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

## Chapter 6: Application Of Derivatives

Solved ExamplesExercise
Solved Examples [Pages 119 - 135]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 6 Application Of DerivativesSolved Examples [Pages 119 - 135]

Solved Examples | Q 1 | Page 119

For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then how fast is the slope of curve changing when x = 3?

Solved Examples | Q 2 | Page 120

Water is dripping out from a conical funnel of semi-vertical angle pi/4 at the uniform rate of 2cm2/sec in the surface area, through a tiny hole at the vertex of the bottom. When the slant height of cone is 4 cm, find the rate of decrease of the slant height of water.

Solved Examples | Q 3 | Page 120

Find the angle of intersection of the curves y2 = x and x2 = y.

Solved Examples | Q 4 | Page 121

Prove that the function f(x) = tanx – 4x is strictly decreasing on ((-pi)/3, pi/3)

Solved Examples | Q 5 | Page 121

Determine for which values of x, the function y = x^4 – (4x^3)/3 is increasing and for which values, it is decreasing.

Solved Examples | Q 6 | Page 122

Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.

Solved Examples | Q 7 | Page 122

Using differentials, find the approximate value of sqrt(0.082)

Solved Examples | Q 8 | Page 123

Find the condition for the curves x^2/"a"^2 - y^2/"b"^2 = 1; xy = c2 to interest orthogonally.

Solved Examples | Q 9 | Page 123

Find all the points of local maxima and local minima of the function f(x) = - 3/4 x^4 - 8x^3 - 45/2 x^2 + 105

Solved Examples | Q 10 | Page 124

Show that the local maximum value of x + 1/x is less than local minimum value.

Solved Examples | Q 11 | Page 124

Water is dripping out at a steady rate of 1 cu cm/sec through a tiny hole at the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the vertical angle of the conical vessel is pi/6

Solved Examples | Q 12 | Page 125

Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.

Solved Examples | Q 13 | Page 126

Find the angle of intersection of the curves y2 = 4ax and x2 = 4by.

Solved Examples | Q 14 | Page 127

Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ

Solved Examples | Q 15 | Page 128

Find the maximum and minimum values of f(x) = secx + log cos2x, 0 < x < 2π

Solved Examples | Q 16 | Page 129

Find the area of greatest rectangle that can be inscribed in an ellipse x^2/"a"^2 + y^2/"b"^2 = 1

Solved Examples | Q 17 | Page 130

Find the difference between the greatest and least values of the function f(x) = sin2x – x, on [- pi/2, pi/2]

Solved Examples | Q 18 | Page 131

An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when θ = pi/6

#### Objective Type Questions from 19 to 23

Solved Examples | Q 19 | Page 132

The abscissa of the point on the curve 3y = 6x – 5x3, the normal at which passes through origin is ______.

• 1

• 1/3

• 2

• 1/2

Solved Examples | Q 20 | Page 132

The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 = 2 ______.

• Touch each other

• Cut at right angle

• Cut at an angle pi/3

• Cut at an angle pi/4

Solved Examples | Q 21 | Page 133

The tangent to the curve given by x = et . cost, y = et . sint at t = pi/4 makes with x-axis an angle ______.

• 0

• pi/4

• pi/3

• pi/2

Solved Examples | Q 22 | Page 133

The equation of the normal to the curve y = sinx at (0, 0) is ______.

• x = 0

• y = 0

• x + y = 0

• x – y = 0

Solved Examples | Q 23 | Page 133

The point on the curve y2 = x, where the tangent makes an angle of pi/4 with x-axis is ______.

• (1/2, 1/4)

• (1/4, 1/2)

• (4, 2)

• (1, 1)

#### Fill in the blanks in the following Examples 24 to 29

Solved Examples | Q 24 | Page 134

The values of a for which y = x2 + ax + 25 touches the axis of x are ______.

Solved Examples | Q 25 | Page 134

If f(x) = 1/(4x^2 + 2x + 1), then its maximum value is ______.

Solved Examples | Q 26 | Page 134

Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.

Solved Examples | Q 27 | Page 134

Minimum value of f if f(x) = sinx in [(-pi)/2, pi/2] is ______.

Solved Examples | Q 28 | Page 134

The maximum value of sinx + cosx is ______.

