# NCERT solutions for Mathematics Class 12 chapter 6 - Application of Derivatives [Latest edition]

## Solutions for Chapter 6: Application of Derivatives

Below listed, you can find solutions for Chapter 6 of CBSE, Karnataka Board PUC NCERT for Mathematics Class 12.

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6
Exercise 6.1 [Pages 197 - 199]

### NCERT solutions for Mathematics Class 12 Chapter 6 Application of Derivatives Exercise 6.1 [Pages 197 - 199]

Exercise 6.1 | Q 1.1 | Page 197

Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm

Exercise 6.1 | Q 1.2 | Page 197

Find the rate of change of the area of a circle with respect to its radius r when r = 4 cm

Exercise 6.1 | Q 2 | Page 197

The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?

Exercise 6.1 | Q 3 | Page 197

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Exercise 6.1 | Q 4 | Page 197

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Exercise 6.1 | Q 5 | Page 197

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

Exercise 6.1 | Q 6 | Page 198

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Exercise 6.1 | Q 7 | Page 198

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Exercise 6.1 | Q 8 | Page 198

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Exercise 6.1 | Q 9 | Page 198

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.

Exercise 6.1 | Q 10 | Page 198

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Exercise 6.1 | Q 11 | Page 198

A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Exercise 6.1 | Q 12 | Page 198

The radius of an air bubble is increasing at the rate  1/2  cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Exercise 6.1 | Q 13 | Page 198

A balloon, which always remains spherical, has a variable diameter  3/2 (2x +   1) Find the rate of change of its volume with respect to x.

Exercise 6.1 | Q 14 | Page 198

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

Exercise 6.1 | Q 15 | Page 198

The total cost C(x) in Rupees associated with the production of x units of an item is given by

C(x) = 0.007x3 – 0.003x2 + 15x + 4000.

Find the marginal cost when 17 units are produced

Exercise 6.1 | Q 16 | Page 198

The total revenue in Rupees received from the sale of x units of a product is given by

R(x) = 13x2 + 26x + 15.

Find the marginal revenue when x = 7.

Exercise 6.1 | Q 17 | Page 198

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

(A) 10π

(B) 12π

(C) 8π

(D) 11π

Exercise 6.1 | Q 18 | Page 199

The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is

(A) 116

(B) 96

(C) 90

(D) 126

Exercise 6.2 [Pages 205 - 206]

### NCERT solutions for Mathematics Class 12 Chapter 6 Application of Derivatives Exercise 6.2 [Pages 205 - 206]

Exercise 6.2 | Q 2 | Page 205

Show that the function given by f(x) = 3x + 17 is strictly increasing on R.

Exercise 6.2 | Q 3 | Page 205

Show that the function given by f(x) = sin x is

(a) strictly increasing in (0, pi/2)

(b) strictly decreasing in (pi/2, pi)

(c) neither increasing nor decreasing in (0, π)

Exercise 6.2 | Q 4 | Page 205

Find the intervals in which the function f given by f(x) = 2x2 − 3x is

(a) strictly increasing

(b) strictly decreasing

Exercise 6.2 | Q 5 | Page 205

Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is

(a) strictly increasing

(b) strictly decreasing

Exercise 6.2 | Q 6.1 | Page 205

Find the intervals in which the following functions are strictly increasing or decreasing:

x2 + 2x − 5

Exercise 6.2 | Q 6.2 | Page 205

Find the intervals in which the following functions are strictly increasing or decreasing:

10 − 6x − 2x2

Exercise 6.2 | Q 6.3 | Page 205

Find the intervals in which the following functions are strictly increasing or decreasing:

−2x3 − 9x2 − 12x + 1

Exercise 6.2 | Q 6.4 | Page 205

Find the intervals in which the following functions are strictly increasing or decreasing:

6 − 9x − x2

Exercise 6.2 | Q 6.5 | Page 205

Find the intervals in which the following functions are strictly increasing or decreasing:

(x + 1)3 (x − 3)3

Exercise 6.2 | Q 7 | Page 205

Show that y = log(1+x) - (2x)/(2+x), x> -  1, is an increasing function of x throughout its domain.

Exercise 6.2 | Q 8 | Page 205

Find the values of x for  y = [x(x - 2)]^2 is an increasing function.

Exercise 6.2 | Q 9 | Page 205

Prove that  y = (4sin theta)/(2 + cos theta) - theta is an increasing function of θ in [0, pi/2]

Exercise 6.2 | Q 10 | Page 206

Prove that the logarithmic function is strictly increasing on (0, ∞).

