Area of Triangle

Q. Let $x$ be the length of one of the equal sides of an isosceles triangle, and let $\theta$ be the angle between them. If $x$ is increasing at the rate $\left(\frac{1}{12}\right)m/hr,$ and $\theta$ is increasing at the rate of $\frac{\pi }{180}rad/hr,$ then the rate in$\frac{m^2}{hr}$ at which the area of the triangle is increasing when $x=12m$ and $\theta =\frac{\pi }{4},$ is

1. $2^{1/2}(1+\frac{2\pi }{5})$
2. $\frac{73}{2}{.2}^{1/2}$
3. $\frac{3^{1/2}}{2}+\frac{\pi }{5}$
4. $2^{1/2}(\frac{1}{2}+\frac{\pi }{5})$

Q.

In a triangle ABC, if b = 2, B = ${30}^0$ then the area of the circumcircle of triangle ABC in square unit is

1. π

2. 2π

3. 4π

4. 6π

Q. In the circle given below, let $OA=1$ unit, $OB=13$ unit and $PQ\perp OB.$ Then, the area of the triangle $PQB$ (in square units) is :

1. $26\sqrt{3}$
2. $24\sqrt{2}$
3. $24\sqrt{3}$
4. $26\sqrt{2}$

Q.

The line $x=\frac{\mathrm{\pi }}{4}$ divides the area of the region bounded by $y=\sin \left(x\right)$, $y=\cos \left(x\right)$ and $x$-axis $\left(0\le x\le \frac{\mathrm{\pi }}{2}\right)$ into two regions of areas $A_1$ and $A_2$. Then, $A_1:A_2$ equals to

1. $4:1$

2. $3:1$

3. $2:1$

4. $1:1$

Q.

$\text{In the adjacent figure, 'P' is any arbitrary interior of the triangle ABC,}$ H_a, H_b, H_c $\text{are the length of altitudes drawn from vertices A, B and C respectively. If}$ x_a, x_b and x_c $\text{represent the distance of 'P' from sides BC, AC and AB respectively, then}$

$\frac{x_a}{H_a}+\frac{x_b}{H_b}+\frac{x_c}{H_c}\text{is always equal to}$

1. 3

2. 2

3. 1

4. 1/3

Q.

$P\left(E\right)+P\left(\overline{E}\right)$ is equal to

1. $\frac{1}{2}$
2. None
3. $0$
4. $1$

Q. In a triangle $ABC,$ altitudes from vertices $A$ and $B$ have lengths $3$ and $6$ respectively. Then the exhaustive set of values of the length of altitude from vertex $C$ is

1. $(3,7)$
2. $(2,6)$
3. $(3,4)$
4. $(1,5)$

Q. The perimeter of an equilateral triangle is $60$ cm, then the area is

1. $10\sqrt{3}{\text{cm}}^2$
2. $15\sqrt{3}{\text{cm}}^2$
3. $20\sqrt{3}{\text{cm}}^2$
4. $100\sqrt{3}{\text{cm}}^2$

Q. From a square with sides of length $5$ unit, triangular pieces from the four corners are removed to from a regular octagon. Then the area removed is
(correct answer + 1, wrong answer - 0.25)

1. $25(3+\sqrt{2})\text{sq. units}$
2. $25(3+2\sqrt{2})\text{sq. units}$
3. $25(3-2\sqrt{2})\text{sq. units}$
4. $25(3-\sqrt{2})\text{sq. units}$

Q.

1. ${\sin }^{2}A+{\sin }^{2}B+={\sin }^{2}C$
2. $2\sin A\sin B\sin C=1$
3. $2\sin A\cos B=\sin C$
4. none of these

Q. In a $\Delta ABC$, let $\tan A=1,\tan B=2,\tan C=3$ and $c=3$. Then the area (in sq. units) of the $\Delta ABC$ is:

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

Q. Find the perimete of a triangle whose sides are $4.5cm$ , $10cm$ and $10.5cm$.

Q. A pair of perpendicular straight lines drawn through the origin form an isosceles triangle with line $2x+3y=6$ then area of the triangle so formed is

1. $\frac{36}{13}$
2. $\frac{12}{17}$
3. $\frac{13}{5}$
4. $\frac{17}{13}$

Q. In a triangle $ABC,$ altitudes from vertices $A$ and $B$ have lengths $3$ and $6$ respectively. Then the exhaustive set of values of the length of altitude from vertex $C$ is

1. $(3,7)$
2. $(2,6)$
3. $(3,4)$
4. $(1,5)$

Q. The lengths of two sides of triangle are $12$ $cm$ and $15$ $cm$. Between what two measurements should the length of the third side fall?

Q. The angles $A,B$ and $C$ of a triangle $ABC$ are in $A.P.$ and $a:b=1:\sqrt{3}.$ If $c=4\text{cm}$, then the area (in sq.cm) of this triangle is:

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

Q. If the three successive vertices of a parallelogram have the position vectors as, $A(-3,-2,0);B(3,-3,1)$ and $C(5,0,2).$ Then find :
position vector of the fourth vertex $D$

Q. The sides of a triangle are $24.7cm$, $20.4cm$ and $10.5cm$ receptively. What is the perimeter of the triangle?

Q. In a triangle $ABC,$ altitudes from vertices $A$ and $B$ have lengths $3$ and $6$ respectively. Then the exhaustive set of values of the length of altitude from vertex $C$ is

1. $(3,7)$
2. $(2,6)$
3. $(3,4)$
4. $(1,5)$

Q. If $r_1,r_2,$ and $r_3$ be the radii of encircles $ABC$, then $\frac{\sum r_1}{\sqrt{\sum r_1{r}_2}}$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. $\parallel \tan \frac{A}{2}$

Q. The angles $A,B$ and $C$ of a triangle $ABC$ are in $A.P.$ and $a:b=1:\sqrt{3}.$ If $c=4\text{cm}$, then the area (in sq.cm) of this triangle is:

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