The expression {{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}} can be written as:

The number of values of $$k$$, for which the system of equations : $\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \cr {kx + \left( {k + 3} \right)y = 3k - 1} \cr }$ has no solution, is

If P = \left[ {\matrix{ 1 & \alpha & 3 \cr 1 & 3 & 3 \cr 2 & 4 & 4 \cr } } \right] is the adjoint of a $$3\times 3$$ matrix $$A$$ and $$\left|A\right|=4,$$ then $$\alpha$$ is equal to :

A multiple choice examination has $$5$$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $$4$$ or more correct answers just by guessing is :

If $$y=\sec \left({\tan }^{-1}x\right),$$ then {{{dy} \over {dx}}} at $$x=1$$ is equal to :

At present, a firm is manufacturing $$2000$$ items. It is estimated that the rate of change of production P w.r.t. additional number of workers $$x$$ is given by {{dp} \over {dx}} = 100 - 12\sqrt x . If the firm employs $$25$$ more workers, then the new level of production of items is

The $$x$$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $$(0,1)(1,1)$$ and $$(1,0)$$ is :

A ray of light along $$x+\sqrt{3}y=\sqrt{3}$$ gets reflected upon reaching $$X$$-axis, the equation of the reflected ray is :

$$ABCD$$ is a trapezium such that $$AB$$ and $$CD$$ are parallel and $$BC\perp CD.$$ If $$\angle ADB=\theta ,BC=p$$ and $$CD=q,$$ then AB is equal to:

The term independent of $$x$$ in expansion of {\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}} is

If $$\int f\left(x\right)dx=\psi \left(x\right),$$ then $$\int x^{5}f\left(x^3\right)dx$$ is equal to

Given : A circle, $$2x^2+2y^2=5$$ and a parabola, $$y^2=4\sqrt{5}x$$. Statement-1 : An equation of a common tangent to these curves is $$y=x+\sqrt{5}$$. Statement-2 : If the line, y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right) is their common tangent, then $$m$$ satiesfies $$m^4-3m^2+2=0$$.

Statement-1 : The value of the integral \int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}} is equal to $$\pi /6$$ Statement-2 : $$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)dx.$$

If z is a complex number of unit modulus and argument $$\theta$$, then arg \left( {{{1 + z} \over {1 + \overline z }}} \right) equals :

The circle passing through $$(1,-2)$$ and touching the axis of $$x$$ at $$(3,0)$$ also passes through the point :

The equation of the circle passing through the foci of the ellipse {{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1, and having centre at $$(0,3)$$ is :

\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}} is equal to

If $$x,y,z$$ are in A.P. and $${\tan }^{-1}x,{\tan }^{-1}y$$ and $${\tan }^{-1}z$$ are also in A.P., then :

Let $$T_n$$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $$T_{n+1}-T_n$$ = 10, then the value of n is :

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A $$\times$$ B having 3 or more elements is :