Mathematics · Textbook solutions

Application of Derivatives

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 143 questions

6.2 Rate of Change of Quantities

25 q

Solved Examples

Worked · 6

6.1 Eg.1 $r$
6.1 Eg.2 $The volume of a cube is increasing at a rate of 9 cubic centimetres per second. How fast is the surface area increasing when the length of an edge is 10 centimetres?$
6.1 Eg.3 $A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant, when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?$
6.1 Eg.4 $x$
6.1 Eg.5 $C(x)$
6.1 Eg.6 $x$

Exercise 6.1

Practice · 19

Find the rate of change of the area of a circle with respect to its radius $r$ when
Ex 6.1 Q1(a) $r = 3$
Ex 6.1 Q1(b) $r = 4$
Ex 6.1 Q2 $8\ \text{cm}^3/\text{s}$
Ex 6.1 Q3 $The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.$
Ex 6.1 Q4 $An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?$
Ex 6.1 Q5 $A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?$
Ex 6.1 Q6 $The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?$
Ex 6.1 Q7 $x$
Ex 6.1 Q8 $A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.$
Ex 6.1 Q9 $A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.$
Ex 6.1 Q10 $A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?$
Ex 6.1 Q11 $6y = x^3 + 2$
Ex 6.1 Q12 $\frac{1}{2}$
Ex 6.1 Q13 $\frac{3}{2}(2x + 1)$
Ex 6.1 Q14 $12\ \text{cm}^3/\text{s}$
Ex 6.1 Q15 $C(x)$
Ex 6.1 Q16 $x$
Ex 6.1 Q17 $r$
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Ex 6.1 Q18 $x$
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6.3 Increasing and Decreasing Functions

30 q

Solved Examples

Worked · 7

6.2 Eg.7 $f(x) = 7x - 3$
6.2 Eg.8 $f$
6.2 Eg.9 $f(x) = \cos x$
6.2 Eg.10 $f$
6.2 Eg.11 $f$
6.2 Eg.12 $f(x) = \sin 3x,\ x \in \left[0, \frac{\pi}{2}\right]$
6.2 Eg.13 $f$

Exercise 6.2

Practice · 23

Ex 6.2 Q1 $f(x) = 3x + 17$
Ex 6.2 Q2 $f(x) = e^{2x}$
Ex 6.2 Q3 $f(x) = \sin x$
Ex 6.2 Q4 $f$
Ex 6.2 Q5 $f$
Find the intervals in which the following functions are strictly increasing or decreasing:
Ex 6.2 Q6(a) $x^2 + 2x - 5$
Ex 6.2 Q6(b) $10 - 6x - 2x^2$
Ex 6.2 Q6(c) $-2x^3 - 9x^2 - 12x + 1$
Ex 6.2 Q6(d) $6 - 9x - x^2$
Ex 6.2 Q6(e) $(x + 1)^3 (x - 3)^3$
Ex 6.2 Q7 $y = \log(1 + x) - \frac{2x}{2 + x},\ x > -1$
Ex 6.2 Q8 $x$
Ex 6.2 Q9 $y = \frac{4\sin\theta}{(2 + \cos\theta)} - \theta$
Ex 6.2 Q10 $(0, \infty)$
Ex 6.2 Q11 $f$
Ex 6.2 Q12 $\left(0, \frac{\pi}{2}\right)$
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Ex 6.2 Q13 $f$
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Ex 6.2 Q14 $a$
Ex 6.2 Q15 $[-1, 1]$
Ex 6.2 Q16 $f$
Ex 6.2 Q17 $f$
Ex 6.2 Q18 $f(x) = x^3 - 3x^2 + 3x - 100$
Ex 6.2 Q19 $y = x^2 e^{-x}$
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6.4 Maxima and Minima

