JEE Advanced Paper 2 - 2013

Consider a triangle $PQR$ having sides of lengths $p,q$ and $r$ opposite to the angles $P,Q$ and $R$, respectively. Then which of the following statements is (are) TRUE?

In a triangle the sum of two sides is$x$ and the product of the same two sides is $y$. If $x^2-c^2=y,$ (where $c$ is the third side of the triangle) then the ratio of the inradius to the circumradius of the triangle is

In a $∆PQR$, $P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $PQ, QR$ and $RP$ at $N, L$ and $M$ respectively, such that the lengths of $PN, QL$ and $RM$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

In a triangle $PQR,$ let $\angle PQR=30°$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3} \mathrm{a}\mathrm{n}\mathrm{d} 10$ units, respectively. Then, which of the following statement(s) is (are) TRUE?

In a $△XYZ$ let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$ respectively and $2s=x+y+z.$ If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle $△XYZ$ is $\frac{8\pi }{3}$, then

In a triangle $ABC$, let $AB=\sqrt{23},BC=3$ and $CA=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$ respectively. If $\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2y}{x+y+z}$, then which of the following statements is/are TRUE?

In a non-right-angled triangle $\mathrm{\Delta }PQR,$ let $p\mathit{,}q\mathit{,}r$ denote the lengths of the sides opposite to the angles at $P,Q,R$ respectively. The median from $R$ meets the side $PQ$ at $S,$ the perpendicular from $P$ meets the side $QR$ at $E$, and $RS$ and $PE$ intersect at $O.$ If $p=\sqrt{3},q=1,$ and the radius of the circumcircle of the $\mathrm{\Delta }PQR$ equals $1,$ then which of the following options is/are correct?