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Mathematics · 2024 · 4 marks
CBSE 2024 · Region 1 · Set 2 · Q38
The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding…
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 $\displaystyle 300$ m and even function in the dark.
A camera is installed on a pole at the height of $\displaystyle 5$ m . It detects a car travelling away from the pole at the speed of $\displaystyle 20 \mathrm{~m} / \mathrm{s}$. At any point, $\displaystyle \times \mathrm{m}$ away from the base of the pole, the angle of elevation of the speed camera from the car $\displaystyle C$ is $\displaystyle \theta$. On the basis of the above information, answer the following questions :
(i)
Express $\displaystyle \theta$ in terms of height of the camera installed on the pole and x .
(ii)
Find $\displaystyle \frac{d \theta}{d x}$.
(iii)
(a)
Find the rate of change of angle of elevation with respect to time at an instant when the car is $\displaystyle 50$ m away from the pole.
OR
(b)
If the rate of change of angle of elevation with respect to time of another car at a distance of $\displaystyle 50$ m from the base of the pole is $\displaystyle \frac{3}{101} \mathrm{rad} / \mathrm{s}$, then find the speed of the car.
Marking-scheme solution
(i)
$\displaystyle \tan \boldsymbol{\theta}=\frac{\mathbf{5}}{\boldsymbol{x}} \Rightarrow \boldsymbol{\theta}=\boldsymbol{\operatorname { t a n }}^{-\mathbf{1}}\left(\frac{\mathbf{5}}{\boldsymbol{x}}\right)$
(ii)
$\displaystyle \frac{d \theta}{d x}=\frac{-5}{5^{2}+x^{2}}$
(iii)(a)
$\displaystyle \left.\frac{d \theta}{d t}=\frac{d \theta}{d x} \times \frac{d x}{d t}=\frac{-5}{5^{2}+x^{2}} \times 20\right]_{x=50}$
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CBSE Class 12 Mathematics past-paper question from the 2024 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.