A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $$m$$ and $$n$$, respectively, then $$m+n^2$$ is equal to

Let $\mathrm{P}$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $\mathrm{P}$ and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point $\mathrm{Q}$ such that $\mathrm{P}$ and $\mathrm{Q}$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $P Q$ such that $P R: R Q=4: 3$ as $P$ moves on the ellipse, is :

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\mathrm{a}>\mathrm{b}$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is :

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1,\frac{2}{5}\right)$, is equal to :

Let $$P$$ be a parabola with vertex $$(2,3)$$ and directrix $$2x+y=6$$. Let an ellipse $$E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$$, of eccentricity $$\frac{1}{\sqrt{2}}$$ pass through the focus of the parabola $$P$$. Then, the square of the length of the latus rectum of $$E$$, is

Let $$A(\alpha ,0)$$ and $$B(0,\beta )$$ be the points on the line $$5x+7y=50$$. Let the point $$P$$ divide the line segment $$AB$$ internally in the ratio $$7:3$$. Let $$3x-25=0$$ be a directrix of the ellipse $$E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ and the corresponding focus be $$S$$. If from $$S$$, the perpendicular on the $$x$$-axis passes through $$P$$, then the length of the latus rectum of $$E$$ is equal to,

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Let $$f(x)=x^2+9,g(x)=\frac{x}{x-9}$$ and $$\mathrm{a}=f\circ g(10),\mathrm{b}=g\circ f(3)$$. If $$\mathrm{e}$$ and $$l$$ denote the eccentricity and the length of the latus rectum of the ellipse $$\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{b}}=1$$, then $$8{\mathrm{e}}^2+l^2$$ is equal to.

Let the line $$2x+3y-\mathrm{k}=0,\mathrm{k}>0$$, intersect the $$x$$-axis and $$y$$-axis at the points $$\mathrm{A}$$ and $$\mathrm{B}$$, respectively. If the equation of the circle having the line segment $$AB$$ as a diameter is $$x^2+y^2-3x-2y=0$$ and the length of the latus rectum of the ellipse $$x^2+9y^2=k^2$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$2\mathrm{m}+\mathrm{n}$$ is equal to

Consider the relations $R_1$ and $R_2$ defined as $a{R}_{1}b\iff a^2+b^2=1$ for all $a,b\in \mathbf{R}$ and $(a,b)R_2(c,d)\iff$ $a+d=b+c$ for all $(a,b),(c,d)\in \mathbf{N}\times \mathbf{N}$. Then :

Let $S=\{1,2,3,\dots ,10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A},\mathrm{B}):\mathrm{A}\cap \mathrm{B}\ne \varphi ;\mathrm{A},\mathrm{B}\in \mathrm{M}\}$ is :

Let $$A$$ and $$B$$ be two finite sets with $$m$$ and $$n$$ elements respectively. The total number of subsets of the set $$A$$ is 56 more than the total number of subsets of $$B$$. Then the distance of the point $$P(m,n)$$ from the point $$Q(-2,-3)$$ is :

Let $$R$$ be a relation on $$Z\times Z$$ defined by $$(a,b)R(c,d)$$ if and only if $$ad-bc$$ is divisible by 5. Then $$R$$ is

If R is the smallest equivalence relation on the set $$\{1,2,3,4\}$$ such that $$\{(1,2),(1,3)\}\subset \mathrm{R}$$, then the number of elements in $$\mathrm{R}$$ is __________.

Let a relation $$\mathrm{R}$$ on $$\mathrm{N}\times \mathbb{N}$$ be defined as: $$\left(x_1,y_1\right)\mathrm{R}\left(x_2,y_2\right)$$ if and only if $$x_1\le x_2$$ or $$y_1\le y_2$$. Consider the two statements: (I) $$\mathrm{R}$$ is reflexive but not symmetric. (II) $$\mathrm{R}$$ is transitive Then which one of the following is true?

Let $$A=\{2,3,6,8,9,11\}$$ and $$B=\{1,4,5,10,15\}$$. Let $$R$$ be a relation on $$A\times B$$ defined by $$(a,b)R(c,d)$$ if and only if $$3ad-7bc$$ is an even integer. Then the relation $$R$$ is

Let $$\mathrm{A}=\{1,2,3,4,5\}$$. Let $$\mathrm{R}$$ be a relation on $$\mathrm{A}$$ defined by $$x\mathrm{R}y$$ if and only if $$4x\le 5\mathrm{y}$$. Let $$\mathrm{m}$$ be the number of elements in $$\mathrm{R}$$ and $$\mathrm{n}$$ be the minimum number of elements from $$\mathrm{A}\times \mathrm{A}$$ that are required to be added to R to make it a symmetric relation. Then m + n is equal to :

Let $$A=\{n\in [100,700]\cap \mathrm{N}:n$$ is neither a multiple of 3 nor a multiple of 4$$\}$$. Then the number of elements in $$A$$ is

Let the relations $$R_1$$ and $$R_2$$ on the set $$X=\{1,2,3,\dots ,20\}$$ be given by $$R_1=\{(x,y):2x-3y=2\}$$ and $$R_2=\{(x,y):-5x+4y=0\}$$. If $$M$$ and $$N$$ be the minimum number of elements required to be added in $$R_1$$ and $$R_2$$, respectively, in order to make the relations symmetric, then $$M+N$$ equals

Let $f(x)=\left|2x^2+5\right|x|-3|,x\in \mathbf{R}$. If $\mathrm{m}$ and $\mathrm{n}$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $\mathrm{m}+\mathrm{n}$ is equal to :