Sine Rule

Q.

Given that $\cos (\alpha -\beta )/2=2\cos (\alpha +\beta )/2,$ then $\tan \alpha /2\tan \beta /2$ is equal to

1. $1/2$

2. $1/3$

3. $1/4$

4. $1/8$

Q. Triangle $ABC$ is isosceles with $AC=BC$ and $\angle ACB={106}^{\circ }.$ Point $M$ is in the interior of the triangle so that $\angle MAC=7^{\circ }$ and $\angle MCA={23}^{\circ }.$ Find the number of degrees in $\angle CMB.$
(correct answer + 5, wrong answer 0)

Q.

In a triangle $ABC\angle A={60}^{\circ },\angle B={40}^{\circ }$ and $\angle C={80}^{\circ }$ If P is the centre of the circumcircle of triangle $ABC$ with radius unity, then the radius of the circumcircle of triangle $BPC$ is

1. 1

2. $\sqrt{3}$

3. 2

4. $\frac{\sqrt{3}}{2}$

Q.

Let $\mathrm{\theta }$ be an angle in quadrant $\mathrm{IV}$ such that $s\mathrm{in\theta }=-\frac{3}{4}$. Find the exact values of $\mathrm{sec\theta }$ and $\mathrm{cot\theta }$.

Q. Sides $AB$ and $AC$ in an equilateral triangle $ABC$ with side length $3$ is extended to form two rays emanating from the point $A$ as shown in the figure. A point $P$ is chosen outside the triangle $ABC$ and between the two rays such that $\angle ABP+\angle BCP={180}^{\circ }.$ If the maximum length of $CP$ is $M,$ then the value of $\frac{M^2}{2}$ is

Q. In a triangle ABC, if $\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}$, then triangle ABC will be:

1. Isosceles triangle
2. Equilateral triangle
3. Right triangle
4. Right Isosceles triangle

Q. In a triangle $a^2+b^2+c^2=ca+ab\sqrt{3}$, then triangle is

1. Equilateral
2. Right angled and isosceles
3. Right angled
4. None of these

Q.

In In $\Delta ABC,$ if $8R^2=a^2+b^2+c^2,$ then the triangle is

1. Right angled

2. Equilateral

3. Acute angled

4. Obtuse angled

Q.

Evaluate $\tan 3\mathrm{A}-\tan 2\mathrm{A}-\tan \mathrm{A}$

1. $\tan 3\mathrm{A}\tan 2\mathrm{A}\tan \mathrm{A}$

2. $-\tan 3\mathrm{A}\tan 2\mathrm{A}\tan \mathrm{A}$

3. $\tan \mathrm{A}\tan 2\mathrm{A}-\tan 2\mathrm{A}\tan 3\mathrm{A}-\tan 3\mathrm{A}\tan \mathrm{A}$

4. None of the above

Q.

In triangle $ABC,a^2+c^2=2002b^2$, then $\frac{cotA+cotC}{cotB}$ is equal to

1. $\frac{1}{2001}$

2. $\frac{2}{2001}$

3. $\frac{3}{2001}$

4. $\frac{4}{2001}$

Q.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{\sin \theta -2{\sin }^3\theta }{2{\cos }^3\theta -\cos \theta }=\tan \theta$

Q.

In $\Delta ABC,if\angle C={90}^{\circ },\angle A={30}^{\circ },c=20$, then the values of a and b are

1. $10,10$

2. $10,10\sqrt{3}$

3. $5,5\sqrt{3}$

4. $8,8\sqrt{3}$

Q. In $△ABC$ with usual notations, $\frac{a\cos A+b\cos B+c\cos C}{a+b+c}=\frac{1}{2},$ then the value of $\frac{\sin A+\sin B+\sin C}{\sin A\sin B\sin C}$ is

Q. If for a triangle $ABC$,
$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$
then ${\sin }^{3}A+{\sin }^{3}B+{\sin }^{3}C=$

1. $\sin A+\sin B+\sin C$
2. $3\sin A\sin B\sin C$
3. $\sin 3A+\sin 3B+\sin 3C$
4. ${\sin }^{3}A{\sin }^{3}B{\sin }^{3}C$

Q. If for a triangle $ABC$,
$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$
then ${\sin }^{3}A+{\sin }^{3}B+{\sin }^{3}C=$

1. ${\sin }^{3}A{\sin }^{3}B{\sin }^{3}C$
2. $\sin A+\sin B+\sin C$
3. $3\sin A\sin B\sin C$
4. $\sin 3A+\sin 3B+\sin 3C$

Q. In $△ABC,\frac{sinB}{sin(A+B)}=$

1. $\frac{b}{a+b}$
2. $\frac{b}{c}$
3. $\frac{c}{b}$
4. None of these

Q.

In $\text{△RST}$, $t=9.5$ inches, $\angle R=149°$ and $\angle S=12°$.

Find the length of $r$, to the nearest ${10}^{th}$ of an inch.

Q. $ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC\perp CD$. If $\angle ADB=\theta ,BC=p$ and $CD=q$, then $AB$ is equal to :

1. $\frac{(p^2+q^2)\sin \theta }{p\cos \theta +q\sin \theta }$
2. $\frac{p^2+q^2\cos \theta }{p\cos \theta +q\sin \theta }$
3. $\frac{p^2+q^2}{p\cos \theta +q\sin \theta }$
4. $\frac{(p^2+q^2)\sin \theta }{(p\cos \theta +q\sin \theta {)}^2}$

Q.

In $\Delta ABC,ccos(A-\alpha )+\alpha cos(C+\alpha )=$

1. $\alpha cos\alpha$

2. $bcos\alpha$

3. $ccos\alpha$

4. $2bcos\alpha$

Q. If in a triangle $ABC,a,b,c$ are in A.P. and $p_1,p_2,p_3$ are the altitudes from the vertices $A,B,C$ respectively then

1. $p_1,p_2,p_3$ are in A.P.
2. $p_1,p_2,p_3$ are in H.P.
3. $p_1+p_2+p_3\le \frac{3R}{\Delta }$
4. $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le \frac{3R}{\Delta }$