

𝟐 −𝟏 𝟏𝟎
If x[ ] + 𝒚 [ ] = [ ] then Find the values of X and Y.
𝟑 𝟏 𝟓

𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝜽
A= [ ] AA’=I
−𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽

𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝜽
If A= [ ] then verify that AA’=I
−𝒄𝒐𝒔𝜽 𝒔𝒊𝒏𝜽

(Long Answers Questions : Marks 6)

𝟑 −𝟐 𝟏 𝟎
A=[ ] I= [ ] 𝐀𝟐 = 𝐊𝐀 − 𝟐𝐈 k
𝟒 −𝟐 𝟎 𝟏

𝟑 −𝟐 𝟏 𝟎
If A=[ ] and I= [ ]and A2=kA-2I then Find the value of k.
𝟒 −𝟐 𝟎 𝟏

13

𝟐 𝟎 𝟏
A=[𝟐 𝟏 𝟑] A2-5A+6I
𝟏 −𝟏 𝟎

𝟐 𝟎 𝟏
If A=[𝟐 𝟏 𝟑] is, then Find the value of A2-5A+6I.
𝟏 −𝟏 𝟎

𝟏 𝟎 𝟐
A=[𝟎 𝟐 𝟏] A3-6A2+7A+2I=0
𝟐 𝟎 𝟑

𝟏 𝟎 𝟐
If A=[𝟎 𝟐 𝟏]then prove that A3-6A2+7A+2I=0
𝟐 𝟎 𝟑

𝟓 𝟑
A= [ ] A2-3A-7I=0 A-1
−𝟏 −𝟐

𝟓 𝟑
If A= [ ]then prove that A2-3A-7I=0 and Find A-1 .
−𝟏 −𝟐

A B AB=BA

(AB)'=A'B'

If A and B are any two square matrices and AB=BA, then prove by

mathematical induction that: (AB)'= A'B'

14

𝟎 𝟏 𝟐
𝟐 𝟏
(i) [𝟏 𝟐 𝟑] (ii) [ ]
𝟕 𝟒
𝟑 𝟏 𝟏
𝟐 −𝟑 𝟑
𝟑 𝟏𝟎
(iii) [ ] (iv) [𝟐 𝟐 𝟑]
𝟐 𝟕
𝟑 −𝟐 𝟐
𝟏 𝟑 −𝟐 𝟐 𝟎 −𝟏
(v) [−𝟑 𝟎 −𝟓] (vi) [𝟓 𝟏 𝟎]
𝟐 𝟓 𝟎 𝟎 𝟏 𝟑

Find the inverse of the following matrices using elementary
operations:
𝟎 𝟏 𝟐
𝟐 𝟏
(i) [𝟏 𝟐 𝟑] (ii) [ ]
𝟕 𝟒
𝟑 𝟏 𝟏
𝟐 −𝟑 𝟑
𝟑 𝟏𝟎
(iii) [ ] (iv) [𝟐 𝟐 𝟑]
𝟐 𝟕
𝟑 −𝟐 𝟐
𝟏 𝟑 −𝟐 𝟐 𝟎 −𝟏
(v) [−𝟑 𝟎 −𝟓] (vi) [𝟓 𝟏 𝟎]
𝟐 𝟓 𝟎 𝟎 𝟏 𝟑
𝟏 𝟎 𝟐 𝒙
[𝒙 −𝟓 −𝟏] [𝟎 𝟐 𝟏] [𝟒] = 𝐨 x
𝟐 𝟎 𝟑 𝟏

𝟏 𝟎 𝟐 𝒙
If [𝒙 −𝟓 −𝟏] [𝟎 𝟐 𝟏] [𝟒] = 𝐨 then Find the value of X.
𝟐 𝟎 𝟑 𝟏
𝟎 𝟐𝒚 𝒛
𝒙, 𝒚 𝒛 A= [𝒙 𝒚 −𝒛]
𝒙 −𝒚 𝒛
A’A=I

15

𝟎 𝟐𝒚 𝒛
Find the values of 𝒙, 𝒚 and 𝒛 if matrix A= [𝒙 𝒚 −𝒛] satisfies the
𝒙 −𝒚 𝒛
equation A’A=I.

𝒄𝒐𝒔𝒙 −𝒔𝒊𝒏𝒙 𝟎
f(𝒙) =[𝒔𝒊𝒏𝒙 𝒄𝒐𝒔𝒙 𝟎]
𝟎 𝟎 𝟏

F(𝒙)F(𝒚) = F(𝒙 + 𝒚)

𝒄𝒐𝒔𝒙 −𝒔𝒊𝒏𝒙 𝟎
If F(𝒙) = [ 𝒔𝒊𝒏𝒙 𝒄𝒐𝒔𝒙 𝟎] , prove that F(𝒙)F(𝒚) = F(𝒙 + 𝒚)
𝟎 𝟎 𝟏

𝟐 −𝟐 −𝟒
B = [−𝟏 𝟑 𝟒]
𝟏 −𝟐 −𝟑

𝟐 −𝟐 −𝟒
Express the matrix B = [−𝟏 𝟑 𝟒 ] as the sum of a
𝟏 −𝟐 −𝟑

symmetric and a skew symmetric matrix.

𝟑𝒙 − 𝟐𝒚 + 𝟑𝒛 = 𝟖

𝟐𝒙 + 𝒚 − 𝒛 = 𝟏

𝟒𝒙 − 𝟑𝒚 + 𝟐𝒛 = 𝟒

16

Solve the following system of equations by matrix method –

𝟑𝒙 − 𝟐𝒚 + 𝟑𝒛 = 𝟖

𝟐𝒙 + 𝒚 − 𝒛 = 𝟏

𝟒𝒙 − 𝟑𝒚 + 𝟐𝒛 = 𝟒

The sum oy three number is 6. If we multiphy third number by 3 and add
second number to it, we get 11. By adding first and third numbers, we, get
double of the second number, Represent it algebraically and Find the
number using matrix method.

17

(Determinant)

(Very Short Answers Questions : Marks 2)

A= [𝟏𝟒 𝟐𝟐] |𝟐𝑨| = 𝟒|𝑨|

If A= [𝟏𝟒 𝟐
𝟐
]then show that|𝟐𝑨| = 𝟒|𝑨|

(3,8),(-4,2) (5,1)

Find the area of a triangle using determinant if vertices of triangle are
(3,8), (4,2) and (5,1).

(k,0),(4,0),(0,2)
k

If the vertices of a triangle of area of 4 square units are (k,0),(4, 0),( 0,2),
then Find the value of K.

𝟏 −𝟐
| |
𝟒 𝟑

Find the minors and cofactor of all the elements of the determinant
𝟏 −𝟐
| |
𝟒 𝟑
𝟏 𝟏 𝟐
A = [𝟐 𝟏 𝟑] |𝐀|
𝟓 𝟒 𝟗

18

𝟏 𝟏 𝟐
if A = [𝟐 𝟏 𝟑] , then Find |𝐀|.
𝟓 𝟒 𝟗

𝟐 𝟒 𝟐𝒙 𝟒
| |=| | 𝒙
𝟓 𝟏 𝟔 𝒙
𝟐 𝟒 𝟐𝒙 𝟒
if | |=| | , then Find the value of 𝒙 .
𝟓 𝟏 𝟔 𝒙

(Short Answers Questions : Marks 4)

𝒙+𝟒 𝟐𝒙 𝟐𝒙
| 𝟐𝒙 𝒙+𝟒 𝟐𝒙 | = (𝟓𝒙 + 𝟒)(𝟒 − 𝒙)𝟐
𝟐𝒙 𝟐𝒙 𝒙+𝟒
𝒙+𝟒 𝟐𝒙 𝟐𝒙
Prove that | 𝟐𝒙 𝒙+𝟒 𝟐𝒙 | = (𝟓𝒙 + 𝟒)(𝟒 − 𝒙)𝟐
𝟐𝒙 𝟐𝒙 𝒙+𝟒

𝟏 𝒙 𝒙𝟐
𝟑 𝟐
|𝒙𝟐 𝟏 𝒙 | = (𝟏 − 𝒙 )
𝒙 𝒙𝟐 𝟏

𝟏 𝒙 𝒙𝟐
𝟑 𝟐
Prove that |𝒙𝟐 𝟏 𝒙 | = (𝟏 − 𝒙 )
𝒙 𝒙𝟐 𝟏

𝒃+𝒄 𝒂 𝒂
| 𝒃 𝒄+𝒂 𝒃 | = 𝟒𝒂𝒃𝒄
𝒄 𝒄 𝒂+𝒃

19

𝒃+𝒄 𝒂 𝒂
Prove that | 𝒃 𝒄+𝒂 𝒃 | = 𝟒𝒂𝒃𝒄
𝒄 𝒄 𝒂+𝒃

𝒙 𝒙𝟐 𝟏 + 𝒙𝟑
x,y,z |𝒚 𝒚𝟐 𝟏 + 𝒚𝟑 | = 𝟎
𝒛 𝒛𝟐 𝟏 + 𝒛𝟑

1+xyz=0

𝒙 𝒙𝟐 𝟏 + 𝒙𝟑
If x,y,z are different and |𝒚 𝒚𝟐 𝟏 + 𝒚𝟑 | = 𝟎 Then show that
𝒛 𝒛𝟐 𝟏 + 𝒛𝟑
1+xyz=0

A(1,3) B(0,0)
k D(k, 0)
Δ(ABC)

Find the equation of the line joining A(1,3) and B(0,0) using
determinants and Find k if D(k,0) is a point such that area of
triangle ABC is 3 square unit.

