Important Questions for CBSE Class 12 Maths Chapter 6 - Application of Derivatives -CoolGyan

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CBSE Class 12 Mathematics Important Questions Chapter 6 – Application of Derivatives

4 Mark Questions

1. The length x of a rectangle is decreasing at the rate of 3 cm/ mint and the width y is increasing at the rate of 2cm/min. when x = 10cm and y = 6cm, find the ratio of change of (a) the perimeter (b) the area of the rectangle.
Ans.
(a) Let P be the perimeter

(b)

2. Find the interval in which the function given by f(x) = 4x³ – 6x²– 72x + 30 is
(a) strictly increasing
(b) strictly decreasing.
Ans.

int	Sign of f’(x)	Result
	+ tive	Increase
	+ tive	Decrease
	+ tive	increase

Hence function is increasing in and decreasing in (-2, 3)

3. Find point on the curveat which the tangents are (i) parallel to x –axis (ii) parallel to y – axis
Ans.
Differentiate side w.r.t. to x

For tangent || to x – axis the slope of tangent is zero

Put x = 0 in equation (1)

Points are (0, 5) and (0, -5) now is tangent is || is to y – axis

4. Use differentiation to approximate
Ans. Let

Let
Then

Put the value of dy in equation (1)

5. The volume of a cube is increasing at a rate of 9cm³/s. How fast is the surface area increasing when the length of on edge is 10cm?
Ans. Let x be the length, V be the volume and S be the surface area of cube

6. Find the interval in which the function is strictly increasing and decreasing. (x+1)³ (x-3)³
Ans.

int	Singh of f’(x)	Result
	-tive	Decrease
	-tive	Decrease
	+tive	Increase
	+tive	Increase

7.Find the equations of the tangent and normal to curve at (1, 1)
Ans.
Differentiate both side w.r.t to x

Slope of tangent = -1
Slope of normal

8. IF the radius of a sphere is measured as 9cm with an error of 0.03cm, then find the approximate error in calculating its volume.
Ans. Let r be radius and be error

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

Ans.

When

10. 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.
Ans.

Put the value of x in equation (1)

11. Find the interval in which increase/decrease.
Ans.

int.	Sign of f’(x)	Result
	+tive	increase
	-tive	Decrease

Hence, f(x) is increasing onand decreasing on

12. Find the intervals in which the function f given by is strictly increasing or decreasing.
Ans.

Int	Singh of f’(x)	Result
	+tive	Increase
	-tive	Decrease
	+tive	increase

13. Find the equation, of the tangent line to the curve y = x² – 2x + 7 which is
(a) Parallels to the line 2x – y + 9 = 0
(b) Perpendicular to the line 5y – 15x = 13
Ans. Let (x, y) be the point a
(a) y = x² – 2x + 7 —–(1)

Slope of line = 2

Equation of tangent

(b)
Slope of Line =

Put x₁ in equation (1)

Equation of tangent

14. Find the equation of the tangent to the hyperbola at the point (x_o, y_o).
Ans.

Equation

Dividing by a²b²

From (1)

15. Find the approximate value of
Ans. Let

16. Using differentiates find the approximate value of
Ans.

We get

17. Sand is pouring from a pipe at the rate of 12cm³/s. the falling sand forms a cone on the ground in much a way that the height of the cone is always one – sixth of the radius of the here. How fast is the height of the sand cone increasing when the height in 4cm.
Ans.

18. The total revenue in RS received from the sale of x units of the product is given by R (x) = 13x² + 26x + 15 find MR when 17 unit are produce.
Ans.

19. Prove that is an increasing function for in
Ans.

20. Prove that the function given by f(x) = log sinx is strictly increasing on and strictly decreasing on
Ans.

and

Hence f(x) = log sinx is strictly increasing on and decreasing on

21. 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)
Ans.
Slope of tangent to curve

Slope of chord

Put x = 3 in equation (1)

Points (3, 1)

22. Find the equation of tangent to the curve given by at a point where
Ans.

When
Equation of tangent

23. Find the approximate value of f(3.02) where f(x) = 3x² + 5x + 3.
Ans.

Put

24. Find the approximate value of
Ans.

25. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900cm³/s. find the rate at which the radius of the balloon increase when the radius is 15cm.
Ans. Let V be the volume of sphere

26. A circular disc of radius 3cm is being heated. Due to expansion, their radius increase at the rate of 0.05 cm/s. find the rate at which its area is increasing when radius is 3.2cm.
Ans.