64 q

Solved Examples

Worked · 16

6.3 Eg.14 $f$
6.3 Eg.15 $f$
6.3 Eg.16 $f(x) = x$
6.3 Eg.17 $f$
6.3 Eg.18 $f$
6.3 Eg.19 $f$
6.3 Eg.20 $f$
[NCERT poses this same function as Example 18 and again as Example 21. Example 18 settles it with the FIRST derivative test; this one is worked to show that the SECOND derivative test FAILS here (f''(1) = 0) and one must fall back to the first. Both are kept because the two teach different things.]
6.3 Eg.21 $f$
6.3 Eg.22 $Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.$
6.3 Eg.23 $(0, c)$
6.3 Eg.24 $\mathrm{RP}^2 + \mathrm{RQ}^2$
6.3 Eg.25 $If length of three sides of a trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.$
6.3 Eg.26 $Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.$
6.3 Eg.27 $f$
6.3 Eg.28 $f$
6.3 Eg.29 $y = x^2 + 7$

Exercise 6.3

Practice · 48

Find the maximum and minimum values, if any, of the following functions given by
Ex 6.3 Q1(i) $f(x) = (2x - 1)^2 + 3$
Ex 6.3 Q1(ii) $f(x) = 9x^2 + 12x + 2$
Ex 6.3 Q1(iii) $f(x) = -(x - 1)^2 + 10$
Ex 6.3 Q1(iv) $g(x) = x^3 + 1$
Find the maximum and minimum values, if any, of the following functions given by
Ex 6.3 Q2(i) $f(x) = |x + 2| - 1$
Ex 6.3 Q2(ii) $g(x) = -|x + 1| + 3$
Ex 6.3 Q2(iii) $h(x) = \sin(2x) + 5$
Ex 6.3 Q2(iv) $f(x) = |\sin 4x + 3|$
Ex 6.3 Q2(v) $h(x) = x + 1$
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
Ex 6.3 Q3(i) $f(x) = x^2$
Ex 6.3 Q3(ii) $g(x) = x^3 - 3x$
Ex 6.3 Q3(iii) $h(x) = \sin x + \cos x$
Ex 6.3 Q3(iv) $f(x) = \sin x - \cos x$
Ex 6.3 Q3(v) $f(x) = x^3 - 6x^2 + 9x + 15$
Ex 6.3 Q3(vi) $g(x) = \frac{x}{2} + \frac{2}{x}$
Ex 6.3 Q3(vii) $g(x) = \frac{1}{x^2 + 2}$
Ex 6.3 Q3(viii) $f(x) = x\sqrt{1 - x}$
Prove that the following functions do not have maxima or minima:
Ex 6.3 Q4(i) $f(x) = e^x$
Ex 6.3 Q4(ii) $g(x) = \log x$
Ex 6.3 Q4(iii) $h(x) = x^3 + x^2 + x + 1$
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
Ex 6.3 Q5(i) $f(x) = x^3$
Ex 6.3 Q5(ii) $f(x) = \sin x + \cos x$
Ex 6.3 Q5(iii) $f(x) = 4x - \frac{1}{2}x^2$
Ex 6.3 Q5(iv) $f(x) = (x - 1)^2 + 3$
Ex 6.3 Q6 $p(x) = 41 - 72x - 18x^2$
Ex 6.3 Q7 $3x^4 - 8x^3 + 12x^2 - 48x + 25$
Ex 6.3 Q8 $[0, 2\pi]$
Ex 6.3 Q9 $\sin x + \cos x$
Ex 6.3 Q10 $2x^3 - 24x + 107$
Ex 6.3 Q11 $x = 1$
Ex 6.3 Q12 $x + \sin 2x$
Ex 6.3 Q13 $Find two numbers whose sum is 24 and whose product is as large as possible.$
Ex 6.3 Q14 $x$
Ex 6.3 Q15 $x$
Ex 6.3 Q16 $Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.$
Ex 6.3 Q17 $A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.$
Ex 6.3 Q18 $A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?$
Ex 6.3 Q19 $Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.$
Ex 6.3 Q20 $Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.$
Ex 6.3 Q21 $Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?$
Ex 6.3 Q22 $A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?$
Ex 6.3 Q23 $\frac{8}{27}$
Ex 6.3 Q24 $\sqrt{2}$
Ex 6.3 Q25 $\tan^{-1}\sqrt{2}$
Ex 6.3 Q26 $\sin^{-1}\left(\frac{1}{3}\right)$
Ex 6.3 Q27 $x^2 = 2y$
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Ex 6.3 Q28 $x$
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Ex 6.3 Q29 $[x(x - 1) + 1]^{\frac{1}{3}}$
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Miscellaneous Exercise on Chapter 6

