The intermolecular interaction that is dependent on the inverse cube of distance between the molecule is:

Which of the vitamins given below is water soluble?

Let f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right) where $$x\in R$$ and $$k\ge 1.$$ Then $$f_4\left(x\right)-f_6\left(x\right)$$ equals :

If $$\alpha ,\beta \ne 0,$$ and $$f\left(n\right)={\alpha }^n+{\beta }^n$$ and $\left| {\matrix{ 3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} \cr {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} & {1 + f\left( 4 \right)} \cr } } \right|$ $$=K{\left(1-\alpha \right)}^2{\left(1-\beta \right)}^2{\left(\alpha -\beta \right)}^2,$$ then $$K$$ is equal to :

If $$A$$ is a $$3\times 3$$ non-singular matrix such that $$A{A}^{\prime }=A^{\prime }A$$ and $$B=A^{-1}{A}^{\prime },$$ then $$B{B}^{\prime }$$ equals:

Let $$A$$ and $$B$$ be two events such that P\left( {\overline {A \cup B} } \right) = {1 \over 6},\,P\left( { {A \cap B} } \right) = {1 \over 4} and P\left( {\overline A } \right) = {1 \over 4}, where $$\overline{A}$$ stands for the complement of the event $$A$$. Then the events $$A$$ and $$B$$ are :

If $$g$$ is the inverse of a function $$f$$ and f'\left( x \right) = {1 \over {1 + {x^5}}}, then $$g^{\prime }\left(x\right)$$ is equal to:

Let the population of rabbits surviving at time $$t$$ be governed by the differential equation {{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200. If $$p(0)=100,$$ then $$p(t)$$ equals:

Let $$a,b,c$$ and $$d$$ be non-zero numbers. If the point of intersection of the lines $$4ax+2ay+c=0$$ and $$5bx+2by+d=0$$ lies in the fourth quadrant and is equidistant from the two axes then :

Let $$PS$$ be the median of the triangle with vertices $$P(2,2)$$, $$Q(6,-1)$$ and $$R(7,3)$$. The equation of the line passing through $$(1,-1)$$ band parallel to PS is :

A bird is sitting on the top of a vertical pole $$20$$ m high and its elevation from a point $$O$$ on the ground is $$45^{\circ }$$. It files off horizontally straight away from the point $$O$$. After one second, the elevation of the bird from $$O$$ is reduced to $$30^{\circ }$$. Then the speed (in m/s) of the bird is :

If the coefficints of $$x^3$$ and $$x^4$$ in the expansion of $$\left(1+ax+b{x}^2\right){\left(1-2x\right)}^{18}$$ in powers of $$x$$ are both zero, then $$\left(a,b\right)$$ is equal to:

The integral \int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx} is equal to

The slope of the line touching both the parabolas $$y^2=4x$$ and $$x^2=-32y$$ is

The integral \int\limits_0^\pi {\sqrt {1 + 4{{\sin }^2}{x \over 2} - 4\sin {x \over 2}{\mkern 1mu} } } dx equals:

If z is a complex number such that $$\left|z\right|\ge 2$$, then the minimum value of \,\,\left| {z + {1 \over 2}} \right| :

Let $$C$$ be the circle with centre at $$(1,1)$$ and radius $$=$$ $$1$$. If $$T$$ is the circle centred at $$(0,y)$$, passing through origin and touching the circle $$C$$ externally, then the radius of $$T$$ is equal to :

The locus of the foot of perpendicular drawn from the centre of the ellipse $$x^2+3y^2=6$$ on any tangent to it is :

\mathop {\lim }\limits_{x \to 0} {{\sin \left( {\pi {{\cos }^2}x} \right)} \over {{x^2}}} is equal to :

Let $$\alpha$$ and $$\beta$$ be the roots of equation $$p{x}^2+qx+r=0,$$ $$p\ne 0.$$ If $$p,q,r$$ in A.P. and {1 \over \alpha } + {1 \over \beta } = 4, then the value of $$\left|\alpha -\beta \right|$$ is :