Question 1 :
The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola {tex} \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1 , {/tex} meets {tex} x {/tex}-axis and {tex} y {/tex}-axis at {tex} A {/tex} and {tex} B {/tex}, respectively. Then {tex} ( O A ) ^ { 2 } - ( O B ) ^ { 2 } , {/tex} where {tex} O {/tex} is origin, equals
Question 2 :
The normal {tex} y = m x - 2 a m - a m ^ { 3 } {/tex} to the parabola {tex} y ^ { 2 } = 4 a x {/tex} subtends a right angle at the vertex if
Question 3 :
If {tex} a > 2 b > 0 {/tex} then the positive value of {tex} m {/tex} for which {tex} y = m x - b \sqrt { 1 + m ^ { 2 } } {/tex} is a common tangent to {tex} x ^ { 2 } + y ^ { 2 } = b ^ { 2 } {/tex} and {tex} ( x - a ) ^ { 2 } + y ^ { 2 } = b ^ { 2 } {/tex} is
Question 4 :
Let {tex} \mathrm E _ { 1 } {/tex} be the ellipse {tex} \frac { x ^ { 2 } } { a ^ { 2 } + 2 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 {/tex} and {tex} \mathrm E _ { 2 } {/tex} be the ellipse {tex} \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } + 1 } = 1 . {/tex} The number of points from which to perpendicular tangents can be drawn to each of {tex}\mathrm E _ { 1 } {/tex} and {tex} \mathrm E _ { 2 } {/tex} is {tex} \ldots {/tex}
Question 5 :
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
Question 6 :
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 7 :
Eccentricity of a hyperbola angle between whose asymptotes is {tex} \frac { \pi } { 6 } {/tex} is
Question 8 :
If two distinct chords, drawn from the point {tex} ( p , q ) {/tex} on the circle {tex} x ^ { 2 } + y ^ { 2 } = p x + q y ( \text { where } p q \neq 0 ) {/tex} are bisected by the {tex} x {/tex} -axis, then
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 :
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 11 :
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 12 :
<strong>Statement 1:</strong> If the parabola <em>y</em> = (<em>a</em>−<em>b</em>)<em>x</em><sup>2</sup> + (<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><sub>1</sub> + <em>λ</em><em>L</em><sub>2</sub> = 0 or <em>μ</em><em>L</em><sub>1</sub> + <em>ν</em><em>L</em><sub>2</sub> = 0 represent a line passing through the intersection of the lines <em>L</em><sub>1</sub> = 0 and <em>L</em><sub>2</sub> = 0</p> <p>Which is a fixed point, when <em>λ</em>, <em>μ</em>, <em>ν</em> are constants</p>
Question 13 :
A circle is given by {tex} x ^ { 2 } + ( y - 1 ) ^ { 2 } = 1 , {/tex} another circle {tex} C {/tex} touches it externally and also the {tex} x {/tex} -axis, then the locus of its centre is
Question 14 :
The coordinates of the point on the circle {tex} x ^ { 2 } + y ^ { 2 } - 2 x - 4 y - 11 = 0 {/tex} farthest from the origin are
Question 15 :
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