Consider three points $P = ( - \sin ( \beta - \alpha ) , - \cos \beta ) , Q = ( \cos ( \beta - \alpha ) , \sin \beta )$ and $R = ( \cos ( \beta - \alpha + \theta ) , \sin ( \beta - \theta ) )$, where $0 < \alpha , \beta , \theta < \frac { \pi } { 4 }$. Then,
(A) $P$ lies on the line segment $R Q$
(B) $Q$ lies on the line segment $P R$
(C) $R$ lies on the line segment $Q P$
(D) $P , Q , R$ are non-collinear

Consider the lines given by
$$\begin{aligned} & L _ { 1 } : x + 3 y - 5 = 0 \\ & L _ { 2 } : 3 x - k y - 1 = 0 \\ & L _ { 3 } : 5 x + 2 y - 12 = 0 \end{aligned}$$
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.
Column I
(A) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ are concurrent, if
(B) One of $L _ { 1 } , L _ { 2 } , L _ { 3 }$ is parallel to at least one of the other two, if
(C) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ form a triangle, if
(D) $L _ { 1 } , L _ { 2 } , L _ { 3 }$ do not form a triangle, if
Column II
(p) $k = - 9$
(q) $k = - \frac { 6 } { 5 }$
(r) $k = \frac { 5 } { 6 }$
(s) $k = 5$

For $a > b > c > 0$, the distance between $( 1,1 )$ and the point of intersection of the lines $a x + b y + c = 0$ and $b x + a y + c = 0$ is less than $2 \sqrt { 2 }$. Then
(A) $a + b - c > 0$
(B) $a - b + c < 0$
(C) $a - b + c > 0$
(D) $a + b - c < 0$

Let $T$ be the line passing through the points $P ( - 2,7 )$ and $Q ( 2 , - 5 )$. Let $F _ { 1 }$ be the set of all pairs of circles ( $S _ { 1 } , S _ { 2 }$ ) such that $T$ is tangent to $S _ { 1 }$ at $P$ and tangent to $S _ { 2 }$ at $Q$, and also such that $S _ { 1 }$ and $S _ { 2 }$ touch each other at a point, say, $M$. Let $E _ { 1 }$ be the set representing the locus of $M$ as the pair ( $S _ { 1 } , S _ { 2 }$ ) varies in $F _ { 1 }$. Let the set of all straight line segments joining a pair of distinct points of $E _ { 1 }$ and passing through the point $R ( 1,1 )$ be $F _ { 2 }$. Let $E _ { 2 }$ be the set of the mid-points of the line segments in the set $F _ { 2 }$. Then, which of the following statement(s) is (are) TRUE?
(A) The point $( - 2,7 )$ lies in $E _ { 1 }$
(B) The point $\left( \frac { 4 } { 5 } , \frac { 7 } { 5 } \right)$ does NOT lie in $E _ { 2 }$
(C) The point $\left( \frac { 1 } { 2 } , 1 \right)$ lies in $E _ { 2 }$
(D) The point $\left( 0 , \frac { 3 } { 2 } \right)$ does NOT lie in $E _ { 1 }$

Let $A ( h , k ) , B ( 1,1 )$ and $C ( 2,1 )$ be the vertices of a right angled triangle with $A C$ as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' k ' can take is given by
(1) $\{ 1,3 \}$
(2) $\{ 0,2 \}$
(3) $\{ - 1,3 \}$
(4) $\{ - 3 , - 2 \}$

Let $P = ( - 1,0 ) , Q = ( 0,0 )$ and $R = ( 3,3 \sqrt { 3 } )$ be three points. The equation of the bisector of the angle PQR
(1) $\sqrt { 3 } x + y = 0$
(2) $x + \frac { \sqrt { 3 } } { 2 } y = 0$
(3) $\frac { \sqrt { 3 } } { 2 } x + y = 0$
(4) $x + \sqrt { 3 } y = 0$

If one of the lines of $m y ^ { 2 } + \left( 1 - m ^ { 2 } \right) x y - m x ^ { 2 } = 0$ is a bisector of the angle between the lines $x y = 0$, then $m$ is
(1) $- 1 / 2$
(2) - 2
(3) 1
(4) 2

The lines $L_{1}: y-x=0$ and $L_{2}: 2x+y=0$ intersect the line $L_{3}: y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$. Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

If the point $(1, a)$ lies between the straight lines $x + y = 1$ and $2(x+y) = 3$ then $a$ lies in interval
(1) $\left(\frac{3}{2}, \infty\right)$
(2) $\left(1, \frac{3}{2}\right)$
(3) $(-\infty, 0)$
(4) $\left(0, \frac{1}{2}\right)$

If the straight lines $x + 3 y = 4,3 x + y = 4$ and $x + y = 0$ form a triangle, then the triangle is
(1) scalene
(2) equilateral triangle
(3) isosceles
(4) right angled isosceles

If two vertices of a triangle are $(5, -1)$ and $(-2, 3)$ and its orthocentre is at $(0, 0)$, then the third vertex is
(1) $(4, -7)$
(2) $(-4, -7)$
(3) $(-4, 7)$
(4) $(4, 7)$

Let $L$ be the line $y = 2 x$, in the two dimensional plane. Statement 1: The image of the point $( 0,1 )$ in $L$ is the point $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$. Statement 2: The points $( 0,1 )$ and $\left( \frac { 4 } { 5 } , \frac { 3 } { 5 } \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(4) Statement 1 is false, Statement 2 is true.

