Application of Derivatives

Previous Years' CBSE Board Questions

6.2 Rate of Change of Quantities

1. (-4.0) (Term I, 2021-22)

VSA. (1 mark)

The radius of a circle is increasing at the uniform rate of 3 cm/sec. At the instant when the radius of the circle is 2 cm, its area increases at the rate of . cm²/s. (2020) (Ap

2 The rate of change of the area of a circle with respect to its radius r, when <math>r = 3</math> cm, is _____ (2020)

SAI (2 marks)

1. The total cost C(x) associated with production of x units of an item is given by <math>C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000</math>. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. (2018) [Ap

1. The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. (Delhi 2017)
2. The volume of a cube is increasing at the rate of 9 cm³/s. How fast is its surface area increasing when the length of an edge is 10 cm? (NCERT, AI 2017) EV

LAI (4 marks)

(2019) U

1. A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

(AI 2019)

1. The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm? (Delhi 2015) EV
2. The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. Find the rate at which the area increases, when the side is 10 cm. (Al 2014C)

6.3 Increasing and Decreasing Functions

MCQ

1. The interval in which the function <math>f(x) = 2x^3 + 9x^2 +</math> 12x - 1 is decreasing, is
2. (-1,∞) (b) (-2,-1) (c) (-∞,-2) (d) [-1,1]

(2023)

1. The function <math>f(x) = x^3 + 3x</math> is increasing in interval
2. (-∞,0)(b) (0,∞)(c) R (d) (0, 1)

(2023)

1. The interval, in which function <math>y = x^3 + 6x^2 + 6</math> is increasing, is
2. <math>(-\infty, -4) \cup (0, \infty)</math> (b) <math>(-\infty, -4)</math>

(d) (-∞,0)∪(4,∞)

The function (x - sin x) decreases for

1. all x (b) <math>x < \frac{\pi}{2}</math>
2. <math>0 < x < \frac{\pi}{-}</math> (d) no value of x (Term I, 2021-22)

VSA (1 mark)

Find the interval in which the function f given by

<math>f(x) = 7 - 4x - x^2</math> is strictly increasing. (2020)

SAI (2 marks)

1. Find the interval in which the function <math>f(x) = 2x^3 - 3x</math> is strictly increasing. (2023)
2. Show that the function <math>f(x) = 4x^3 - 18x^2 + 27x - 7</math> is always increasing on R. (Delhi 2017)
3. Show that the function <math>f(x) = x^3 - 3x^2 + 6x - 100</math> is increasing on R. (Al 2017) Ap

LAI (4 marks)

17. Find whether the function <math>f(x) = \cos\left(2x + \frac{\pi}{4}\right)</math>; is

increasing or decreasing in the interval <math>\frac{3\pi}{\alpha} < x < \frac{5\pi}{\alpha}</math>.

Find the intervals in which the function

<math>f(x) = \frac{x^4}{4} - x^3 - 5x^2 + 24x + 12</math> is

(a) strictly increasing (b) Find the intervals in which the function

<math>f(x) = 3x^4 - 4x^3 - 12x^2 + 5</math> is

1. (b) strictly decreasing (Delhi 2014)

Find the value(s) of x for which y = [x (x-2)]² is an

increasing function. (Al 2014)

Find the 21. intervals in which the function

<math>f(x) = \frac{3}{2}x^4 - 4x^3 - 45x^2 + 51</math> is

1. (ii) strictly decreasing (Foreign 2014) Ap

Find the intervals in which the function

<math>f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11</math> is

1. (b) strictly decreasing. (NCERT, AI 2014C) [EV]

LAII (5/6 marks)

1. Find the intervals on which the function <math>f(x) = (x - 1)^3 (x - 2)^2</math> is (a) strictly increasing
2. strictly decreasing.

Find the intervals in which the function f defined as <math>f(x) = \sin x + \cos x</math>, <math>0 \le x \le 2\pi</math> is strictly increasing or decreasing. (2020) U

Find the intervals in which f(x) = sin 3x - cos 3x, 0 < x < π,</li> is strictly increasing or strictly decreasing. (Delhi 2016) An

(1 mark) VSA.

