Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board chapter 2 - Applications of Derivatives [Latest edition]

Find the equation of tangent and normal to the curve at the point on it.

y = x² + 2e^x + 2 at (0, 4)

Find the equations of tangents and normals to the following curves at the indicated points on them : x³ + y³ – 9xy = 0 at (2, 4)

Find the equations of tangents and normals to the following curves at the indicated points on them: `x^2 - sqrt(3)xy + 2y^2 = 5  "at"  (sqrt(3),2)`

Find the equations of tangents and normals to the following curves at the indicated points on them : 2xy + π sin y = `2pi  "at" (1, pi/2)`

Find the equations of tangents and normals to the following curves at the indicated points on them : x sin 2y = y cos 2x at `(pi/4, pi/2)`

Find the equations of tangents and normals to the following curve at the indicated points on them:

x = sin θ and y = cos 2θ at θ = `pi/(6)`

Find the equations of tangents and normals to the following curves at the indicated points on them : `x = sqrt(t), y = t  - (1)/sqrt(t)` at = 4.

Find the point on the curve y = `sqrt(x - 3)` where the tangent is perpendicular to the line 6x + 3y – 5 = 0.

Find the points on the curve y = x³ – 2x² – x where the tangents are parllel to 3x – y + 1 = 0.

Find the equation of the tangents to the curve x² + y² – 2x – 4y + 1 =0 which a parallel to the X-axis.

Find the equations of the normals to the curve 3x² – y² = 8, which are parallel to the line x + 3y = 4.

If the line y = 4x – 5 touches the curves y² = ax³ + b at the point (2, 3), find a and b.

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

A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?

The surface area of a spherical balloon is increasing at the rate of 2cm²/sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?

If each side of an equilateral triangle increases at the rate of `(sqrt(2)"cm")/sec`, find the rate of increase of its area when its side of length 3 cm.

The volume of a sphere increases at the rate of 20 cm³/sec. Find the rate of change of its surface area, when its radius is 5 cm

The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.

A man of height 2 metres walks at a uniform speed of 6 km/hr away from a lamp post of 6 metres high. Find the rate at which the length of the shadow is increasing.

A man of height 1.5 meters walks towards a lamp post of height 4.5 meters, at the rate of `(3/4)` meter/sec. Find the rate at which (i) his shadow is shortening (ii) the tip of shadow is moving.

A ladder 10 metres long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 metres per second, find how fast the top of the ladder is sliding down the wall, when the bottom is 6 metres away from the wall.

If water is poured into an inverted hollow cone whose semi-vertical angle is 30°, so that its depth (measured along the axis) increases at the rate of`( 1"cm")/sec`. Find the rate at which the volume of water increasing when the depth is 2 cm.

Find the approximate values of : `sqrt(8.95)`

Find the approximate values of : `root(3)(28)`

Find the approximate values of : `root(5)(31.98)`

Find the approximate values of : (3.97)⁴ 

Find the approximate values of (4.01)³ 

Find the approximate values of : sin 61° , given that 1° = 0.0174^c, `sqrt(3) = 1.732`

Find the approximate values of : sin (29° 30'), given that 1°= 0.0175°, `sqrt(3) = 1.732`

Find the approximate values of : cos(60° 30°), given that 1° = 0.0175°, `sqrt(3) = 1.732`

Find the approximate values of : tan (45° 40'), given that 1° = 0.0175°.

Find the approximate values of : tan^–1(0.999)

Find the approximate values of : cot^–1 (0.999)

Find the approximate values of : tan^–1 (1.001)

Find the approximate values of : e^0.995, given that e = 2.7183.

Find the approximate values of : e^2.1, given that e² = 7.389

Find the approximate values of : 3^2.01, given that log 3 = 1.0986

Find the approximate values of : log_e(101), given that log_e10 = 2.3026.

Find the approximate values of : log_e(9.01), given that log 3 = 1.0986.