24 q

Solved Examples

Worked · 8

Misc Eg.30 $t = 0$
Misc Eg.31 $\tan^{-1}(0.5)$
Misc Eg.32 $A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post which is 6 metres high. Find the rate at which the length of his shadow increases.$
Misc Eg.33 $f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11$
Misc Eg.34 $f$
Misc Eg.35 $A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s. Find the rate at which its area is increasing when radius is 3.2 cm.$
Misc Eg.36 $An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.$
Misc Eg.37 $x$

Miscellaneous Exercise

Practice · 16

Misc Q1 $f(x) = \frac{\log x}{x}$
Misc Q2 $b$
Misc Q3 $f$
Misc Q4 $f$
Misc Q5 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Misc Q6 $^3$
Misc Q7 $k$
Misc Q8 $A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.$
Misc Q9 $a$
Misc Q10 $f$
Misc Q11 $f$
Misc Q12 $r$
Misc Q13 $f$
Misc Q14 $\frac{2\mathrm{R}}{\sqrt{3}}$
Misc Q15 $h$
Misc Q16 $A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of$
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What the board has asked from this chapter

Questions per paper · 21 papers · 2023 to 2026

Bars average the full papers of each year. 51 further questions come from set variants and are listed below.

2026

44 q

6 $^3$
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Rate of Change of Quantities
6 $S = 72t + 3t^2 - t^3$
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Rate of Change of Quantities
6 $The surface area of a sphere when its volume changes at the same rate as its radius is :$
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Rate of Change of Quantities
7 $f(x) = x^3 - 12x$
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Absolute Maximum and Minimum on a Closed Interval
8 $f(x) = x + \dfrac{1}{x}$
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Maxima and Minima
9 $f(x) = (x - 2)^2 + 5$
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Absolute Maximum and Minimum on a Closed Interval
9 $The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :$
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Rate of Change of Quantities
11 $f(x) = e^{-x}$
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Absolute Maximum and Minimum on a Closed Interval
18 $f(x) = x^2 + 1$
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Absolute Maximum and Minimum on a Closed Interval
22 $\left(0, \dfrac{\pi}{2}\right)$
Increasing and Decreasing Functions
23 $f(x) = \dfrac{x}{3} + \dfrac{3}{x}$
Increasing and Decreasing Functions
22 $^3$
Rate of Change of Quantities
23 $f(x) = \dfrac{x - 3}{x + 1}$
Increasing and Decreasing Functions
23 (a) $f(x) = \cos x + \sin^2 x$
Absolute Maximum and Minimum on a Closed Interval
23 $f(x) = x^x$
Increasing and Decreasing Functions
23 $(0, \infty)$
Increasing and Decreasing Functions
23 $f(x) = 5x^{3/2} - 3x^{5/2}$
Increasing and Decreasing Functions
23 $f(x) = \dfrac{x}{\log x}$
Increasing and Decreasing Functions
23 (b) $If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.$
Rate of Change of Quantities
23 $f(x) = \dfrac{x}{3} + \dfrac{3}{x}$
Increasing and Decreasing Functions
25 $(0, \infty)$
Increasing and Decreasing Functions
26 $The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate of change of its surface area varies inversely as the length of edge of the cube.$
Rate of Change of Quantities
26 $A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.$
Rate of Change of Quantities
26 $A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that the rate of change of circumference varies inversely as the radius.$
Rate of Change of Quantities
32 (a) $\left(0, \dfrac{\pi}{2}\right)$
Increasing and Decreasing Functions
32 (b) $A rectangle of perimeter 30 cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$
Maxima and Minima
33 (a) $f(x) = \cot^{-1}(\sin x + \cos x)$
Increasing and Decreasing Functions
33 (b) $A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$
Maxima and Minima
35 (a) $(0, \pi)$
Increasing and Decreasing Functions
35 (b) $A rectangle of perimeter 24 cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$
Maxima and Minima
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
36 (i) $x$
Maxima and Minima
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
36 (i) $\alpha$
Rate of Change of Quantities
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
36 (i) $p(x) = l_1 + l_2$
Maxima and Minima
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
36 (ii) $R(x)$
Maxima and Minima
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
36 (ii) $p'(x)$
Maxima and Minima
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
36 (ii) $At what rate is the juice level in the cup rising when the juice is 6 cm deep ?$
Rate of Change of Quantities
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
36 (iii) (a) $l_1^2 + l_2^2$
Maxima and Minima
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
36 (iii) (a) $When the juice is 6 cm deep, then find at what rate is the upper surface area of juice increasing ?$
Rate of Change of Quantities
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
36 (iii) (a) $x$
Maxima and Minima
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹ 300 per subscriber for unlimited free deliveries. The company wishes to increase the annual subscription fee. It is predicted that, for every increase of ₹ 1, ten subscribers will discontinue. Assume that the company increased the annual fee by ₹ $x$ . Based on the given information, answer the following questions :
36 (iii) (b) $R(x)$
Increasing and Decreasing Functions
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
36 (iii) (b) $If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum ?$
Maxima and Minima
At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of $0{\cdot}1$ cm $^3$ /s. On the basis of the above information, answer the following questions :
36 (iii) (b) $When the juice is 6 cm deep, then find the rate at which the wetted surface area of the cup is increasing.$
Rate of Change of Quantities
A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. Based on the above information, answer the following questions : If for a tumbler, V is its volume, h the height and r the radius of the circular base, then :
38 (i) $Differentiate its volume with respect to radius of the base, where the surface area is constant.$
Maxima and Minima
A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. Based on the above information, answer the following questions : If for a tumbler, V is its volume, h the height and r the radius of the circular base, then :
38 (ii) $If the company wants to maximize the volume of each tumbler, then establish a relation between its height and the radius of the base.$
Maxima and Minima