1. Prove that the function f defined by <math>f(x) = x^2 - x + 1</math> is neither increasing nor decreasing in (-1, 1). Hence, find the intervals in which f(x) is (i) strictly increasing
2. strictly decreasing. (Delhi 2014C)

6.4 Maxima and Minima

MCQ

The value of x for which (x - x²) is maximum, is

1. 3/4 (b) 1/2 (c) 1/3 (d) 1/4 (Term I, 2021-22) (U

A wire of length 20 cm is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is

1. 20 sq.cm (b) 25 sq.cm
2. 10 sq.cm (d) 30 sq.cm (Term I, 2021-22)

Case study-Some young entrepreneur started a industry "young achievers" for casting metal into various shapes. They put up an advertisement online stating the same and expecting order to cast metal for toys, sculptures, decorative pieces and more.

A group of friends wanted to make innovative toys and hence contacted the "young achievers" to order them to cast metal into solid half cylinders with a rectangular base and semi-circular ends.

Based on the above information, answer the following questions (29 to 33):

The volume (V) of the casted half cylinder will be

1. <math>\pi r^2 h</math> (b) <math>\frac{1}{2}\pi r^2 h</math>
2. <math>\frac{1}{2}\pi r^2 h</math> (d) <math>\pi r^2 (r + h)</math> (Term I, 2021-22)

The total surface area (S) of the casted half cylinder will be

1. <math>\pi r h + 2\pi r^2 + r h</math> (b) <math>\pi r h + \pi r^2 + 2r h</math>
2. <math>2\pi rh + \pi r^2 + 2rh</math> (d) <math>\pi rh + \pi r^2 + rh</math> (Term I, 2021-22) [EV]

The total surface area S can be expressed in terms of V and r as <math display="block">2\pi r + \frac{2V(\pi+2)}{\pi r}</math> <math>\pi r + \frac{2V}{}</math>

1. <math>\pi r^2 + \frac{2V(\pi+2)}{\pi}</math> (d) <math>2\pi r^2 + \frac{2V(\pi+2)}{\pi}</math> (Term I, 2021-22)

32. For the given half-cylinder of volume V, the total

surface area S is minimum, when

(a) <math>(\pi + 2) V = \pi^2 r^3</math> (b) <math>(\pi + 2) V = \pi^2 r^2</math>

(c) <math>2(\pi + 2) V = \pi^2 r^3</math> (d) <math>(\pi + 2) V = \pi^2 r</math>

(Term I. 2021-22)

The ratio h: 2r for which 5 to be minimum will be equal to

1. <math>2\pi : \pi + 2</math> (b) <math>2\pi : \pi + 1</math>
2. <math>\pi:\pi+1</math> (d) <math>\pi:\pi+2</math> (Term I, 2021-22) (Ci

1. The absolute minimum value of <math>f(x) = 2 \sin x</math> in <math>\left[0, \frac{3\pi}{2}\right]</math> is (2020)
2. The least value of the function <math>f(x) = ax + \frac{b}{a}</math> (a > 0, b > 0, x > 0) is ______. (2020)

(4 marks)

Case-study: Sooraj's father wants to construct a

rectangular garden using a brick wall on one side of

the garden and wire fencing for the other three sides

as shown in the figure. He has 200 metres of fencing

wire.

Based on the above information, answer the

following questions: (i) (ii) Determine the maximum value of A(x). (2023)

An open tank with a square base and vertical sides is

to be constructed from a metal sheet so as to hold a

given quantity of water. Show that the cost of material

will be least when depth of the tank is half of its width.

If the cost is to be borne by nearby settled lower

income families, for whom water will be provided,

what kind of value is hidden in this question?

(2018) Ap

LAII (5 / 6 marks)

1. The median of an equilateral triangle is increasing at the rate of <math>2\sqrt{3}</math> cm/s. Find the rate at which its side is increasing. (2023)
2. Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers. (2023)
3. Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right