Exercise 6.2 | Q 11 | Page 206

Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).

Exercise 6.2 | Q 12 | Page 206

Which of the following functions are strictly decreasing on (0, pi/2)?

(A) cos x

(B) cos 2x

(C) cos 3x

(D) tan x

Exercise 6.2 | Q 13 | Page 206

On which of the following intervals is the function f given byf(x) =x100 + sin x –1 strictly decreasing?

(A) (0,1)

(B) (pi/2, pi)

(C) (0, pi/2)

(D) None of these

Exercise 6.2 | Q 14 | Page 206

Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].

Exercise 6.2 | Q 15 | Page 206

Let I be any interval disjoint from (−1, 1). Prove that the function f given by f(x) = x + 1/x is strictly increasing on I.

Exercise 6.2 | Q 16 | Page 206

Prove that the function f given by f(x) = log sin x is strictly increasing on (0. pi/2) and strictly decreasing on (pi/2, pi)

Exercise 6.2 | Q 17 | Page 206

Prove that the function f given by f(x) = log cos x is strictly decreasing on (0, pi/2) and strictly increasing on (pi/2, pi)

Exercise 6.2 | Q 18 | Page 206

Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.

Exercise 6.2 | Q 19 | Page 206

The interval in which y = x2 e–x is increasing is

(A) (– ∞, ∞)

(B) (– 2, 0)

(C) (2, ∞)

(D) (0, 2)

Exercise 6.3 [Pages 211 - 213]

### NCERT solutions for Mathematics Class 12 Chapter 6 Application of Derivatives Exercise 6.3 [Pages 211 - 213]

Exercise 6.3 | Q 1 | Page 211

Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.

Exercise 6.3 | Q 2 | Page 211

Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.

Exercise 6.3 | Q 3 | Page 211

Find the slope of the tangent to curve y = x3 − + 1 at the point whose x-coordinate is 2.

Exercise 6.3 | Q 4 | Page 211

Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.

Exercise 6.3 | Q 5 | Page 211

Find the slope of the normal to the curve x = acos3θy = asin3θ at theta = pi/4

Exercise 6.3 | Q 7 | Page 211

Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.

Exercise 6.3 | Q 8 | Page 211

Find a point on the curve y = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Exercise 6.3 | Q 9 | Page 212

Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.

Exercise 6.3 | Q 10 | Page 212

Find the equation of all lines having slope −1 that are tangents to the curve  y = 1/(x -1), x != 1

Exercise 6.3 | Q 11 | Page 212

Find the equation of all lines having slope 2 which are tangents to the curve y =   1/(x- 3), x != 3

Exercise 6.3 | Q 12 | Page 212

Find the equations of all lines having slope 0 which are tangent to the curve  y =   1/(x^2-2x + 3)

Exercise 6.3 | Q 13.1 | Page 212

Find points on the curve x^2/9 + "y"^2/16 = 1 at which the tangent is parallel to x-axis.

Exercise 6.3 | Q 13.2 | Page 212

Find points on the curve x^2/9 + "y"^2/16 = 1 at which the tangent is parallel to y-axis.

Exercise 6.3 | Q 14.1 | Page 212

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)

Exercise 6.3 | Q 14.2 | Page 212

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x4 − 6x3 + 13x2 − 10x + 5 at (1, 3)

Exercise 6.3 | Q 14.3 | Page 212

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x3 at (1, 1)

Exercise 6.3 | Q 14.4 | Page 212

Find the equations of the tangent and normal to the given curves at the indicated points:

y = x2 at (0, 0)

Exercise 6.3 | Q 14.5 | Page 212

Find the equations of the tangent and normal to the given curves at the indicated points:

x = cos ty = sin t at  t = pi/4

Exercise 6.3 | Q 15. (a) | Page 212

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0.

Exercise 6.3 | Q 15. (b) | Page 212

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is perpendicular to the line 5y − 15x = 13.

Exercise 6.3 | Q 16 | Page 212

Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.

Exercise 6.3 | Q 17 | Page 212

Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

Exercise 6.3 | Q 18 | Page 212

For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.

Exercise 6.3 | Q 19 | Page 212

Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.

Exercise 6.3 | Q 20 | Page 212

Find the equation of the normal at the point (am2am3) for the curve ay2 = x3.

Exercise 6.3 | Q 21 | Page 213

Find the equation of the normals to the curve y = x3 + 2+ 6 which are parallel to the line x + 14y + 4 = 0.