20

(Unit-3)

CHAPTER 5 – CONTINUTIY AND DEFFERENTIABILITY

(Very Short Answers Questions : Marks 2)

f ( x) = sin x + cos x 𝑥=𝜋
Show that the function f ( x) = sin x + cos x is continuous at the point 𝑥 = 𝜋
f ( x) = x − 5 x=5

Show that the function f ( x) = x − 5 is continuous at the point x = 5 but
not differentiable.
𝒇(𝒙) = 𝒕𝒂𝒏𝒙. 𝒔𝒆𝒄𝒙
Check the continuity of the function 𝒇(𝒙) = 𝒕𝒂𝒏𝒙. 𝒔𝒆𝒄𝒙
f ( x) = sin x. cos x

Check the continuity of the function f ( x) = sin x. cos x
𝒇(𝒙) = 𝒔𝒊𝒏(𝒙𝟐 )
Show that the function defined by 𝒇(𝒙) = 𝒔𝒊𝒏(𝒙𝟐 ) is a continuous
function.
f ( x) = 2 x 2 − 1 x=3

Check the continuity of the function f ( x) = 2 x 2 − 1 at the point x = 3 .
x 2 − 25
f ( x) = x −5
x+5 x 2 − 25
Check the continuity of the function f ( x) = at the point x  − 5.
x+5
f ( x) = x  x 
x = 1 .5
Show that the function f ( x) = x  where, x  is a greatest integer
function, is continuous on. x = 1.5

21

f ( x) =
1 y = f  f (x)
x −1

𝟏
if 𝒇(𝒙) = then Find the discontinuous points of the
𝒙−𝟏

composition of the function 𝒚 = 𝒇[𝒇(𝒙)].

 1
 x sin , x  0
f ( x) =  x

 0 , x=0

x=0
 1
 x sin , x  0
Show that the function f ( x) =  x

 0 , x=0

is continuous at the point x = 0.
𝒇(𝒙) = 𝒙|𝒙|, ∀𝒙 ∈ 𝑹 x=0

Check the differentiability of the function at the point x = 0
𝒇(𝒙) = 𝒙|𝒙|, ∀𝒙 ∈ 𝑹
 x2
 2 x 2 − 3x − 2

f ( x) = 
 x−2

 5 x=2

x=2
For the defined function  If x  2
 2 x 2 − 3x − 2

f ( x) = 
 x−2

 5
If x = 2

check the continuity of the function of the point x = 2

22

f ( x) = sin x + cos x x =
Show that the function f ( x) = sin x + cos x is continuous at point x =  .
f f

2 x + 3 x2
f ( x) =  x2
2 x − 3
Find all the points of discontinuity of the function f where f is defined by

2 x + 3 If x2
f ( x) =  x2
2 x − 3 If

𝒇
𝒇
𝒙
यदि 𝒙 < 𝟐
𝒇(𝒙) = { |𝒙|
−𝟏 यदि 𝒙 ≥ 𝟐

Find all the points of discontinuity of F
Whereas the function F is defined as follows.
𝒙
𝐈𝐟 𝒙 < 𝟐
𝒇(𝒙) = { |𝒙|
−𝟏 𝐈𝐟 𝒙 ≥ 𝟐

f


sin x x0
f ( x) =  x
 x0
 x +1
Find all the points of discontinuity of function f
while


sin x If x2
f ( x) =  x
 If x2
 x +1

23

f f

𝒔𝒊𝒏𝒙 − 𝒄𝒐𝒔𝒙 x0
𝒇(𝒙) = {
−𝟏
x=0

Examine the continuity of f, where f is defiened by
sin x − cos x If x  0
f ( x) = 
 −1 If x = 0

𝒌𝒙𝟐
𝒇(𝒙) = { x2
𝟑
x2

x=2 k

Given function
K x 2 If x  2
f ( x) = 
 3 If x  2
is continuous at the point x = 2 then Find the value of k.
𝒌𝒙+𝟏
𝒇(𝒙) = { 𝒄𝒐𝒔𝒙 𝒙≤ 𝝅

𝒙 > 𝝅
𝒙= 𝝅 k

If the function f is defined by

 Kx +1 If 𝒙≤ 𝝅
f ( x) = 
 cos x If 𝒙 > 𝝅
is continuous at 𝒙 = 𝝅 then Find the value of k.

24

𝒇(𝒙) = 𝒄𝒐𝒔(𝒙𝟐 )
Show that the function 𝒇(𝒙) = 𝒄𝒐𝒔(𝒙𝟐 ) is a continuous function.
a
ax +1, x  1
f ( x) = 
 x + 2, x  1
ax +1, x  1
If the function f ( x) =  is a continuous function then
 x + 2, x  1
what will be the value of a.

𝒇(𝒙) = |𝒙| − |𝒙 + 𝟏|

Find all the points of discontinuity of defined by 𝒇(𝒙) = |𝒙| − |𝒙 + 𝟏|
𝒅𝒚
𝒄𝒐𝒔𝒚 = 𝒙𝒄𝒐𝒔(𝒂 + 𝒚) 𝒄𝒐𝒔𝒂 ≠ ±𝟏 =
𝒅𝒙
𝒄𝒐𝒔𝟐 (𝒂+𝒚)
𝒔𝒊𝒏𝒂
𝒅𝒚
If 𝒄𝒐𝒔𝒚 = 𝒙𝒄𝒐𝒔(𝒂 + 𝒚), with 𝒄𝒐𝒔𝒂 ≠ ±𝟏, Prove that =
𝒅𝒙
𝒄𝒐𝒔𝟐 (𝒂+𝒚)
𝒔𝒊𝒏𝒂

x

Find the derivative of the given functions with respect to 𝒙:-
(i) sin(cos( x 2 )) (ii)
sin x 2 + sin 2 x + sin 2 ( x 2 )
(iii) log (log (log( x 5 )) (iv) 𝒔𝒆𝒄(𝒕𝒂𝒏(√𝒙))
(v) (vi)
2 cot( x 2 ) e x

(vii) 𝒄𝒐𝒕−𝟏 𝒙 (viii) cos x
log x
(ix) .8 x
x8
(xi) −1
esin x

25

(xiii) sin(tan −1 (e − x )

2
(xv) 2cos x

𝒅𝒚
𝒅𝒙
𝒅𝒚
Find the value of
𝒅𝒙
If

1. 𝒚 + 𝒔𝒊𝒏𝒚 = 𝒄𝒐𝒔𝒙

2. 𝒔𝒊𝒏𝟐 𝒚 + 𝒄𝒐𝒔𝒙𝒚 = 𝑲

3. (𝒙𝟐 + 𝒚𝟐 ) = 𝒙𝒚

4. 𝒕𝒂𝒏−𝟏 (𝒙𝟐 + 𝒚𝟐 ) = 𝒂

(Short Answers Questions : Marks 4)
x
(i) 𝟐𝒙
𝒚 = 𝐜𝐨𝐬 −𝟏 ( ) 𝟏<𝒙<𝟏
𝟏+𝒙𝟐

(ii) 𝟑𝒙−𝒙𝟐 −𝟏 𝟏
𝒚 = 𝐭𝐚𝐧−𝟏 ( ) <𝒙<
𝟏+𝒙𝟐 √𝟑 √𝟑

−𝝅 𝝅
(iii) 𝒚 = 𝐭𝐚𝐧−𝟏 (𝒔𝒆𝒄 𝒙 + 𝒕𝒂𝒏 𝒙) <𝒙<
𝟐 𝟐

(iv) 𝟏−𝒄𝒐𝒔𝒙 𝝅 𝝅
𝒕𝒂𝒏−𝟏 (√ ) − <𝒙<
𝟏+𝒄𝒐𝒔𝒙 𝟐 𝟐

(v) 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙 −𝝅 𝝅
𝐜𝐨𝐬−𝟏 ( ) <𝒙<
√𝟐 𝟐 𝟐

