Given: 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.To do: We have to find whether the given statement is true or false.Solution:Let the height of the tower$=h$ and the distance of the point of observation from its foot$=x$Let the angle of elevation$=\alpha$$\therefore tan\alpha=\frac{h}{x}\ \ \ \ ...........\ ( i)$Again, if the height of the tower and the distance of the point of observation from its foot are increased by $10\ %$, Then the ... Read More

Given: The angle of elevation of the top of a tower is \( 30^{\circ} \). If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.To do:  We have to find whether the given statement is true or false.Solution:The given angle of elevation $=30^o$. Let the height of the tower$=h$, and the viewer be at a distance of $x$ from the foot of the tower.Then,   $\frac{h}{x}=tan30^o=\frac{1}{3}\ ........\ ( i)$If the height of the tower is doubled then the new height $=2h$.Let the angle of elevation of the top be $\theta$. Then, $tan\theta =\frac{2h}{x}=2\times ... Read More

Given:\( \cos \theta=\frac{a^{2}+b^{2}}{2 a b} \), where \( a \) and \( b \) are two distinct numbers such that \( a b>0 \).To do:We have to find whether the given statement is true or false.Solution:$a$ and $b$ are two distinct numbers such that $ab>0$.This implies, $AM>GM$AM and GM of two number $a$ and $b$ are $\frac{a+b}{2}$ and $\sqrt{a b}$Therefore, $\frac{a^{2}+b^{2}}{2}>\sqrt{a^{2} \times b^{2}}$$a^{2}+b^{2}>2 a b$$\frac{a^{2}+b^{2}}{2 a b}>1$$\cos \theta=\frac{a^{2}+b^{2}}{2 a b}$$\cos \theta>1$ which is not possible.        [Since $-1 \leq \cos \theta \leq 1$]Hence, $\cos \theta≠\frac{a^{2}+b^{2}}{2 a b}$.Read More

Given:The value of \( 2 \sin \theta \) can be \( a+\frac{1}{a} \), where \( a \) is a positive number, and \( a ≠ 1 \).To do:We have to find whether the given statement is true or false.Solution:$a$ is a positive number and $a ≠1$This implies, $AM>GM$AM and GM of two numbers $a$ and $b$ are $\frac{(a+b)}{2}$ and $\sqrt{a b}$.Therefore, $\frac{a+\frac{1}{a}}{2}>\sqrt{a \times \frac{1}{a}}$$(a+\frac{1}{a})>2$$2 \sin \theta>2$      ($2 \sin \theta=a+\frac{1}{a}$)$\sin \theta>1$ which is not possible.        [Since $-1 \leq \sin \theta \leq 1$]Hence, the value of $2 \sin \theta$ cannot be $a+\frac{1}{a}$.Read More

Given: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.To do:We have to find whether the given statement is true or false.Solution: We draw a figure according to the question, after observation we find that the height $( h)$ of the cloud from the surface is much more than the depth of the reflection of clouds in the lake.$\Rightarrow h>d$ [$h=$height of the of the cloud and $d=$ depth ... Read More

Given: If the length of the shadow of a tower is increasing, then the angle of elevation of the sun is also increasing.To do: We have to find whether the given statement is true or false.Solution:We can express the angle of elevation as $tan\theta=\frac{height of the tower}{length of the shadow}$$\therefore tan\theta$ get decreased as the the length of the shadow increases i.e $\theta$ get decreased.In figure it has been shown that how the angle of elevation is decreasing with the increment in the length of the shadow.

Given:\( (\tan \theta+2)(2 \tan \theta+1)=5 \tan \theta+\sec ^{2} \theta \).To do:We have to find whether the given statement is true or false.Solution:We know that,$\sec ^{2} \theta-\tan ^{2} \theta=1$Therefore,$(\tan \theta+2)(2 \tan \theta+1)=2 \tan ^{2} \theta+4 \tan \theta+\tan \theta+2 $$=2(\sec ^{2} \theta-1)+5 \tan \theta+2$$=2 \sec ^{2} \theta+5 \tan \theta$The given statement is true.

Given:If \( \cos \mathrm{A}+\cos ^{2} \mathrm{~A}=1 \), then \( \sin ^{2} \mathrm{~A}+\sin ^{4} \mathrm{~A}=1 \).To do:We have to find whether the given statement is true or false.Solution:We know that,$\sin ^{2} A+\cos ^{2} A=1$$\cos ^{2} A=1-\sin ^{2} A$Therefore,$\cos A+\cos ^{2} A=1$$\cos A=1-\cos ^{2} A$$\cos A=\sin ^{2} A$Squaring on both sides, we get,$\cos ^{2} A=\sin ^{4} A$$1-\sin ^{2} A=\sin ^{4} A$$\sin ^{2} A+\sin ^{4} A=1$The given statement is true.

Given:\( \sqrt{\left(1-\cos ^{2} \theta\right) \sec ^{2} \theta}=\tan \theta \) To do:We have to find whether the given statement is true or false.Solution:We know that,$\sin ^{2} \theta+\cos ^{2} \theta=1$$\operatorname{sec}^{2} \theta = \frac{1}{\cos ^{2} \theta}$$\sec \theta =\frac{1}{\cos \theta}$$\tan \theta=\frac{\sin \theta}{\cos \theta}$Therefore,$\sqrt{(1-\cos ^{2} \theta) \sec ^{2} \theta} =\sqrt{\sin ^{2} \theta . \sec ^{2} \theta}$$=\sqrt{\sin ^{2} \theta . \frac{1}{\cos ^{2} \theta}}$$=\sqrt{\tan ^{2} \theta}$$=\tan \theta$The given statement is true.

Given:The value of the expression \( \left(\sin 80^{\circ}-\cos 80^{\circ}\right) \) is negative.To do:We have to find whether the given statement is true or false.Solution:We know that,$\sin \theta$ is increasing when, $0^{\circ}