2025

51 q

2 $f(x) = x^2 - 4x + 6$
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Increasing and Decreasing Functions
5 $\lambda$
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Increasing and Decreasing Functions
22 $The radius of a cylinder is decreasing at a rate of 2 cm/s and the altitude is increasing at the rate of 3 cm/s. Find the rate of change of volume of this cylinder when its radius is 4 cm and altitude is 6 cm.$
Rate of Change of Quantities
22 $Let the volume of a metallic hollow sphere be constant. If the inner radius increases at the rate of 2 cm/s, find the rate of increase of the outer radius when the radii are 2 cm and 4 cm respectively.$
Rate of Change of Quantities
26 $f(x) = \tan^{-1}(\sin x + \cos x)$
Increasing and Decreasing Functions
8 $f(x) = x^3 - 3x + 2$
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Absolute Maximum and Minimum on a Closed Interval
9 $f(x) = x^x$
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Maxima and Minima
9 $f(x) = x^2$
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Maxima and Minima
26 $f(x) = \sin 3x - \cos 3x$
Increasing and Decreasing Functions
10 $y = -x^3 + 3x^2 + 8x - 20$
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Maxima and Minima
10 $Edge of a variable cube increases at the rate of 5 cm/s. The rate at which the surface area of the cube increases when the edge is 2 cm long is :$
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Rate of Change of Quantities
10 $\dfrac{5}{2}(3x + 1)$
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Rate of Change of Quantities
11 $f : R \to R$
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Increasing and Decreasing Functions
11 $100\,\pi$
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Rate of Change of Quantities
13 $f(x) = 2x + \cos x$
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Increasing and Decreasing Functions
22 $f(x) = \sqrt{3} \sin x - \cos x - 2ax + b$
Increasing and Decreasing Functions
22 (a) $f(x) = 2x^2 - ax + 3$
Increasing and Decreasing Functions
22 $^2$
Rate of Change of Quantities
22 $f(x) = x^2 - 2ax + b$
Increasing and Decreasing Functions
22 (b) $f(x) = x + \frac{1}{x}$
Increasing and Decreasing Functions
23 $f(x) = \dfrac{x - 4}{x + 1}$
Increasing and Decreasing Functions
23 $f(x) = x + \dfrac{1}{x}$
Increasing and Decreasing Functions
23 $f(x) = \sin x - ax + b$
Increasing and Decreasing Functions
23 $f(x) = \dfrac{2}{x} - 5$
Increasing and Decreasing Functions
24 $f(x) = 5x^{\frac{3}{2}} - 3x^{\frac{5}{2}}$
Increasing and Decreasing Functions
25 $y = 5x - 2x^3$
Rate of Change of Quantities
25 $f(x) = \frac{8}{3}x^3 - 12x^2 + 18x + 5$
Maxima and Minima
25 $5\ \text{cm}^3/\text{s}$
Rate of Change of Quantities
26 $f(x) = \sqrt{3}\sin x - \cos x - 2ax + 6$
Increasing and Decreasing Functions
26 $The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm ?$
Rate of Change of Quantities
28 $^2$
Rate of Change of Quantities
28 $A spherical medicine ball when dropped in water dissolves in such a way that the rate of decrease of volume at any instant is proportional to its surface area. Calculate the rate of decrease of its radius.$
Rate of Change of Quantities
29 $Show that of all the rectangles with a fixed perimeter, the square has the greatest area.$
Maxima and Minima
29 $Find dimensions of a rectangle of perimeter 12 cm which will generate maximum volume when swept along a circular rotation keeping the shorter side fixed as the axis.$
Maxima and Minima