(vi) 𝒅𝟐 𝒚
−1
y = tan x 𝒅𝒙𝟐
y

26

−1 𝒅𝟐 𝒚
If y = tan x yes then Find the value of in terms of y.
𝒅𝒙𝟐

x
(vii)
x=e y

𝒅𝒚 𝒙−𝒚
=
𝒅𝒙 𝒙𝒍𝒐𝒈𝒙

x
dy x− y
If x=e y
is then prove that. =
dx x log x

(viii) 𝒅𝒚
(𝒄𝒐𝒔𝒙)𝒚 = (𝒄𝒐𝒔𝒚)𝒙
𝒅𝒙

dy
Find the value of for the function (𝒄𝒐𝒔𝒙)𝒚 = (𝒄𝒐𝒔𝒚)𝒙
dx
.
(ix) y x = e y−x dy (1 + log y ) 2
=
dx log y

dy (1 + log y ) 2
If y x = e y−x then prove that =
dx log y

(x) x(sin ( x + y ) + sin a cos( a + y ) = 0 dy sin 2 (a + y )
=
dx sin x

dy sin 2 (a + y )
If x(sin ( x + y ) + sin a cos( a + y ) = 0 then prove that =
dx sin x
dy
(xi) x2/3 + y2/3 = a2/3
dx

dy
If x 2 / 3 + y 2 / 3 = a 2 / 3 then Find the value of
dx
(xii) −1 −1 dy − y
x= a sin t , y = a cos t =
dx x
dy − y
−1 −1 =
If x = a sin t , y = a cos t is then show that dx x

27

(xiii) x
sin x
sin x

x
Differentiate the function with respect to 𝐒𝐢𝐧𝒙.
sin x
(xiv)  1 + x2 −1  tan −1 x x 0
tan 
−1 
 x 
 
 1 + x2 −1 
Differentiate the function tan −1   with respect to tan −1 x
 x 
 
while x  0 .
(xv) e x ( x + 1) = 1
2
d 2 y  dy 
= 
dx 2  dx 
2
d 2 y  dy 
If e ( x + 1) = 1
x
then show dx 2 =  dx 
 

(Long Answers Questions : Marks 6)

y = sin −1 x

𝟐
𝒅𝟐 𝒚 𝒅𝒚
(𝟏 − 𝒙 ) 𝟐 − 𝒙 =𝟎
𝒅𝒙 𝒅𝒙
𝒅𝟐 𝒚 𝒅𝒚
if 𝒚 = 𝐬𝐢𝐧−𝟏 𝒙 then show that (𝟏 − 𝒙𝟐 ) −𝒙 =𝟎
𝒅𝒙𝟐 𝒅𝒙

𝒚 = (𝐭𝐚𝐧−𝟏 𝒙)𝟐 ( x 2 + 1) 2 y2 + 2 x( x 2 + 1) y1 = 2

If 𝒚 = (𝐭𝐚𝐧−𝟏 𝒙)𝟐 exists then show that ( x 2 + 1)2 y2 + 2 x( x 2 + 1) y1 = 2 .

1, 4 f ( x) = x 2 − 4 x − 3

Verify the mean value theorem for f ( x) = x 2 − 4 x − 3 in the interval 1, 4 .
𝒇(𝒙) = 𝒍𝒐𝒈(𝒙𝟐 + 𝟐) − 𝒍𝒐𝒈𝟑; 𝒙 ∈ [−𝟏, 𝟏]

Verify Rolle's theorem for the function 𝒇(𝒙) = 𝒍𝒐𝒈(𝒙𝟐 + 𝟐) − 𝒍𝒐𝒈𝟑; 𝒙 ∈ [−𝟏, 𝟏]
f ( x) = 4 − x 2 ; x  − 2,2

Verify Rolle's theorem for the function f ( x) = 4 − x 2 ; x  − 2,2 .

28

1
f ( x) = x  (1,4)
4x −1

1
Verify the mean value theorem for the function f ( x) = x  (1,4) .
4x −1

0,   f ( x) = sin x − sin 2 x

Verify the mean value theorem for f ( x) = sin x − sin 2 x in the interval 0,  

29

(Application of Derivatives)

(Short Answers Questions : Marks 4)

𝟑
(𝟐𝒙 + 𝟏)
𝟐

𝒙
A balloon, which always remains spherical, has a variable diameter
𝟑
(𝟐𝒙 + 𝟏) Find the rate of change of its volume with respect to 𝒙.
𝟐

A ladder 5 meter 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 it’s height on the wall decreasing when the foot of the ladder is 4
meter away from.

The length of a rectangle is decreasing at the rate of 3 cm per minute and
the width is increasing at the rate of 2 cm per minute when the length is 10
cm. and width is 6 cm then find the rate of change in both perimeter and
area of the rectangle.

𝟓 𝒄𝒎/𝒔

A stone is dropped into a quiet lake and waves move in circles at the
speed of 5 cm/s. At the instant whe the radius of the circular wave is 8
cm, how fast is the enclosed area increasing.

30

f ( x) = 2 x 3 − 3 x 2 − 36 x + 7

Find the intervals in which the given function is increasing and decreasing
. f ( x) = 2 x 3 − 3x 2 − 36 x + 7
(0,  )
Prove that the logarithmic function is an increasing function. in (𝟎, ∞)
y = ( x( x − 2)]2
x

Find those values of 𝒙 for which y = ( x( x − 2)]2 is an increasing function.
R f ( x) = 3 x 2 + 3 x − 100
Prove that the function f ( x) = 3x 2 + 3x − 100 given in R is increasing.

𝒇(𝒙) = 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙, 𝟎 ≤ 𝒙𝒙 ≤ 𝟐𝝅
f
Find the interval in which the function F given
by 𝒇(𝒙) = 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙, 𝟎 ≤ 𝒙𝒚 ≤ 𝟐𝝅 is increasing or decreasing.
𝟏
0 𝒚= 𝟐
𝒙 −𝟐𝒙+𝟑

𝟏
Find the equation of all lines of slope 0 which touch the curve 𝒚 =
𝒙𝟐 −𝟐𝒙+𝟑
x2 y2
+ =1
9 16
(i) x- (ii) y-

2 2
Find the points on the curve x + y = 1 at which the tangents are
9 16

(ii) Parallel to the axis; (iii) paralled to y-axis.
y 2 = 4ax (𝒂𝒕𝟐 , 𝟐𝒂𝒕)

Find the equation of tangent and normal at point (𝒂𝒕𝟐 , 𝟐𝒂𝒕) of parabola
𝒚𝟐 = 𝟒𝒂𝒙

𝒚 = √𝟓𝒙 − 𝟑 − 𝟐
𝟒𝒙 − 𝟐𝒚 + 𝟑 = 𝟎
Find the equations of the tangents to the curve 𝒚 = √𝟓𝒙 − 𝟑 − 𝟐 which
is paralled to the line 𝟒𝒙 − 𝟐𝒚 + 𝟑 = 𝟎

31

y = x 4 − 6 x 3 + 13 x 2 − 10 x + 5 (0, 5)

Find the equation of tangent and normal to the given curve
𝒚 = 𝒙𝟐 − 𝟔𝒙𝟐 + 𝟏𝟑𝒙𝟐 − 𝟏𝟎𝒙 + 𝟓
at point (0, 5)
2
y+ =0
x −3

Find the equation of all lines having slope 2 and being tongent to the
curve y + 2 = 0 .
x −3
𝟏
(𝟐𝟓)𝟑
𝟏
Using Differential, find the approximate value of (𝟐𝟓)𝟑

√𝟎. 𝟔

Using Differentials, find the approximate value of √𝟎. 𝟔 upto three places of
decimal.
m m

The radius of a sphere is measured to be 7 mm with an error of 0.02 mm.
Find the approrimate error in calculating its volume.
f (2.001) f ( x) = 4 x 2 + 5 x + 2
Find the approximate value of 𝒇(𝒙) = 𝟒𝒙𝟐 + 𝟓𝒙 + 𝟐 where f (2.001) is.
[0,3] 3x 4 − 8 x 3 + 12 x 2 − 48 x + 25

Find the maximum value and minimum value in 3x 4 − 8 x 3 + 12 x 2 − 48 x + 25 on
the interval [0,3] .
sin x + cos x
Find the maximum value of the function .
[𝟎, 𝟐𝝅] x + sin 2 x
Find the maximum and minimum value of the function x + sin 2 x
on [𝟎, 𝟐𝝅] .

