Question 1 :
A square is inscribed in the circle {tex} x ^ { 2 } + y ^ { 2 } - 2 x + 4 y + 3 = 0 . {/tex} Its sides are parallel to the coordinate axes. The one vertex of the square is

Question 2 :
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 3 :
If the two circles {tex} ( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 } {/tex} and {tex} x ^ { 2 } + y ^ { 2 } - 8 x + 2 y + 8 = 0 {/tex} intersect in two distinct points, then

Question 4 :
Let {tex} A B {/tex} be a chord of the circle {tex} x ^ { 2 } + y ^ { 2 } = r ^ { 2 } {/tex} subtending a right angle at the centre. Then the locus of the centroid of the triangle {tex} PAB {/tex} as {tex}P{/tex} moves on the circle is

Question 5 :
Equation of the normal at a point on the parabola {tex} y ^ { 2 } = 36 x , {/tex} whose ordinate is three times its abscissa is

Question 6 :
An equation of a tangent common to the parabolas {tex} y ^ { 2 } = 4 x {/tex} and {tex} x ^ { 2 } = 4 y {/tex} is

Question 7 :
The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line {tex} 4 x - 5 y {/tex} {tex} = 20 {/tex} to the circle {tex} x ^ { 2 } + y ^ { 2 } = 9 {/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 :
If {tex}\mathrm P {/tex} is a point on the ellipse {tex}\mathrm { \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = }{/tex} with foci {tex}\mathrm S {/tex} and {tex}\mathrm S {/tex} and eccentricity {tex} \mathrm e , {/tex} then locus of the incentre of the triangle {tex}\mathrm { P S S ^ { \prime } }{/tex} is an ellipse of eccentricity

Question 10 :
Consider a branch of the hyperbola \[ x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0 \]<br>with vertex at the point {tex} A {/tex}. Let {tex} B {/tex} be one of the end points of its latus rectum. If {tex} C {/tex} is the focus of the hyperbola nearest to the point {tex} A , {/tex} then the area of the triangle {tex} A B C {/tex} is<br>

Question 11 :
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 12 :
The point of contact of the tangent to the parabola {tex} y ^ { 2 } = 9 x {/tex} which passes through the point ( 4,10 ) and makes an angle {tex} \theta {/tex} with the axis of the parabola such that tan {tex} \theta > 2 {/tex} is

Question 13 :
If {tex} a , b > 0 , {/tex} then the angle of intersection of two parabolas {tex} y ^ { 2 } = a ^ { 3 } x {/tex} and {tex} x ^ { 2 } = b ^ { 3 } y {/tex} at a point other than the origin is

Question 14 :
Let length of transverse axis of a hyperbola with eccentricity {tex} \sqrt { 5 } {/tex} be {tex} 4 . {/tex} The difference between length of latus rectum and conjugate axis is

Question 15 :
Each of the four inequalties given below defines a region in the {tex} x y {/tex} plane. One of these four regions does not have the following property. For any two points {tex} \left( x _ { 1 } , y _ { 1 } \right) {/tex} and {tex} \left( x _ { 2 } , y _ { 2 } \right) {/tex} in the region, the point {tex} \left( \frac { x _ { 1 } + x _ { 2 } } { 2 } , \frac { y _ { 1 } + y _ { 2 } } { 2 } \right) {/tex} is also in the region. The inequality defining this region is