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NCERT Solutions Class 12 Maths (Inverse Trigonometric Functions) Miscellaneous Exercise

NCERT Solutions Class 12 Maths (Inverse Trigonometric Functions) Miscellaneous Exercise

NCERT Solutions Class 12 Maths from class 12th Students will get the answers of Chapter-2 (Inverse Trigonometric Functions) Miscellaneous Exercise This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.
Solutions Class 12 Maths (Inverse Trigonometric Functions) Miscellaneous Exercise

Set-1

Question 1. Find the value of Miscellaneous Exercise on Chapter 2

Solution: 


We know that Miscellaneous Exercise on Chapter 2 

Here, Miscellaneous Exercise on Chapter 2

Now, Miscellaneous Exercise on Chapter 2 can be written as :

Miscellaneous Exercise on Chapter 2, where Miscellaneous Exercise on Chapter 2

Hence, the value of Miscellaneous Exercise on Chapter 2 = π/6

Question 2. Find the value of Miscellaneous Exercise on Chapter 2

Solution: 


We know that Miscellaneous Exercise on Chapter 2

Here, Miscellaneous Exercise on Chapter 2

Now, Miscellaneous Exercise on Chapter 2 can be written as:

Miscellaneous Exercise on Chapter 2      -[\tan(2\pi-x)=-\tan x]

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2 where Miscellaneous Exercise on Chapter 2

Hence, the value of Miscellaneous Exercise on Chapter 2 = π/6

Question 3. Prove Miscellaneous Exercise on Chapter 2

Solution: 


Let Miscellaneous Exercise on Chapter 2          -(1)

sin x = 3/5

So,Miscellaneous Exercise on Chapter 2= 4/5

tan x = 3/4

Hence, Miscellaneous Exercise on Chapter 2

Now put the value of x from eq(1), we get

Miscellaneous Exercise on Chapter 2

Now, we have

L.H.S Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2                –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Hence, proved.

Question 4. ProvMiscellaneous Exercise on Chapter 2

Solution: 


Let Miscellaneous Exercise on Chapter 2 

Then sin x = 8/17 

 cos x =Miscellaneous Exercise on Chapter 2 = 15/17

Therefore, Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2         -(1)

Now, let Miscellaneous Exercise on Chapter 2

Then, sin y = 3/5 

Miscellaneous Exercise on Chapter 2 = 4/5

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2     -(2)

Now, we have:

L.H.S.Miscellaneous Exercise on Chapter 2

 From equation(1) and (2), we get

Miscellaneous Exercise on Chapter 2                         

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2                                 –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Hence proved

Question 5. Prove Miscellaneous Exercise on Chapter 2

Solution: 


Let Miscellaneous Exercise on Chapter 2

Then, cos x = 4/5

Miscellaneous Exercise on Chapter 2 = 3/5

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2        -(1)

Now let Miscellaneous Exercise on Chapter 2

Then, cos y = 3/4 

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2            -(2)

Let Miscellaneous Exercise on Chapter 2 

Then, cos z = 33/65

sin z = 56/65 

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2    -(3)

Now, we will prove that : 

L.H.S. Miscellaneous Exercise on Chapter 2

From equation (1) and equation (2)

Miscellaneous Exercise on Chapter 2                                                                 

Miscellaneous Exercise on Chapter 2         –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Using equation(3)

Miscellaneous Exercise on Chapter 2                                                                                                  

Hence proved

Question 6. Prove  Miscellaneous Exercise on Chapter 2

Solution: 


Let Miscellaneous Exercise on Chapter 2  

Thensin x = 3/5 

Miscellaneous Exercise on Chapter 2 = 4/5

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2           -(1)

Now, let Miscellaneous Exercise on Chapter 2  

Then, cos y = 12/13 and sin y = 5/13

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2        -(2)                                                             

Let Miscellaneous Exercise on Chapter 2

Then, sin z = 56/65 and cos z = 33/65

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2           -(3)

Now, we have:

L.H.S.=Miscellaneous Exercise on Chapter 2

From equation(1) and equation(2)

=Miscellaneous Exercise on Chapter 2                                                                           

Miscellaneous Exercise on Chapter 2               –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

From equation (3)

Miscellaneous Exercise on Chapter 2    

Hence proved                                                                                                           

Question 7. Prove  Miscellaneous Exercise on Chapter 2

Solution: 


Let Miscellaneous Exercise on Chapter 2   

Then, sin x = 5/13 and cos x = 12/13.