32

show that of all the rectangles inscribed in a given fixed circle, the square
has the maximum area.

Find two numbers whose sum is 24 and product is maximum .

Find two positive numbers whose sum is 16 and the sum of whose cubes
is minimum.

33

(Integration)
(Very Short Answers Questions : Marks 1)

 sin x dx
2

Find the value of  sin 2 x dx

 sec x dx
2

Find the value of  sec x dx
2

−1
e tan x
 1 + x 2 dx −1
e tan x
Find the value of.  1 + x 2 dx

𝒔𝒊𝒏(𝐭𝐚𝐧−𝟏 𝒙)
∫ 𝒅𝒙
𝟏+𝒙𝟐

𝒔𝒊𝒏(𝐭𝐚𝐧−𝟏 𝒙)
Find the value of ∫ 𝒅𝒙
𝟏+𝒙𝟐

 1 + cos 2 x dx
Find the value of  1 + cos 2 x dx
(cos x + 1)
 (sin x + x) dx
(cos x + 1)
Find the value of  dx
(sin x + x)

∫ 𝒄𝒐𝒕𝒙𝒅𝒙

Find the value of ∫ 𝒄𝒐𝒕𝒙𝒅𝒙

𝒅𝒙
∫ 𝒂𝒙+𝒃

𝒅𝒙
Find the value of ∫
𝒂𝒙+𝒃

34

(Very Short Answers Questions : Marks 2)

2 − 3 sin x
 cos 2 x
2 − 3 sin x
Integrate  cos 2 x

sec2 (log x)
 x sec2 (log x)
Find the value of 
x
cos x
 x dx
cos x
Find the value of  dx
x
𝟏
∫ 𝟏−𝒄𝒐𝒔𝒙

𝟏
Find the value of ∫
𝟏−𝒄𝒐𝒔𝒙
1
 16 − 9 x dx
2

1
Find the value of  16 − 9 x
dx
2

(1 − cos 2 x)
 (1 + cos 2 x) dx
(1 − cos 2 x)
Find the value of  (1 + cos 2 x) dx
𝝅
∫𝟎 𝒔𝒊𝒏𝟐𝒙𝒅𝒙
𝟒

𝝅
Find the value of ∫ 𝒔𝒊𝒏𝟐𝒙𝒅𝒙 𝟒
𝟎

∫ 𝐬𝐢𝐧−𝟏 (𝒄𝒐𝒔𝒙)𝒅𝒙

Find the value of ∫ 𝐬𝐢𝐧−𝟏 (𝒄𝒐𝒔𝒙)𝒅𝒙

35

(Short Answers Questions : Marks 4)
1
 x − 6 x + 13 dx
2

1
Find the value of  x − 6 x + 13
dx 2

 2x 
 sin  1 + x dx
−1
2

 2x 
Find the value of 
sin −1  2 
dx
1+ x 
dx
 5 + 4 sin x
dx
Find the value of  5 + 4 sin x
1 1
 e  x − x  dx
x
2

1 1
Find the value of  e  x − x 2  dx
x

 
e (1 + sin x)
x

 1 + cos x dx

e x (1 + sin x)
Find the value of
 1 + cos x dx
𝒄𝒐𝒔𝒙
∫ 𝒔𝒊𝒏𝟐𝒙+𝟒𝒔𝒊𝒏𝒙+𝟓 𝒅𝒙
cos x
Find the value of
 sin x + 4 sin x + 5 dx
2

 x + 2 x + 5 dx
2

Find the value of 
x 2 + 2 x + 5 dx

−𝟏 𝒙
𝒆𝒙𝒕𝒂𝒏
∫ (𝟏+𝒙𝟐)𝟑/𝟐 𝒅𝒙

−𝟏 𝒙
𝒆𝒙𝒕𝒂𝒏
Find the value of ∫ 𝒅𝒙
(𝟏+𝒙𝟐 )𝟑/𝟐

36

dx
 sin x − cos x
dx
Find the value of  sin x − cos x

1
 1 + cot x dx
1

Find the value of 1 + cot x
dx

𝟏
∫ 𝒙(𝒙𝒏+𝟏) 𝒅𝒙

𝟏
Find the value of ∫ 𝒅𝒙
𝒙(𝒙𝒏 +𝟏)

37

(Long Answers Questions : Marks 6)

