If $$\alpha ,\beta$$ are the roots of the equation $$x^2-\left(5+3^{\sqrt{{\log }_{3}5}}-5^{\sqrt{{\log }_{5}3}}\right)x+3\left(3^{{\left({\log }_{3}5\right)}^{\frac{1}{3}}}-5^{{\left({\log }_{5}3\right)}^{\frac{2}{3}}}-1\right)=0$$, then the equation, whose roots are $$\alpha +\frac{1}{\beta }$$ and $$\beta +\frac{1}{\alpha }$$, is :

$$\text{ Let }S=\left\{x\in [-6,3]-\{-2,2\}:\frac{|x+3|-1}{|x|-2}\ge 0\right\}\text{ and }$$ $$T=\left\{x\in \mathbb{Z}:x^2-7|x|+9\le 0\right\}\text{. }$$ Then the number of elements in $$\mathrm{S}\cap \mathrm{T}$$ is :

Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^2-\sqrt{2}x+\sqrt{6}=0$$ and $$\frac{1}{{\alpha }^2}+1,\frac{1}{{\beta }^2}+1$$ be the roots of the equation $$x^2+ax+b=0$$. Then the roots of the equation $$x^2-(a+b-2)x+(a+b+2)=0$$ are :

If $$\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\dots +\frac{1}{(180-a)(200-a)}=\frac{1}{256}$$, then the maximum value of $$\mathrm{a}$$ is :

Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola {{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1. Let e' and l' respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If {e^2} = {{11} \over {14}}l and {\left( {e'} \right)^2} = {{11} \over 8}l', then the value of $$77a+44b$$ is equal to :

Let the eccentricity of the hyperbola H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1 be \sqrt {{5 \over 2}} and length of its latus rectum be $$6\sqrt{2}$$. If $$y=2x+c$$ is a tangent to the hyperbola H, then the value of c2 is equal to :

The normal to the hyperbola {{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1 at the point $$\left(8,3\sqrt{3}\right)$$ on it passes through the point :

Let the foci of the ellipse $$\frac{x^2}{16}+\frac{y^2}{7}=1$$ and the hyperbola $$\frac{x^2}{144}-\frac{y^2}{\alpha }=\frac{1}{25}$$ coincide. Then the length of the latus rectum of the hyperbola is :

Let the tangent drawn to the parabola $$y^2=24x$$ at the point $$(\alpha ,\beta )$$ is perpendicular to the line $$2x+2y=5$$. Then the normal to the hyperbola $$\frac{x^2}{{\alpha }^2}-\frac{y^2}{{\beta }^2}=1$$ at the point $$(\alpha +4,\beta +4)$$ does NOT pass through the point :

If the line $$x-1=0$$ is a directrix of the hyperbola $$k{x}^2-y^2=6$$, then the hyperbola passes through the point :

Let the hyperbola $$H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ pass through the point $$(2\sqrt{2},-2\sqrt{2})$$. A parabola is drawn whose focus is same as the focus of $$\mathrm{H}$$ with positive abscissa and the directrix of the parabola passes through the other focus of $$\mathrm{H}$$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $$\mathrm{H}$$, where e is the eccentricity of H, then which of the following points lies on the parabola?

The lengths of the sides of a triangle are 10 + x2, 10 + x2 and 20 $$-$$ 2x2. If for x = k, the area of the triangle is maximum, then 3k2 is equal to :

Let a, b and c be the length of sides of a triangle ABC such that {{a + b} \over 7} = {{b + c} \over 8} = {{c + a} \over 9}. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of {R \over r} is equal to :

If the mirror image of the point (2, 4, 7) in the plane 3x $$-$$ y + 4z = 2 is (a, b, c), then 2a + b + 2c is equal to :

Let {{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}} lie on the plane $$px-qy+z=5$$, for some p, q $$\in$$ R. The shortest distance of the plane from the origin is :

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line $$\overrightarrow{r}=-\hat{k}+\lambda \left(\hat{i}+\hat{j}+2\hat{k}\right),\lambda \in R$$. Then, which of the following points lies on T?

Let the plane ax + by + cz = d pass through (2, 3, $$-$$5) and is perpendicular to the planes 2x + y $$-$$ 5z = 10 and 3x + 5y $$-$$ 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20d is equal to :

If two distinct point Q, R lie on the line of intersection of the planes $$-x+2y-z=0$$ and $$3x-5y+2z=0$$ and $$PQ=PR=\sqrt{18}$$ where the point P is (1, $$-$$2, 3), then the area of the triangle PQR is equal to :

The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes $$5x+8y+13z-29=0$$ and $$8x-7y+z-20=0$$ and the points (2, 1, 3) and (0, 1, 2), respectively, is :

Let the plane $$P:\overrightarrow{r}.\overrightarrow{a}=d$$ contain the line of intersection of two planes $$\overrightarrow{r}.\left(\hat{i}+3\hat{j}-\hat{k}\right)=6$$ and $$\overrightarrow{r}.\left(-6\hat{i}+5\hat{j}-\hat{k}\right)=7$$. If the plane P passes through the point \left( {2,3,{1 \over 2}} \right), then the value of {{|13\overrightarrow a {|^2}} \over {{d^2}}} is equal to :