Question 1 :
The triangle {tex} P Q R {/tex} is inscribed in the circle {tex} x ^ { 2 } + y ^ { 2 } = 25 {/tex}. If {tex} Q {/tex} and {tex} R {/tex} have co-ordinates {tex} ( 3,4 ) {/tex} and {tex} ( - 4,3 ) {/tex} respectively, then {tex} \angle Q P R {/tex} is equal to

Question 2 :
Let {tex} x , y {/tex} be real variable satisfying the {tex} x ^ { 2 } + y ^ { 2 } + 8 x - 10 y - 40 = 0. {/tex} If {tex} a = \max \left\{ ( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } \right\} {/tex} and {tex} b = \min \left\{ ( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } \right\} , {/tex} then

Question 3 :
<strong>Statement 1:</strong> If the parabola <em>y</em> = (<em>a</em>−<em>b</em>)<em>x</em>² + (<em>b</em>−<em>c</em>) + (<em>c</em> − <em>a</em>) touches the <em>x</em>-axis in the interval (0, 1), then the line ax + by + <em>c</em> = 0 always passes through a fixed point <strong><br> Statement 2:</strong> <p>The equation <em>L</em>₁ + <em>λ</em><em>L</em>₂ = 0 or <em>μ</em><em>L</em>₁ + <em>ν</em><em>L</em>₂ = 0 represent a line passing through the intersection of the lines <em>L</em>₁ = 0 and <em>L</em>₂ = 0</p> <p>Which is a fixed point, when <em>λ</em>, <em>μ</em>, <em>ν</em> are constants</p>

Question 4 :
The equations of the tangents drawn from the origin to the circle {tex} x ^ { 2 } + y ^ { 2 } - 2 r x - 2 h y + h ^ { 2 } = 0 , {/tex} are

Question 5 :
The locus of the point of intersection of the tangents at the extremities of the chord of the ellipse {tex} x ^ { 2 } + 2 y ^ { 2 } {/tex} {tex} = 6 {/tex} which touches the ellipse {tex} x ^ { 2 } + 4 y ^ { 2 } = 4 {/tex} is

Question 6 :
Let {tex} P Q {/tex} and {tex} R S {/tex} be tangents at the extremities of the diameter {tex} P R {/tex} of a circle of radius {tex} r {/tex}. If {tex} P S {/tex} and {tex} R Q {/tex} intersect at a point {tex} X {/tex} on the circumference of the circle, then {tex} 2 r {/tex} equals

Question 7 :
Let {tex} P ( a \cos \theta , b \sin \theta ) {/tex} and {tex} Q ( a \cos \phi , b \sin \phi ) {/tex} where {tex} \theta + \phi = \frac { \pi } { 2 } {/tex} be two points on the ellipse {tex} \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 {/tex} The locus of point of intersections of normals at {tex} P {/tex} and {tex} Q {/tex} is

Question 8 :
The centre of the circle passing through the point {tex} ( 0,1 ) {/tex} and touching the curve {tex} y = x ^ { 2 } {/tex} at {tex} ( 2,4 ) {/tex} is

Question 9 :
The locus of the orthocentre of the triangle formed by the lines \[ \begin{array} { l } { ( 1 + p ) x - p y + p ( 1 + p ) = 0, } \\ { ( 1 + q ) x - q y + q ( 1 + q ) = 0, } \\ \end{array} \]{tex}{ \text { and } y = 0 , \text { where } p \neq q , \text { is } } {/tex}

Question 10 :
The curve described parametrically by {tex} x = t ^ { 2 } + t + 1 {/tex}, {tex} y = t ^ { 2 } - t + 1 {/tex} represents

Question 11 :
The coordinates of the end point of the latus rectum of the parabola {tex} ( y - 1 ) ^ { 2 } = 2 ( x + 2 ) , {/tex} which does not lie on the line {tex} 2 x + y + 3 = 0 {/tex} are

Question 12 :
The angle between a pair of tangents drawn from a point {tex} P {/tex} to the circle {tex} x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 9 \sin ^ { 2 } \alpha + 13 \cos ^ { 2 } \alpha = 0 {/tex} is {tex} 2 \alpha {/tex}. The equation of the locus of the point {tex} P {/tex} is

Question 13 :
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola {tex} y ^ { 2 } = 4 a x {/tex} is another parabola with directrix

Question 14 :
If {tex} x = 9 {/tex} is the chord of contact of the hyperbola {tex} x ^ { 2 } - y ^ { 2 } = 9 {/tex} then the equation of the corresponding pair of tangents is

Question 15 :
The locus of the mid-point of a chord of the circle {tex} x ^ { 2 } + y ^ { 2 } = 4 {/tex} which subtends a right angle at the origin is