## EXERCISE 1.11

#### Page No 347:

#### Question 1:

#### Answer:

Adding (1) and (2), we obtain

#### Question 2:

#### Answer:

Adding (1) and (2), we obtain

#### Question 3:

#### Answer:

Adding (1) and (2), we obtain

#### Question 4:

#### Answer:

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].

#### Question 6:

#### Answer:

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

#### Question 7:

#### Answer:

#### Question 8:

#### Answer:

#### Question 9:

#### Answer:

#### Question 10:

#### Answer:

Adding (1) and (2), we obtain

#### Question 11:

#### Answer:

As sin^{2 }(−*x*) = (sin (−*x*))^{2} = (−sin *x*)^{2} = sin^{2}*x*, therefore, sin^{2}*x *is an even function.

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

#### Question 12:

#### Answer:

Adding (1) and (2), we obtain

#### Question 13:

#### Answer:

As sin^{7 }(−*x*) = (sin (−*x*))^{7} = (−sin *x*)^{7} = −sin^{7}*x*, therefore, sin^{2}*x *is an odd function.

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

#### Question 14:

#### Answer:

It is known that,

#### Question 15:

#### Answer:

Adding (1) and (2), we obtain

#### Question 16:

#### Answer:

Adding (1) and (2), we obtain

sin (π − *x*) = sin *x*

Adding (4) and (5), we obtain

Let 2*x* = *t* ⇒ 2*dx* = *dt*

When *x* = 0, *t *= 0 and when

x=π2, t=π∴

I=12∫0πlog sin tdt-π2log 2

⇒I=I2-π2log 2 [from 3]

⇒I2=-π2log 2

⇒I=-πlog 2

#### 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

#### Question 19:

Show that if *f* and *g* are defined as and

#### Answer:

Adding (1) and (2), we obtain

#### Question 20:

The value of is

**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.**

#### Answer:

Adding (1) and (2), we obtain

Hence, the correct answer is C.