MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2024: question 36

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedApplication of Derivatives

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 $θ$ .

On the basis of the above information, answer the following questions :

(i)
Express $θ$ in terms of height of the camera installed on the pole and x.
1 mark
(ii)
Find $\frac{d θ}{d x}$ .
1 mark
(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.
2 marks
OR
(iii)
(b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is $\frac{3}{101}$ rad/s, then find the speed of the car.
2 marks

Figure in the paper: Illustration of speed cameras on poles: average speed detection with Point A and Point B cameras (Speed = Distance / (Time B - Time A)), and RADAR speed detection where radar measures change in frequency of returned radio waves (Doppler effect), with cars passing below. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Draw the right triangle: the pole height is the opposite side and $x$ is the adjacent side to $θ$ .

Hint 2

Use $\frac{d}{d x} tan^{- 1} u = \frac{1}{1 + u ^{2}} \frac{d u}{d x}$ with $u = 5/ x$ .

Hint 3

For (iii), use the chain rule $\frac{d θ}{d t} = \frac{d θ}{d x} \cdot \frac{d x}{d t}$ and put $x = 50$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) In the right triangle formed by the pole (height 5 m) and the car at distance $x$ m from its base, $tan θ = \frac{5}{x}$ , so $θ = tan^{- 1} (\frac{5}{x})$ .
1 mark
(ii) $\frac{d θ}{d x} = \frac{1}{1 + \frac{25}{x ^{2}}} \cdot (- \frac{5}{x ^{2}}) = \frac{- 5}{25 + x ^{2}}$
1 mark
(iii)(a) The car moves away, so $\frac{d x}{d t} = 20$ m/s.
$\frac{d θ}{d t} = \frac{d θ}{d x} \cdot \frac{d x}{d t} = \frac{- 5}{25 + x ^{2}} \times 20$ . At $x = 50$ : $\frac{d θ}{d t} = \frac{- 100}{25 + 2500}$
1½ marks
(iii)(a) $= \frac{- 100}{2525} = - \frac{4}{101}$ rad/s. The angle of elevation decreases at $\frac{4}{101}$ rad/s.
½ mark
(iii)(b) $\frac{d θ}{d t} = \frac{d θ}{d x} \cdot \frac{d x}{d t}$ . At $x = 50$ : $\frac{3}{101} = \frac{- 5}{2525} \cdot \frac{d x}{d t}$
1½ marks
(iii)(b) $\frac{d x}{d t} = \frac{3}{101} \times \frac{2525}{- 5} = - 15$ m/s. The negative sign means the car is moving towards the pole. Speed of the car = 15 m/s.
½ mark

Answer: (i) $θ = tan^{- 1} (5/ x)$ ; (ii) $\frac{d θ}{d x} = \frac{- 5}{25 + x ^{2}}$ ; (iii)(a) $- \frac{4}{101}$ rad/s; (iii)(b) 15 m/s

Where the marks are

(i) Uses $tan θ = 5/ x$ to write $θ = tan^{- 1} (5/ x)$ (1 mark)
(ii) Differentiates correctly to get $\frac{- 5}{25 + x ^{2}}$ (1 mark)
(iii)(a) Applies the chain rule with $d x / d t = 20$ at $x = 50$ (1½ marks)
(iii)(a) Gets $- \frac{4}{101}$ rad/s (½ mark)
(iii)(b) Sets up $\frac{3}{101} = \frac{- 5}{2525} \cdot \frac{d x}{d t}$ (1½ marks)
(iii)(b) Gets $d x / d t = - 15$ , so speed = 15 m/s (½ mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2024: question 36

Hints

Step-by-step solution

Revise the chapter

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