If the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the circle $$x^2+y^2-2x+y=5$$ meet at the point $$R\left(\frac{9}{4},2\right)$$, then the area of the triangle $$\mathrm{PQR}$$ is :

If the maximum distance of normal to the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b

Let the tangent and normal at the point $$(3\sqrt{3},1)$$ on the ellipse $$\frac{x^2}{36}+\frac{y^2}{4}=1$$ meet the $$y$$-axis at the points $$A$$ and $$B$$ respectively. Let the circle $$C$$ be drawn taking $$AB$$ as a diameter and the line $$x=2\sqrt{5}$$ intersect $$C$$ at the points $$P$$ and $$Q$$. If the tangents at the points $$P$$ and $$Q$$ on the circle intersect at the point $$(\alpha ,\beta )$$, then $${\alpha }^2-{\beta }^2$$ is equal to :

Let $$\mathrm{P}\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right),\mathrm{Q},\mathrm{R}$$ and $$\mathrm{S}$$ be four points on the ellipse $$9x^2+4y^2=36$$. Let $$\mathrm{PQ}$$ and $$\mathrm{RS}$$ be mutually perpendicular and pass through the origin. If $$\frac{1}{(PQ{)}^2}+\frac{1}{(RS{)}^2}=\frac{p}{q}$$, where $$p$$ and $$q$$ are coprime, then $$p+q$$ is equal to :

If the radius of the largest circle with centre (2,0) inscribed in the ellipse $$x^2+4y^2=36$$ is r, then 12r$${}^2$$ is equal to :

Consider ellipses $${\mathrm{E}}_k:k{x}^2+k^2{y}^2=1,k=1,2,\dots ,20$$. Let $${\mathrm{C}}_k$$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $${\mathrm{E}}_k$$. If $$r_k$$ is the radius of the circle $${\mathrm{C}}_k$$, then the value of \sum_\limits{k=1}^{20} \frac{1}{r_{k}^{2}} is :

Let a circle of radius 4 be concentric to the ellipse $$15x^2+19y^2=285$$. Then the common tangents are inclined to the minor axis of the ellipse at the angle :

Let the ellipse $$E:x^2+9y^2=9$$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is {m \over n}, where m and n are coprime, then $$m-n$$ is equal to :

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $$\alpha$$ and the number of persons who speak only Hindi is $$\beta$$, then the eccentricity of the ellipse $$25\left({\beta }^2{x}^2+{\alpha }^2{y}^2\right)={\alpha }^2{\beta }^2$$ is :

Let $$P(S)$$ denote the power set of $$S=\{1,2,3,\dots .,10\}$$. Define the relations $$R_1$$ and $$R_2$$ on $$P(S)$$ as $${\mathrm{AR}}_1\mathrm{B}$$ if $$\left(\mathrm{A}\cap {\mathrm{B}}^{\mathrm{c}}\right)\cup \left(\mathrm{B}\cap {\mathrm{A}}^{\mathrm{c}}\right)=\varnothing$$ and $${\mathrm{AR}}_2\mathrm{B}$$ if $$\mathrm{A}\cup {\mathrm{B}}^{\mathrm{c}}=\mathrm{B}\cup {\mathrm{A}}^{\mathrm{c}},\forall \mathrm{A},\mathrm{B}\in \mathrm{P}(\mathrm{S})$$. Then :

Among the relations $\mathrm{S}=\left\{(\mathrm{a},\mathrm{b}):\mathrm{a},\mathrm{b}\in \mathbb{R}-\{0\},2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$ and $\mathrm{T}=\left\{(\mathrm{a},\mathrm{b}):\mathrm{a},\mathrm{b}\in \mathbb{R},{\mathrm{a}}^2-{\mathrm{b}}^2\in \mathbb{Z}\right\}$,

Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R=\{(a,b):3a-3b+\sqrt{7}$$ is an irrational number $$\}$$. Then $$R$$ is

Let $$\mathrm{R}$$ be a relation on $$\mathrm{N}\times \mathbb{N}$$ defined by $$(a,b)\mathrm{R}(c,d)$$ if and only if $$ad(b-c)=bc(a-d)$$. Then $$\mathrm{R}$$ is

The minimum number of elements that must be added to the relation $$\mathrm{R}=\{(\mathrm{a},\mathrm{b}),(\mathrm{b},\mathrm{c})\}$$ on the set $$\{a,b,c\}$$ so that it becomes symmetric and transitive is :

Let R be a relation defined on $$\mathbb{N}$$ as $$a\mathrm{R}b$$ if $$2a+3b$$ is a multiple of $$5,a,b\in \mathbb{N}$$. Then R is

The relation $$\mathrm{R}=\{(\mathrm{a},\mathrm{b}):\gcd (\mathrm{a},\mathrm{b})=1,2a\ne \mathrm{b},\mathrm{a},\mathrm{b}\in \mathbb{Z}\}$$ is :

Let $$\mathrm{A}=\{1,3,4,6,9\}$$ and $$\mathrm{B}=\{2,4,5,8,10\}$$. Let $$\mathrm{R}$$ be a relation defined on $$\mathrm{A}\times \mathrm{B}$$ such that $$\mathrm{R}=\{\left(\left(a_1,b_1\right),\left(a_2,b_2\right)\right):a_1\le b_2$$ and $$b_1\le a_2\}$$. Then the number of elements in the set R is :

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

Let $$\mathrm{A}=\{2,3,4\}$$ and $$\mathrm{B}=\{8,9,12\}$$. Then the number of elements in the relation $$\mathrm{R}=\{\left(\left(a_1,{\mathrm{b}}_1\right),\left(a_2,{\mathrm{b}}_2\right)\right)\in (A\times B,A\times B):a_1$$ divides $${\mathrm{b}}_2$$ and $${\mathrm{a}}_2$$ divides $${\mathrm{b}}_1\}$$ is :

Let $$\mathrm{A}=\{1,2,3,4,5,6,7\}$$. Then the relation $$\mathrm{R}=\{(x,y)\in \mathrm{A}\times \mathrm{A}:x+y=7\}$$ is :