Let the area of the triangle with vertices $A ( 1 , \alpha ) , B ( \alpha , 0 )$ and $C ( 0 , \alpha )$ be 4 sq. units. If the points $( \alpha , - \alpha ) , ( - \alpha , \alpha )$ and $\left( \alpha ^ { 2 } , \beta \right)$ are collinear, then $\beta$ is equal to
(1) 64
(2) - 8
(3) - 64
(4) 512

In an isosceles triangle $ABC$, the vertex $A$ is $( 6,1 )$ and the equation of the base $BC$ is $2x + y = 4$. Let the point $B$ lie on the line $x + 3y = 7$. If $( \alpha , \beta )$ is the centroid of the triangle $ABC$, then $15( \alpha + \beta )$ is equal to

The distance between the two points $A$ and $A ^ { \prime }$ which lie on $y = 2$ such that both the line segments $A B$ and $A ^ { \prime } B$ (where $B$ is the point $( 2,3 )$ ) subtend angle $\frac { \pi } { 4 }$ at the origin, is equal to
(1) 10
(2) $\frac { 48 } { 5 }$
(3) $\frac { 52 } { 5 }$
(4) 3

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x - y - 2 = 0$ at the point $B$. If the length of the line segment $AB$ is $\frac { \sqrt { 29 } } { 3 }$, then $B$ also lies on the line
(1) $2x + y = 9$
(2) $3x - 2y = 7$
(3) $x + 2y = 6$
(4) $2x - 3y = 3$

Let $A ( 1,1 ) , B ( - 4,3 ) , C ( - 2 , - 5 )$ be vertices of a triangle $A B C , P$ be a point on side $B C$, and $\Delta _ { 1 }$ and $\Delta _ { 2 }$ be the areas of triangle $A P B$ and $A B C$ respectively. If $\Delta _ { 1 } : \Delta _ { 2 } = 4 : 7$, then the area enclosed by the lines $A P , A C$ and the $x$-axis is
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 1 } { 2 }$
(4) 1

Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L_1: 3x - 4y + 12 = 0$, and $L_2: 8x + 6y + 11 = 0$. If $P$ lies below $L_1$ and above $L_2$, then $100(\alpha + \beta)$ is equal to
(1) $-14$
(2) 42
(3) $-22$
(4) 14

For $t \in (0, 2\pi)$, if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$, $B(\cos t, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $a^2 - b^2$ is equal to
(1) $\frac{8}{3}$
(2) $8$
(3) $\frac{77}{9}$
(4) $\frac{80}{9}$

The equations of the sides $A B , B C$ and $C A$ of a triangle $A B C$ are $2 x + y = 0 , x + p y = 39$ and $x - y = 3$ respectively and $P ( 2,3 )$ is its circumcentre. Then which of the following is NOT true
(1) $( A C ) ^ { 2 } = 9 p$
(2) $( A C ) ^ { 2 } + p ^ { 2 } = 136$
(3) $32 <$ area $( \triangle A B C ) < 36$
(4) $34 <$ area $( \triangle A B C ) < 38$

Let $m_1, m_2$ be the slopes of two adjacent sides of a square of side $a$ such that $a^2 + 11a + 3(m_1^2 + m_2^2) = 220$. If one vertex of the square is $(10\cos\alpha - \sin\alpha, 10\sin\alpha + \cos\alpha)$, where $\alpha \in \left(0, \frac{\pi}{2}\right)$ and the equation of one diagonal is $(\cos\alpha - \sin\alpha)x + (\sin\alpha + \cos\alpha)y = 10$, then $72(\sin^4\alpha + \cos^4\alpha) + a^2 - 3a + 13$ is equal to
(1) 119
(2) 128
(3) 145
(4) 155

The equations of the sides $AB , BC$ and $CA$ of a triangle $ABC$ are $2 x + y = 0 , x + p y = 15 a$ and $x - y = 3$ respectively. If its orthocentre is $( 2 , a ) , - \frac { 1 } { 2 } < a < 2$, then $p$ is equal to $\_\_\_\_$.

Let $( \alpha , \beta )$ be the centroid of the triangle formed by the lines $15 x - y = 82$, $6 x - 5 y = - 4$ and $9 x + 4 y = 17$. Then $\alpha + 2 \beta$ and $2 \alpha - \beta$ are the roots of the equation
(1) $x ^ { 2 } - 7 x + 12 = 0$
(2) $x ^ { 2 } - 14 x + 48 = 0$
(3) $x ^ { 2 } - 13 x + 42 = 0$
(4) $x ^ { 2 } - 10 x + 25 = 0$

If $( \alpha , \beta )$ is the orthocenter of the triangle $ABC$ with vertices $A ( 3 , - 7 ) , B ( - 1,2 )$ and $C ( 4,5 )$, then $9 \alpha - 6 \beta + 60$ is equal to
(1) 25
(2) 35
(3) 30
(4) 40

If the orthocentre of the triangle, whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha, \beta)$, then the quadratic equation whose roots are $\alpha + 4\beta$ and $4\alpha + \beta$, is
(1) $x^2 - 19x + 90 = 0$
(2) $x^2 - 18x + 80 = 0$
(3) $x^2 - 22x + 120 = 0$
(4) $x^2 - 20x + 99 = 0$