Find the approximate value of log₁₀ (1016), given that log₁₀e = 0⋅4343.

Find the approximate values of : f(x) = x³ – 3x + 5 at x = 1.99.

Find the approximate values of : f(x) = x³ + 5x² – 7x + 10 at x = 1.12.

Check the validity of the Rolle’s theorem for the following functions : f(x) = x² – 4x + 3, x ∈ [1, 3]

Check the validity of the Rolle’s theorem for the following functions : f(x) = e^–x sin x, x ∈ [0, π].

Check the validity of the Rolle’s theorem for the following functions : f(x) = 2x² – 5x + 3, x ∈ [1, 3].

Check the validity of the Rolle’s theorem for the following functions : f(x) = sin x – cos x + 3, x ∈ [0, 2π].

Check the validity of the Rolle’s theorem for the following function:

f(x) = x², if 0 ≤ x ≤ 2

= 6 – x, if 2 < x ≤ 6.

Check the validity of the Rolle’s theorem for the following function:

f(x) = `x^(2/3), x ∈ [ - 1, 1]`

Given an interval [a, b] that satisfies hypothesis of Rolle's theorem for the function f(x) = x⁴ + x² – 2. It is known that a = – 1. Find the value of b.

Verify Rolle’s theorem for the following functions : f(x) = sin x + cos x + 7, x ∈ [0, 2π]

Verify Rolle’s theorem for the following functions  : f(x) = `sin(x/2), x ∈ [0, 2pi]` 

Verify Rolle’s theorem for the following functions : f(x) = x² – 5x + 9, x ∈ 1, 4].

If Rolle's theorem holds for the function f(x) = x³ + px² + qx + 5, x ∈ [1, 3] with c = `2 + (1)/sqrt(3)`, find the values of p and q.

If Rolle’s theorem holds for the function f(x) = (x –2) log x, x ∈ [1, 2], show that the equation x log x = 2 – x is satisfied by at least one value of x in (1, 2).

The function f(x) = `x(x + 3)e^(-(x)/2)` satisfies all the conditions of Rolle's theorem on [– 3, 0]. Find the value of c such that f'(c) = 0.

Verify Lagrange’s mean value theorem for the following function:

f(x) = log x on [1, e]

Verify Lagrange’s mean value theorem for the following functions : f(x) = (x – 1)(x – 2)(x – 3) on [0, 4].

Verify Lagrange’s mean value theorem for the following function:

`f(x) = x^2 - 3x - 1, x ∈ [(-11)/7 , 13/7]`.

Verify Lagrange’s mean value theorem for the following functions : f(x) = 2x – x², x ∈ [0, 1].

Verify Lagrange’s mean value theorem for the following functions : f(x) = `(x - 1)/(x - 3)` on [4, 5].

Test whether the following functions are increasing or decreasing : f(x) = x³ – 6x² + 12x – 16, x ∈ R.

Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x² – x³, x ∈ R.

Test whether the following functions are increasing or decreasing : f(x) = `(1)/x`, x ∈ R , x ≠ 0.

Find the values of x for which the following functions are strictly increasing : f(x) = 2x³ – 3x² – 12x + 6

Find the values of x for which the following functions are strictly increasing:

f(x) = 3 + 3x – 3x² + x³

Find the values of x for which the following func- tions are strictly increasing : f(x) = x³ – 6x² – 36x + 7

Find the values of x for which the following functions are strictly decreasing:

f(x) = 2x³ – 3x² – 12x + 6

Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`

Find the values of x for which the following functions are strictly decreasing : f(x) = x³ – 9x² + 24x + 12

Find the values of x for which the function f(x) = x³ – 12x² – 144x + 13 (a) increasing (b) decreasing

Find the values of x for which f(x) = 2x³ – 15x² – 144x – 7 is

(a) Strictly increasing
(b) strictly decreasing

Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.

show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.

Show that f(x) = x – cos x is increasing for all x.