29 $Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.$
Maxima and Minima
33 $y = 4x - \frac{1}{2}x^2$
Maxima and Minima
33 $f(x) = 2x^3 - 15x^2 + 36x + 1$
Absolute Maximum and Minimum on a Closed Interval
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
36 (i) $x$
Maxima and Minima
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum. On the basis of the above information, answer the following questions :
36 (i) $Taking length = breadth = x m and height = y m, express the surface area (S) of the box in terms of x and its volume (V), which is constant.$
Maxima and Minima
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum. On the basis of the above information, answer the following questions :
36 (ii) $\dfrac{dS}{dx}$
Maxima and Minima
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
36 (ii) $x$
Maxima and Minima
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
36 (iii) (a) $Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.$
Maxima and Minima
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum. On the basis of the above information, answer the following questions :
36 (iii) (a) $Find a relation between x and y such that the surface area (S) is minimum.$
Maxima and Minima
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be y metres. Based on this information, answer the following questions :
36 (iii) (b) $Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.$
Maxima and Minima
A carpenter needs to make a wooden cuboidal box, closed from all sides, which has a square base and fixed volume. Since he is short of the paint required to paint the box on completion, he wants the surface area to be minimum. On the basis of the above information, answer the following questions :
36 (iii) (b) $(V) = \dfrac{1}{4}(Sx - 2x^3)$
Maxima and Minima
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
37 (i) $Express the distance (y) between the wall and foot of the ladder in terms of ‘h’ and height (x) on the wall at a certain instant. Also, write an expression in terms of h and x for the area (A) of the right triangle, as seen from the side by an observer.$
Maxima and Minima
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
37 (ii) $Find the derivative of the area (A) with respect to the height on the wall (x), and find its critical point.$
Maxima and Minima
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
37 (iii) (a) $Show that the area (A) of the right triangle is maximum at the critical point.$
Maxima and Minima
A ladder of fixed length ‘h’ is to be placed along the wall such that it is free to move along the height of the wall. Based upon the above information, answer the following questions :
37 (iii) (b) $If the foot of the ladder whose length is 5 m, is being pulled towards the wall such that the rate of decrease of distance (y) is 2 m/s, then at what rate is the height on the wall (x) increasing, when the foot of the ladder is 3 m away from the wall ?$
Rate of Change of Quantities
A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point $x$ metres from the start of the street can be modelled by $f(x) = e^x \sin x$ , where $x$ is in metres. Based on the above, answer the following :
38 (i) $f(x)$
Increasing and Decreasing Functions
A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point $x$ metres from the start of the street can be modelled by $f(x) = e^x \sin x$ , where $x$ is in metres. Based on the above, answer the following :
38 (ii) $x \in [0, \pi]$
Maxima and Minima

