If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

A regular pentagon is inscribed in a circle of radius $r$ and another regular pentagon is circumscribed about the same circle. Find the ratio of the area of the inscribed pentagon to the area of the circumscribed pentagon.
(A) $\sin^2 36^\circ$ (B) $\cos^2 36^\circ$ (C) $\tan^2 36^\circ$ (D) $\cos^2 54^\circ$

In a triangle $ABC$, the circumradius is $r$ and $BC = r/2$. The circumcentre $O$ lies on $AD$ where $D$ is the midpoint of $BC$. Find the ratio $BC : AD$.
(A) $\sqrt{3} : \sqrt{2}$ (B) $\sqrt{2} : \sqrt{3}$ (C) $1 : \sqrt{3}$ (D) $\sqrt{3} : 1$

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is
(A) $75 ^ { \circ }$
(B) $\left( \frac { x } { x + 1 } \pi \right)$ radians
(C) $120 ^ { \circ }$
(D) $135 ^ { \circ }$

Two poles, $AB$ of length two metres and $CD$ of length twenty metres are erected vertically with bases at $B$ and $D$. The two poles are at a distance not less than twenty metres. It is observed that $\tan \angle ACB = 2/77$. The distance between the two poles is
(A) $72 m$
(B) 68 m
(C) 24 m
(D) 24.27 m

Two poles, $A B$ of length two metres and $C D$ of length twenty metres are erected vertically with bases at $B$ and $D$. The two poles are at a distance not less than twenty metres. It is observed that $\tan \angle A C B = 2 / 77$. The distance between the two poles is
(A) $72 m$
(B) 68 m
(C) 24 m
(D) 24.27 m

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

Suppose $ABCD$ is a quadrilateral such that $\angle BAC = 50^\circ, \angle CAD = 60^\circ, \angle CBD = 30^\circ$ and $\angle BDC = 25^\circ$. If $E$ is the point of intersection of $AC$ and $BD$, then the value of $\angle AEB$ is
(A) $75^\circ$
(B) $85^\circ$
(C) $95^\circ$
(D) $110^\circ$

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(A) $75 ^ { \circ }$
(B) $85 ^ { \circ }$
(C) $95 ^ { \circ }$
(D) $110 ^ { \circ }$

In a triangle $ABC$, $3\sin A + 4\cos B = 6$ and $4\sin B + 3\cos A = 1$ hold. Then the angle $C$ equals
(A) $30^\circ$
(B) $60^\circ$
(C) $120^\circ$
(D) $150^\circ$.

Consider a paper in the shape of an equilateral triangle $A B C$ with circumcenter $O$ and perimeter 9 units. If we fold the paper in such a way that each of the vertices $A , B , C$ gets identified with $O$, then the area of the resulting shape in square units is:
(A) $\frac { 3 \sqrt { 3 } } { 4 }$
(B) $\frac { 4 } { \sqrt { 3 } }$
(C) $\frac { 3 \sqrt { 3 } } { 2 }$
(D) $3 \sqrt { 3 }$.

A triangle has sides of lengths $\sqrt { 5 } , 2 \sqrt { 2 } , \sqrt { 3 }$ units. Then, the radius of its inscribed circle is:
(A) $\frac { \sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 } } { 2 }$
(B) $\frac { \sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 } } { 3 }$
(C) $\sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 }$
(D) $\frac { \sqrt { 5 } + \sqrt { 3 } - 2 \sqrt { 2 } } { 2 }$

Consider a right angled triangle $\triangle A B C$ whose hypotenuse $A C$ is of length 1. The bisector of $\angle A C B$ intersects $A B$ at $D$. If $B C$ is of length $x$, then what is the length of $CD$?
(A) $\sqrt { \frac { 2 x ^ { 2 } } { 1 + x } }$
(B) $\frac { 1 } { \sqrt { 2 + 2 x } }$
(C) $\sqrt { \frac { x } { 1 + x } }$
(D) $\frac { x } { \sqrt { 1 - x ^ { 2 } } }$

In $\triangle ABC$, $CD$ is the median and $BE$ is the altitude. Given that $\overline{CD} = \overline{BE}$, what is the value of $\angle ACD$?
(A) $\pi/3$
(B) $\pi/4$
(C) $\pi/5$
(D) $\pi/6$

The number of triplets $( a , b , c )$ of integers such that $a < b < c$ and $a , b , c$ are sides of a triangle with perimeter 21 is
(a) 7 .
(B) 8.
(C) 11 .
(D) 12 .

Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is
(a) $75 ^ { \circ }$.
(B) $85 ^ { \circ }$.
(C) $95 ^ { \circ }$.
(D) $110 ^ { \circ }$.

In a triangle $A B C$ with fixed base $B C$, the vertex $A$ moves such that
$$\cos B + \cos C = 4 \sin ^ { 2 } \frac { A } { 2 }$$
If $a , b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A , B$ and $C$, respectively, then
(A) $b + c = 4 a$
(B) $b + c = 2 a$
(C) locus of point $A$ is an ellipse
(D) locus of point $A$ is a pair of straight lines

Let $ABC$ and $ABC^{\prime}$ be two non-congruent triangles with sides $AB=4$, $AC=AC^{\prime}=2\sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is

Consider a triangle ABC and let $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to vertices $\mathrm { A } , \mathrm { B }$ and C respectively. Suppose $\mathrm { a } = 6 , \mathrm {~b} = 10$ and the area of the triangle is $15 \sqrt { 3 }$. If $\angle \mathrm { ACB }$ is obtuse and if r denotes the radius of the incircle of the triangle, then $r ^ { 2 }$ is equal to

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) $\frac{3y}{4c(x+c)}$