Find the maximum and minimum of the following functions : y = 5x³ + 2x² – 3x.

Find the maximum and minimum of the following functions : f(x) = 2x³ – 21x² + 36x – 20

Find the maximum and minimum of the following functions : f(x) = x³ – 9x² + 24x

Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`

Find the maximum and minimum of the following functions : f(x) = x log x

Find the maximum and minimum of the following functions : f(x) = `logx/x`

Divide the number 30 into two parts such that their product is maximum.

Divide the number 20 into two parts such that sum of their squares is minimum.

A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.

A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t². Find the maximum height it can reach.

Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.

An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.

The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?

A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?

The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.

Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.

Show that among rectangles of given area, the square has least perimeter.

Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.

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

Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`

Choose the correct option from the given alternatives :  

If the function f(x) = ax³ + bx² + 11x – 6 satisfies conditions of Rolle's theoreem in [1, 3] and  `f'(2 + 1/sqrt(3))` = 0, then values of a and b are respectively

Choose the correct option from the given alternatives : 

If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.

Choose the correct option from the given alternatives :

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of `(1.5 "m")/sec`. The length of the higher point of ladder when the foot of the ladder is 4.0 m away from the wall decreases at the rate of

Choose the correct option from the given alternatives:

Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.

Choose the correct option from the given alternatives :

Let f(x) = x³ – 6x² + 9x + 18, then f(x) is strictly decreasing in ______.

Choose the correct option from the given alternatives :

If x = –1 and x = 2 are the extreme points of y = αlogx + βx² + x`, then ______.

Choose the correct option from the given alternatives :

The normal to the curve x² + 2xy – 3y² = 0 at (1, 1)

Choose the correct option from the given alternatives :

The equation of the tangent to the curve y = `1 - e^(x/2)` at the point of intersection with Y-axis is

Choose the correct option from the given alternatives :

If the tangent at (1, 1) on y² = x(2 – x)² meets the curve again at P, then P is

The approximate value of tan (44° 30°), given that 1° = 0.0175, is ______.

Solve the following : If the curves ax² + by² = 1 and a'x² + b'y² = 1, intersect orthogonally, then prove that `(1)/a - (1)/b = (1)/a' - (1)/b'`.

Solve the following : Determine the area of the triangle formed by the tangent to the graph of the function y = 3 – x² drawn at the point (1, 2) and the coordinate axes.

Solve the following : Find the equation of the tangent and normal drawn to the curve y⁴ – 4x⁴ – 6xy = 0 at the point M (1, 2).

Solve the following:

A water tank in the farm of an inverted cone is being emptied at the rate of 2 cubic feet per second. The height of the cone is 8 feet and the radius is 4 feet. Find the rate of change of the water level when the depth is 6 feet.

Solve the following : Find all points on the ellipse 9x² + 16y² = 400, at which the y-coordinate is decreasing and the coordinate is increasing at the same rate.

Verify Rolle’s theorem for the function f(x)  `(2)/(e^x + e^-x)` on [– 1, 1].

Solve the following : The position of a particle is given by the function s (t) = 2t² + 3t – 4. Find the time t = c in the interval 0 ≤ t ≤ 4 when the instantaneous velocity of the particle equal to its average velocity in this interval.

Find the approximate value of the function f(x) = `sqrt(x^2 + 3x)` at x = 1.02.

Solve the following : Find the approximate value of cos^–1 (0.51), given π = 3.1416, `(2)/sqrt(3)` = 1.1547.

Solve the following : Find the intervals on which the function y = x^x, (x > 0) is increasing and decreasing.

Solve the following:

Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.

Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a². Show that the maximum volume of the box is `a^3/(6sqrt(3)`.

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

Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.

Solve the following :  A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.

Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.

Solve the following:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

Solve the following:

A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.

Solve the following : Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is  `(4r)/(3)`.

Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.

Solve the following: 

Find the maximum and minimum values of the function f(x) = cos²x + sinx.