2024

37 q

4 $Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if$
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Increasing and Decreasing Functions
5 $f(x) = x^3 - 3x^2 + 12x - 18$
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Increasing and Decreasing Functions
5 $1{\cdot}5$
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Rate of Change of Quantities
9 $f(x) = \dfrac{x}{2} + \dfrac{2}{x}$
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Maxima and Minima
10 $y = 7x - x^3$
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Rate of Change of Quantities
11 $f(x)$
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Maxima and Minima
11 $'r'$
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Rate of Change of Quantities
18 $f(x) = kx - \sin x$
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Increasing and Decreasing Functions
21 $f(x) = x^2 - 6x + 3$
Increasing and Decreasing Functions
22 $f(x) = 4x^3 - 18x^2 + 27x - 7$
Maxima and Minima
23 $f(x) = \dfrac{4 \sin x}{2 + \cos x} - x$
Increasing and Decreasing Functions
23 $^3$
Rate of Change of Quantities
23 $^2$
Rate of Change of Quantities
23 $f(x) = 4x^2 + \frac{1}{x}$
Maxima and Minima
23 $^2$
Rate of Change of Quantities
23 $f(x) = \sin x + \cos x$
Increasing and Decreasing Functions
23 $f(x) = x^{4} - 4x^{3} + 10$
Increasing and Decreasing Functions
23 $f(x) = x + \dfrac{1}{x}$
Maxima and Minima
24 $^{3}$
Rate of Change of Quantities
25 $f(x) = e^x - e^{-x} + x - \tan^{-1} x$
Increasing and Decreasing Functions
25 $f(x) = 12x^{4/3} - 6x^{1/3}$
Absolute Maximum and Minimum on a Closed Interval
27 (a) $f(x) = \dfrac{\log x}{x}$
Increasing and Decreasing Functions
27 (b) $f(x) = \dfrac{x}{2} + \dfrac{2}{x}$
Absolute Maximum and Minimum on a Closed Interval
33 (a) $f(x) = x^4 - 62x^2 + ax + 9$
Maxima and Minima
33 (b) $The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.$
Maxima and Minima
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
36 (i) $Find F, when V = 40 km/h.$
Rate of Change of Quantities
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
36 (i) $\theta$
Rate of Change of Quantities
A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units ( $x$ ) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \dfrac{1}{2}x$ . Based on the above information, answer the following questions :
36 (i) $x$
Maxima and Minima
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
36 (ii) $\dfrac{d\theta}{dx}$
Rate of Change of Quantities
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
36 (ii) $\dfrac{dF}{dV}$
Rate of Change of Quantities
A store has been selling calculators at ₹ 350 each. A market survey indicates that a reduction in price (p) of calculator increases the number of units ( $x$ ) sold. The relation between the price and quantity sold is given by the demand function $p = 450 - \dfrac{1}{2}x$ . Based on the above information, answer the following questions :
36 (ii) $What rebate in price of calculator should the store give to maximise the revenue ?$
Maxima and Minima
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
36 (iii) (a) $Find the speed V for which fuel consumption F is minimum.$
Maxima and Minima
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
36 (iii) (a) $Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.$
Rate of Change of Quantities
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, x m away from the base of the pole, the angle of elevation of the speed camera from the car C is $\theta$ . On the basis of the above information, answer the following questions :
36 (iii) (b) $\dfrac{3}{101}$
Rate of Change of Quantities
Overspeeding increases fuel consumption and decreases fuel economy as a result of tyre rolling friction and air resistance. While vehicles reach optimal fuel economy at different speeds, fuel mileage usually decreases rapidly at speeds above 80 km/h. The relation between fuel consumption F (l/100 km) and speed V (km/h) under some constraints is given as $F = \dfrac{V^2}{500} - \dfrac{V}{4} + 14$ . On the basis of the above information, answer the following questions :
36 (iii) (b) $\dfrac{dF}{dV} = -0{\cdot}01$
Rate of Change of Quantities
A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm as shown below : On the basis of the above information, answer the following questions :
37 (i) $x$
Maxima and Minima
A rectangular visiting card is to contain 24 sq.cm. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be 1½ cm as shown below : On the basis of the above information, answer the following questions :
37 (ii) $Obtain the dimensions of the card of minimum area.$
Maxima and Minima