𝝅/𝟐 √𝒔𝒊𝒏 𝒙 𝝅
∫𝟎 𝒅𝒙 =
√𝒔𝒊𝒏 𝒙+√𝒄𝒐𝒔 𝒙 𝟒

𝝅/𝟐 √𝒔𝒊𝒏 𝒙 𝝅
Prove that ∫
𝟎
𝒅𝒙 =
√𝒔𝒊𝒏 𝒙+√𝒄𝒐𝒔 𝒙 𝟒

𝝅/𝟐 𝒔𝒊𝒏𝟒 𝒙 𝝅
∫𝟎 𝒔𝒊𝒏𝟒 𝒙+ 𝒄𝒐𝒔𝟒 𝒙 𝒅𝒙 = 𝟒

𝝅/𝟐 𝒔𝒊𝒏𝟒 𝒙 𝝅
Prove that ∫ 𝒅𝒙 =
𝟎 𝒔𝒊𝒏𝟒 𝒙+ 𝒄𝒐𝒔𝟒 𝒙 𝟒

𝝅 𝒙 𝒔𝒊𝒏 𝒙 𝝅𝟐
∫𝟎 𝟏+𝒄𝒐𝒔𝟐 𝒙 𝒅𝒙 = 𝟒

𝝅 𝒙 𝒔𝒊𝒏 𝒙 𝝅𝟐
Prove that ∫
𝟎 𝟏+𝒄𝒐𝒔𝟐 𝒙
𝒅𝒙 =
𝟒

𝝅 𝒙𝒅𝒙 𝝅𝟐
∫𝟎 𝒂𝟐 𝒄𝒐𝒔𝟐 𝒙+𝒃𝟐 𝒔𝒊𝒏𝟐 𝒙 = 𝟐𝒂𝒃

𝝅 𝒙𝒅𝒙 𝝅𝟐
Prove that ∫
𝟎 𝒂𝟐 𝒄𝒐𝒔𝟐 𝒙+𝒃𝟐 𝒔𝒊𝒏𝟐 𝒙
=
𝟐𝒂𝒃

𝝅 𝒙
∫𝟎 𝟏+𝒔𝒊𝒏 𝒙 𝒅𝒙 = 𝝅

𝝅 𝒙
Prove that ∫
𝟎 𝟏+𝒔𝒊𝒏 𝒙
𝒅𝒙 = 𝝅

𝝅
𝒙 𝒔𝒊𝒏 𝒙.𝒄𝒐𝒔 𝒙 𝝅
∫ 𝟐
𝟎 𝒔𝒊𝒏𝟒 𝒙+𝒄𝒐𝒔𝟒 𝒙
𝒅𝒙 =
𝟏𝟔

𝝅
𝒙 𝒔𝒊𝒏 𝒙.𝒄𝒐𝒔 𝒙 𝝅
Prove that ∫ 𝟐
𝟎 𝒔𝒊𝒏𝟒 𝒙+𝒄𝒐𝒔𝟒 𝒙
𝒅𝒙 =
𝟏𝟔

𝝅
𝟏 𝝅
∫𝝅𝟑 𝟏+ 𝒕𝒂𝒏 𝒙 𝒅𝒙 = 𝟏𝟐
𝟔
√

𝝅
𝟏 𝝅
Prove that ∫𝝅𝟑 𝟏+ 𝒕𝒂𝒏 𝒙 𝒅𝒙 = 𝟏𝟐
𝟔
√

38

𝝅 𝝅
𝝅
∫𝟎 𝒍𝒐𝒈 𝒔𝒊𝒏𝒙 𝒅𝒙 = ∫𝟎 𝒍𝒐𝒈 𝒄𝒐𝒔𝒙 𝒅𝒙 = − 𝟐 𝒍𝒐𝒈𝒆 𝟐
𝟐 𝟐

𝝅 𝝅
𝝅
Prove that ∫𝟎 𝒍𝒐𝒈 𝒔𝒊𝒏𝒙 𝒅𝒙 = ∫𝟎 𝒍𝒐𝒈 𝒄𝒐𝒔𝒙 𝒅𝒙 = − 𝟐 𝒍𝒐𝒈𝒆 𝟐
𝟐 𝟐

𝝅
𝝅
∫𝟎 𝒍𝒐𝒈(𝟏 + 𝒕𝒂𝒏𝒙)𝒅𝒙 = 𝟖 𝒍𝒐𝒈𝒆 𝟐
𝟒

𝝅
𝝅
Prove that ∫
𝟎
𝒍𝒐𝒈(𝟏 + 𝒕𝒂𝒏𝒙)𝒅𝒙 = 𝒍𝒐𝒈𝒆 𝟐
𝟒
𝟖

𝝅
𝟒+𝟑 𝒔𝒊𝒏 𝒙
∫𝟎 𝒍𝒐𝒈 [𝟒+𝟑 𝒄𝒐𝒔 𝒙] 𝒅𝒙 = 𝟎
𝟒

𝝅
𝟒+𝟑 𝒔𝒊𝒏 𝒙
Prove that ∫ 𝟒
𝟎
𝒍𝒐𝒈 [ ] 𝒅𝒙 = 𝟎
𝟒+𝟑 𝒄𝒐𝒔 𝒙

39

(Application of Integrals)

(Long Answers Questions : Marks 6)

x2 + y2 = a2
Find the area of circle x + y = a 2 using integration method.
2 2

x 2 + y 2 = 32 y=x x–

Find the area of the region enclosed by circle x + y = 32 , line 𝒚 = 𝒙 and x–
2 2

axis in the first quadrant using integration method.

𝒂
𝒙= x2 + y2 = a2
√𝟐

Find the area of the smaller part of the circle x 2 + y 2 = a 2 by the secant line
𝒂
𝒙= Using integration method.
√𝟐

𝒙 = 𝒚𝟐 x=4 x=a

a

If the area enclosed by curve x = y 2 and line x=4 is divided into two

equal parts by the line x=a, then Find the value of a using integration

method.

(1,0), (2,2) (3,1)

Using integration, find the area of a triangle whose vertices (1,0), (2,2)

And (3,1)

40

x2 + y2 = 4 ( x − 2)2 + y 2 = 4

x 2 + y 2 = 4 and ( x − 2) + y = 4
2 2
Find the area of the region between two circles

y = 2 x + 1, y = 3 x + 1 x=4

Using integration, Find the area of a triangular region whose equations

of sides are y = 2 x + 1, y = 3x + 1 and x = 4.

𝒚𝟐 = 𝟒𝒂𝒙 𝒙𝟐 = 𝟒𝒂𝒚 𝒂>𝟎

Using Integration method find the area of the region includes between the

curves 𝒚𝟐 = 𝟒𝒂𝒙 and 𝒙𝟐 = 𝟒𝒂𝒚 where 𝒂 > 𝟎

41

(Differential Equations)

Very short answer questions (1 mark)
4
 ds  d 2s
  + 3s =0
 dt  dt 2 4
 ds  d 2s
Find the order and degree of the differential equation   + 3s 2 = 0
 dt  dt
2
 dy  dy
x3  2  + x =0
 dx  dx 2
3  dy  dy
Find the order and degree of the differential equation x  2  + x =0
 dx  dx
2
 dy  d y
2
5 x  − 2 − 6 y = log x
 dx  dx 2
 dy  d y
2
Find the order and degree of the differential equation 5 x  − 2 − 6 y = log x
 dx  dx
3
d2y
4
 dy 
x 2  + y  +x =0
3

 dx   dx  3
d2y
4
 dy 
Find the order and degree of the differential equation x 2  + y  + x 3 = 0
 dx   dx 
dy
x − y = 2x 2
dx
dy
Find the integration factor of the differential equationx − y = 2x 2
dx

(tan −1 y − x)dy = (1 + y 2 )dx
−1
Find the integration factor of the differential equation (tan y − x)dy = (1 + y 2 )dx

Very short answer questions (2 mark)
x y
+ =1
a b
𝒙 𝒚
Find the defferential equation representing family of curve 𝒂 + 𝒃 = 𝟏
dy 1 − cos x
=
dx 1 + cos x
dy 1 − cos x
Find the general solution of dx = 1 + cos x

42

y = a cos x + b sin x 𝒂, 𝒃 ∈ 𝑹, d2y
+ y= 0
dx 2

Is 𝒚 = 𝒂𝒄𝒐𝒔𝒙 + 𝒃𝒔𝒊𝒏𝒙 where 𝒂, 𝒃 ∈ 𝑹, a solution of the differential
2
equation d y + y = 0 ?
dx 2

y = ex +1 𝒚" + 𝒚′ = 𝟎

Prove that y = e x + 1 is a solution of the differential equation y " + y , = 0 .

(−2, 3)

( x, y )
2x
y2

Find the equation of a curve passing through point (−2, 3) whose slope

of the tangent line at any point ( x, y ) is 2 x2 .
y

(Short Answers Questions : Marks 4)
dy
(1 + x 2 ) + y = tan −1 x
dx

Find the general solution of the differential equation (1 + x 2 ) dy + y = tan−1 x
dx
𝒅𝒚 𝟐
𝒙𝒍𝒐𝒈𝒙 + 𝒚 = 𝒍𝒐𝒈𝒙
𝒅𝒙 𝒙

𝒅𝒚
Find the general solution of the differential equation 𝒙𝒍𝒐𝒈𝒙 +𝒚=
𝒅𝒙

𝟐
𝒍𝒐𝒈𝒙
𝒙

dy
xy = ( x + 2) ( y + 2) (1, − 1)
dx

If xy dy = ( x + 2) ( y + 2) then Find the solution of the curve passing through
dx

point (1, − 1) .

43

dy y  y 𝒙=𝟏
− + cos ec  = 0, y = 0
dx x x

Find the solution of the differential equation dy y  y
− + cos ec  = 0, y = 0
dx x x
when x=1.

(x + xy )dy = ( x + y )dx
2 2 2

Find the solution of the differential equation (x 2 + xy )dy = ( x 2 + y 2 )dx

𝒅𝒚
+ 𝒚𝒄𝒐𝒕𝒙 = 𝟐𝒙 + 𝒙𝟐 𝒄𝒐𝒕𝒙(𝒙 ≠ 𝟎)
𝒅𝒙

y =o 𝒙 = 𝝅
𝟐

𝒅𝒚
Find the specific solution of the differential equation 𝒅𝒙 + 𝒚𝒄𝒐𝒕𝒙 = 𝟐𝒙 + 𝒙𝟐 𝒄𝒐𝒕𝒙(𝒙 ≠ 𝟎)

𝝅
if 𝒚 = 𝟎; 𝒙 = 𝟐

𝒅𝒚
− 𝒚 = 𝒄𝒐𝒔𝒙
𝒅𝒙

𝒅𝒚
Find the solution of the differential equation. − 𝒚 = 𝒄𝒐𝒔𝒙
𝒅𝒙

ydx − xdy
=0 x 1
y
ydx − xdy
Find the general solution of the differential equation =0
y

while x  1 .
dy 1− y2
+
dx 1− x2
dy 1− y2
Find the general solution of the differential equation +
dx 1− x2
𝒅𝒚 𝝅
𝒄𝒐𝒔𝟐 𝒙 + 𝒚 = 𝒕𝒂𝒏𝒙(𝟎 ≤ 𝒙 < )
𝒅𝒙 𝟐

Find the general Solution of defferential 𝒄𝒐𝒔𝟐𝒙 𝒅𝒚
𝒅𝒙
𝝅
+ 𝒚 = 𝒕𝒂𝒏𝒙(𝟎 ≤ 𝒙 < 𝟐 )

44

(Vector Algebra)

Very short answer questions (1 mark)

;fn a = iˆ + ˆj + 2kˆ gks rks 𝑎⃗. 𝑎⃗ dk eku Kkr dhft,A

If a = iˆ + ˆj + 2kˆ then Find the value of 𝑎⃗. 𝑎⃗ .

lfn'k a = iˆ + ˆj + 2kˆ dk ekiakd Kkr dhft,A

Find the modulus of vector a = iˆ + ˆj + 2kˆ .
;fn dksbZ lfn'k v{kksa ox, oy, oz ds lkFk Øe'k%  ,  ,  dks.k cukrh gS] rc fl)
dhft, fd Sin 2 + Sin 2  + Sin 2 = 2

If a vector makes angles  ,  ,  with the axes ox, oy, oz respectively,
then prove that Sin  + Sin  + Sin  = 2
2 2 2

fl) dhft, fd lfn'k 2iˆ − 3 ˆj + 5kˆ rFkk − 2iˆ + 2 ˆj + 2kˆ ijLij yac gSA
Prove that vectors 2iˆ − 3 ˆj + 5kˆ and − 2iˆ + 2 ˆj + 2kˆ are mutually
perpendicular.

