Solution of Triangle

Q. If the incircle of the triangle $ABC(AB\ne BC\ne CA)$, passes through it's circumcentre, then the $(\cos A+\cos B+\cos C{)}^2$ is

Q. Consider an acute angle triangle $\Delta ABC$ with area $S$. Let the area of its pedal triangle is '$p$', satisfies by the relation $\cos B=\frac{2p}{S}$ and $\sin B=\frac{2\sqrt{3}p}{S}$. Then

1. $\Delta ABC$ is a right angled triangle.
   
2. $\cos A\cdot \cos B\cdot \cos C=\frac{1}{8}$
3. $\cos (A-C)=1$
4. ${\cos }^{2}A\cos B+{\cos }^{2}B\cos C+{\cos }^{2}C\cos A=\frac{3}{8}$

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

Q. Consider a triangular plot $ABC$ with sides $AB=7m,BC=5m$ and $CA=6m$. A vertical lamp-post at the mid point $D$ of $AC$ subtends an angle ${30}^{\circ }$ at $B$. The height (in $m$) of the lamp-post is :

1. $\frac{3}{2}\sqrt{21}$
2. $7\sqrt{3}$
3. $2\sqrt{21}$
4. $\frac{2}{3}\sqrt{21}$

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

Q. In a right angled triangle, medians drawn from the acute angles make an angle of $\theta$ which each other and L is the length of the hypotenuse. Then the area of the triangle is equal to:

1. $\Delta =\frac{L^{2}tan\theta }{6}$
2. $\Delta =\frac{L^{2}tan\theta }{3}$
3. $\Delta =\frac{L^{2}cot\theta }{3}$
4. $\Delta =\frac{L^{2}cot\theta }{6}$

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. Match $\text{List I}$ with $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{In a}△ABC,\text{if}{a}^2+b^2+c^2=ab+bc+ca,\text{then}& \text{(P)}& △ABC\text{is an equilateral triangle}\\ \text{(B)}& \text{In a}△ABC,\text{if}{a}^2+2b^2+c^2=2bc+2ab,\text{then}& \text{(Q)}& △ABC\text{is a right angled triangle}\\ \text{(C)}& \text{In a}△ABC,\text{if}{a}^2+b^2+c^2=\sqrt{2}a(b+c),\text{then}& \text{(R)}& △ABC\text{is a scalene triangle}\\ \text{(D)}& \text{In a}△ABC,\text{if}{a}^2+b^2+c^2=ca+\sqrt{3}ab,\text{then}& \text{(S)}& A={90}^{\circ },B={45}^{\circ },C={45}^{\circ }\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(S)},\text{(D)}\to \text{(R)}$
2. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(P)}$
3. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(S)}$
4. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(R)}$

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