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

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?

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.

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

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

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

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

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

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

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`

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

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`

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.

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

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

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

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

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

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`

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

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

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

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

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

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

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

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

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

The maximum value of sinx + cosx is ______.

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

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

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

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.

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.

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

Find the approximate value of (1.999)⁵.

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

A man, 2m tall, walks at the rate of `1 2/3` m/s towards a street light which is `5 1/3`m 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 1/3`m from the base of the light?

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?

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

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.

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

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

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

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

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

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

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

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

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

Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R

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

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

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

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`

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.

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?

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.

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

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.

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?

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.

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.

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.

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 ______.

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 ______.

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

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

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

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

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

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

The tangent to the curve y = e^2x at the point (0, 1) meets x-axis at ______.

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

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

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

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

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

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

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

The function f(x) = tanx – x ______.

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

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

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

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

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

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

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

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

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

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

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

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

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