The equations of the sides $AB , BC \& CA$ of a triangle $ABC$ are $2x + y = 0 , x + py = 21a ( a \neq 0 )$ and $x - y = 3$ respectively. Let $P ( 2 , a )$ be the centroid of the triangle $ABC$, then $( BC ) ^ { 2 }$ is equal to

If the point $\left( \alpha , \frac { 7 \sqrt { 3 } } { 3 } \right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta + y \sin \theta = 7 , \theta \in \left( 0 , \frac { \pi } { 2 } \right)$ between the co-ordinates axes, then $\alpha$ is equal to
(1) - 7
(2) $- 7 \sqrt { 3 }$
(3) $7 \sqrt { 3 }$
(4) 7

Let $A ( 0,1 ) , B ( 1,1 )$ and $C ( 1,0 )$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y ^ { 2 } = 4 a x$ passing through $D$ is $( \alpha + \beta \sqrt { 2 } , 0 )$, where $\alpha$ and $\beta$ are rational numbers, then $\frac { \alpha } { \beta ^ { 2 } }$ is equal to
(1) 8
(2) 12
(3) 6
(4) $\frac { 9 } { 2 }$

The distance of the point $( 6 , - 2 \sqrt { 2 } )$ from the common tangent $y = m x + c , m > 0$, of the curves $x = 2 y ^ { 2 }$ and $x = 1 + y ^ { 2 }$ is
(1) $\frac { 1 } { 3 }$
(2) 5
(3) $\frac { 14 } { 3 }$
(4) $5 \sqrt { 3 }$

A light ray emits from the origin making an angle $30 ^ { \circ }$ with the positive $x$-axis. After getting reflected by the line $\mathrm { x } + \mathrm { y } = 1$, if this ray intersects x-axis at Q , then the abscissa of Q is
(1) $\frac { 2 } { ( \sqrt { 3 } - 1 ) }$
(2) $\frac { 2 } { 3 + \sqrt { 3 } }$
(3) $\frac { 2 } { 3 - \sqrt { 3 } }$
(4) $\frac { \sqrt { 3 } } { 2 ( \sqrt { 3 } + 1 ) }$

Let $C ( \alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4 x + 3 y = 69$, $4 y - 3 x = 17$, and $x + 7 y = 61$. Then $( \alpha - \beta ) ^ { 2 } + \alpha + \beta$ is equal to
(1) 18
(2) 17
(3) 15
(4) 16

The equations of sides $AB$ and $AC$ of a triangle $ABC$ are $( \lambda + 1 ) x + \lambda y = 4$ and $\lambda x + ( 1 - \lambda ) y + \lambda = 0$ respectively. Its vertex $A$ is on the $y$-axis and its orthocentre is $( 1,2 )$. The length of the tangent from the point $C$ to the part of the parabola $y ^ { 2 } = 6 x$ in the first quadrant is
(1) $\sqrt { 6 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) 4

Let $B$ and $C$ be the two points on the line $y + x = 0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $\mathrm { y } - 2 \mathrm { x } = 2$ such that $\triangle A B C$ is an equilateral triangle. Then, the area of the $\triangle A B C$ is
(1) $3 \sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) $\frac { 8 } { \sqrt { 3 } }$
(4) $\frac { 10 } { \sqrt { 3 } }$

A line passing through the point $A ( 9,0 )$ makes an angle of $30 ^ { \circ }$ with the positive direction of $x$-axis. If this line is rotated about $A$ through an angle of $15 ^ { \circ }$ in the clockwise direction, then its equation in the new position is
(1) $\frac { y } { \sqrt { 3 } - 2 } + x = 9$
(2) $\frac { x } { \sqrt { 3 } - 2 } + y = 9$
(3) $\frac { x } { \sqrt { 3 } + 2 } + y = 9$
(4) $\frac { y } { \sqrt { 3 } + 2 } + x = 9$

Let $\alpha , \quad \beta , \quad \gamma , \quad \delta \in Z$ and let $A(\alpha , \beta)$, $B(1, 0)$, $C(\gamma , \delta)$ and $D(1, 2)$ be the vertices of a parallelogram $ABCD$. If $AB = \sqrt { 10 }$ and the points $A$ and $C$ lie on the line $3 y = 2 x + 1$, then $2 \alpha + \beta + \gamma + \delta$ is equal to
(1) 10
(2) 5
(3) 12
(4) 8

Let a variable line of slope $m > 0$ passing through the point $( 4 , - 9 )$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is
(1) 30
(2) 25
(3) 15
(4) 10

The portion of the line $4 x + 5 y = 20$ in the first quadrant is trisected by the lines $\mathrm { L } _ { 1 }$ and $\mathrm { L } _ { 2 }$ passing through the origin. The tangent of an angle between the lines $L _ { 1 }$ and $L _ { 2 }$ is:
(1) $\frac { 8 } { 5 }$
(2) $\frac { 25 } { 41 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 30 } { 41 }$