Solved Examples | Q 29 | Page 135

The rate of change of volume of a sphere with respect to its surface area, when the radius is 2 cm, is ______.

Exercise [Pages 135 - 142]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 6 Application Of DerivativesExercise [Pages 135 - 142]

Exercise | Q 1 | Page 135

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate

Exercise | Q 2 | Page 135

If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius

Exercise | Q 3 | Page 135

A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.

Exercise | Q 4 | Page 135

Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being seperated.

Exercise | Q 5 | Page 135

Find an angle θ, 0 < θ < pi/2, which increases twice as fast as its sine.

Exercise | Q 6 | Page 135

Find the approximate value of (1.999)5.

Exercise | Q 7 | Page 135

Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm respectively

Exercise | Q 8 | Page 135

A man, 2m tall, walks at the rate of 1 2/3 m/s towards a street light which is 5 1/3m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is 3 1/3m from the base of the light?

Exercise | Q 9 | Page 136

A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and L = 200 (10 – t)2. How fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during the first 5 seconds?

Exercise | Q 10 | Page 136

The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side

Exercise | Q 11 | Page 136

x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.

Exercise | Q 12 | Page 136

Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.

Exercise | Q 13 | Page 136

Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.

Exercise | Q 14 | Page 136

Find the co-ordinates of the point on the curve sqrt(x) + sqrt(y) = 4 at which tangent is equally inclined to the axes

Exercise | Q 15 | Page 136

Find the angle of intersection of the curves y = 4 – x2 and y = x2.

Exercise | Q 16 | Page 136

Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2)

Exercise | Q 17 | Page 136

Find the equation of the normal lines to the curve 3x2 – y2 = 8 which are parallel to the line x + 3y = 4.

Exercise | Q 18 | Page 136

At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?

Exercise | Q 19 | Page 136

Show that the line x/"a" + y/"b" = 1, touches the curve y = b · e– x/a at the point where the curve intersects the axis of y

Exercise | Q 20 | Page 136

Show that f(x) = 2x + cot–1x + log(sqrt(1 + x^2) - x) is increasing in R

Exercise | Q 21 | Page 137

Show that for a ≥ 1, ∈ is decreasing in R

Exercise | Q 22 | Page 137

Show that f(x) = tan–1(sinx + cosx) is an increasing function in (0, pi/4)

Exercise | Q 23 | Page 137

At what point, the slope of the curve y = – x3 + 3x2 + 9x – 27 is maximum? Also find the maximum slope.

Exercise | Q 24 | Page 137

Prove that f(x) = sinx + sqrt(3) cosx has maximum value at x = pi/6

Exercise | Q 25 | Page 137

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is pi/3

Exercise | Q 26 | Page 137

Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.

Exercise | Q 27 | Page 137

A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

Exercise | Q 28 | Page 137

If the straight line x cosα + y sinα = p touches the curve x^2/"a"^2 + y^2/"b"^2 = 1, then prove that a2 cos2α + b2 sin2α = p2.

Exercise | Q 29 | Page 137

An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is "c"^3/(6sqrt(3)) cubic units

Exercise | Q 30 | Page 137

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.

Exercise | Q 31 | Page 138

If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?

Exercise | Q 32 | Page 138

AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.

Exercise | Q 33 | Page 138

A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.

Exercise | Q 34 | Page 138

The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and x/3 and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.

#### Objective Type Questions from 35 to 39

Exercise | Q 35 | Page 138

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is ______.

• 10 "cm"^(2/"s")

• sqrt(3) "cm"^(2/"s")

• 10sqrt(3) "cm"^(2/"s")

• 10/3 "cm"^(2/"s")

Exercise | Q 36 | Page 138

A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is ______.

• 1/10 radian/sec

Exercise | Q 37 | Page 138

The curve y = x^(1/5) has at (0, 0) ______.

• A vertical tangent (parallel to y-axis)

• A horizontal tangent (parallel to x-axis)

• An oblique tangent

• No tangent

Exercise | Q 38 | Page 139

The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.

• 3x – y = 8

• 3x + y + 8 = 0

• x + 3y ± 8 = 0

• x + 3y = 0

Exercise | Q 39 | Page 139

If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is ______.

