If the angles $\mathrm { A } , \mathrm { B }$ and C of a triangle are in an arithmetic progression and if $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively, then the value of the expression $\frac { a } { c } \sin 2 C + \frac { c } { a } \sin 2 A$ is
A) $\frac { 1 } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) 1
D) $\sqrt { 3 }$

Let ABC be a triangle such that $\angle \mathrm { ACB } = \frac { \pi } { 6 }$ and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively. The value(s) of x for which $\mathrm { a } = \mathrm { x } ^ { 2 } + \mathrm { x } + 1 , \mathrm {~b} = \mathrm { x } ^ { 2 } - 1$ and $\mathrm { c } = 2 \mathrm { x } + 1$ is (are)
A) $- ( 2 + \sqrt { 3 } )$
B) $1 + \sqrt { 3 }$
C) $2 + \sqrt { 3 }$
D) $4 \sqrt { 3 }$

In a triangle $P Q R$, $P$ is the largest angle and $\cos P = \frac { 1 } { 3 }$. Further the incircle of the triangle touches the sides $P Q , Q R$ and $R P$ at $N , L$ and $M$ respectively, such that the lengths of $P N , Q L$ and $R M$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)
(A) 16
(B) 18
(C) 24
(D) 22

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 in-radius to the circum-radius of the triangle is
(A) $\frac{3y}{2x(x+c)}$
(B) $\frac{3y}{2c(x+c)}$
(C) $\frac{3y}{4x(x+c)}$
(D) $\frac{3y}{4c(x+c)}$

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) $\sin^2\left(\frac{X+Y}{2}\right) = \frac{3}{5}$

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

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) $\frac{p^2 + q^2\cos\theta}{p\cos\theta + q\sin\theta}$

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) $2 : 3$

The angle of elevation of the top of a vertical tower from a point A , due east of it is $45 ^ { \circ }$. The angle of elevation of the top of the same tower from a point B , due south of A is $30 ^ { \circ }$. If the distance between A and B is $54 \sqrt { 2 } m$, then the height of the tower (in meters), is:
(1) 108
(2) $36 \sqrt { 3 }$
(3) $54 \sqrt { 3 }$
(4) 54

Let a vertical tower $AB$ have its end $A$ on the level ground. Let $C$ be the mid-point of $AB$ and $P$ be a point on the ground such that $AP = 2AB$. If $\angle BPC = \beta$, then $\tan\beta$ is equal to:
(1) $\dfrac{6}{7}$
(2) $\dfrac{1}{4}$
(3) $\dfrac{2}{9}$
(4) $\dfrac{4}{9}$

An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it, is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 1500
(2) 750
(3) 720
(4) 1440

$P Q R$ 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) $400 \sqrt { 3 }$