True False[1 Marks ]

Question 11 Mark

Write ‘True’ or ‘False’ and justify your answer.
If a man standing on a platform 3 metres above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.

Answer

False. The observer is at the platform (P) 3m above the surface LK of the lake. He observes the angle of elevation of cloud C from P and its reflection image in the lake is formed at I. The observer measures in the angle of depression of image (I) $\theta_2.$ Draw PM $\perp$ on the vertical line passing through the cloud and its image.

CK = KI = x by the prop. of reflection. CM = CK - MK = x - 3 MI = KI + MK = x + 3 Now, $\tan\theta_1=\frac{\text{x}-3}{\text{y}}\text{ and }\tan\theta_2=\frac{\text{x}+3}{\text{y}}$ $\Rightarrow\ \text{y}=\frac{\text{x}-3}{\tan\theta_1}\text{ and }\text{y}=\frac{\text{x}+3}{\tan\theta_2}$ $\Rightarrow\ \frac{\text{x}+3}{\tan\theta_2}=\frac{\text{x}-3}{\tan\theta_1}$ $\Rightarrow\ \tan\theta_2=\Big(\frac{\text{x}+3}{\text{x}-3}\Big)\tan\theta_1$ $\Rightarrow\ \tan\theta_1\neq\tan\theta_2$or $\theta_1\neq\theta_2$
Alternate Answer
By the property of image formation, the distance of image and the object are equal from the reflecting surface. So, $\text{KC}=\text{KI}$ $\Rightarrow\ \text{MI}\neq\text{MC}$ $\Rightarrow\ \triangle\text{MPC}\neq\triangle\text{MPI}$ So $\theta_1\neq\theta_2$

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Question 21 Mark

Write ‘True’ or ‘False’ and justify your answer.
The angle of elevation of the top of a tower is 30°. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

Answer

False.Let the height of the tower is h. For the observer at A the angle of elevation is equal to 30º.

$\tan30^\circ=\frac{\text{h}}{\text{y}}$
$\Rightarrow\ \frac{1}{\sqrt{3}}=\frac{\text{h}}{\text{y}}$
$\Rightarrow\ \text{y}=\text{h}\sqrt{3}$
Now, the height of the tower increases to 2h.
Now, let the new angle of elevation at A becomes $\theta$ then
$\tan\theta=\frac{2\text{h}}{\text{y}}$
$\Rightarrow\ \tan\theta=\frac{2\text{h}}{\text{h}\sqrt{3}}$
$\Rightarrow\ \tan\theta=\frac{2}{\sqrt{3}}$
But, $\tan60^\circ=\sqrt{3}$
$\Rightarrow\ \tan60^\circ=\sqrt{3}\neq\frac{2}{\sqrt{3}}$
So, $\theta\neq60^\circ$
Hence, angle of elevation will not be doubled or the given statement is false.

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Question 31 Mark

Write ‘True’ or ‘False’ and justify your answer.
If the height of a tower and the distance of the point of observation from its foot, both, are increased by 10%, then the angle of elevation of its top remains unchanged.

Answer

True.Let height h of tower TW makes an angle of elevation to observer at A and the distance from foot of tower to the observer is x.

$\therefore\ \tan\theta=\frac{\text{h}}{\text{x}}$
Now, h and increases by 10%
$\therefore\ \text{h'=h+10% of h}=\text{h}+\frac{10}{100}\times\text{h}=\text{h}+0.1\text{h}$
⇒ h' = 1.1h
Similarly, x' = 1.1x

$\therefore\ \tan\theta'=\frac{1.1\text{h}}{1.1\text{x}}=\frac{\text{h}}{\text{x}}\ \ (\text{II})$
From (I) and (II), we get
$\tan\theta=\tan\theta'$
$\Rightarrow\ \theta=\theta'$
Hence, the given statement is true.

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Question 41 Mark

Write ‘True’ or ‘False’ and justify your answer.
If the length of the shadow of a tower is increasing, then the angle of elevation of the sun is also increasing.

Answer

False.The shadow of a tower on the ground increases from x to (x + y) when angle of elevation of the sun change from $\theta_1\text{ to }\theta_2.$

$\therefore\ \theta_1$ is the exterior angle of $\triangle\text{TSD}$
So $\theta_1>\theta_2$
So, on increasing the length of shadow the angle of elevation decreases.
Hence, the given statement is false.

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Maths True False[1 Marks ]

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