MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives

MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives

Limits and Derivatives Important Questions

Limits and Derivatives Short Answer Type Questions

Evaluate the following limits :

Question 1.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 1
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 2

Question 2.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 3
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 4

Question 3.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 5
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 6

Question 4.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 7
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 8

Question 5.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 9
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 10

Question 6.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 11
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 12

Question 7.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 13
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 14

Question 8.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 15
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 16

Question 9.
If MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 17= 405, then find the value of n.
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 18

Question 10.
Find the value of :
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 19
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 20

Question 11.
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 21
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 22

Question 12.
If y = ex cos x, then find the value of \(\frac { dy }{ dx }\)
Solution:
y = ex.cos x
I II
\(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)(ex.cos x)
= ex.\(\frac { d }{ dx }\)cos x + cos x\(\frac { d }{ dx }\)ex
= ex( – sin x) + cos x\(\frac { d }{ dx }\)ex
= ex[cos x – sin x].

Question 13.
If y = ex cos x, then find the value of \(\frac { dy }{ dx }\)
Solution:
y = ex.sin x
I II
\(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)(ex.sin x)
= ex.\(\frac { d }{ dx }\)sin x + sin x\(\frac { d }{ dx }\)ex
= excos x + x.ex
= ex(cos x + sin x).

MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives

Question 14.
Differentiate sin(x + a) w.r.t. x.
Solution:
Let y = sin(x + a)
y = sin x cos a + cos x sin a
∴\(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)(sin x cos a + cos x sin a)
= cos a\(\frac { d }{ dx }\) sin x + sin\(\frac { d }{ dx }\)cos x
= cos a cos x – sin a sin x
⇒ \(\frac { dy }{ dx }\) = cos(x + a).

Question 15.
Differentiate cosecx.cotx w.r.t. x.
Solution:
Let y = cosec x. cot x
⇒ \(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)[cosec x. cot x]
⇒ \(\frac { dy }{ dx }\) = cot x\(\frac { d }{ dx }\) cosec x + cosec x\(\frac { d }{ dx }\) cot x
= – cot xcosec x cot x – cosec x.cosec2 x
\(\frac { dy }{ dx }\) = – cosec x[cot2 x + cosec2 x].

Question 16.
Differentiate \(\frac { cos x }{ 1 + sin x }\) w.r.t. x.
Solution:
Let y = \(\frac { cos x }{ 1 + sin x }\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 23

Question 17.
Differentiate \(\frac { sec c – 1 }{ sec x + 1 }\) w.r.t. x.
Solution:
Let y = \(\frac { sec c – 1 }{ sec x + 1 }\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 24

Question 18.
Differentiate sinn x w.r.t. x.
Solution:
Let y = sinn x
\(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)(sinn x )
Put sin x = t,
\(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)tn = \(\frac { d }{ dt }\)tn. \(\frac { dt }{ dx }\) = n sinn – 1\(\frac { d }{ dx }\)(sin x)
⇒ \(\frac { dy }{ dx }\) = n sinn – 1x cos x.

Limits and Derivatives Long Answer Type Questions

Question 1.
Let f(x) =
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 25
, then find the value of a and b.
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 26

Question 2.
Find the value of MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 27, where
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 28
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 29

Question 3.
f(x) is defined such that
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 30
and MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 31is exist x = 2, then find the value of k.
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 32

Question 4.
If the function f(x) satisfiesMP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 33= π, then find the value of
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 34
Solution:
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 35

Question 5.
Find the differential coefficient of the following functions by using first principle method :
(i) sin(x + 1), (ii) cos(x – \(\frac { π }{ 8 }\),
Solution:
(i) Let f(x) = sin(x + 1)
∴ f(x + h) = sin[x + h + 1]
By definition of first principle,
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 37
By definition of first principle,
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 38

Find the differential coefficient of the following functions

Question 6.
(ax2 + sin x)(p + q cos x). (NCERT)
Solution:
Let y = (ax2 + sin x)(p + q cos x).
∴ \(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)[(ax2 + sin x)(p + q cos x)]
⇒ \(\frac { dy }{ dx }\) = (p + q cos x)\(\frac { d }{ dx }\)(ax2 + sin x) + (ax2 + sin x)\(\frac { d }{ dx }\) (p + q cos x)
⇒ \(\frac { dy }{ dx }\) = (p + q cos x)(2a + cos x) + (ax2 + sin x)(0 – q sin x)
⇒ \(\frac { dy }{ dx }\) = – q sin x(ax2 + sin x) + (p + q cos x)(2a + cos x)

MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives

Question 7.
(x + cos x)(x – tan x) (NCERT)
Solution:
Let y = (x + cos x)(x – tan x)
∴ \(\frac { dy }{ dx }\) = \(\frac { d }{ dx }\)[(x + cos x)(x – tan x)]
⇒ \(\frac { dy }{ dx }\) = (x – tan x)\(\frac { d }{ dx }\)(x + cos x) + (x + cos x)\(\frac { d }{ dx }\)(x – tan x)
⇒ \(\frac { dy }{ dx }\) = (x – tan x)(1 – sinx) + (x + cos x)(1 – sec2 x)
= (x – tan x)(1 – sinx) + (x + cos x)(sec2 x – 1)
⇒ \(\frac { dy }{ dx }\) = (x – tan x)(1 – sinx) – tan2 x(x + cos x)

Question 8.
\(\frac { 4x + 5sin x }{ 3x + 7 cos x}\)
Solution:
Let y = \(\frac { 4x + 5sin x }{ 3x + 7 cos x}\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 39

Question 9.
\(\frac{x^{2} \cos \frac{\pi}{4}}{\sin x}\)
Solution:
Let y = \(\frac{x^{2} \cos \frac{\pi}{4}}{\sin x}\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 40

Question 10.
\(\frac { x }{ 1 + tan x}\)
Solution:
Let y = \(\frac { x }{ 1 + tan x}\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 41

Question 11.
Find the differential coefficient of \(\frac { sec x – 1 }{ sec x + 1}\)
Solution:
Let y = \(\frac { sec x – 1 }{ sec x + 1}\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 42

Question 12.
If y = \(\frac { x }{ x + 4 }\), then prove that:
x\(\frac { dy }{ dx}\) = y(1 – y)
Solution:
Given : y = \(\frac { x }{ x + 4 }\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 43

Question 13.
If y = \(\sqrt {x}\) + \(\frac{1}{\sqrt{x}}\), then prove that : 2x\(\frac { dy }{ dx}\) + y – 2 \(\sqrt {x}\) = 0
Solution:
Given : y = \(\sqrt {x}\) + \(\frac{1}{\sqrt{x}}\)
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 44

Question 14.
Find the differential coefficient of \(\frac { sin(x + a) }{ cos x }\)
Solution:
Let y = \(\frac { sin(x + a) }{ cos x }\)
⇒ y = \(\frac { sin x + cos a + cos x sin a }{ cos x }\)
⇒ y = \(\frac { sin x + cos a }{ cos x }\) + \(\frac { cos x + sin a }{ cos x }\)
⇒ y = cos a tan x + sin a
∴ \(\frac { dy }{ dx}\) = \(\frac { d }{ dx}\)(cos a tan x + sin a)
= cos a \(\frac { d }{ dx}\) tan x + \(\frac { d }{ dx}\) sin a
= cos a x sec2 x + 0
= sec2x. cos a.

MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives

Question 15.
If f(x) = \(\frac { { x }^{ 100 } }{ 100 } +\frac { { x }^{ 99 } }{ 100 }\)+……..\(\frac { { x }^{ 2 } }{ 2 }\) + x +1, then prove that:
f'(1) = 100 f'(0). (NCERT)
Solution:
Put x = 1, we get
f'(1) = 1 + 1 + ………. 1 + 1 (100 times)
f’(1) = 100 …. (1)
Put x = 0, we get
f'(0) = 0 + 0 + ……… 0 + 1
f’(0) = 1 …. (2)
From equation (1) and (2),
f'(1) = 100 f'(0).

Question 16.
Find the differential coefficient of cos x by first principle method. (NCERT)
Solution:
Let f(x) = cosx
∴ f(x + h) = cos(x + h)
By definition of first principle
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 45

Question 17.
Find the differential coefficient of f(x) = \(\frac { x + 1 }{ x – 1 }\) by first principle method.
Solution:
Given : f(x) = \(\frac { x + 1 }{ x – 1 }\)
f(x + h) = \(\frac { x + h + 1}{ x + h – 1 }\)
By definition of first principle,
MP Board Class 11th Maths Important Questions Chapter 13 Limits and Derivatives 46

MP Board Class 11th Maths Important Questions

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