2023

34 q

9 $f(x) = 2x^3 + 9x^2 + 12x - 1$
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Increasing and Decreasing Functions
9 $f(x) = x^3 + 3x$
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Increasing and Decreasing Functions
16 $f(x) = a(x - \cos x)$
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Increasing and Decreasing Functions
21 $f(x) = 2x^3 - 3x$
Increasing and Decreasing Functions
21 $If the product of two positive numbers is 9, find the numbers so that the sum of their squares is minimum.$
Maxima and Minima
21 $b \in \mathbb{R}$
Increasing and Decreasing Functions
23 $-$
Increasing and Decreasing Functions
22 $3y = ax^3 + 1$
Rate of Change of Quantities
23 $If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal ?$
Rate of Change of Quantities
23 $f(x) = 5 + \sin 2x$
Maxima and Minima
23 $f(x) = \log (2 + x) - \frac{x}{2 + x}$
Increasing and Decreasing Functions
24 $y^2 = 8x$
Rate of Change of Quantities
24 $6y = x^3 + 2$
Rate of Change of Quantities
25 $f(x) = \frac{16 \sin x}{4 + \cos x} - x$
Increasing and Decreasing Functions
25 $If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.$
Rate of Change of Quantities
32 (a) $2\sqrt{3}$
Rate of Change of Quantities
32 (b) $Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.$
Maxima and Minima
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area $75\pi$ cm $^2$ . Based on the above information, answer the following questions :
36 (i) $If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r.$
Maxima and Minima
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area $75\pi$ cm $^2$ . Based on the above information, answer the following questions :
36 (ii) $\frac{dV}{dr}$
Maxima and Minima
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area $75\pi$ cm $^2$ . Based on the above information, answer the following questions :
36 (iii)(a) $Find the radius of cylinder when its volume is maximum.$
Maxima and Minima
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area $75\pi$ cm $^2$ . Based on the above information, answer the following questions :
36 (iii)(b) $For maximum volume, h > r. State true or false and justify.$
Maxima and Minima
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
37 (i) $Find the volume of water in the tank in terms of its radius r.$
Rate of Change of Quantities
In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
37 (i) $Find the total cost C of digging the tank in terms of x.$
Maxima and Minima
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
37 (ii) $r = 2\sqrt{2}$
Rate of Change of Quantities
In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
37 (ii) $\frac{dC}{dx}$
Maxima and Minima
In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
37 (iii) (a) $Find the value of x for which cost C is minimum.$
Maxima and Minima
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
37 (iii) (a) $r = 2\sqrt{2}$
Rate of Change of Quantities
In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 $\text{m}^3$ . The cost of land is ₹ 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is ₹ 40,000 $h^2$ , where h is the depth of the tank in metres. x is the side of the square base of the tank in metres.
37 (iii) (b) $Check whether the cost function C(x) expressed in terms of x is increasing or not, where x > 0.$
Increasing and Decreasing Functions
A tank, as shown in the figure below, formed using a combination of a cylinder and a cone, offers better drainage as compared to a flat bottomed tank. A tap is connected to such a tank whose conical part is full of water. Water is dripping out from a tap at the bottom at the uniform rate of $2 \text{ cm}^3/\text{s}$ . The semi-vertical angle of the conical tank is $45^\circ$ . On the basis of given information, answer the following questions :
37 (iii) (b) $Find the rate of change of height 'h' at an instant when slant height is 4 cm.$
Rate of Change of Quantities
The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time t is given by the function V : $V(t) = \frac{1}{5}t^3 - \frac{5}{2}t^2 + 25t - 2$ where t represents the time and t = 1, 2, 3.... corresponds to year 2001, 2002, 2003, ....... respectively. Based on the above information, answer the following questions :
38 (i) $Can the above function be used to estimate number of vehicles in the year 2000 ? Justify.$
Increasing and Decreasing Functions
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 :
38 (i) $Let 'x' metres denote the length of the side of the garden perpendicular to the brick wall and 'y' metres denote the length of the side parallel to the brick wall. Determine the relation representing the total length of fencing wire and also write A(x), the area of the garden.$
Maxima and Minima
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 :
38 (ii) $Determine the maximum value of A(x).$
Maxima and Minima
A volleyball player serves the ball which takes a parabolic path given by the equation $h(t) = -\frac{7}{2} t^2 + \frac{13}{2} t + 1$ , where h(t) is the height of ball at any time t (in seconds), $(t \geq 0)$ .
38 (ii) $Find the time at which the height of the ball is maximum.$
Maxima and Minima
The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and estimated electric vehicles in use at any time t is given by the function V : $V(t) = \frac{1}{5}t^3 - \frac{5}{2}t^2 + 25t - 2$ where t represents the time and t = 1, 2, 3.... corresponds to year 2001, 2002, 2003, ....... respectively. Based on the above information, answer the following questions :
38 (ii) $Prove that the function V(t) is an increasing function.$
Increasing and Decreasing Functions