  →' →'
;fn nks lfn'k a vkSj b gS bl izdkj gS] fd a = 2 , b = 3 , vkSj 𝑎⃗. 𝑏
⃗⃗ = 4,
→ →
a −' b
rks Kkr dhft,A

  →' →'
If two vectors a and b are such that a = 2 , b = 3 , and 𝑎⃗. 𝑏⃗⃗ = 4,
→ →
Then Find a − b '
.

lfn'k iˆ + ˆj ij lfn'k iˆ − ˆj dk iz{ksi Kkr dhft,A

Find the projection of vector iˆ + ˆj on iˆ − ˆj .

45

(Short Answers Questions : Marks 4)
 
nks lfn'kksa a vkSj b ds ifjek.k Kkr dhft,] ;fn muds ifjek.k leku gS vkSj

buds chp dk dks.k 600 rFkk a . b = 8 

Find the magnitude of two vectors a and b , if their magnitudes are equal
 
and the angle between them is 600 and a . b = 8.

 
;fn nks bdkbZ lfn'k aa vkSj b ds chp dk dks.k 𝜃 gks rks] fl) dhft, &
𝜃
𝑠𝑖𝑛 = |𝑎⃗ − 𝑏⃗⃗|
2
 
If the angle between two unit vectors a and b is 𝜃 , then prove –
𝜃
𝑠𝑖𝑛 = |𝑎⃗ − 𝑏⃗⃗|
2

n”kkZb, fd fcanq 𝐴(1,2,7), 𝐵(2,6,3) vkSj 𝐶(3,10, −1) lajs[k gSaA
Show that (using vector method) points A(1,2,7),B(2,6,3) and
C(3,10,-1) are co-linear.

;fn 𝑎⃗ = (2𝑖̂ + 2𝑗̂ + 3𝑘̂ ), 𝑏⃗⃗ = (−𝑖̂ + 2𝑗̂ + 𝑘̂ ) vkSj 𝑐⃗ = (3𝑖̂ + 𝑗̂) bl
izdkj gSa fd 𝑎⃗ + 𝜆𝑏⃗⃗, 𝑐⃗ ij yac gS rks 𝜆 dk eku Kkr dhft,A
If 𝑎⃗ = (2𝑖̂ + 2𝑗̂ + 3𝑘̂ ), 𝑏⃗⃗ = (−𝑖̂ + 2𝑗̂ + 𝑘̂ ) and 𝑐⃗ = (3𝑖̂ + 𝑗̂) such
that 𝑎⃗ + 𝜆𝑏⃗⃗, 𝑐⃗ perpendiculor on 𝑐⃗, find out the value of 𝜆.


cy F = 4iˆ − 3 ˆj + 2kˆ }kjk ,d d.k dks ljy js[kk ds vuqfn'k fcUnq ¼3] 2]&1½ ls
fcUnq ¼2]&1]4½ rd foLFkkfir djus esa fd;k x;k dk;Z Kkr dhft,A
A particle is displaced along a strainght line from point (3,2,-1) to point
(2,-1,4) by applying force 𝐹⃗ = 4𝑖̂ − 3𝑗̂ + 2𝑘̂. Find the work done by
force.

fl) dhft, fd 3iˆ + ˆj + 2kˆ vkSj 2iˆ − 2 ˆj + 4kˆ ls izR;sd ij yac ek=d lfn'k
 iˆ − ˆj − kˆ  gSA
 
 
 3 

46

̂
𝑖̂−𝑗̂ −𝑘
Prove that is perpendicular unit vector on each 3𝑖̂ + 𝑗̂ + 2𝑘̂ and
√3
2𝑖̂ − 2𝑗̂ + 4𝑘̂

ml lekarj prqHkqZt dk {ks=Qy Kkr dhft, ftldh nks vklUu Hkqtk,¡ lfn'kksa
 
a = iˆ − ˆj + 3kˆ vkSj b = 2 iˆ − 7 ˆj + kˆ ls fu:fir gSA
Find the area of parallelogram whose adjacent sides are 𝑎⃗ = 𝑖̂ − 𝑗̂ + 3𝑘̂
and 𝑏⃗⃗ = 2𝑖̂ − 7𝑗
̂ + 𝑘̂.

 dk eku Kkr dhft, ;fn nks lfn'k 2 iˆ + 3 ˆj − kˆ vkSj − 4iˆ − 6 ˆj + kˆ
(i) yEcor~] (ii) lekUrj gSA

Find the value of 𝜆 if two vectors 2𝑖̂ + 3𝑗̂ − 𝑘̂ and −4𝑖̂ − 6𝑗̂ + 𝜆𝑘̂
(i) Perpendiculor
(ii) Parallel
 
lfn'k a = iˆ + ˆj − kˆ vkSj b = 3 iˆ − 2 ˆj + kˆ ds chp dk dksT;k (cosine) Kkr dhft,A
Find cosine between vector 𝑎⃗ = 𝑖̂ + 𝑗̂ − 𝑘̂ and 𝑏⃗⃗ = 3𝑖̂ − 2𝑗̂ + 𝑘̂.

fl) dhft, fd [𝑎⃗ + 𝑏⃗⃗ 𝑏⃗⃗ + 𝑐⃗ 𝑐⃗ + 𝑎⃗] = 2[𝑎⃗ ⃗⃗⃗⃗
𝑏 𝑐⃗]
Prove that [𝑎⃗ + 𝑏⃗⃗ 𝑏⃗⃗ + 𝑐⃗ 𝑐⃗ + 𝑎⃗] = 2[𝑎⃗ ⃗⃗⃗⃗
𝑏 𝑐⃗]

𝑖̂ + 3𝑗̂ + 7𝑘̂ 2iˆ − 3 ˆj + 6kˆ

Find the projection of vector 𝑖̂ + 3𝑗̂ + 7𝑘̂ on vector 2iˆ − 3 ˆj + 6kˆ

⃗⃗, ⃗𝒃⃗, 𝒄
𝒂 ⃗⃗ ⃗⃗ + ⃗𝒃⃗ + 𝒄
𝒂 ⃗⃗ = 𝟎 ⃗⃗. ⃗𝒃⃗ +
𝒂
⃗⃗⃗⃗ 𝒄
𝒃. ⃗⃗ + 𝒄.
⃗⃗⃗ 𝒂
⃗⃗
⃗⃗, ⃗𝒃⃗, 𝒄
If unit vectors 𝒂 ⃗⃗ + ⃗𝒃⃗ + 𝒄
⃗⃗ are such that 𝒂 ⃗⃗ = 𝟎 then find out
⃗⃗. ⃗𝒃⃗ + 𝒃.
𝒂 ⃗⃗⃗⃗ 𝒄
⃗⃗ + 𝒄.
⃗⃗⃗ 𝒂
⃗⃗.
𝑨(𝟏, 𝟏, 𝟏), 𝑩(𝟏, 𝟐, 𝟑)
𝑪(𝟐, 𝟑, 𝟏)
If vertices of a triangle are 𝑨(𝟏, 𝟏, 𝟏), 𝑩(𝟏, 𝟐, 𝟑) and 𝑪(𝟐, 𝟑, 𝟏)
then find the area of triangle using vector method.

