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) $- \frac { 12 } { 5 }$

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) $x + 6 y + 6 = 0$

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

Let $A$ be the point of intersection of the lines $3 x + 2 y = 14,5 x - y = 6$ and $B$ be the point of intersection of the lines $4 x + 3 y = 8,6 x + y = 5$. The distance of the point $P ( 5 , - 2 )$ from the line $A B$ is
(1) $\frac { 13 } { 2 }$
(2) 8
(3) $\frac { 5 } { 2 }$
(4) 6

Let $A(a, b)$, $B(3, 4)$ and $(-6, -8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3, 7b+5)$ from the line $2x + 3y - 4 = 0$ measured parallel to the line $x - 2y - 1 = 0$ is
(1) $\dfrac{15\sqrt{5}}{7}$
(2) $\dfrac{17\sqrt{5}}{6}$
(3) $\dfrac{17\sqrt{5}}{7}$
(4) $\dfrac{\sqrt{5}}{17}$

In a $\triangle \mathrm { ABC }$, suppose $\mathrm { y } = \mathrm { x }$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x - y = 2$. If $2 A B = B C$ and the point $A$ and $B$ are respectively $( 4,6 )$ and $( \alpha , \beta )$, then $\alpha + 2 \beta$ is equal to
(1) - 4
(2) 42
(3) 2
(4) - 1

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

If the sum of squares of all real values of $\alpha$, for which the lines $2 x - y + 3 = 0,6 x + 3 y + 1 = 0$ and $\alpha x + 2 y - 2 = 0$ do not form a triangle is $p$, then the greatest integer less than or equal to $p$ is $\_\_\_\_$ .

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 $\mathrm { A } , \mathrm { B } , \mathrm { C }$ be three points in $xy$-plane, whose position vector are given by $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\mathrm { a } \hat { i } + ( 1 - \mathrm { a } ) \hat { j }$ respectively with respect to the origin O. If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow { \mathrm { OA } }$ and $\overrightarrow { \mathrm { OB } }$ is $\frac { 9 } { \sqrt { 2 } }$, then the sum of all the possible values of $a$ is :
(1) 2
(2) $9/2$
(3) 1
(4) 0

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 points $\left( \frac{11}{2}, \alpha \right)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:
(1) 44
(2) 22
(3) 33
(4) 55

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

If the line $3 x - 2 y + 12 = 0$ intersects the parabola $4 y = 3 x ^ { 2 }$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $A B$ subtends an angle equal to
(1) $\tan ^ { - 1 } \left( \frac { 4 } { 5 } \right)$
(2) $\tan ^ { - 1 } \left( \frac { 9 } { 7 } \right)$
(3) $\tan ^ { - 1 } \left( \frac { 11 } { 9 } \right)$
(4) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 3 } { 2 } \right)$

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 $A ( 6,8 ) , B ( 10 \cos \alpha , - 10 \sin \alpha )$ and $C ( - 10 \sin \alpha , 10 \cos \alpha )$, be the vertices of a triangle. If $L ( a , 9 )$ and $G ( h , k )$ be its orthocenter and centroid respectively, then $( 5 a - 3 h + 6 k + 100 \sin 2 \alpha )$ is equal to $\_\_\_\_$

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 $\_\_\_\_$

Let $m$ be a real number. On a plane with the coordinate system, in which the origin is denoted by O, consider the parabola $y = x^2$ and the two points on it,
$$\mathrm{A}(a,\, ma+1), \quad \mathrm{B}(b,\, mb+1) \quad (a < 0 < b)$$
(1) The $x$-coordinates $a$ and $b$ of the two points A and B can be expressed in terms of $m$ as
$$a = \frac{m - \sqrt{D}}{\mathbf{A}}, \quad b = \frac{m + \sqrt{D}}{\mathbf{B}},$$
where the expression $D$ is
$$D = m^2 + \mathbf{C}.$$
(2) Let the coordinates of the point of intersection of the segment AB and the $y$-axis be denoted by $(0, c)$. Then $c = \mathbf{D}$.
(3) Further, when the area $S$ of the triangle OAB with the three vertices O, A and B is expressed in terms of $a$ and $b$, we have
$$S = \frac{1}{2}\mathbf{E},$$
where $E$ is the appropriate choice from among (0) $\sim$ (5). (0) $a + b$
(1) $a - b$
(2) $b - a$
(3) $a^2 + b^2$
(4) $a^2 - b^2$
(5) $b^2 - a^2$
Also, when $S$ is represented in terms of $m$, we have
$$S = \frac{\mathbf{F}}{\mathbf{G}} \sqrt{m^2 + \mathbf{H}}.$$
Hence the value of $S$ is minimalized when $m = \mathbf{I}$, and its minimum value is $S = \mathbf{J}$.

Consider a figure made by cutting two corners from a rectangle, as in the diagram to the right. The lengths of the sides are
$$\begin{array}{lll} \mathrm{AB} = 11, & \mathrm{BC} = 4, & \mathrm{CD} = 2\sqrt{13} \\ \mathrm{DE} = 5, & \mathrm{EF} = 2\sqrt{5}, & \mathrm{FA} = 6 \end{array}$$
We are to find the area of this figure.
First, extend the sides of the figure as in the diagram and denote the sides forming the right angles by $x, y, u$ and $v$. Then
$$u = \mathbf{A} - y, \quad v = x + \mathbf{B}.$$
Substituting these expressions in the equation $u^2 + v^2 = \mathbf{CD}$ and also using the equation $x^2 + y^2 = \mathbf{EF}$, we obtain
$$x = \mathbf{G}.\, y - \mathbf{H}.$$
Then, since
$$\mathbf{I}y^2 - \mathbf{J} = \mathbf{J}y - \mathbf{K} = 0,$$
we obtain $y = \mathbf{L}$.
From this we have $x = \mathbf{M}$, and further $u = \mathbf{N}$ and $v = \mathbf{O}$. Finally we conclude that the area of this figure is $\mathbf{PQ}$.