The words you are searching are inside this book. To get more targeted content, please make full-text search by clicking here.

NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

Discover the best professional documents and content resources in AnyFlip Document Base.
Search
Published by rajusingh79, 2019-07-31 10:20:56

NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

Keywords: IIT JEE study material, NEET Study Material, JEE mains Study Material, JEE advanced study material, AIIMS Study Material, IIT JEE Foundation study material, NEET Foundation study material, CBSE Study Material, Test Series, Question Bank, Editable Study Material, School Exams study material, board exams study material, XII board exams Study Material, X board exams Study Material, Study Material, JEE mains, JEE advanced, Video Lectures, Study Innovations, online tuition, home tuition, online tutors, coaching & tutorials for English, Mathematics, Science, Physics, Chemistry, Biology, Soft copy study material, customized study material provider for coaching institutes, how to make study material for coaching institute, study material for coaching classes

Class XII Chapter 7 – Integrals Maths

Question 5:
Answer
Let cos x = t ⇒ −sinx dx = dt
When x = 0, t = 1 and when

Page 151 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:
Answer

Let ⇒ dx = dt
Page 152 of 216

Class XII Chapter 7 – Integrals Maths

Page 153 of 216

Class XII Chapter 7 – Integrals Maths
Question 7:
Answer

Let x + 1 = t ⇒ dx = dt
When x = −1, t = 0 and when x = 1, t = 2

Question 8:
Answer
Let 2x = t ⇒ 2dx = dt
When x = 1, t = 2 and when x = 2, t = 4

Page 154 of 216

Class XII Chapter 7 – Integrals Maths

Question 9: is

The value of the integral
A. 6
B. 0
C. 3
D. 4
Answer

Page 155 of 216

Class XII Chapter 7 – Integrals Maths

Let cotθ = t ⇒ −cosec2θ dθ= dt
Page 156 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is A.

Question 10:
If
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x
Answer

Integrating by parts, we obtain

Page 157 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is B.

Page 158 of 216

Class XII Chapter 7 – Integrals Maths
Question 1: Exercise 7.11
Answer

Adding (1) and (2), we obtain

Question 2:
Answer

Page 159 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 3:
Answer

Page 160 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 4:
Answer

Page 161 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 5:
Answer
It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].

Page 162 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:
Answer
It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].

Question 7:
Answer

Page 163 of 216

Class XII Chapter 7 – Integrals Maths

Question 8:
Answer

Page 164 of 216

Class XII Chapter 7 – Integrals Maths

Question 9:
Answer

Page 165 of 216

Class XII Chapter 7 – Integrals Maths

Question 10:
Answer

Page 166 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 11:
Answer
As sin2 (−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2x is an even function.

Page 167 of 216

Class XII Chapter 7 – Integrals Maths

It is known that if f(x) is an even function, then

Question 12:
Answer
Adding (1) and (2), we obtain

Page 168 of 216

Class XII Chapter 7 – Integrals Maths

Question 13:

Answer
As sin7 (−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2x is an odd function.
It is known that, if f(x) is an odd function, then

Question 14:
Answer

It is known that,

Page 169 of 216

Class XII Chapter 7 – Integrals Maths

Question 15:
Answer

Adding (1) and (2), we obtain
Question 16:
Answer

Page 170 of 216

Class XII Chapter 7 – Integrals Maths
Adding (1) and (2), we obtain

sin (π − x) = sin x
Adding (4) and (5), we obtain

Let 2x = t ⇒ 2dx = dt
When x = 0, t = 0 and when

Page 171 of 216

Class XII Chapter 7 – Integrals Maths
Question 17:

Answer

It is known that,
Adding (1) and (2), we obtain

Question 18:
Answer
It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4

Page 172 of 216

Class XII Chapter 7 – Integrals Maths

Question 19: if f and g are defined as and
Show that
Answer

Adding (1) and (2), we obtain

Question 20:

Page 173 of 216

Class XII Chapter 7 – Integrals Maths
is
The value of
A. 0
B. 2
C. π
D. 1
Answer

It is known that if f(x) is an even function, then and
if f(x) is an odd function, then

Hence, the correct Answer is C.
Question 21:

The value of is
A. 2

B.
C. 0
D.

Page 174 of 216

Class XII Chapter 7 – Integrals Maths
Answer

Adding (1) and (2), we obtain

Hence, the correct Answer is C.

