## Rajasthan Board RBSE Class 11 Maths Chapter 10 Limits and Derivatives Miscellaneous Exercise

Question 1.

The value of

is:

(A) 1/3

(B) -1/3

(C) 1

(D) – 1

Solution:

Hence, option (B) is correct.

Question 2.

The value of

(A) 0

(B) ∞

(C) 1

(D) – 1

Solution:

Hence, option (A) is correct.

Question 3.

The value of

(A) 2/3

(B) 1/3

(C) 1/2

(D) 3/2

Solution:

Hence, option (D) is correct.

Question 4.

The value of

(A) 3

(B) 2

(C) 1

(D) – 1

Solution:

Hence, option (C) is correct.

Question 5.

The value of

(A) π/4

(B) π/2

(C) 0

(D) ∞

Solution:

Hence, option (A) is correct.

Question 6.

The value of

(A) 0

(B) 1

(C) log_{e} (ab)

(D) log_{e} (a/b)

Solution:

Hence, option (D) is correct.

Question 7.

The value of

(A) 0

(B) 1

(C) π/180

(D) π

Solution:

Hence, option (C) is correct.

Question 8.

The value of

(A) 0

(B) 1/2

(C) -1/2

(D) -1

Solution:

Hence, option (B) is correct.

Question 9.

The value of

(A) 0

(B) 81

(C) 4

(D) 1

Solution:

Hence, option (B) is correct.

Question 10.

The value of

(A) 0

(B) ∞

(C) – 1

(D) 1

Solution:

Hence, option (C) is correct.

Question 11.

If y is function of x, then derivative of y with respect to x is :

Solution:

y = ax^{2} + bx + c (Let)

Hence, option (C) is correct.

Question 12.

Derivative of x^{n} is :

(A) x^{n – 1}

(B) (n – 1)x^{n – 2}

(C) nx^{n – 1}

(D) x^{n + 1}/n + 1

Solution:

Let y = x^{n}

Hence, option (C) is correct.

Question 13.

Derivative of [latex]\frac { 1 }{ \sqrt { x } } [/latex] is:

Solution:

Hence, option (B) is correct.

Question 14.

[latex]\frac { d }{ dx }[/latex](5^{x}) is equal to :

(A) 5^{x}

(B) 10^{x}

(C) 10^{x} log_{e} 5

(D) 5^{x} log_{e} 5

Solution:

[latex]\frac { d }{ dx }[/latex] (5^{x}) = 5^{x} log_{e}5 ( ∵[latex]\frac { d }{ dx }[/latex] (a^{x} .log a)

Hence, option (D) is correct.

Question 15.

[latex]\frac { d }{ dx }[/latex] (log_{a} x) is equal to :

Solution:

Hence, option (A) is correct.

Question 16.

If f(x) = x^{3} + 6x^{2} – 5 then f'(1) is equal to :

(A) 0

(B) 9

(C) 4

(D) 15

Solution:

f(x) = x^{3} + 6x^{2} – 5

⇒ f'(x)= 3x^{2} + 12x – 0

⇒ f'(x)= 3x^{2} + 12x

⇒ f'(1)= 3(1)^{2} + 12(1)

⇒ f'(1)= 3 + 12 = 15

Hence, option (D) is correct

Question 17.

Derivative of sec x° is :

Solution:

Hence, option (C) is correct.

Question 18.

Derivative of log_{x} a is :

Solution:

Hence, option (B) is correct.

Question 19.

and f'(0) = 0, then value of c is :

(A) 0

(B) 1

(C) 2

(D) -2

Solution:

Hence, option (D) is correct.

Question 20.

Derivative of log_{e}√x is :

Solution:

Hence, option (A) is correct

Question 21.

then find the value of a, b and c.

Solution:

Question 22.

Evaluate

Solution:

Question 23.

Evaluate

Solution:

Question 24.

Evaluate

Solution:

Question 25.

Evaluate

Solution:

Question 26.

Evaluate

Solution:

Question 27.

Evaluate

Solution:

Question 28.

Solution:

Question 29.

If y = x^{3}. e^{x} sin x, then find [latex]\frac { dy }{ dx }[/latex].

Solution:

Given, y = x^{3}. e^{x} sin x

On differentiating