The distance of the point $( 2,3 )$ from the line $2 x - 3 y + 28 = 0$, measured parallel to the line $\sqrt { 3 } x - y + 1 = 0$, is equal to
(1) $4 \sqrt { 2 }$
(2) $6 \sqrt { 3 }$
(3) $3 + 4 \sqrt { 2 }$
(4) $4 + 6 \sqrt { 3 }$

If the line segment joining the points $( 5,2 )$ and $( 2 , a )$ subtends an angle $\frac { \pi } { 4 }$ at the origin, then the absolute value of the product of all possible values of $a$ is : (1) 6 (2) 8 (3) 2 (4) - 4

Consider a triangle ABC having the vertices $\mathrm { A } ( 1,2 ) , \mathrm { B } ( \alpha , \beta )$ and $\mathrm { C } ( \gamma , \delta )$ and angles $\angle A B C = \frac { \pi } { 6 }$ and $\angle B A C = \frac { 2 \pi } { 3 }$. If the points B and C lie on the line $y = x + 4$, then $\alpha ^ { 2 } + \gamma ^ { 2 }$ is equal to $\_\_\_\_$

Let a ray of light passing through the point $( 3,10 )$ reflects on the line $2 x + y = 6$ and the reflected ray passes through the point $( 7,2 )$. If the equation of the incident ray is $a x + b y + 1 = 0$, then $a ^ { 2 } + b ^ { 2 } + 3 a b$ is equal to $\_\_\_\_$

If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0 , x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $( 3,4 )$ and $( - 6 , - 8 )$, then the value of $| a - b |$ is $\_\_\_\_$

Let the triangle PQR be the image of the triangle with vertices $( 1,3 ) , ( 3,1 )$ and $( 2,4 )$ in the line $x + 2 y = 2$. If the centroid of $\triangle \mathrm { PQR }$ is the point $( \alpha , \beta )$, then $15 ( \alpha - \beta )$ is equal to:
(1) 19
(2) 24
(3) 21
(4) 22

Let the line $x + y = 1$ meet the axes of $x$ and $y$ at A and B , respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines $O B$ and $A B$, respectively. If the area of the triangle $A M N$ is $\frac { 4 } { 9 }$ of the area of the triangle $O A B$ and $\mathrm { AN } : \mathrm { NB } = \lambda : 1$, then the sum of all possible value(s) of $\lambda$ is:
(1) 2
(2) $\frac { 5 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 13 } { 6 }$

Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to
(1) 84
(2) 113
(3) 91
(4) 101

Let the range of the function $f ( x ) = 6 + 16 \cos x \cdot \cos \left( \frac { \pi } { 3 } - x \right) \cdot \cos \left( \frac { \pi } { 3 } + x \right) \cdot \sin 3 x \cdot \cos 6 x , x \in \mathbf { R }$ be $[ \alpha , \beta ]$. Then the distance of the point $( \alpha , \beta )$ from the line $3 x + 4 y + 12 = 0$ is :
(1) 11
(2) 8
(3) 10
(4) 9

If $A$ and $B$ are the points of intersection of the circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ and a point P moves on the line $2 x - 3 y + 4 = 0$, then the centroid of $\triangle \mathrm { PAB }$ lies on the line :
(1) $x + 9 y = 36$
(2) $4 x - 9 y = 12$
(3) $6 x - 9 y = 20$
(4) $9 x - 9 y = 32$

Let the area of a $\triangle P Q R$ with vertices $P ( 5,4 ) , Q ( - 2,4 )$ and $R ( a , b )$ be 35 square units. If its orthocenter and centroid are $O \left( 2 , \frac { 14 } { 5 } \right)$ and $C ( c , d )$ respectively, then $c + 2 d$ is equal to
(1) $\frac { 8 } { 3 }$
(2) $\frac { 7 } { 3 }$
(3) 2
(4) 3

Let $ABC$ be a triangle formed by the lines $7x - 6y + 3 = 0$, $x + 2y - 31 = 0$ and $9x - 2y - 19 = 0$. Let the point $(h, k)$ be the image of the centroid of $\triangle ABC$ in the line $3x + 6y - 53 = 0$. Then $h^2 + k^2 + hk$ is equal to:
(1) 47
(2) 37
(3) 36
(4) 40

Two equal sides of an isosceles triangle are along $- x + 2 y = 4$ and $x + y = 4$. If m is the slope of its third side, then the sum, of all possible distinct values of $m$, is :
(1) $- 2 \sqrt { 10 }$
(2) 12
(3) 6
(4) $-6$

