Updated By SaraNextGen

On March 11, 2024, 11:35 AM

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 sin² (−x) = (sin (−x))² = (−sin x)² = sin²x, therefore, sin²x 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 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,

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 2x = t ⇒ 2dx = 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.