The point (2, 1) is translated parallel to the line L : x− y = 4 by $$2\sqrt{3}$$ units. If the newpoint Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is :

If a variable line drawn through the intersection of the lines {x \over 3} + {y \over 4} = 1 and {x \over 4} + {y \over 3} = 1, meets the coordinate axes at A and B, (A $$\ne$$ B), then the locus of the midpoint of AB is :

The angle of elevation of the top of a vertical tower from a point A, due east of it is 45o. The angle of elevation of the top of the same tower from a point B, due south of A is 30o. If the distance between A and B is $$54\sqrt{2}m,$$ then the height of the tower (in metres), is :

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30o. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60o. Then the time taken (in minutes) by him, from B to reach the pillar, is :

If the coefficients of x−2 and x−4 in the expansion of {\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}},\left( {x > 0} \right), are m and n respectively, then {m \over n} is equal to :

For x $$\in$$ R, x $$\ne$$ -1, if (1 + x)2016 + x(1 + x)2015 + x2(1 + x)2014 + . . . . + x2016 = $$\sum _{i=0}^{2016}{a}_i{x}^i,$$ then a17 is equal to :

If the number of terms in the expansion of {\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0, is 28, then the sum of the coefficients of all the terms in this expansion, is :

The integral \int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx is equal to :

The integral \int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}} is equal to : (where C is a constant of integration.)

If \int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k, where k is a constant of integration, then A + B +C equals :

For x $$\in$$ R, x $$\ne$$ 0, Let f0(x) = {1 \over {1 - x}} and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . . Then the value of f100(3) + f1\left( {{2 \over 3}} \right) + f2\left( {{3 \over 2}} \right) is equal to :

If $f(x)+2f\left(\frac{1}{x}\right)=3x,x\ne 0$, and $\mathrm{S}=\{x\in \mathbf{R}:f(x)=f(-x)\}$; then $\mathrm{S}:$

Let $$P$$ be the point on the parabola, $$y^2=8x$$ which is at a minimum distance from the centre $$C$$ of the circle, $$x^2+{\left(y+6\right)}^2=1$$. Then the equation of the circle, passing through $$C$$ and having its centre at $$P$$ is:

P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P passes through Q, then the minimum value of $$t_1^2$$ is :

\mathop {\lim }\limits_{n \to \infty } {\left( {{{\left( {n + 1} \right)\left( {n + 2} \right)...3n} \over {{n^{2n}}}}} \right)^{{1 \over n}}} is equal to:

For x $$\in$$ R, x $$\ne$$ 0, if y(x) is a differentiable function such that x $$\underset{1}{\overset{x}{\int }}y$$ (t) dt = (x + 1) $$\underset{1}{\overset{x}{\int }}ty$$ (t) dt, then y (x) equals : (where C is a constant.)

The value of the integral \int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} , where [x] denotes the greatest integer less than or equal to x, is :

If $$2\underset{0}{\overset{1}{\int }}{\tan }^{-1}xdx=\underset{0}{\overset{1}{\int }}{\cot }^{-1}\left(1-x+x^2\right)dx,$$ then $$\underset{0}{\overset{1}{\int }}{\tan }^{-1}\left(1-x+x^2\right)dx$$ is equalto :

A value of $$\theta$$ for which {{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}} is purely imaginary, is :

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $$2\sqrt{2}$$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by :