# NCERT Solutions | Class 11 Maths Chapter 13 | Limits and derivatives

## CBSE Solutions | Maths Class 11

Check the below NCERT Solutions for Class 11 Maths Chapter 13 Limits and derivatives Pdf free download. NCERT Solutions Class 11 Maths  were prepared based on the latest exam pattern. We have Provided Limits and derivatives Class 11 Maths NCERT Solutions to help students understand the concept very well.

### NCERT | Class 11 Maths

Book: National Council of Educational Research and Training (NCERT) Central Board of Secondary Education (CBSE) 11th Maths 13 Limits and derivatives English

#### Limits and derivatives | Class 11 Maths | NCERT Books Solutions

You can refer to MCQ Questions for Class 11 Maths Chapter 13 Limits and derivatives to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.

## NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.1

Evaluate the following limits in Exercises 1 to 22.

Ex 13.1 Class 11 Maths Question 1.

Solution.

Ex 13.1 Class 11 Maths Question 2.

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Ex 13.1 Class 11 Maths Question 3.

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Ex 13.1 Class 11 Maths Question 4.

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Ex 13.1 Class 11 Maths Question 5.

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Ex 13.1 Class 11 Maths Question 6.

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Ex 13.1 Class 11 Maths Question 7.

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Ex 13.1 Class 11 Maths Question 8.

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Ex 13.1 Class 11 Maths Question 9.

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Ex 13.1 Class 11 Maths Question 10.

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Ex 13.1 Class 11 Maths Question 11.

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Ex 13.1 Class 11 Maths Question 12.

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Ex 13.1 Class 11 Maths Question 13.

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Ex 13.1 Class 11 Maths Question 14.

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Ex 13.1 Class 11 Maths Question 15.

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Ex 13.1 Class 11 Maths Question 16.

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Ex 13.1 Class 11 Maths Question 17.

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Ex 13.1 Class 11 Maths Question 18.

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Ex 13.1 Class 11 Maths Question 19.

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Ex 13.1 Class 11 Maths Question 20.

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Ex 13.1 Class 11 Maths Question 21.

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Ex 13.1 Class 11 Maths Question 22.

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Ex 13.1 Class 11 Maths Question 23.

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Ex 13.1 Class 11 Maths Question 24.

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Ex 13.1 Class 11 Maths Question 25.

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Ex 13.1 Class 11 Maths Question 26.

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Ex 13.1 Class 11 Maths Question 27.

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Ex 13.1 Class 11 Maths Question 28.

Solution.

Adding (ii) and (iii), we get 2b = 8 ⇒ b = 4
Subtituting the value of b in (iii), we get
4 – a = 4 ⇒ a = 0
Thus a = 0 and b = 4.

Ex 13.1 Class 11 Maths Question 29.

Solution.

Ex 13.1 Class 11 Maths Question 30.

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Ex 13.1 Class 11 Maths Question 31.

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Ex 13.1 Class 11 Maths Question 32.

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## NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives Ex 13.2

Ex 13.2 Class 11 Maths Question 1.

Find the derivative of x22 at x = 10.
Solution.
let f(x) = x2 – 2
Differentiating (i) with respect to x, we get
f'(x) = 2x
At x = 10, f'(10) = 2(10) = 20.

Ex 13.2 Class 11 Maths Question 2.

Find the derivative of 99x at x = 10.
Solution.
let f(x) = 99x
Differentiating (i) with respect to x, we get
f'(x) = 90
At x = 100, f'(100) = 99.

Ex 13.2 Class 11 Maths Question 3.

Find the derivative of x at x = 10.
Solution.
let f(x) = x
Differentiating (i) with respect to x, we get
f'(x) = 1
At x = 1, f'(1) = 1.

Ex 13.2 Class 11 Maths Question 4.

Find the derivative of the following functions from first principle.
(i) x3 – 27
(ii) (x – 1)(x – 2)
(iii) $$\frac { 1 }{ { x }^{ 2 } }$$
(iv) $$\frac { x+1 }{ x-1 }$$
Solution.

Ex 13.2 Class 11 Maths Question 5.

For the function

Prove that f'(1) = 100f'(0)
Solution.
We have

Ex 13.2 Class 11 Maths Question 6.

Find the derivative of xn + axn-1 + a2xn-2+
…. + an-1x + an for some fixed real number a.
Solution.
Let f(x) = xn + axn-1 + a2xn-2+
…. + an-1x + an
Differentiating (i) with respect to x, we get
f'(x) = nxn-1 + (n – 1)axn-2 + …… + an-1

Ex 13.2 Class 11 Maths Question 7.

For some constants a and b, find the derivative of
(i) (x – a)(x – b)
(ii) (ax2 + b)2
(iii) $$\frac { x-a }{ x-b }$$
Solution.
(i) Let f(x) = (x – a)(x – b) ….(1)
Differentiating (1) with respect to x, we get
f'(x) = (x – a)(x – b)’ + (x – a)’ (x – b)
⇒ f'(x) = (x – a) + (x – b) = 2x – a – b

Ex 13.2 Class 11 Maths Question 8.

Find the derivative $$\frac { { x }^{ n }-{ a }^{ n } }{ x-a }$$ for some constant a.
Solution.
Let f(x) = $$\frac { { x }^{ n }-{ a }^{ n } }{ x-a }$$ ….(i), where a is a constant.
Differentiating (i) with respect to x, we get

Ex 13.2 Class 11 Maths Question 9.

Find the derivative of

Solution.
(i) Let f(x) = $$2x-\frac { 3 }{ 4 }$$ …(1)
Differentiating (i) with respect to x, we get
f'(x) = 2·1 – 0 ⇒ f'(x) = 2.
(ii) Let f(x) = 5x3 + 3x – 1)(x – 1)

Ex 13.2 Class 11 Maths Question 10.

Find the derivative of cos x from first principle.
Solution.
Let f(x) = cos x

Ex 13.2 Class 11 Maths Question 11.

Find the derivative of the following functions:
(i) sin x cos x
(ii) secx
(iii) 5 secx + 4 cosx
(iv) cosecx
(v) 3 cotx + 5 cosecx
(vi) 5sinx – 6 cosx + 7
(vii) 2 tanx – 7 secx.
Solution.
(i) Let f(x) = sin x cos x … (1)
Differentiating (1) with respect to x, we get