Exercise 6.3 | Q 22 | Page 213

Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).

Exercise 6.3 | Q 23 | Page 213

Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

Exercise 6.3 | Q 24 | Page 213

Find the equations of the tangent and normal to the hyperbola x^2/a^2 - y^2/b^2 at the point (x_0, y_0)

Exercise 6.3 | Q 25 | Page 213

Find the equation of the tangent to the curve y = sqrt(3x-2)  which is parallel to the line 4x − 2y + 5 = 0.

Exercise 6.3 | Q 26 | Page 213

The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is

(A) 3

(B) 1/3

(C) −3

(D) -1/3

Exercise 6.3 | Q 27 | Page 213

The line y = x + 1 is a tangent to the curve y2 = 4x at the point

(A) (1, 2)

(B) (2, 1)

(C) (1, −2)

(D) (−1, 2)

Exercise 6.4 [Page 216]

### NCERT solutions for Mathematics Class 12 Chapter 6 Application of Derivatives Exercise 6.4 [Page 216]

Exercise 6.4 | Q 1.1 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

sqrt(25.3)

Exercise 6.4 | Q 1.1 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(401)^(1/2)

Exercise 6.4 | Q 1.11 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(0.0037)^(1/2)

Exercise 6.4 | Q 1.12 | Page 2016

Using differentials, find the approximate value of the following up to 3 places of decimal

(26.57)^(1/3)

Exercise 6.4 | Q 1.13 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(81.5)^(1/4)

Exercise 6.4 | Q 1.14 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(3.968)^(3/2)

Exercise 6.4 | Q 1.15 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(32.15)^(1/5)

Exercise 6.4 | Q 1.2 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

sqrt(49.5)

Exercise 6.4 | Q 1.3 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

sqrt(0.6)

Exercise 6.4 | Q 1.4 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(0.009)^(1/3)

Exercise 6.4 | Q 1.5 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(0.999)^(1/10)

Exercise 6.4 | Q 1.6 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(15)^(1/4)

Exercise 6.4 | Q 1.7 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(26)^(1/3)

Exercise 6.4 | Q 1.8 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(255)^(1/4)

Exercise 6.4 | Q 1.9 | Page 216

Using differentials, find the approximate value of the following up to 3 places of decimal

(82)^(1/4)

Exercise 6.4 | Q 2 | Page 216

Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2

Exercise 6.4 | Q 3 | Page 216

Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15.

Exercise 6.4 | Q 4 | Page 216

Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1%.

Exercise 6.4 | Q 5 | Page 216

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%

Exercise 6.4 | Q 6 | Page 216

If the radius of a sphere is measured as 7 m with an error of 0.02m, then find the approximate error in calculating its volume.

Exercise 6.4 | Q 7 | Page 216

If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating in surface area

Exercise 6.4 | Q 8 | Page 216

If f (x) = 3x2 + 15x + 5, then the approximate value of (3.02) is

A. 47.66

B. 57.66

C. 67.66

D. 77.66

Exercise 6.4 | Q 9 | Page 216

The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

A. 0.06 x3 m3

B. 0.6 x3 m3

C. 0.09 x3 m3

D. 0.9 x3 m3

Exercise 6.5 [Pages 231 - 234]

### NCERT solutions for Mathematics Class 12 Chapter 6 Application of Derivatives Exercise 6.5 [Pages 231 - 234]

Exercise 6.5 | Q 1.1 | Page 231

Find the maximum and minimum values, if any, of the following functions given by

f(x) = (2x − 1)2 + 3

Exercise 6.5 | Q 1.2 | Page 231

Find the maximum and minimum values, if any, of the following functions given by

f(x) = 9x2 + 12x + 2

Exercise 6.5 | Q 1.3 | Page 231

Find the maximum and minimum values, if any, of the following functions given by

f(x) = −(x − 1)2 + 10

Exercise 6.5 | Q 1.4 | Page 231

Find the maximum and minimum values, if any, of the following functions given by

g(x) = x3 + 1

Exercise 6.5 | Q 2.1 | Page 232

Find the maximum and minimum values, if any, of the functions given by

f(x) = |x + 2| − 1

Exercise 6.5 | Q 2.2 | Page 232

Find the maximum and minimum values, if any, of the following functions given by

g(x) = − |x + 1| + 3

Exercise 6.5 | Q 2.3 | Page 232

Find the maximum and minimum values, if any, of the following functions given by

h(x) = sin(2x) + 5

Exercise 6.5 | Q 2.4 | Page 232

Find the maximum and minimum values, if any, of the following functions given by

f(x) = |sin 4x + 3|

Exercise 6.5 | Q 2.5 | Page 232

Find the maximum and minimum values, if any, of the following functions given by

h(x) = + 4, x ∈ (−1, 1)

