Semi Perimeter

Q. If $\tan \left(\frac{\pi }{9}\right),$ $x,$ $\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),$ $y,$ $\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to

1. $0$
2. $1$
3. $3$
4. $4$

Q. In a $\Delta ABC$. If $\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}$ are in $H.P$, then $a,b,c$ are in:

1. $H.P$
2. $G.P$
3. $A.P$
4. $A.G.P$

Q. If in a triangle $PQR,\sin P,\sin Q,\sin R$ are in $A.P,$ then:

1. The altitudes are in $A.P.$
2. The altitudes are in $H.P.$
3. The altitudes are in $G.P.$
4. Nothing can be said.

Q. A triangle $ABC$ is such that sides $a,b,c$ are in $G.P.$ and $\sin A,\sin B,\sin C$ are in $A.P.,$ then the $△ABC$ is
(where $a,b,c$ are sides (in $\text{units}$) of $△ABC$ opposite to $\angle A,\angle B$ and $\angle C$ respectively)

1. Equilateral triangle
2. Right angled triangle
3. scalene triangle
4. None of these

Q.

In any $\Delta$ ABC, $2R^2$ sin A sin B sin C is equal to

1. 2∆

2. ∆

3. 3∆

4. ∆/2

Q. If $\tan \left(\frac{\pi }{9}\right),$ $x,$ $\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),$ $y,$ $\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to

1. $1$
2. $0$
3. $3$
4. $4$

Q. If $3^{2\sin 2\alpha -1}$, $14$ and $3^{4-2\sin 2\alpha }$ are the first three terms of an A.P. for some $\alpha$, then the sixth term of this A.P. is:

1. $65$
2. $81$
3. $78$
4. $66$

Q. In triangle $ABC$, if $\cot A,\cot B,\cot C$ are in $A.P.$, then $a^2,b^2,c^2$ are in:

1. $A.P.$
2. $G.P.$
3. $H.P.$
4. $A.G.P.$

Q. In $\Delta ABC$, if ${\sin }^2\frac{A}{2},{\sin }^2\frac{B}{2},{\sin }^2\frac{C}{2}$ are in $\text{H.P.}$, then $a,b,c$ are in

1. $\text{H.P.}$
2. $\text{A.P.}$
3. $\text{G.P.}$
4. None of the above

Q. If the angles $A,B$ and $C$ of a triangle are in arithmetic progression and if $a,b$ and $c$ denote the lengths of the sides opposite to $A,B$ and $C$ respectively, then the value of the expression $\frac{a}{c}\sin 2C+\frac{c}{a}\sin 2A$ is:

1. $\frac{1}{2}$
2. $\frac{\sqrt{3}}{2}$
3. $1$
4. $\sqrt{3}$

Q. In $\Delta ABC$, if ${\sin }^2\frac{A}{2},{\sin }^2\frac{B}{2},{\sin }^2\frac{C}{2}$ are in $\text{H.P.}$, then $a,b,c$ are in

1. $\text{H.P.}$
2. $\text{A.P.}$
3. $\text{G.P.}$
4. None of the above

Q. If the angles $A,B$ and $C$ of a triangle are in an arithmetic progression and if $a,b$ and $c$ denote the lengths of the sides opposite to $A,B$ and $C$ respectively, then the value of the expression $\frac{a}{2}\sin 2C+\frac{c}{a}\sin 2A$ is

1. $1$
2. $\frac{\sqrt{3}}{2}$
3. $\frac{1}{2}$
4. $\sqrt{3}$

Q. In $\Delta ABC$, if ${\sin }^2\frac{A}{2},{\sin }^2\frac{B}{2},{\sin }^2\frac{C}{2}$ are in $\text{H.P.}$, then $a,b,c$ are in

1. $\text{H.P.}$
2. $\text{A.P.}$
3. $\text{G.P.}$
4. None of the above