Basics of Geometry

Q.

Question 15
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of ${30}^{\circ }$, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be ${60}^{\circ }$. Find the time taken by the car to reach the foot of the tower from this point.

Q.

If $A+B=45°$ , then $(cotA-1)(cotB-1)$ is equal to

1. $3$

2. $\frac{1}{2}$

3. $-1$

4. $-2$

5. $2$

Q. What is the relationship between range and maximum height when one angle is theta and other is 90-theta

Q.

If $\tan \left(\frac{x}{2}\right)=\mathrm{cosec}\left(x\right)-\sin \left(x\right)$, then the value of ${\tan }^2\left(\frac{x}{2}\right)$ is

1. $2-\sqrt{5}$

2. $2+\sqrt{5}$

3. $-2-\sqrt{5}$

4. $-2+\sqrt{5}$

Q. Question 14
In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse.

Q.

If we consider only the principal values of the inverse trigonometric functions, then the value of $\tan \left({\cos }^{-1}\frac{1}{5\sqrt{2}}-{\sin }^{-1}\frac{4}{\sqrt{17}}\right)$.

1. $\sqrt{\frac{29}{3}}$

2. $\frac{29}{3}$

3. $\sqrt{\frac{3}{29}}$

4. $\frac{3}{29}$

Q.

Is the value of cosA = $\frac{5}{4}$ true for some angle A?

1. True

2. False

Q.

If $u={\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ and $v={\tan }^{-1}x$, then $\frac{du}{dv}$is equal to

1. $4$

2. $1$

3. $\frac{1}{4}$

4. $-\frac{1}{4}$

Q.

If x is an acute angle and tan x = 5, find the exact value of the trigonometric function sin x.

1. $3/\sqrt{26}$

2. $5/\sqrt{26}$

3. $1/\sqrt{26}$

4. $5/\sqrt{27}$

Q.

In $∆ABC,\angle A=\frac{\mathrm{\pi }}{2},b=4,c=3$, then the value of $\frac{R}{r}$ is

1. $\frac{5}{2}$

2. $\frac{7}{2}$

3. $\frac{9}{2}$

4. $\frac{35}{24}$

Q.

Is the triangle with sides $25cm,5cm$ and $24cm$ a right triangle? Give a reason for your answer.

Q.

Rohan went for trekking in the Himalayas. At the campsite, he saw the tent in which he would be staying. The slanted sides of the tent were 15 feet long and the base was 24 feet long. If Rohan’s height is 5 feet 12 inches, would he be able to stand erectly anywhere inside the tent?

1. No

2. Yes

Q. Which is the longest side in a right angled triangle?

Q.

In $△ABC$, right angled at C, AC=4cm and $\angle BAC={60}^{\circ }$, find the length of side BC

1. 3 cm

2. $4\sqrt{3}$ cm

3. $\sqrt{3}$ cm

4. $5\sqrt{3}$ cm

Q. In a right $\Delta ABC$, right angled at $B$, the ratio of $AB$ to $AC$ is $1:\sqrt{2}$, then $\frac{2\tan A}{1-ta{n}^{2}A}$ is equal to

1. $2$
2. $undefined$
3. $1$
4. $3$

Q.

In triangle ABC, right angled at B, if $\angle$A is increased till it becomes ${90}^{\circ }$, what can you say about AC and AB if BC = 1 unit

1. 0, 0

2. 0, 1

3. 1, 1

4. 1 , 0

Q. Relation between circumRadius and Sine Rule is given by

1. $R=\frac{a}{SinA}=\frac{b}{SinB}=\frac{c}{SinC}$
2. $R=\frac{a}{2SinA}=\frac{b}{2SinB}=\frac{c}{2SinC}$
3. $R=\frac{ab}{SinASinB}=\frac{bc}{SinBSinC}=\frac{ca}{SinCSinA}$
4. $R=\frac{abc}{SinASinBSinC}$

Q. Lines PQ and RS are parallel. It is given that $\angle RSP={52}^{\circ }$, then find $\angle SPQ$.

1. ${38}^{\circ }$
2. ${45}^{\circ }$
3. ${48}^{\circ }$
4. ${52}^{\circ }$

Q. In $△PQR$, right-angled at $Q,PR+QR=25$ cm and $PQ=5$ cm. Determine the values of $\sin P,\cos P$ and $\tan P$.

Q.

In $△$ PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Find the value of sin P.

1. $\frac{12}{13}$
2. $\frac{12}{14}$
3. $0.9231$
4. 0.8571

Q.

If $x=sin({90}^{\circ }-a)cot({90}^{\circ }-a)$, then $x$ is equal to .................

1. $tana$
2. $sina$
3. $cota$
4. $cosa$

Q.

In $△$ PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the value of cos R.

1. 5/13

2. 12/13

3. 15/12

4. 5/12

Q.

A triangle has sides with lengths of $6$ $\mathrm{feet}$, $8$ $\mathrm{feet}$, and $10$ $\mathrm{feet}$. Is it a right triangle?

Q. Prove that $\frac{(secA-1)}{(secA+1)}=\frac{(1-cosA)}{(1+cosA)}$

Q. In $△ABC$, if $\cos C=\frac{\sin A}{2\sin B}$, then show that triangle is right angled triangle.

Q.

_{Show that in a right angle traingle , the hypotenuse is the longest side.}

Q. Lines PQ and RS are parallel. It is given that $\angle RSP={52}^{\circ }$, then find $\angle SPQ$.

1. ${52}^{\circ }$
2. ${45}^{\circ }$
3. ${48}^{\circ }$
4. ${38}^{\circ }$

Q. If $\frac{{\cos }^{4}x}{{\cos }^{2}y}+\frac{{\sin }^{4}x}{{\sin }^{2}y}=1$ then show that $\frac{{\cos }^{4}y}{{\cos }^{2}x}+\frac{{\sin }^{4}y}{{\sin }^{2}x}=1$ .

Q. The side opposite to the right angle in a right angled triangle is called

1. Base
2. Hypotenuse
3. Perpendicualr
4. None of these

Q. If angle $C$ of $△$ $ABC$ is ${90}^{\circ }$, then $\tan A+\tan B=...........$(where $a,b,c$ are sides opposite to angles A, B, C respectively)

1. $\frac{c^2}{ab}$
2. $\frac{a^2}{bc}$
3. $\frac{b^2}{ac}$
4. $\frac{c^3}{ab}$