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QUESTION: 1

The value of ‘a’ for which x^{3} - 3x + a = 0 has two distinct roots in [0, 1] is given by

Solution:

Let α, β ∈[0,1].f (x) is continuous on [a,b] & differentiable on (a,b) and f (α) = f (β) = 0

∴ c ∈ (α, β) such that f' (c) = 0 ⇒ c = ±1∉ (0,1)

QUESTION: 2

The value of ‘c’ in Lagrange’s mean value theorem for f (x) = x (x- 2)^{2} in [0, 1]

Solution:

f '(c) = 0 2c(c - 2) + (c - 2)^{2} = 0

c = 2,2/3

∴ c = 2/3 (c ≠2)

QUESTION: 3

For the function f (x) = x^{3} - 6x^{2} + ax + b, if Roll’s theorem holds in [1, 3] with

Solution:

f (1) = f (3) ⇒ a = 11

QUESTION: 4

Find Value of ‘c’ by using Rolle’s theorem for f (x) = log (x^{2} + 2) - log 3 on [-1,1]

Solution:

QUESTION: 5

The chord joining the points where x = p and x = q on the curve y = ax^{2} + bx + c is parallel to the tangent at the point on the curve whose abscissa is

Solution:

Apply Lagrange’s theorm

QUESTION: 6

The least value of k for which the function f(x) = x^{2} + kx + 1 is a increasing function in the interval 1 __<__ x __<__ 2

Solution:

QUESTION: 7

The interval in which f (x) = x^{3} - 3x^{2} - 9x + 20 is strictly decreasing

Solution:

Given f (x) = x^{3} - 3x^{2} - 9x + 20

⇒ f '(x) = 3x^{2} -6x -9

⇒ f '(x) = 3(x - 3)(x +1)

Thus, f (x) is strictly increasing for

x ∈ (-∞,-1) U (3, ∞) and strictly decreasing for x ∈ (-1, 3)

QUESTION: 8

The critical points of

Solution:

f ' (x) = 0 ⇒ x = 1; f^{1} (x) does not exist at x = 2

∴ x = 1 and x = 2 are two critical points

QUESTION: 9

The number of stationary points of f (x) = sin x in [0,2π] are

Solution:

f (x) = sinx ⇒ f '(x) = cosx ⇒ f '(x) = 0

Therefore number of stationary points of f (x) in [0, 2π] is 2.

QUESTION: 10

Local minimum values of the function

Solution:

AM > GM

QUESTION: 11

If the function has maximum at x =-3, then the value of ‘a’ is

Solution:

since f (x) has local maximum at x = -3 ⇒ f ' (-3) = 0 and f ^{11} (-3)< 0

QUESTION: 12

The point at which f (x) = (x- 1)^{4} assumes local maximum or local minimum value are

Solution:

Therefore n = iv is even and f^{iv} (1) = 24> 0

Therefore f (x) has local minimum at x = 1.

QUESTION: 13

The global maximum and global minimum of f (x) = 2x^{3} - 9x^{2} + 12x + 6 in [0, 2]

Solution:

Therefore global maximum M_{1} = max{f (0), f (1), f (2)}= 11

Global minimum

M_{2} = max{f (0), f (1), f (2)}= 6

QUESTION: 14

The approximate value of

Solution:

QUESTION: 15

The approximate value of

Solution:

QUESTION: 16

If the percentage error in the surface area of sphere is k, then the percentage error in its volume is

Solution:

QUESTION: 17

If an error of is made in measuring the radius of a sphere then percentage error in its volume is

Solution:

V% = 3(S%)

QUESTION: 18

The height of a cylinder is equal to its radius. If an error of 1 % is made in its height. Then the percentage error in its volume is

Solution:

h = r and v = ph^{3}; V% = 3( h%)

QUESTION: 19

The slope of the normal to the curve given by

Solution:

QUESTION: 20

The line is a tangent to the curve then n ∈

Solution:

Calculate slope

QUESTION: 21

The points on the curve at which the tangent is perpendicular to x-axis are

Solution:

dy/dx is not defined.

QUESTION: 22

The point on the curve at which the tangent drawn is

Solution:

QUESTION: 23

The sum of the squares of the intercepts on the axes of the tangent at any point on the curve x ^{2/3} + y^{2/3}= a^{2/3} is

Solution:

Equation of the tangent at p (θ) to

QUESTION: 24

If the straight line x cos α + y sinα = p touches the curve at the point (a, b) on it, then

Solution:

Find dy/dx and the equation of the tangent

QUESTION: 25

If the curves x = y² and xy = k cut each other orthogonally then k² =

Solution:

m_{1}.m_{2} =-1

QUESTION: 26

The angle between the curves y = x³ and

Solution:

Find dy/dx to the two curves at (1, 1) they are m_{1} and m_{2}. Then

QUESTION: 27

If the curves ay + x² = 7 and x³ = y cut orthogonally at (1, 1) then a =

Solution:

Slope of the first curve at (1, 1) is Slope of the second curve at (1, 1) is m_{2} = 3

QUESTION: 28

A particle moves along a line is given by then the distance travelled by the particle before it first comes to rest is

Solution:

QUESTION: 29

A particle is moving along a line such that s = 3t^{3} - 8t + 1. Find the time ‘t’ when the distance ‘S’ travelled by the particle increases.

Solution:

QUESTION: 30

A particle moves along a line by S = t^{3} - 9t^{2} + 24t the time when its velocity decreases.

Solution:

### NCERT Solution - Application of Derivative (Part -2)

Doc | 26 Pages

### NCERT Solution - Application of Derivative

Doc | 52 Pages

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