The line parallel to $x$-axis and passing through the point of intersection of lines $a x + 2 b y + 3 b = 0$ and $b x - 2 a y - 3 a = 0$, where $( a , b ) \neq ( 0,0 )$ is
(1) above $x$-axis at a distance $2/3$ from it
(2) above $x$-axis at a distance $3/2$ from it
(3) below $x$-axis at a distance $3/2$ from it
(4) below $x$-axis at a distance $2/3$ from it

Consider the straight lines $$\begin{aligned} & L _ { 1 } : x - y = 1 \\ & L _ { 2 } : x + y = 1 \\ & L _ { 3 } : 2 x + 2 y = 5 \\ & L _ { 4 } : 2 x - 2 y = 7 \end{aligned}$$ The correct statement is
(1) $L _ { 1 } \left\| L _ { 4 } , L _ { 2 } \right\| L _ { 3 } , L _ { 1 }$ intersect $L _ { 4 }$.
(2) $L _ { 1 } \perp L _ { 2 } , L _ { 1 } \| L _ { 3 } , L _ { 1 }$ intersect $L _ { 2 }$.
(3) $L _ { 1 } \perp L _ { 2 } , L _ { 2 } \| L _ { 3 } , L _ { 1 }$ intersect $L _ { 4 }$.
(4) $L _ { 1 } \perp L _ { 2 } , L _ { 1 } \perp L _ { 3 } , L _ { 2 }$ intersect $L _ { 4 }$.

A line is drawn through the point $(1,2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle of area $\frac{9}{2}$ sq. units with the coordinate axes. The equation of the line $PQ$ is
(1) $x+2y=5$
(2) $3x+y=5$
(3) $x+2y=4$
(4) $2x+y=4$

If two vertical poles 20 m and 80 m high stand apart on a horizontal plane, then the height (in m) of the point of intersection of the lines joining the top of each pole to the foot of other is
(1) 16
(2) 18
(3) 50
(4) 15

A light ray emerging from the point source placed at $\mathrm { P } ( 1,3 )$ is reflected at a point Q in the axis of $x$. If the reflected ray passes through the point $R ( 6,7 )$, then the abscissa of $Q$ is:
(1) 1
(2) 3
(3) $\frac { 7 } { 2 }$
(4) $\frac { 5 } { 2 }$

Let $\theta _ { 1 }$ be the angle between two lines $2 x + 3 y + c _ { 1 } = 0$ and $- x + 5 y + c _ { 2 } = 0$ and $\theta _ { 2 }$ be the angle between two lines $2 x + 3 y + c _ { 1 } = 0$ and $- x + 5 y + c _ { 3 } = 0$, where $c _ { 1 } , c _ { 2 } , c _ { 3 }$ are any real numbers: Statement-1: If $c _ { 2 }$ and $c _ { 3 }$ are proportional, then $\theta _ { 1 } = \theta _ { 2 }$. Statement-2: $\theta _ { 1 } = \theta _ { 2 }$ for all $c _ { 2 }$ and $c _ { 3 }$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation of Statement-1.
(2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation of Statement-1.
(3) Statement-1 is false; Statement-2 is true.
(4) Statement-1 is true; Statement-2 is false.

If the image of point $\mathrm { P } ( 2,3 )$ in a line L is $\mathrm { Q } ( 4,5 )$, then the image of point $\mathrm { R } ( 0,0 )$ in the same line is:
(1) $( 2,2 )$
(2) $( 4,5 )$
(3) $( 3,4 )$
(4) $( 7,7 )$

If the three lines $x - 3 y = p , a x + 2 y = q$ and $a x + y = r$ form a right-angled triangle then :
(1) $a ^ { 2 } - 9 a + 18 = 0$
(2) $a ^ { 2 } - 6 a - 12 = 0$
(3) $a ^ { 2 } - 6 a - 18 = 0$
(4) $a ^ { 2 } - 9 a + 12 = 0$

A ray of light along $x + \sqrt{3}y = \sqrt{3}$ gets reflected upon reaching $X$-axis, the equation of the reflected ray is
(1) $y = \sqrt{3}x - \sqrt{3}$
(2) $\sqrt{3}y = x - 1$
(3) $y = x + \sqrt{3}$
(4) $\sqrt{3}y = x - \sqrt{3}$

The $x$-coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as $(0,1)$, $(1,1)$ and $(1,0)$ is
(1) $1 + \sqrt{2}$
(2) $1 - \sqrt{2}$
(3) $2 + \sqrt{2}$
(4) $2 - \sqrt{2}$

Equation of the line passing through the points of intersection of the parabola $x ^ { 2 } = 8 y$ and the ellipse $\frac { x ^ { 2 } } { 3 } + y ^ { 2 } = 1$ is :
(1) $y - 3 = 0$
(2) $y + 3 = 0$
(3) $3 y + 1 = 0$
(4) $3 y - 1 = 0$

If the extremities of the base of an isosceles triangle are the points $( 2 a , 0 )$ and $( 0 , a )$ and the equation of one of the sides is $x = 2 a$, then the area of the triangle, in square units, is :
(1) $\frac { 5 } { 4 } a ^ { 2 }$
(2) $\frac { 5 } { 2 } a ^ { 2 }$
(3) $\frac { 25 a ^ { 2 } } { 4 }$
(4) $5 a^2$

If a line $L$ is perpendicular to the line $5 x - y = 1$, and the area of the triangle formed by the line $L$ and the coordinate axes is 5 sq units, then the distance of the line $L$ from the line $x + 5 y = 0$ is
(1) $\frac { 7 } { \sqrt { 13 } }$ units
(2) $\frac { 7 } { \sqrt { 5 } }$ units
(3) $\frac { 5 } { \sqrt { 13 } }$ units
(4) $\frac { 5 } { \sqrt { 7 } }$ units