Area Using Sine Rule

Q. In triangle $ABC$ given $9a^2+9b^2=17c^2$
If $\frac{\cot A+\cot B}{\cot C}=\frac{m}{n}$ then the value of $(m+n)$ equals

1. $13$
2. $5$
3. $7$
4. $9$

Q. If $y=\frac{4\sin x}{2x+\cos x},$ then $\frac{dy}{dx}=$

1. $\frac{4(x\cos x-\sin x)+4}{(2x+\cos x{)}^2}$
2. $\frac{8(x\cos x-\sin x)}{(2x+\cos x{)}^2}$
3. $\frac{8(x\cos x-\sin x)+4}{(2x+\cos x{)}^2}$
4. $\frac{4(x\cos x-\sin x)}{(2x+\cos x{)}^2}$

Q. The incircle touches side $BC$ of triangle $ABC$ at $D$ and $ID$ is produced to $H$ so that $DH=s,$ where $s$ and $I$ are the semi-perimeter and incentre of triangle $ABC$ respectively. If $HBIC$ is cyclic, then $\cot \left(\frac{A}{4}\right)$ is equal to

1. $2-\sqrt{3}$
2. $\sqrt{2}+1$
3. $\sqrt{2}-1$
4. $2+\sqrt{3}$

Q.

If $r_1,r_2,r_3$ be the radii of excircles of the triangle $ABC$ then $\sum _{}\frac{r_1}{\sqrt{r_1{r}_2}}$ is equal to.

1. $\sum _{}cot\frac{A}{2}$

2. $\sum _{}cot\frac{A}{2}cot\frac{B}{2}$

3. $\sum _{}\tan \frac{A}{2}$

4. $\prod _{}\tan \frac{A}{2}$

Q. Two sides of a triangle have lengths 'a' and 'b' and the angle between them is $\theta$. What value of $\theta$ will maximize the area of the triangle? Find the maximum area of the triangle also. [CBSE 2002 C]

Q. Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.

Q. If $x=\frac{2at}{1+t^2}$ and $y=\frac{b(1-t^2)}{1+t^2}$ where $a,b$ are non-zero constants then $\frac{dy}{dx}$ is equal to

1. $-\frac{x}{y}$
2. $\frac{y}{x}$
3. $\frac{b^{2}x}{a^{2}y}$
4. $-\frac{b^{2}x}{a^{2}y}$

Q. Let $ABC$ be a triangle with sides $a,b,c$ and corresponding angles $A,B,C$ respectively. If angle $A=3B$, then $(a^2-b^2)(a-b)$ is equal to

1. $b{c}^2$
2. $a^{2}b$
3. $a{b}^2$
4. $b^{2}c$

Q. 40. If the altitude of an equilateral triangle is x cmthen the area is equal to

Q. The transformed equation of $\frac{dy}{dx}+x\sin 2y=x^3{\cos }^{2}y$ is

1. $\frac{du}{dx}+\frac{u}{x^2}=x$
2. $\frac{du}{dx}+xu=\frac{x^3}{2}$
3. $\frac{du}{dx}+2xu=x^3$
4. $\frac{du}{dx}+\frac{u}{x}=x^2$

Q. If $A=\left[\begin{array}{cc}0& -\tan \frac{\alpha }{2}\\ \tan \frac{\alpha }{2}& 0\end{array}\right]$ and $I$ is the identity matrix of order $2$, show that $I+A=(I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$

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. $\frac{4}{2001}$

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

Q. In a triangle $ABC,$ the value of $r\cot \frac{B}{2}\cdot \cot \frac{C}{2}$ is equal to
(where $r$ is inradius )

1. the radius of excircle touching side $AB$
2. the radius of excircle touching side $BC$
3. the radius of excircle touching side $AC$
4. radius of incircle

Q.

$\text{If in a}\Delta ABC,\angle A={45}^{\circ },\angle B={60}^{\circ },\text{and}\angle C={75}^{\circ };\text{find the ratio of its sides.}$

Q. In radius of a circle which is inscribed in a isosceles triangle one of whose angle is $\frac{2\pi }{3}$ is $\sqrt{3}$, then area of triangle (in sq units) is

1. $4\sqrt{3}$
2. $12-7\sqrt{3}$
3. $12+7\sqrt{3}$
4. None of the above

Q. area bounded by x=a-sina; y=1-cosa; 0