Column I
(A) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $2 \left( a ^ { 2 } - b ^ { 2 } \right) = c ^ { 2 }$ and $\lambda = \frac { \sin ( X - Y ) } { \sin Z }$, then possible values of $n$ for which $\cos ( n \pi \lambda ) = 0$ is (are)
(B) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $1 + \cos 2 X - 2 \cos 2 Y = 2 \sin X \sin Y$, then possible value(s) of $\frac { a } { b }$ is (are)
(C) In $\mathbb { R } ^ { 2 }$, let $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\beta \hat { i } + ( 1 - \beta ) \hat { j }$ be the position vectors of $X , Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow { O X }$ with $\overrightarrow { O Y }$ is $\frac { 3 } { \sqrt { 2 } }$, then possible value(s) of $| \beta |$ is (are)
(D) Suppose that $F ( \alpha )$ denotes the area of the region bounded by $x = 0 , x = 2 , y ^ { 2 } = 4 x$ and $y = | \alpha x - 1 | + | \alpha x - 2 | + \alpha x$, where $\alpha \in \{ 0,1 \}$. Then the value(s) of $F ( \alpha ) + \frac { 8 } { 3 } \sqrt { 2 }$, when $\alpha = 0$ and $\alpha = 1$, is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5

In a triangle $XYZ$, 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
(A) area of the triangle $XYZ$ is $6\sqrt{6}$
(B) the radius of circumcircle of the triangle $XYZ$ is $\frac{35}{6}\sqrt{6}$
(C) $\sin\frac{X}{2}\sin\frac{Y}{2}\sin\frac{Z}{2} = \frac{4}{35}$
(D) $\sin^2\left(\frac{X+Y}{2}\right) = \frac{3}{5}$

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

In a triangle $P Q R$, let $\angle P Q R = 30 ^ { \circ }$ and the sides $P Q$ and $Q R$ have lengths $10 \sqrt { 3 }$ and 10, respectively. Then, which of the following statement(s) is (are) TRUE?
(A) $\angle Q P R = 45 ^ { \circ }$
(B) The area of the triangle $P Q R$ is $25 \sqrt { 3 }$ and $\angle Q R P = 120 ^ { \circ }$
(C) The radius of the incircle of the triangle $P Q R$ is $10 \sqrt { 3 } - 15$
(D) The area of the circumcircle of the triangle $P Q R$ is $100 \pi$

In a non-right-angled triangle $\triangle P Q R$, let $p , q , r$ denote the lengths of the sides opposite to the angles at $P , Q , R$ respectively. The median from $R$ meets the side $P Q$ at $S$, the perpendicular from $P$ meets the side $Q R$ at $E$, and $R S$ and $P E$ intersect at $O$. If $p = \sqrt { 3 } , q = 1$, and the radius of the circumcircle of the $\triangle P Q R$ equals 1, then which of the following options is/are correct?
(A) Length of $R S = \frac { \sqrt { 7 } } { 2 }$
(B) Area of $\triangle S O E = \frac { \sqrt { 3 } } { 12 }$
(C) Length of $O E = \frac { 1 } { 6 }$
(D) Radius of incircle of $\triangle P Q R = \frac { \sqrt { 3 } } { 2 } ( 2 - \sqrt { 3 } )$

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
$$\tan \frac { X } { 2 } + \tan \frac { Z } { 2 } = \frac { 2 y } { x + y + z }$$
then which of the following statements is/are TRUE?
(A) $2 Y = X + Z$
(B) $Y = X + Z$
(C) $\tan \frac { X } { 2 } = \frac { x } { y + z }$
(D) $x ^ { 2 } + z ^ { 2 } - y ^ { 2 } = x z$

Consider a triangle $P Q R$ 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 ?
(A) $\cos P \geq 1 - \frac { p ^ { 2 } } { 2 q r }$
(B) $\cos R \geq \left( \frac { q - r } { p + q } \right) \cos P + \left( \frac { p - r } { p + q } \right) \cos Q$
(C) $\frac { q + r } { p } < 2 \frac { \sqrt { \sin Q \sin R } } { \sin P }$
(D) If $p < q$ and $p < r$, then $\cos Q > \frac { p } { r }$ and $\cos R > \frac { p } { q }$

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 $P Q R S$ be a quadrilateral in a plane, where $Q R = 1 , \angle P Q R = \angle Q R S = 70 ^ { \circ } , \angle P Q S = 15 ^ { \circ }$ and $\angle P R S = 40 ^ { \circ }$. If $\angle R P S = \theta ^ { \circ } , P Q = \alpha$ and $P S = \beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta ^ { \circ }$ is/are
(A) $( 0 , \sqrt { 2 } )$
(B) $( 1,2 )$
(C) $( \sqrt { 2 } , 3 )$
(D) $( 2 \sqrt { 2 } , 3 \sqrt { 2 } )$

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac { \pi } { 2 }$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.
Let $a$ be the area of the triangle $ABC$. Then the value of $( 64a ) ^ { 2 }$ is

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac { \pi } { 2 }$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.
Then the inradius of the triangle $ABC$ is

In a $\triangle PQR$, if $3\sin P + 4\cos Q = 6$ and $4\sin Q + 3\cos P = 1$, then the angle $R$ is equal to
(1) $\frac{5\pi}{6}$
(2) $\frac{\pi}{6}$
(3) $\frac{\pi}{4}$
(4) $\frac{3\pi}{4}$