47

(Three Dimensional Geometry)
Very short answer questions (2 mark)

;fn fdlh ljy js[kk dh fnd~&dksT;k,¡ cos  , cos  , cos  gks rks fl) dhft, fd
𝑐𝑜𝑠2 ∝ +𝑐𝑜𝑠2𝛽 + 𝑐𝑜𝑠2𝛾 = −1
x y z x y z
fl) dhft, fd js[kk,¡ = = rFkk = = ijLij yacor gSA
1 2 1 1 −1 1
𝑥 𝑦 𝑧 𝑥 𝑦 𝑧
Prove that lines = = and = = are perpendicular
1 2 1 1 −1 1

ml lery dk lehdj.k Kkr dhft, tks x, y rFkk z& v{kksa ij Øe'k% 2] 3 vkSj 4
vUr%[k.M dkVrk gSA
Find the equation of the plane which intercepts 2, 3 and 4 on X,Y and Z
axes respectively.
;fn ,d js[kk ds fnd~ vuqikr 2] &1] &2 gS rks bldh fnd~ dkslkbu Kkr dhft,A
If the direction ratio of a line is 2, -1, -2 then Find its direction cosine.
ml ljy js[kk dh fnd~ dksT;k,¡ Kkr dhft, tks v{kksa ls leku dks.k cukrh gSaA
Find the direction cosines of the straight line which makes equal angles
with the axes.

;fn fdlh ljy js[kk dh fnd~ dksT;k,¡ cos  , cos  , cos  gks rks fl) djks
fd 𝑠𝑖𝑛2 ∝ +𝑠𝑖𝑛2 𝛽 + 𝑠𝑖𝑛2 𝛾 = 2
If direction cosines of a strainght line are 𝑐𝑜𝑠 ∝, 𝑐𝑜𝑠𝛽, 𝑎𝑛𝑑 𝑐𝑜𝑠𝛾
Then prove that 𝑠𝑖𝑛2 ∝ +𝑠𝑖𝑛2 𝛽 + 𝑠𝑖𝑛2 𝛾 = 2

𝟐𝒙 − 𝒚 + 𝒛 = 𝟔 𝒙 + 𝒚 + 𝟐𝒛 = 𝟕

Find angle between planes 2𝑥 − 𝑦 + 𝑧 = 6 and 𝑥 + 𝑦 + 2𝑧 = 7.
fl) dhft, fd lery 𝑥 + 2𝑦 + 3𝑧 = 6 vkSj 3𝑥 − 3𝑦 + 𝑧 = 1 ijLij
yEcor~ gSaA
Prove that planes 2𝑥 − 𝑦 + 𝑧 = 6 and 3𝑥 − 3𝑦 + 𝑧 = 1 are
perpendicular.

48

(Long Answers Questions : Marks 6)

fcUnqvksa (2, 2,−1), (3, 4, 2) vkSj (7, 0, 6) ls tkus okys lery dk lehdj.k Kkr

dhft,A

Find the equation of plane passing through points (2,2,-1), (3,4,2) and

(7,0,6)

fl) dhft, fd js[kk,¡ r = iˆ + ˆj − kˆ + d (3iˆ − ˆj ) rFkk r = 4iˆ − kˆ + a(2iˆ + 3 kˆ)

izfrPNsn djrh gSaA izfrPNsn fcUnq Hkh Kkr dhft,A
̂ + 𝒅(𝟑𝒊̂ − 𝒋̂) and 𝒓
⃗⃗ = 𝒊̂ + 𝒋̂ − 𝒌
Prove that lines 𝒓 ̂ + 𝒂(𝟐𝒊̂ + 𝟑𝒌
⃗⃗ = 𝟒𝒊̂ − 𝒌 ̂)

Intersect at a point Find intersecting point also.

js[kkvksa 𝑟⃗ = (1 + 2𝜆)𝑖̂ + (2 + 3𝜆)𝑗̂ + (3 + 4𝜆) rFkk 𝑟⃗ = (2 + 3𝜇)𝑖̂ + (3 + 4𝜇)𝑗̂ +

(4 + 5𝜇) ds chp dh U;wure nwjh lfn'k fof/k ls Kkr dhft,A

Find the shorlest distance between given lines using vector method.

𝑟⃗ = (1 + 2𝜆)𝑖̂ + (2 + 3𝜆)𝑗̂ + (3 + 4𝜆)𝑘̂

𝑟⃗ = (2 + 3𝜇)𝑖̂ + (3 + 4𝜇)𝑗̂ + (4 + 5𝜇)𝑘̂

⃗⃗. (𝟐𝒊 + 𝟐𝒋 − 𝟑𝒌) = 𝟕, 𝒓.
𝒓 ⃗⃗⃗ (𝟐𝒊 + 𝟓𝒋 + 𝟑𝒌) = 𝟗 (2, 1,

3)

Find the vector equation of plane which passess through intersection

line of planes and 𝒓⃗⃗. (𝟐𝒊 + 𝟐𝒋 − 𝟑𝒌) = 𝟕 𝒂𝒏𝒅 𝒓.
⃗⃗⃗ (𝟐𝒊 + 𝟓𝒋 + 𝟑𝒌) = 𝟗 points

(2, 1, 3)

49

𝟑𝒙 − 𝒚 + 𝟐𝒛 − 𝟒 =

𝟎 𝒙+𝒚+𝒛−𝟐=𝟎 (2, 2, 1)

Find the equation of a plane which passess through intersecting

point of planes 𝟑𝒙 − 𝒚 + 𝟐𝒛 − 𝟒 = 𝟎 and 𝒙 + 𝒚 + 𝒛 − 𝟐 = 𝟎 and the

points (2, 2, 1)

𝒚+𝟏 𝒛+𝟏 𝒙−𝟑 𝒚−𝟓 𝒛−𝟕
= = = =
𝟕 −𝟔 𝟏 −𝟐 𝟏

𝒚+𝟏 𝒛+𝟏 𝒙−𝟑
Find the shortest distance between lines = = and =
𝟕 −𝟔 𝟏

𝒚−𝟓 𝒛−𝟕
=
−𝟐 𝟏

50

(linear programming)
(Long Answers Questions : Marks 6)

1) fuEu vojks/kksa ds varxZr 𝑧 = 3𝑥 + 4𝑦 dk jsf[kd izksxzkeu xzkQh; fof/k ls
vf/kdrehdj.k dhft,A
𝑥 + 𝑦 ≤ 4; 𝑥 ≥ 0; 𝑦 ≥ 0
Maximize given objective function 𝑧 = 3𝑥 + 4𝑦 under given constraints
using linear programming graphically
𝑥 + 𝑦 ≤ 4; 𝑥 ≥ 0; 𝑦 ≥ 0
2) fuEu vojks/kksa ds varxZr 𝑧 = 5𝑥 + 10𝑦 dk jSf[kd izksxzkeu xzkQh; fof/k ls
U;wurehdj.k ,oa vf/kdrehdj.k dhft,A
𝑥 + 2𝑦 ≤ 120; 𝑥 + 𝑦 ≥ 60; 𝑥 − 2𝑦 ≥ 0; 𝑥, 𝑦 ≥ 0
Minimize and maximize the function 𝑧 = 5𝑥 + 10𝑦 under given
constraints using linear programming graphically Type equation here.
𝑥 + 2𝑦 ≤ 120; 𝑥 + 𝑦 ≥ 60; 𝑥 − 2𝑦 ≥ 0; 𝑥, 𝑦 ≥ 0
3) js'kek nks izdkj ds HkksT; P vkSj Q dks bl izdkj feykuk pkgrh gS fd feJ.k esa
foVkfeu vo;oksa esa 8 ek=d foVkfeu A rFkk 11 ek=d foVkfeu B gksAa HkksT; P
dh ykxr Rs 60/kg vkSj HkksT; Q dh ykxr Rs 80/kg gSA HkksT; P esa 3 ek=d/kg
foVkfeu A vkSj 5 ek=d/kg foVkfeu B gSA tcfd HkksT; Q esa 4 ek=d/kg foVkfeu
A vkSj 2 ek=d/kg foVkfeu B gSA feJ.k dh U;wure ykxr Kkr dhft,A
Reshma wishes to mix two type of food P and Q in such a way that the
vitamin contents of the mixture contain at least 8 unit of vitamins A and 11
units of vitamin B. Food P costs Rs 60/kg and food Q costs Rs 80/kg. Food
P containts 3 unit/kg of vitamin A and 5 unit/kg vitmins while Food Q
contains 4 unit/kg of vitamin A and 2 unit/kg of vitamin B. Determine the
minimum cost mixture.

