The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2x + 1$ 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

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

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) $110 ^ { \circ }$

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) $\pi/6$

1. A triangle $A B C$ is given with sides $A B = a$ and $B C = \sqrt { 3 } a$. Which of the following statements is correct? – If $A \hat { C } B = \frac { \pi } { 6 }$, then the triangle is right-angled; – If the triangle is right-angled, then $A \hat { C } B = \frac { \pi } { 6 }$. Justify your answers.

24. Let A0 A1 A2 A3 A4 A5 be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A0 A1 A0 and A0 A4 is :
(A) $3 / 4$
(B) $3 \sqrt { } 3$
(C) 3
(D) $( 3 \sqrt { } 3 ) / 2$

10. The triangle $P Q R$ is inscribed in the circle $x 2 + y 2 = 25$. If $Q$ and $R$ have coordinates ( 3 , $4 )$ and $( - 4,3 )$ respectively, then $\angle P Q R$ is equal to:
(A) $\pi / 2$
(B) $\pi / 3$
(C) $\pi / 4$
(D) $\pi / 6$

14. Let $\mathrm { g } ( \mathrm { x } ) = \int 0 \mathrm { x } f ( t ) \mathrm { dt }$, where f is such that $1 / 2 \leq f ( t ) \leq 1$ for $\mathrm { t } \in [ 0,1 ]$ and $0 \leq f ( t ) \leq 1 / 2$ for $\mathrm { t } \in [ 1,2 ]$. Then $\mathrm { g } ( 2 )$ satisfies the inequality:
(A) $- 3 / 2 \leq g ( 2 ) < 1 / 2$
(B) $0 \leq g ( 2 ) < 2$
(C) $3 / 2 < g ( 2 ) \leq 5 / 2$
(D) $2 < g ( 2 ) < 4$

10. Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $A B C$ ( $R$ being the radius of the circumcircle)?
(A) $a \sin A , \sin B$
(B) $a , b , c$
(C) $a , \sin B , R$

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(D) $a , \sin A , R$

8. If the angles of a triangle are in the ratio $4 : 1 : 1$, then the ratio of the longest side to the perimeter is:
(a) $\sqrt { } 3 : ( 2 + \sqrt { } 3 )$
(b) $1 : 6$
(c) $1 : 2 + \sqrt { } 3$
(d) $2 : 3$

24. In a $\triangle \mathrm { ABC }$, among the following which one is true?
(a) $( b + c ) \cos A / 2 = a \sin ( ( B + C ) / 2 )$
(b) $( b + c ) \cos ( ( B + C ) / 2 ) = a \sin A / 2$
(c) $( b - c ) \cos ( ( B - C ) / 2 ) = a \cos ( A / 2 )$
(d) $( b - c ) \cos A / 2 = a \cos ( ( B - C ) / 2 )$

3. Given an isosceles triangle, whose one angle is $120 ^ { \circ }$ and radius of its incircle $= \sqrt { 3 }$. Then the area of the triangle in sq. units is
(A) $7 + 12 \sqrt { 3 }$
(B) $12 - 7 \sqrt { 3 }$
(C) $12 + 7 \sqrt { 3 }$
(D) $4 \pi$
Sol. (C)
$$\Delta = \frac { \sqrt { 3 } } { 4 } \mathrm {~b} ^ { 2 }$$
Also $\frac { \sin 120 ^ { \circ } } { \mathrm { a } } = \frac { \sin 30 ^ { \circ } } { \mathrm { b } } \Rightarrow \mathrm { a } = \sqrt { 3 } \mathrm {~b}$ and $\Delta = \sqrt { 3 } \mathrm {~s}$ and $\mathrm { s } = \frac { 1 } { 2 } ( \mathrm { a } + 2 \mathrm {~b} )$ $\Rightarrow \quad \Delta = \frac { \sqrt { 3 } } { 2 } ( \mathrm { a } + 2 \mathrm {~b} )$
From (1) and (2), we get $\Delta = ( 12 + 7 \sqrt { 3 } )$.

17. Internal bisector of $\angle \mathrm { A }$ of triangle ABC meets side BC at D . A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F . If $\mathrm { a } , \mathrm { b } , \mathrm { c }$ represent sides of $\triangle \mathrm { ABC }$ then
(A) AE is HM of b and c
(B) $\mathrm { AD } = \frac { 2 \mathrm { bc } } { \mathrm { b } + \mathrm { c } } \cos \frac { \mathrm { A } } { 2 }$
(C) $\mathrm { EF } = \frac { 4 \mathrm { bc } } { \mathrm { b } + \mathrm { c } } \sin \frac { \mathrm { A } } { 2 }$
(D) the triangle AEF is isosceles
Sol. (A), (B), (C), (D). We have $\triangle \mathrm { ABC } = \triangle \mathrm { ABD } + \triangle \mathrm { ACD }$ $\Rightarrow \quad \frac { 1 } { 2 } \mathrm { bc } \sin \mathrm { A } = \frac { 1 } { 2 } \mathrm { cAD } \sin \frac { \mathrm { A } } { 2 } + \frac { 1 } { 2 } \mathrm {~b} \times \mathrm { AD } \sin \frac { \mathrm { A } } { 2 }$ $\Rightarrow \quad \mathrm { AD } = \frac { 2 \mathrm { bc } } { \mathrm { b } + \mathrm { c } } \cos \frac { \mathrm { A } } { 2 }$ Again $\mathrm { AE } = \mathrm { AD } \sec \frac { \mathrm { A } } { 2 }$ $= \frac { 2 b c } { b + c } \Rightarrow A E$ is HM of $b$ and $c$. [Figure] $\mathrm { EF } = \mathrm { ED } + \mathrm { DF } = 2 \mathrm { DE } = 2 \times \mathrm { AD } \tan \frac { \mathrm { A } } { 2 } = \frac { 2 \times 2 \mathrm { bc } } { \mathrm { b } + \mathrm { c } } \times \cos \frac { \mathrm { A } } { 2 } \times \tan \frac { \mathrm { A } } { 2 }$ $= \frac { 4 \mathrm { bc } } { \mathrm { b } + \mathrm { c } } \sin \frac { \mathrm { A } } { 2 }$ As $\mathrm { AD } \perp \mathrm { EF }$ and $\mathrm { DE } = \mathrm { DF }$ and AD is bisector ⇒ AEF is isosceles. Hence A, B, C and D are correct answers.

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