Exercise 6.5 | Q 3.1 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be

f(x) = x2

Exercise 6.5 | Q 3.2 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be

g(x) = x3 − 3x

Exercise 6.5 | Q 3.3 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

h(x) = sinx + cosx, 0 < x < pi/2

Exercise 6.5 | Q 3.5 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be

f(x) = x3 − 6x2 + 9x + 15

Exercise 6.5 | Q 3.6 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

g(x) = x/2 + 2/x, x > 0

Exercise 6.5 | Q 3.7 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

g(x) = 1/(x^2 + 2)

Exercise 6.5 | Q 3.8 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

f(x) = xsqrt(1-x), x > 0

Exercise 6.5 | Q 4 | Page 232

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

f(x) = sinx − cos x, 0 < x < 2π

Exercise 6.5 | Q 4.1 | Page 232

Prove that the following functions do not have maxima or minima:

f(x) = ex

Exercise 6.5 | Q 4.2 | Page 232

Prove that the following functions do not have maxima or minima:

g(x) = logx

Exercise 6.5 | Q 4.3 | Page 232

Prove that the following functions do not have maxima or minima:

h(x) = x3 + x2 + x + 1

Exercise 6.5 | Q 4.3 | Page 232

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

f(x) = 4x - 1/x x^2, x in [-2 ,9/2]

Exercise 6.5 | Q 5.1 | Page 232

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

f(x) =x^3, x in [-2,2]

Exercise 6.5 | Q 5.2 | Page 232

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals

f (x) = sin x + cos x , x ∈ [0, π]

Exercise 6.5 | Q 5.4 | Page 232

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

f (x) = (x −1)2 + 3, x ∈[−3,1]

Exercise 6.5 | Q 6 | Page 232

Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2

Exercise 6.5 | Q 7 | Page 232

Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3]

Exercise 6.5 | Q 8 | Page 232

At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?

Exercise 6.5 | Q 9 | Page 232

What is the maximum value of the function sin x + cos x?

Exercise 6.5 | Q 10 | Page 232

Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].

Exercise 6.5 | Q 11 | Page 233

It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Exercise 6.5 | Q 12 | Page 233

Find the maximum and minimum values of x + sin 2x on [0, 2π].

Exercise 6.5 | Q 13 | Page 233

Find two numbers whose sum is 24 and whose product is as large as possible.

Exercise 6.5 | Q 14 | Page 233

Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

Exercise 6.5 | Q 15 | Page 233

Find two positive numbers and such that their sum is 35 and the product x2y5 is a maximum

Exercise 6.5 | Q 16 | Page 233

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Exercise 6.5 | Q 17 | Page 233

A square piece of tin of side 18 cm is to made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Exercise 6.5 | Q 18 | Page 233

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?

Exercise 6.5 | Q 19 | Page 233

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Exercise 6.5 | Q 20 | Page 233

Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.

Exercise 6.5 | Q 21 | Page 233

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

Exercise 6.5 | Q 22 | Page 233

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Exercise 6.5 | Q 23 | Page 233

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

Exercise 6.5 | Q 24 | Page 233

Show that the right circular cone of least curved surface and given volume has an altitude equal to sqrt2 time the radius of the base.

Exercise 6.5 | Q 25 | Page 233

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan^(-1) sqrt(2)

Exercise 6.5 | Q 26 | Page 233

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is  Sin^(-1) (1/3)

Exercise 6.5 | Q 27 | Page 234

The point on the curve x2 = 2y which is nearest to the point (0, 5) is

(A) (2sqrt2,4)

(B) (2sqrt2,0)

(C) (0, 0)

(D) (2, 2)

Exercise 6.5 | Q 28 | Page 234

For all real values of x, the minimum value of (1 - x + x^2)/(1+x+x^2) is

(A) 0

(B) 1

(C) 3

(D) 1/3

Exercise 6.5 | Q 29 | Page 234

The maximum value of [x(x −1) +1]^(1/3) , 0 ≤ x ≤ 1 is

(A) (1/3)^(1/3)

(B) 1/2

(C) 1

(D) 0

Exercise 6.6 [Pages 242 - 244]

### NCERT solutions for Mathematics Class 12 Chapter 6 Application of Derivatives Exercise 6.6 [Pages 242 - 244]

Exercise 6.6 | Q 1.1 | Page 242

Using differentials, find the approximate value of each of the following.