51

4) ,d çdkj ds dsd ds fy, 200 xzke vkVk rFkk 25 xzke olk dh vko';drk gksrh
gS rFkk nwljh çdkj ds dsd ds fy, 100 xzke vkVs rFkk 50 xzke olk dh vko';drk
gksrh gS dsdksa dh vfèkdre la[;k Kkr dhft, tks 5 fdyks xzke vkVs rFkk 1
fdyksxzke olk ls cu ldrs gSa ;g eku fy;k x;k gS fd dsdksa dks cukus ds fy,
vU; inkFkksZa dh deh ugha jgsxhA
One type of cake requires 200 grams of flour and 25 grams of fat and the
other type of cake requires 100 grams of flour and 50 grams of fat. Find the
maximum number of cakes that can be made from 5 kilogram of flour and
1 kilogram of fat. It can be assumed that there will be no shortage of other
ingredients to make the cakes.

5) ,d fuekZ.kdrkZ uV vkSj cksYV dk fuekZ.k djrk gS ,d iSdsV uVksa ds fuekZ.k esa
e'khu A ij 1 ?kaVk vkSj e'khu B ij 3 ?kaVs dke djuk iM+rk gS tcfd ,d
iSdsV cksYV ds fuekZ.k esa 3 ?kaVs e'khu A ij vkSj ,d ?kaVk e'khu B ij dke
djuk iM+rk gS og uVkas ls 17-50 #i, çfr iSdsV vkSj cksMZ ij 7 #i, çfr iSdsV
ykHk dekrk gS ;g çfrfnu e'khuksa dk vfèkdre mi;ksx 12 ?kaVs fd;k tk, rks
çR;sd uV vkSj cksYV ds fdrus iSdsV mRikfnr fd;k tk, rkfd vfèkdre ykHk
dek;k tk ldsA
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine
A and 3 hours of machine B to product 1 packet of nuts. It takes 3 hour on
machine A and 1 hour on machine B to produce 1 packet of bolts. He earns
profit of Rs 17.50 packets of each should be produces each day 50 as to
maximize his profit, If he operates his machines for at the most 12 hour of
the day.

52

UNIT 6 (PROBABILITY)

Very short answer questions (2 mark)

iz- 1 ;fn 𝑷(𝑨) = 𝟏 , 𝑷(𝑩) = 𝟏 rFkk P( A  B) = 1 gks rks P A ,oa B
P  Kkr
𝟐 𝟑 4 B  A
dhft,A
𝟏 𝟏  A B
If 𝑷(𝑨) = , 𝑷(𝑩) = or P( A  B) = 1 Find out P  and P 
𝟐 𝟑 4 B  A

iz- 2 ;fn P( A) = 0.8 , P( B) = 0.5 vkSj B gks rks Kkr dhft,A
P  = 0.4 P ( AUB )
 A

B
If P( A) = 0.8 , P( B) = 0.5 and P  = 0.4 then find out P ( AUB ) Find out.
 A

iz- 3 nks ?kukdkj ikls ds lkFk&lkFk mNkys tkrs gSaA igys ikls ij fo"ke la[;k vFkok ;ksx 9
vkus dh izkf;drk Kkr dhft,A

Two cuboids die are tossed together. Find the probability of getting an odd number

on first die or sum 9.

iz- 4 nks Lora= ?kVuk,¡ A o B dh izkf;drk,¡ Øe'k% P(A) = 0.30, P(B) = 0.75 rks P( A B )

rFkk P ( AB ) Kkr dhft,A

Probability of Two independent events. A and B are respectively P(A) = 0.30, P(B) =

0.75 then find out P ( AB )

iz-5 fl) dhft, fd ;fn A vkSj B Lora= ?kVuk,a gS rks A vkSj B esa ls U;wure ,d ds gksus

dh izkf;drk & 𝐼 − 𝑃(𝐴′ )𝑃(𝐵′ )

Prove that, If A and B are independent events then probability of

occurrence of at least one of A and B is givenby 1 − 𝑃(𝐴′ )𝑃(𝐵′ )

53

(Short Answers Questions : Marks 4)

iz-1 nks FkSys A vkSj B esa Øe'k% 8 gjh vkSj 9 lQsn rFkk 5 gjh vkSj 4 lQsn xsansa
j[kh gSA fdlh ,d FkSys esa ls ;n`PN;k ,d xsna fudkyh tkrh gS tks fd gjh
jax dh gSA izkf;drk Kkr dhft, fd ;g xsan FkSys B ls fudkyh x;h gSA
In two bags there are 8 green and 9 white balls and 5 green and 4 white
balls respectively. A ball is taken out at random from one of the bags,
which is green in colour. Find the probability that this ball is drawn from
bag A.

iz-2 rk'k ds 52 iÙkksa dh ,d xM~Mh lss nks iÙks mÙkjksÙkj izfrLFkkiuk ds lkFk fudkys
tkrs gSaA bDdksa dh la[;k dk izkf;drk caVu Kkr dhft,A
Two cards are drawn successively with replacement from a well-shuffled
deck of 52 cards. Find the probability distribution of the number of aces.
iz-3 ,d ;kn`fPNd pj X ds laxr izkf;drk caVu fuEukuqlkj gS &

X 0 1 2 3 4 5

P(X) 0 K 2K K2 2K2 2K2 + K

rc Kkr dhft,A
The corresponding probability distribution of a random variable g is as
follows –

X 0 1 2 3 4 5
P(X) 0 K 2K K2 2K2 2K2 + K

Then Find the value of is K
(i) 𝐾

(ii) 𝑃(𝑋 < 2)

(iii) 𝑃(0 𝑋 < 3)

(iv) 𝑃(𝑋 > 4)

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iz-4 nks FkSys I vkSj II fn, gSA FkSys I esa 3 yky vkSj 4 dkyh xsans gS tcfd FkSys II esa
5 yky vkSj 6 dkyh xsans gSA fdlh ,d FkSys esa ls ;kn`PN;k ,d xsan fudkyh
x;h gSA gS tks fd yky jax dh gSA bl ckr dh D;k izkf;drk gS fd ;g xsna
FkSys II ls fudkyh x;h gSA
Bag I contains 3 red and 4 black balls while in another bag II contains 5
red and 6 black balls. One ball is drawn random from one of the bags and
it is found to be red. Find the probability that it was drawn from bag II.
iz-5 ,d O;fDr ds ckjs esa Kkr gS fd og 4 esa ls 3 ckj lR; cksyrk gSA og ,d
ikls dks mNkyrk gS vkSj crykrk gS fd ml ij vkus okyh la[;k 6 gS bldh
izkf;drk Kkr dhft, fd ikls ij vkus okyh la[;k okLro esa gSA
A man is known to speak truth 3 out of 4 times. He throuws a die and
reports that it is a 6. Find the probability that it is actually a 6.
iz-6 ,d dy'k esa 5 yky vkSj 5 dkyh xsna sa gSaA ;n`PN;k ,d xsan fudkyh tkrh gS]
bldk jax uksV djus ds ckn iqu% dy'k esa j[k nh tkrh gSA iqu% fudkys x,
jax dh nks vfrfjDr xsnsa dy'k esa j[k nh tkrh gSA rFkk dy'k esa ls ,d xsan
fudkyh tkrh gSA nwljh xsan dh yky gksus dh izkf;drk Kkr dhft,A
An urn contains 5 red and 5 black balls. A ball is drawn at random; Its
colour is noted and is returned to the urn. Moreover, 2 additional balls of
the colour drawn are put in the urn and then a ball is drawn at random.
What is the probability that second ball is red.
iz-7 ,d flDds dh nks mNkyksa esa fprksa dh la[;k dk izkf;drk caVu Kkr dhft,A
lkFk gh ek/; vkSj izlj.k Hkh Kkr dhft,A
Find the probability distribution of the number of heads in two tosses of
a coin. Also, Find the mean and variance.
iz-8 52 iÙkksa dh vPNh rjg QsaVh xbZ xM~Mh esa ls ,d ds ckn ,d rhu iÙks fcuk
izfrLFkkfir fd;s x;s fudkys x,A igys nks iÙkksa dk ckn’kkg vkSj rhljs dk
bDdk gksus dh D;k izkf;drk gSA
Three cards are drawn successively without replacement from a pack of
52 well-shuffled cards. What is the probability that first two cards are
kings and the third card drawn is an ace?

ll

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