• 1

• 0

• – 6

• 6

Exercise | Q 40 | Page 139

If y = x4 – 10 and if x changes from 2 to 1.99, what is the change in y ______.

• 0.32

• 0.032

• 5.68

• 5.968

Exercise | Q 41 | Page 139

The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is ______.

• x + 5y = 2

• x – 5y = 2

• 5x – y = 2

• 5x + y = 2

Exercise | Q 42 | Page 139

The points at which the tangents to the curve y = x3 – 12x + 18 are parallel to x-axis are ______.

• (2, –2), (–2, –34)

• (2, 34), (–2, 0)

• (0, 34), (–2, 0)

• (2, 2), (–2, 34)

Exercise | Q 43 | Page 139

The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.

• (0, 1)

• (- 1/2, 0)

• (2, 0)

• (0, 2)

Exercise | Q 44 | Page 139

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

• 22/7

• 6/7

• (-6)/7

• – 6

Exercise | Q 45 | Page 140

The two curves x3 – 3xy2 + 2 = 0 and 3x2y – y3 – 2 = 0 intersect at an angle of ______.

• pi/4

• pi/3

• pi/2

• pi/6

Exercise | Q 46 | Page 140

The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.

• [–1, oo)

• [– 2, – 1]

• (-oo, -2]

• [– 1, 1]

Exercise | Q 47 | Page 140

Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.

• has a minimum at x = π

• has a maximum, at x = 0

• is a decreasing function

• is an increasing function

Exercise | Q 48 | Page 140

y = x(x – 3)2 decreases for the values of x given by : ______.

• 1 < x < 3

• x < 0

• x > 0

• 0 < x < 3/2

Exercise | Q 49 | Page 140

The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.

• Increasing in (pi, (3pi)/2)

• Decreasing in (pi/2, pi)

• Decreasing in [(-pi)/2, pi/2]

• Decreasing in [0, pi/2]

Exercise | Q 50 | Page 140

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

• sin2x

• tanx

• cosx

• cos 3x

Exercise | Q 51 | Page 140

The function f(x) = tanx – x ______.

• Always increases

• Always decreases

• Never increases

• Sometimes increases and sometimes decreases

Exercise | Q 52 | Page 141

If x is real, the minimum value of x2 – 8x + 17 is ______.

• – 1

• 0

• 1

• 2

Exercise | Q 53 | Page 141

The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.

• 126

• 0

• 135

• 160

Exercise | Q 54 | Page 141

The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.

• Two points of local maximum

• Two points of local minimum

• One maxima and one minima

• No maxima or minima

Exercise | Q 55 | Page 141

The maximum value of sin x . cos x is ______.

• 1/4

• 1/2

• sqrt(2)

• 2sqrt(2)

Exercise | Q 56 | Page 141

At x = (5pi)/6, f(x) = 2 sin3x + 3 cos3x is ______.

• Maximum

• Minimum

• Zero

• Neither maximum nor minimum

Exercise | Q 57 | Page 141

Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.

• 0

• 12

• 16

• 32

Exercise | Q 58 | Page 141

f(x) = xx has a stationary point at ______.

• x = e

• x = 1/"e"

• x = 1

• x = sqrt("e")

Exercise | Q 59 | Page 141

The maximum value of (1/x)^x is ______.

• e

• ex

• "e"^(1/"e")

• (1/"e")^(1/"e")

#### Fill in the blanks 60 to 64:

Exercise | Q 60 | Page 142

The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.

Exercise | Q 61 | Page 142

The equation of normal to the curve y = tanx at (0, 0) is ______.

Exercise | Q 62 | Page 142

The values of a for which the function f(x) = sinx – ax + b increases on R are ______.

Exercise | Q 63 | Page 142

The function f(x) = (2x^2 - 1)/x^4, x > 0, decreases in the interval ______.

Exercise | Q 64 | Page 142

The least value of the function f(x) = "a"x + "b"/x` (where a > 0, b > 0, x > 0) is ______.

## Chapter 6: Application Of Derivatives

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

NCERT solutions for Mathematics Exemplar Class 12 chapter 6 (Application Of Derivatives) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar 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.

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

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