Solution of Triangle

Q. Which of the following expressions have value equal to four times the area of the triangle $ABC$?
(All symbols used have their usual meaning in a triangle)

1. $b^2\sin 2C+c^2\sin 2B$
2. $\frac{(a+b+c{)}^2}{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}$
3. $rs+r_1(s-a)+r_2(s-b)+r_3(s-c)$
4. $(a^2+b^2-c^2)\tan B$

Q. If b > c sin B, b < c and B is acute angle then number of triangles possible following the given conditions is 1.

1. True
2. False

Q.

Given angle $A={60}^{\circ },c=\sqrt{3}-1,b=\sqrt{3}+1$. Solve the triangle

1. $a=\sqrt{6},B={15}^{\circ },C={100}^{\circ }$
2. $a=\sqrt{12},B={15}^{\circ },C={105}^{\circ }$
3. $a=\sqrt{6},B={105}^{\circ },C={15}^{\circ }$
4. $a=\sqrt{6},B={15}^{\circ },C={105}^{\circ }$

Q. Number of triangles possible for a given b, c and B(acute angle) under the condition that b < c sin B. Where b, c are the sides and B is the angle opposite to b.

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

Q.

If $co{s}^{2}A+co{s}^{2}C=si{n}^{2}B,$ then $△$ ABC is
[MP PET 1991]

1. Isosceles

2. Right angled

3. None of these

4. Equilateral

Q.

If the sides of a right angled triangle be in A. P. , then their ratio will be

1. 1: 2: 3

2. 2 : 3 : 4

3. 3: 4: 5

4. 4 : 5 : 6

Q. If two sides of a triangle are "a" units & 10 units and the angles opposite of them are ${30}^{\circ }$ and ${60}^{\circ }$ respectively then find the value of a.

1. 10
2. 5
3. $\frac{10}{\sqrt{3}}$
4. $10\sqrt{3}$

Q. In a triangle $ABC,a:b:c=4:5:6$. The ratio of the radius of the circumcircle to that to the incircle is

1. $15/4$
2. $11/5$
3. $16/7$
4. $16/3$

Q. In $\Delta$ ABC having vertices $A(acos{\theta }_1,asin{\theta }_1),B(acos{\theta }_2,asin{\theta }_2)$ and $C(acos{\theta }_3,asin{\theta }_3)$ is equilateral, then which of the followings is/are true?

1. $cos{\theta }_1+cos{\theta }_2+cos{\theta }_3=0$
2. $sin{\theta }_1+sin{\theta }_2+sin{\theta }_3=0$
3. $cos({\theta }_1-{\theta }_2)+cos({\theta }_2-{\theta }_3)+cos({\theta }_3-{\theta }_1)=\frac{-3}{2}$
4. $\left|sin\left(\frac{{\theta }_1-{\theta }_2}{2}\right)\right|=sin\left|\left(\frac{{\theta }_2-{\theta }_3}{2}\right)\right|=\left|sin\left(\frac{{\theta }_1-{\theta }_3}{2}\right)\right|$

Q. Let ABC be a triangle with incentre I and r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB respectively, If $r_1,r_2,and{r}_3$ are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that
$\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1{r}_2{r}_3}{(r-r_1)(r-r_2)(r-r_3)}$

1. True
2. False