Page 175 of 216

Class XII Chapter 7 – Integrals Maths
Miscellaneous Solutions
Question 1:
Answer

Equating the coefficients of x2, x, and constant term, we obtain
−A + B − C = 0
B+C=0
A=1
On solving these equations, we obtain

From equation (1), we obtain

Page 176 of 216

Class XII Chapter 7 – Integrals Maths

Question 2:
Answer

Page 177 of 216

Class XII Chapter 7 – Integrals Maths

Question 3:

[Hint: Put ]

Answer

Page 178 of 216

Class XII Chapter 7 – Integrals Maths

Question 4:
Answer

Page 179 of 216

Class XII Chapter 7 – Integrals Maths

Page 180 of 216

Class XII Chapter 7 – Integrals Maths
Question 5:

Answer

On dividing, we obtain

Question 6:

Page 181 of 216

Class XII Chapter 7 – Integrals Maths
Answer

Equating the coefficients of x2, x, and constant term, we obtain
A+B=0
B+C=5
9A + C = 0
On solving these equations, we obtain

From equation (1), we obtain

Question 7:
Answer

Page 182 of 216

Class XII Chapter 7 – Integrals Maths

Let x − a = t ⇒ dx = dt

Question 8:
Answer

Question 9:
Answer

Page 183 of 216

Class XII Chapter 7 – Integrals Maths

Let sin x = t ⇒ cos x dx = dt

Question 10:
Answer

Question 11:
Answer

Page 184 of 216

Class XII Chapter 7 – Integrals Maths

Question 12:
Answer
Let x4 = t ⇒ 4x3 dx = dt

Page 185 of 216

Class XII Chapter 7 – Integrals Maths

Question 13:
Answer
Let ex = t ⇒ ex dx = dt

Question 14:
Answer

Page 186 of 216

Class XII Chapter 7 – Integrals Maths

Equating the coefficients of x3, x2, x, and constant term, we obtain
A+C=0
B+D=0
4A + C = 0
4B + D = 1
On solving these equations, we obtain

From equation (1), we obtain

Question 15:
Answer

= cos3 x × sin x
Let cos x = t ⇒ −sin x dx = dt

Page 187 of 216

Class XII Chapter 7 – Integrals Maths

Question 16:
Answer

Question 17:
Answer

Page 188 of 216

Class XII Chapter 7 – Integrals Maths

Question 18:
Answer

Page 189 of 216

Class XII Chapter 7 – Integrals Maths

Question 19:
Answer

Page 190 of 216

Class XII Chapter 7 – Integrals Maths

From equation (1), we obtain
Page 191 of 216

Class XII Chapter 7 – Integrals Maths

Question 20:
Answer

Page 192 of 216

Class XII Chapter 7 – Integrals Maths

Question 21:
Answer

Page 193 of 216

Class XII Chapter 7 – Integrals Maths

Question 22:

Answer

Equating the coefficients of x2, x,and constant term, we obtain
A+C=1
3A + B + 2C = 1
2A + 2B + C = 1
On solving these equations, we obtain
A = −2, B = 1, and C = 3
From equation (1), we obtain

Page 194 of 216

Class XII Chapter 7 – Integrals Maths

Question 23:
Answer

Page 195 of 216

Class XII Chapter 7 – Integrals Maths
Question 24:

Answer

Integrating by parts, we obtain
Page 196 of 216

Class XII Chapter 7 – Integrals Maths

Question 25:
Answer

Page 197 of 216

Class XII Chapter 7 – Integrals Maths

Question 26:
Answer

Page 198 of 216

Class XII Chapter 7 – Integrals Maths

When x = 0, t = 0 and

Question 27:
Answer

Page 199 of 216

Class XII Chapter 7 – Integrals Maths

When and when

Page 200 of 216


Click to View FlipBook Version