NCERT Solutions for Class 11 Maths of Chapter 3 Miscellaneous Exercise - Future Study Point

# NCERT Solutions for Class 11 Maths of Chapter 3 Miscellaneous Exercise

The NCERT  maths solutions of chapter 3  miscellaneous exercise containing 10 questions is the extract of the lesson ‘Trigonometry’ . The questions of this exercise are very important for the purpose of your academic exams and entrance exams of engineering.The solutions are explained discretely in order to clear the doubts generally faced by  11 th class students .All questions are explained readily by the teacher expertise in maths. We hope that these solutions by Future Study Point will help you in your preparations of the exam. After you go through each solution write your comment in the form of suggestions, doubts and your requirement.

## NCERT Solutions for Class 11 Maths of Chapter 3 Miscellaneous Exercise

Exercise 3.1 and Exercise 3.2

Exercise 3.3

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## Class 11 th CBSE NCERT Maths Solution of Chapter 3 Miscellaneous Exercise

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Q1. Prove that.

L.H.S

cosA + cosB =2cos(A+B)/2.cos(A–B)/2

As we know cos(–x) = cosx

= 0 = R.H.S

Q2. Prove that

(sin3x + sinx)sinx  + (cos3x – cosx)cosx = 0

∵sinA + sinB = 2sin(A+B)/2.cos(A–B)/2

∵cosA –cosB = –2sin(A+B)/2.sin(A–B)/2

=2sin2x.cosx.sinx – 2sin2x.sinx.cosx  = 0= R.H.S

Q3.Prove that

L.H.S

=cos²x + cos²y + 2cosx.cosy + sin²x + sin²y –2sinx.siny

=(cos²x + sin²x  )+ (cos²y + sin²y) + 2(cosx.cosy – sinx.siny)

=1 + 1 + 2(cosx.cosy – sinx.siny)

=2 + 2(cosx.cosy – sinx.siny)

∵cos(A +B) = cosA.cosB – sinA.sinB

=2 + 2cos(x + y)

=2[1 + cos(x + y)]

∵  cos2A = 2cos²A – 1

⇒1+cos2A  = 2cos²A

Q4.Prove that

L.H.S     (cosx – cosy)² + (sinx – siny)²

=cos²x + cos²y – 2cosx.cosy + sin²x + sin²y –2sinx.siny

=(cos²x  +  sin²x) + (cos²y  + sin²y) –2(cosx.cosy + sinx.siny)

= 1 + 1 – 2cos(x – y)  [∵ cos(A – B) = cosA.cosB + sinA.sinB]

= 2 – 2cos(x– y)

=2[1 – cos(x –y)]

∵ cos2A = 1 – 2sin²A⇒ 1 – cos2A  = 2sin²A

= R.H.S

Q5. Prove that  sinx + sin3x + sin5x + sin7x = 4cosx.cos2x.sin4x

L.H.S

=sinx + sin3x + sin5x + sin7x

= (sinx + sin3x )+ (sin5x + sin7x)

∵ sinA + sinB = 2sin(A + B)/2 .cos(A –B)/2

=2sin2x.cos(–x)  + 2sin6x.cos(–x)

∵ cos(–x) = cosx

=2sin2x.cosx + 2sin6x.cosx

=2cosx(sin2x + sin6x)

∵ sinA + sinB = 2sin(A + B)/2 .cos(A –B)/2

= 4cosx.sin4x.cos(–2x)

∵ cos(–θ) = cosθ

= 4cosx.cos2x.sin4x = R.H.S

Q6. Prove that

L.H.S

= tan6x = R.H.S

Q7. Prove that

Answer. L.H.S = sin3x + (sin2x – sinx)

∵sin2A = 2sinA.cosA

Since x is in 2nd quadrant, so the value of x would be estimated as follows

Dividing all the terms of this equation of inequality

The value of will lye in l quadrant.

We are given that

Placing the value of tanx =(–4/5)

As we have shown above that x/2 lyes in 1st quadrant,tan(x/2) will be positive

x/2 lies in 1 st quadrant so sec(x/2) =√5

sinx/2 = ±√(1–cos²x/2)

x/2 lies in 1st quadrant so sin(x/2) will be positive

Answer.  x is in lll quadrant, so its value would be estimated as follows

π < x < 3π/2, dividing this eq. by 2,we shall have

π/2 < x/2<3π/4, which reveals that x/2  lyes in ll quadrant .

∵cos2x = 2cos²x –1

As we have shown above that x/2 lies in ll quadrant and it is known to us ,the value of cos  is negative in 2nd quadrant.

As we have shown above that x/2 lies in ll quadrant and it is known to us ,the value of sin x/2 is positive in 2nd quadrant.

Since ,x is in ll quadrant ,its value would be estimated as follows

π/2 < x < π

⇒π/4 < x/2 < π/2, implies that  x/2 lies in l st quadrant, which shows that all sinx/2, cos x/2 and tan x/2 will lye in 1st quadrant.

x is in ll quadrant ,so cosx will be negative

cos2x = 2cos²x – 1

We have shown that x/2 lyes in l quadrant and value of cos is positive in l quadrant.

As it is already known to us that x/2 is  in l st quadrant and value of sin x/2 is postitive in lst quadrant.

Multiplying the denominator and numerator by the conjugate of denominator .

## NCERT Solutions of Science and Maths for Class 9,10,11 and 12

### NCERT Solutions for class 9 maths

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### NCERT Solutions for class 10 maths

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 Chapter 1- Chemical reactions and equations Chapter 9- Heredity and Evolution Chapter 2- Acid, Base and Salt Chapter 10- Light reflection and refraction Chapter 3- Metals and Non-Metals Chapter 11- Human eye and colorful world Chapter 4- Carbon and its Compounds Chapter 12- Electricity Chapter 5-Periodic classification of elements Chapter 13-Magnetic effect of electric current Chapter 6- Life Process Chapter 14-Sources of Energy Chapter 7-Control and Coordination Chapter 15-Environment Chapter 8- How do organisms reproduce? Chapter 16-Management of Natural Resources

### NCERT Solutions for class 11 maths

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