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
33 (b) $A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.$
Question 2
Asked 2 times: 2024*, 2026 (* = numbers or wording changed that year)
8 $f(x) = x + \dfrac{1}{x}$
Tap an option to check your answer.
Question 3
Asked 2 times: 2025, 2026
23 $f(x) = 5x^{3/2} - 3x^{5/2}$
Question 4
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
23 $f(x) = \dfrac{x - 3}{x + 1}$
Question 5
Asked 2 times: 2025, 2026
26 $A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.$
Question 6
Asked 2 times: 2025*, 2026 (* = numbers or wording changed that year)
6 $^3$
Tap an option to check your answer.
Question 7
Asked 2 times: 2024*, 2025 (* = numbers or wording changed that year)
25 $y = 5x - 2x^3$
Question 8
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
23 $f(x) = \dfrac{4 \sin x}{2 + \cos x} - x$
Question 9 (case study, 4 parts)
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
Two vertical light poles of height 22 m and 16 m stand on the opposite sides of a 20 m wide road as shown below in the figure. Two ladders of length $l_1$ and $l_2$ are placed from a common point R on the road at a distance of x m from the smaller pole. Based on the above information, answer the following questions :
36 (i) $p(x) = l_1 + l_2$
36 (ii) $p'(x)$
36 (iii) (a) $l_1^2 + l_2^2$
36 (iii) (b) $If the 22 m long pole is also replaced by a 16 m long pole, at what distance from either pole should the ladders be kept so that the sum of squares of lengths of ladders needed to reach the top of the pole is minimum ?$
Question 10
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
22 $\left(0, \dfrac{\pi}{2}\right)$

6.2 Rate of Change of Quantities

Solved Examples

Exercise 6.1

6.3 Increasing and Decreasing Functions

Solved Examples

Exercise 6.2

6.4 Maxima and Minima

Solved Examples

Exercise 6.3

Miscellaneous Exercise on Chapter 6

Solved Examples

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9 (case study, 4 parts)

Question 10