Let two straight lines drawn from the origin $$\mathrm{O}$$ intersect the line $$3x+4y=12$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ such that $$△\mathrm{OPQ}$$ is an isosceles triangle and $$\angle \mathrm{POQ}=90^{\circ }$$. If $$l={\mathrm{OP}}^2+{\mathrm{PQ}}^2+{\mathrm{QO}}^2$$, then the greatest integer less than or equal to $$l$$ is :

Let $$\mathrm{A}(-1,1)$$ and $$\mathrm{B}(2,3)$$ be two points and $$\mathrm{P}$$ be a variable point above the line $$\mathrm{AB}$$ such that the area of $$△\mathrm{PAB}$$ is 10. If the locus of $$\mathrm{P}$$ is $$\mathrm{a}x+\mathrm{by}=15$$, then $$5\mathrm{a}+2\mathrm{b}$$ is :

If the locus of the point, whose distances from the point $$(2,1)$$ and $$(1,3)$$ are in the ratio $$5:4$$, is $$a{x}^2+b{y}^2+cxy+dx+ey+170=0$$, then the value of $$a^2+2b+3c+4d+e$$ is equal to :

Let a variable line of slope $$m>0$$ passing through the point $$(4,-9)$$ intersect the coordinate axes at the points $$A$$ and $$B$$. The minimum value of the sum of the distances of $$A$$ and $$B$$ from the origin is

If two vectors $$\vec{A}$$ and $$\vec{B}$$ having equal magnitude $$R$$ are inclined at angle $$\theta$$, then

The angle between vector $$\vec{Q}$$ and the resultant of $$(2\vec{Q}+2\vec{P})$$ and $$(2\vec{Q}-2\vec{P})$$ is :

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of ${\left(\frac{1}{3}{x}^{\frac{1}{3}}+\frac{1}{2x^{\frac{2}{3}}}\right)}^{18}$. Then ${\left(\frac{\mathrm{n}}{\mathrm{m}}\right)}^{\frac{1}{3}}$ is :

${}^{n-1}{C}_r=\left(k^2-8\right){}^n{C}_{r+1}$ if and only if :

If A denotes the sum of all the coefficients in the expansion of ${\left(1-3x+10x^2\right)}^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of ${\left(1+x^2\right)}^n$, then :

Let $$a$$ be the sum of all coefficients in the expansion of $${\left(1-2x+2x^2\right)}^{2023}{\left(3-4x^2+2x^3\right)}^{2024}$$ and b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right). If the equation $$c{x}^2+dx+e=0$$ and $$2b{x}^2+ax+4=0$$ have a common root, where $$c,d,e\in \mathbb{R}$$, then $$\mathrm{d}:\mathrm{c}:$$ e equals

The sum of the coefficient of $$x^{2/3}$$ and $$x^{-2/5}$$ in the binomial expansion of $${\left(x^{2/3}+\frac{1}{2}{x}^{-2/5}\right)}^9$$ is

The coefficient of $$x^{70}$$ in $$x^2(1+x{)}^{98}+x^3(1+x{)}^{97}+x^4(1+x{)}^{96}+\dots +x^{54}(1+x{)}^{46}$$ is $${}^{99}{\mathrm{C}}_{\mathrm{p}}-{}^{46}{\mathrm{C}}_{\mathrm{q}}$$. Then a possible value of $$\mathrm{p}+\mathrm{q}$$ is :

The sum of all rational terms in the expansion of $${\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)}^{15}$$ is equal to :

If the coefficients of $$x^4,x^5$$ and $$x^6$$ in the expansion of $$(1+x{)}^n$$ are in the arithmetic progression, then the maximum value of $$n$$ is:

If the term independent of $$x$$ in the expansion of $${\left(\sqrt{\mathrm{a}}{x}^2+\frac{1}{2x^3}\right)}^{10}$$ is 105 , then $${\mathrm{a}}^2$$ is equal to :

If the constant term in the expansion of $${\left(\frac{\sqrt[5]{3}}{x}+\frac{2x}{\sqrt[3]{5}}\right)}^{12},x\ne 0$$, is $$\alpha \times 2^8\times \sqrt[5]{3}$$, then $$25\alpha$$ is equal to :

For $$x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$$, if $$y(x)=\int \frac{\mathrm{cosec}x+\sin x}{\mathrm{cosec}x\sec x+\tan x{\sin }^{2}x}dx$$, and \lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0 then $$y\left(\frac{\pi }{4}\right)$$ is equal to

If $$\int \frac{{\sin }^{\frac{3}{2}}x+{\cos }^{\frac{3}{2}}x}{\sqrt{{\sin }^{3}x{\cos }^{3}x\sin (x-\theta )}}dx=A\sqrt{\cos \theta \tan x-\sin \theta }+B\sqrt{\cos \theta -\sin \theta \cot x}+C$$, where $$C$$ is the integration constant, then $$AB$$ is equal to

Let $$\int \frac{2-\tan x}{3+\tan x}\mathrm{d}x=\frac{1}{2}\left(\alpha x+{\log }_e|\beta \sin x+\gamma \cos x|\right)+C$$, where $$C$$ is the constant of integration. Then $$\alpha +\frac{\gamma }{\beta }$$ is equal to :