27. Find the intervals in which the function f given by is
(i) increasing
(ii) decreasing
Ans.

Hence

28. Find the interval in which the function f given by is
(i) increasing
(ii) decreasing.
Ans.

For increasing

So f(x) is increase on and
For decreasing

f(x) is decrease on (-1, 0) (0, 1)

29. Find the equation of the normal to the curve which passes through the point (1, 2)
Ans.

Let (x₁ y₁) be the point

Slope of normal
Equation

Passes through —————-(1, 2)

lies on

Now repeat equation

X + y = 3

30. Show that the normal at any point to the curve is at a constant distance from origin.
Ans.

Slope of normal
Equation of normal

Proved

31. 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.

Ans.

R=3x

= – tive maximum
Altitude

Prove.

32. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is Also find the maximum volume.

Ans.

For maximum/minimum

= – tive maximum
Height of cylinder

33. The two equal side of an isosceles with fixed base b are decreasing at the rate of 3cm/s. How fast is the area decreasing when the two equal sides are equal to the base?

Ans.
Let A be area of

34. A men of height 2m walks at a uniform speed of 5km/h away from a lamp, past which is 6m high. Find the rate at which the lengths of his shadow increase.
Ans. AB is lamp post DC is man

35. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is tan^-1 (0.5) water is poured into it at a constant rate of 5cm³/hr. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4m.

Ans.

36. Find the interval in which the function given by is (a) Strictly increasing (b) Strictly decreasing
Ans.

int	Sign of f’(x)	Result
	-tive	Decrease
	+tive	Increase
	-tive	Decrease
	+tive	increase

37. Show that is always an increasing function in
Ans.

Hence f(x) is strictly increasing on

38. For the curve y = 4x³ – 2x⁵, find all the point at which the tangent passes through the origin.
Ans.

Equation

[Passes through (0, 0)]

39. Prove that the curves x = y², and xy = K cut at right angles if 8k² = 1
Ans.

40. Find the maximum area of an isoscelesinscribed in the ellipse
with its vertex at one end of the major axis.
Ans.

Let A be the area of ABC

For maximum/minimum

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6 Marks Questions

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

Ans.

On again differentiate equation (1)

Hence maximum
Now

Value of sphere
Value of cone of value of sphere.

2. A wire of length 28m 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 areas of the square and the circle is minimum.
Ans. Let 1^st length = x
2^nd length = 28-x

A = area of circle + area of square

+tive hence minimum
1^st length =
2^nd length =

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

For maximum/minimum

-tive maximum

4. Show that semi – vertical angle of right circular cone of given surface area and maximum volume is
Ans.

Let v be the volume

Hence minimum
Now

5. A square piece of tin of side 18cm is to be made into a box without top by cutting a square from each corner and folding of 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?
Ans.

For maximum/minimum

6. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Ans.

For maximum/minimum

7. Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and having semi-vertical angle is one third that of the cone and the greatest volume of cylinder is

Ans.

for maximum/minimum

– tive maximum

8. Show that the right circular cone of least curved surface and given volume has an altitude equal totimes the radius of the base.
Ans.

When K is constant

-tive maximum

9. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2m and volume is 8m³. If building of tank costs Rs 70 per sq. metres for the base and Rs 45 per sq. metres for sides what is the cost of least expansive tank?
Ans. Let x and y be the length and width of rectangular base, v be the volume.

For maximum/minimum

Minimum

10. The sum of the perimeter of a circle and square is k, where K is some constant. Prove that the sum of their area is least when the side of square is double the radius of circle.
Ans. r is the radius of circle and x be side of sq.

For maximum/minimum

Hence maximum

11. A window is the form of a rectangle surmounted by a semi circular opening the total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening.
Ans. Let P be the perimeter of window

Let A be area of window

Length of rectangle

Width

12. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is.
Ans.

For maximum/minimum

L is minimum