(17/81)^(1/4)

Exercise 6.6 | Q 1.2 | Page 242

Using differentials, find the approximate value of each of the following.

(33)^(1/5)

Exercise 6.6 | Q 2 | Page 242

Show that the function given by f(x) = (log x)/x has maximum at e.

Exercise 6.6 | Q 3 | Page 242

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

Exercise 6.6 | Q 4 | Page 242

Find the equation of the normal to curve y2 = 4x at the point (1, 2).

Exercise 6.6 | Q 5 | Page 242

Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.

Exercise 6.6 | Q 6 | Page 242

Find the intervals in which the function given by f(x) = (4sin x - 2x - x cos x)/(2 + cos x) is

(i) increasing

(ii) decreasing

Exercise 6.6 | Q 7 | Page 242

Find the intervals in which the function f given by f(x) = x^3 + 1/x^3 x != 0, is

(i) increasing

(ii) decreasing

Exercise 6.6 | Q 8 | Page 242

Find the maximum area of an isosceles triangle inscribed in the ellipse  x^2/ a^2 + y^2/b^2 = 1 with its vertex at one end of the major axis.

Exercise 6.6 | Q 9 | Page 242

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

Exercise 6.6 | Q 10 | Page 242

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Exercise 6.6 | Q 11 | Page 243

A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Exercise 6.6 | Q 12 | Page 243

A point on the hypotenuse of a triangle is at distance and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is (a^(2/3) + b^(2/3))^(3/2)

Exercise 6.6 | Q 13 | Page 243

Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has

(i) local maxima

(ii) local minima

(iii) point of inflexion

Exercise 6.6 | Q 14 | Page 243

Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π]

Exercise 6.6 | Q 15 | Page 243

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r  (4r)/3

Exercise 6.6 | Q 16 | Page 243

Let f be a function defined on [ab] such that f '(x) > 0, for all x ∈ (ab). Then prove that f is an increasing function on (ab).

Exercise 6.6 | Q 17 | Page 243

Show that the height of the cylinder of maximum volume, which can be inscribed in a sphere of radius R is (2R)/sqrt3.  Also find the maximum volume.

Exercise 6.6 | Q 18 | Page 243

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one-third that of the cone and the greatest volume of cylinder is 4/27 pih^3 tan2α.

Exercise 6.6 | Q 19 | Page 243

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of

(A) 1 m/h

(B) 0.1 m/h

(C) 1.1 m/h

(D) 0.5 m/h

Exercise 6.6 | Q 20 | Page 243

The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is

(A) 22/7

(B) 6/7

(C) 7/6

(D) (-6)/7

Exercise 6.6 | Q 21 | Page 244

The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is

(A) 1

(B) 2

(C) 3

(D) 1/2

Exercise 6.6 | Q 22 | Page 244

The normal at the point (1, 1) on the curve 2y + x2 = 3 is

(A) x + y = 0

(B) x − = 0

(C) x + y + 1 = 0

(D) − y = 1

Exercise 6.6 | Q 23 | Page 244

The normal to the curve x2 = 4y passing (1, 2) is

(A) x + y = 3

(B) x − y = 3

(C) x + = 1

(D) x − = 1

Exercise 6.6 | Q 24 | Page 244

The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are

(A)(4, +- 8/3)

(B) (4,(-8)/3)

(C)(4, +- 3/8)

(D) (+-4, 3/8)

## Solutions for Chapter 6: Application of Derivatives

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6

## NCERT solutions for Mathematics Class 12 chapter 6 - Application of Derivatives

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics Class 12 CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Mathematics Class 12 CBSE, Karnataka Board PUC 6 (Application of Derivatives) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

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Concepts covered in Mathematics Class 12 chapter 6 Application of Derivatives are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies or Quantities, Introduction to Applications of Derivatives, Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies or Quantities, Introduction to Applications of Derivatives.

Using NCERT Mathematics Class 12 solutions Application of Derivatives exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Mathematics Class 12 students prefer NCERT Textbook Solutions to score more in exams.

Get the free view of Chapter 6, Application of Derivatives Mathematics Class 12 additional questions for Mathematics Mathematics Class 12 CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.

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