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$ lies on the other. Then $( QR ) ^ { 2 }$ is equal to $\_\_\_\_$

Q64. Let two straight lines drawn from the origin O intersect the line $3 x + 4 y = 12$ at the points P and Q such that $\triangle \mathrm { OPQ }$ is an isosceles triangle and $\angle \mathrm { POQ } = 90 ^ { \circ }$. If $l = \mathrm { OP } ^ { 2 } + \mathrm { PQ } ^ { 2 } + \mathrm { QO } ^ { 2 }$, then the greatest integer less than or equal to $l$ is :
(1) 42
(2) 46
(3) 44
(4) 48

Q64. 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

Q65. Let $A ( - 1,1 )$ and $B ( 2,3 )$ be two points and $P$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm { PAB }$ is 10 . If the locus of P is $\mathrm { a } x + \mathrm { b } y = 15$, then $5 \mathrm { a } + 2 \mathrm {~b}$ is :
(1) 6
(2) $- \frac { 6 } { 5 }$
(3) 4
(4) $- \frac { 12 } { 5 }$

Q65. The equations of two sides AB and AC of a triangle ABC are $4 x + y = 14$ and $3 x - 2 y = 5$, respectively. The point $\left( 2 , - \frac { 4 } { 3 } \right)$ divides the third side BC internally in the ratio $2 : 1$. the equation of the side BC is
(1) $x + 3 y + 2 = 0$
(2) $x - 6 y - 10 = 0$
(3) $x - 3 y - 6 = 0$
(4) $x + 6 y + 6 = 0$

Q65. A ray of light coming from the point $P ( 1,2 )$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R ( 4,3 )$. If the point $S ( h , k )$ is such that PQRS is a parallelogram, then $h k ^ { 2 }$ is equal to :
(1) 70
(2) 80
(3) 60
(4) 90

Q65. Two vertices of a triangle ABC are $\mathrm { A } ( 3 , - 1 )$ and $\mathrm { B } ( - 2,3 )$, and its orthocentre is $\mathrm { P } ( 1,1 )$. If the coordinates of the point C are $( \alpha , \beta )$ and the centre of the of the circle circumscribing the triangle PAB is $( \mathrm { h } , \mathrm { k } )$, then the value of $( \alpha + \beta ) + 2 ( \mathrm {~h} + \mathrm { k } )$ equals
(1) 5
(2) 81
(3) 15
(4) 51

Q66. The vertices of a triangle are $\mathrm { A } ( - 1,3 ) , \mathrm { B } ( - 2,2 )$ and $\mathrm { C } ( 3 , - 1 )$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :
(1) $x + y + ( 2 - \sqrt { 2 } ) = 0$
(2) $- x + y - ( 2 - \sqrt { 2 } ) = 0$
(3) $x + y - ( 2 - \sqrt { 2 } ) = 0$
(4) $x - y - ( 2 + \sqrt { 2 } ) = 0$

Q67. If the line segment joining the points $( 5,2 )$ and $( 2 , a )$ subtends an angle $\frac { \pi } { 4 }$ at the origin, then the absolute value of the product of all possible values of $a$ is :
(1) 6
(2) 8
(3) 2
(4) - 4

Q83. Consider a triangle ABC having the vertices $\mathrm { A } ( 1,2 ) , \mathrm { B } ( \alpha , \beta )$ and $\mathrm { C } ( \gamma , \delta )$ and angles $\angle A B C = \frac { \pi } { 6 }$ and $\angle B A C = \frac { 2 \pi } { 3 }$. If the points B and C lie on the line $y = x + 4$, then $\alpha ^ { 2 } + \gamma ^ { 2 }$ is equal to $\_\_\_\_$

Q83. Let a ray of light passing through the point $( 3,10 )$ reflects on the line $2 x + y = 6$ and the reflected ray passes through the point $( 7,2 )$. If the equation of the incident ray is $a x + b y + 1 = 0$, then $a ^ { 2 } + b ^ { 2 } + 3 a b$ is equal to $\_\_\_\_$

Q84. If the orthocentre of the triangle formed by the lines $2 x + 3 y - 1 = 0 , x + 2 y - 1 = 0$ and $a x + b y - 1 = 0$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $( 3,4 )$ and $( - 6 , - 8 )$, then the value of $| a - b |$ is $\_\_\_\_$