ABCD is a trapezium such that AB and CD are parallel and $BC \perp CD$. If $\angle ADB = \theta$, $BC = p$ and $CD = q$, then $AB$ is equal to
(1) $\frac{(p^{2}+q^{2})\sin\theta}{p\cos\theta+q\sin\theta}$
(2) $\frac{p^{2}+q^{2}\cos\theta}{p\cos\theta+q\sin\theta}$
(3) $\frac{p^{2}+q^{2}}{p^{2}\cos\theta+q^{2}\sin\theta}$
(4) $\frac{(p^{2}+q^{2})\sin\theta}{(p\cos\theta+q\sin\theta)^{2}}$

If in a triangle $ABC$, $\frac{b+c}{11} = \frac{c+a}{12} = \frac{a+b}{13}$, then $\cos A$ is equal to
(1) $5/7$
(2) $1/5$
(3) $35/19$
(4) $19/35$

$ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC \perp CD$. If $\angle ADB = \theta$, $BC = p$ and $CD = q$, then $AB$ is equal to
(1) $\frac{p^2 + q^2}{p^2 \cos\theta + q^2 \sin\theta}$
(2) $\frac{\left(p^2 + q^2\right)\sin\theta}{(p\cos\theta + q\sin\theta)^2}$
(3) $\frac{\left(p^2 + q^2\right)\sin\theta}{p\cos\theta + q\sin\theta}$
(4) $\frac{p^2 + q^2\cos\theta}{p\cos\theta + q\sin\theta}$

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point $O$ on the ground is $45 ^ { \circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to $30 ^ { \circ }$. Then the speed (in m/s) of the bird is
(1) $20 \sqrt { 2 }$
(2) $20 ( \sqrt { 3 } - 1 )$
(3) $40 ( \sqrt { 2 } - 1 )$
(4) $40 ( \sqrt { 3 } - \sqrt { 2 } )$

If the angles of elevation of the top of a tower from three collinear points $A , B$ and $C$ on a line leading to the foot of the tower are $30 ^ { \circ } , 45 ^ { \circ }$ and $60 ^ { \circ }$ respectively, then the ratio $AB : BC$, is
(1) $2 : 3$
(2) $\sqrt { 3 } : 1$
(3) $\sqrt { 3 } : \sqrt { 2 }$
(4) $1 : \sqrt { 3 }$

If the angles of elevation of the top of a tower from three collinear points A, B and C, on a line leading to the foot of the tower, are $30^\circ$, $45^\circ$ and $60^\circ$ respectively, then the ratio $AB : BC$, is:
(1) $\sqrt{3} : 1$
(2) $\sqrt{3} : \sqrt{2}$
(3) $1 : \sqrt{3}$
(4) $2 : 3$

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min for the angle of depression of the car to change from $30 ^ { \circ }$ to $45 ^ { \circ }$, then the time taken (in $\min$ ) by the car to reach the foot of the tower is
(1) $\frac { 9 } { 2 } ( \sqrt { 3 } + 1 )$
(2) $9 ( \sqrt { 3 } + 1 )$
(3) $18 ( \sqrt { 3 } - 1 )$
(4) $9 ( \sqrt { 3 } - 1 )$

$P Q R$ is a triangular park with $P Q = P R = 200 \mathrm {~m}$. A T.V. tower stands at the mid-point of $Q R$. If the angles of elevation of the top of the tower at $P , Q$ and $R$ are respectively, $45 ^ { \circ } , 30 ^ { \circ }$ and $30 ^ { \circ }$, then the height of the tower (in m) is:
(1) $50 \sqrt { 2 }$
(2) 100
(3) 50
(4) $100 \sqrt { 3 }$

Consider a triangular plot $A B C$ with sides $A B = 7 m , B C = 5 m$ and $C A = 6 m$. A vertical lamp-post at the mid-point $D$ of $A C$ subtends an angle $30 ^ { \circ }$ at $B$. The height (in $m$) of the lamp-post is:
(1) $2 \sqrt { 21 }$
(2) $\frac { 2 } { 3 } \sqrt { 21 }$
(3) $\frac { 3 } { 2 } \sqrt { 21 }$
(4) $7 \sqrt { 3 }$

Two poles standing on a horizontal ground are of heights $5 m$ and $10 m$ respectively. The line joining their tops makes an angle of $15 ^ { \circ }$ with the ground. Then the distance (in $m$) between the poles, is
(1) $10 ( \sqrt { 3 } - 1 )$
(2) $\frac { 5 } { 2 } ( 2 + \sqrt { 3 } )$
(3) $5 ( 2 + \sqrt { 3 } )$
(4) $5 ( \sqrt { 3 } + 1 )$

The angle of elevation of a cloud $C$ from a point $P$, $200$ m above a still lake is $30 ^ { \circ }$. If the angle of depression of the image of $C$ in the lake from the point $P$ is $60 ^ { \circ }$, then $PC$ (in m) is equal to
(1) 100
(2) $200 \sqrt { 3 }$
(3) 400
(4) $400 \sqrt { 3 }$

The angle of elevation of the summit of a mountain from a point on the ground is $45^{\circ}$. After climbing up one km towards the summit at an inclination of $30^{\circ}$ from the ground, the angle of elevation of the summit is found to be $60^{\circ}$. Then the height (in km) of the summit from the ground is:
(1) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$
(2) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$
(3) $\frac{1}{\sqrt{3}-1}$
(4) $\frac{1}{\sqrt{3}+1}$

The angle of elevation of a jet plane from a point $A$ on the ground is $60 ^ { \circ }$. After a flight of 20 seconds at the speed of 432 km / hour, the angle of elevation changes to $30 ^ { \circ }$. If the jet plane is flying at a constant height, then its height is:
(1) $1200 \sqrt { 3 } \mathrm {~m}$
(2) $2400 \sqrt { 3 } \mathrm {~m}$
(3) $1800 \sqrt { 3 } \mathrm {~m}$
(4) $3600 \sqrt { 3 } \mathrm {~m}$