A circle $$C_1$$ passes through the origin $$\mathrm{O}$$ and has diameter 4 on the positive $$x$$-axis. The line $$y=2x$$ gives a chord $$\mathrm{OA}$$ of circle $${\mathrm{C}}_1$$. Let $${\mathrm{C}}_2$$ be the circle with $$\mathrm{OA}$$ as a diameter. If the tangent to $${\mathrm{C}}_2$$ at the point $$\mathrm{A}$$ meets the $$x$$-axis at $$\mathrm{P}$$ and $$y$$-axis at $$\mathrm{Q}$$, then $$\mathrm{QA}:\mathrm{AP}$$ is equal to :

For $$\mathrm{t}\in (0,2\pi )$$, if $$\mathrm{ABC}$$ is an equilateral triangle with vertices $$\mathrm{A}(\sin t,-\cos \mathrm{t}),\mathrm{B}(\mathrm{cost},\sin t)$$ and $$C(a,b)$$ such that its orthocentre lies on a circle with centre $$\left(1,\frac{1}{3}\right)$$, then $$\left(a^2-b^2\right)$$ is equal to :

Let $$C$$ be the centre of the circle $$x^2+y^2-x+2y=\frac{11}{4}$$ and $$P$$ be a point on the circle. A line passes through the point $$\mathrm{C}$$, makes an angle of $$\frac{\pi }{4}$$ with the line $$\mathrm{CP}$$ and intersects the circle at the points $$Q$$ and $$R$$. Then the area of the triangle $$PQR$$ (in unit $${}^2$$ ) is :

Let the tangents at two points $$\mathrm{A}$$ and $$\mathrm{B}$$ on the circle $$x^2+{\mathrm{y}}^2-4x+3=0$$ meet at origin $$\mathrm{O}(0,0)$$. Then the area of the triangle $$\mathrm{OAB}$$ is :

Let the eccentricity of an ellipse {{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1, $$a>b$$, be {1 \over 4}. If this ellipse passes through the point \left( { - 4\sqrt {{2 \over 5}} ,3} \right), then $$a^2+b^2$$ is equal to :

If m is the slope of a common tangent to the curves {{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1 and $$x^2+y^2=12$$, then $$12m^2$$ is equal to :

The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $$x^2+2y^2=4$$ is an ellipse with eccentricity :

The line y = x + 1 meets the ellipse {{{x^2}} \over 4} + {{{y^2}} \over 2} = 1 at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :

Let the maximum area of the triangle that can be inscribed in the ellipse {{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $$6\sqrt{3}$$. Then the eccentricity of the ellipse is :

Let the eccentricity of the ellipse $$x^2+a^2{y}^2=25a^2$$ be b times the eccentricity of the hyperbola $$x^2-a^2{y}^2=5$$, where a is the minimum distance between the curves y = ex and y = logex. Then {a^2} + {1 \over {{b^2}}} is equal to :

If the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ meets the line $$\frac{x}{7}+\frac{y}{2\sqrt{6}}=1$$ on the $$x$$-axis and the line $$\frac{x}{7}-\frac{y}{2\sqrt{6}}=1$$ on the $$y$$-axis, then the eccentricity of the ellipse is :

The acute angle between the pair of tangents drawn to the ellipse $$2x^2+3y^2=5$$ from the point $$(1,3)$$ is :

Let a line L pass through the point of intersection of the lines $$bx+10y-8=0$$ and $$2x-3y=0,\mathrm{b}\in \mathbf{R}-\left\{\frac{4}{3}\right\}$$. If the line $$\mathrm{L}$$ also passes through the point $$(1,1)$$ and touches the circle $$17\left(x^2+y^2\right)=16$$, then the eccentricity of the ellipse $$\frac{x^2}{5}+\frac{y^2}{{\mathrm{b}}^2}=1$$ is :

Let a set A = A1 $$\cup$$ A2 $$\cup$$ ..... $$\cup$$ Ak, where Ai $$\cap$$ Aj = $$\varphi$$ for i $$\ne$$ j, 1 $$\le$$ j, j $$\le$$ k. Define the relation R from A to A by R = {(x, y) : y $$\in$$ Ai if and only if x $$\in$$ Ai, 1 $$\le$$ i $$\le$$ k}. Then, R is :

Let R1 = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\le$$ 13} and R2 = {(a, b) $$\in$$ N $$\times$$ N : |a $$-$$ b| $$\ne$$ 13}. Then on N :

Let $$R_1$$ and $$R_2$$ be two relations defined on $$\mathbb{R}$$ by $$a{R}_{1}b\iff ab\ge 0$$ and $$a{R}_{2}b\iff a\ge b$$ Then,

For $$\alpha \in \mathbf{N}$$, consider a relation $$\mathrm{R}$$ on $$\mathbf{N}$$ given by $$\mathrm{R}=\{(x,y):3x+\alpha y$$ is a multiple of 7$$\}$$. The relation $$R$$ is an equivalence relation if and only if :

Let R be a relation from the set $$\{1,2,3,\dots ,60\}$$ to itself such that $$R=\{(a,b):b=pq$$, where $$p,q⩾3$$ are prime numbers}. Then, the number of elements in R is :

The value of \mathop {\lim }\limits_{x \to 1} {{({x^2} - 1){{\sin }^2}(\pi x)} \over {{x^4} - 2{x^3} + 2x - 1}} is equal to:

Let f, g : R $$\to$$ R be functions defined by $$f(x) = \left\{ {\matrix{ {[x]} & , & {x