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2                   -(1)

Let Miscellaneous Exercise on Chapter 2     

Then, cos y = 3/5 and sin y = 4/5

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2                -(2)

From equation(1) and (2), we have

R.H.S.Miscellaneous Exercise on Chapter 2

=Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2          –Miscellaneous Exercise on Chapter 2

=Miscellaneous Exercise on Chapter 2

=Miscellaneous Exercise on Chapter 2

L.H.S = R.H.S

Hence proved

Question 8. Prove Miscellaneous Exercise on Chapter 2

Solution: 


L.H.S.Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2     –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

π/4 

L.H.S = R.H.S

Hence proved

Question 9. Prove Miscellaneous Exercise on Chapter 2

Solution: 


Let x = tan2θ

Then,Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Now, we have

R.H.S = Miscellaneous Exercise on Chapter 2

L.H.S = R.H.S

Hence proved

Question 10. Prove Miscellaneous Exercise on Chapter 2

Solution: 


ConsiderMiscellaneous Exercise on Chapter 2

By rationalizing

=Miscellaneous Exercise on Chapter 2                         

=Miscellaneous Exercise on Chapter 2

=Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

L.H.S = Miscellaneous Exercise on Chapter 2= x/2

L.H.S = R.H.S

Hence proved

Set-2

Question 11. Prove Miscellaneous Exercise on Chapter 2

Solution: 


Put x=\cos2\theta so that, Miscellaneous Exercise on Chapter 2

Then, we have :

LHS = Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2      – Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2  

L.H.S = R.H.S

Hence Proved

Question 12. Prove Miscellaneous Exercise on Chapter 2

Solution: 


L.H.S. = Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Using Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2      -(1)

Now, let Miscellaneous Exercise on Chapter 2 Then, Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2 

Using equation(1), we get,

Miscellaneous Exercise on Chapter 2

L.H.S = R.H.S

Hence Proved

Question 13. Solve Miscellaneous Exercise on Chapter 2

Solution:

Miscellaneous Exercise on Chapter 2

=Miscellaneous Exercise on Chapter 2                      –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

= cos x/sin x

= cot x =1

Therefore, = π/4

Question 14. Solve Miscellaneous Exercise on Chapter 2

Solution:

Miscellaneous Exercise on Chapter 2

Let x = tanθ

Miscellaneous Exercise on Chapter 2        

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

π/4 – θ = θ/2

θ = π/6

So, x = tan(π/6) = 1/√3

Question 15. Solve Miscellaneous Exercise on Chapter 2 is equal to
(A) Miscellaneous Exercise on Chapter 2    (B) Miscellaneous Exercise on Chapter 2 (C) Miscellaneous Exercise on Chapter 2(D) Miscellaneous Exercise on Chapter 2

Solution:


Let tan y = x, Miscellaneous Exercise on Chapter 2

Let \tan^{-1} x=y Then,

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

So, the correct answer is D.

Question 16. Solve Miscellaneous Exercise on Chapter 2, then x is equal to
(A) 0, 1/2      (B) 1, 1/2      (C) 0     (D) 1/2 

Solution:

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2        -(1)

Let Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Therefore, from equation(1), we have

Miscellaneous Exercise on Chapter 2

Put x = siny then, we have:

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

-2 y=\cos^{-1}(1-\sin y)

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

sin y = 0 or 1/2

x = 0 or x = 1/2

But, when x = 1/2 it can be observed that:

L.H.S. = Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

x = 1/2 is not the solution of given equation.

Thus, x = 0 

Hence, the correct answer is C

Question 17. Solve Miscellaneous Exercise on Chapter 2 is equal to

(A) π​/2      (B) π​/3     (C) π​/4      (D) -3π​/4  

Solution:

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2                –Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Miscellaneous Exercise on Chapter 2

Hence, the correct answer is C

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