Consider the hydrogen atom to be a proton embedded in a cavity of radius a0 (Bohr radius) whose charge is neutralised by

A 3.00 g sample containing Fe3O4, Fe2O3 and an inert impure substance is treated with excess of KI solution in presence

The orbital angular momentum of an electron in 2s orbital is

Calculate the wave number for the shortest wavelength transition in the Balmer series of atomic hydrogen.

Which of the following has the maximum number of unpaired electrons?

When N2, the N-N bond distance _____ and when O2 goes to $$O_2^+$$ the O-O bond distance ___.

The number and type of bonds between two carbon atoms in CaC2 are:

Among the following species, identity the isostructural pairs NF3, $$NO_3^-$$, BF3, H3O+ , HN3

A mixture of ideal gas is cooled upto liquid helium temperature (4.22K) to form an ideal solution

The following compounds have been arranged in order of their increasing thermal stabilities. Identify the correct order

Explain the difference in the nature of bonding in LiF and LiI.

The molar volume of liquid benzene (density = 0.877 g mL-1) increases by a factor of 2750 as it vaporises at 20oC and th

The standard reduction potential for Cu2+|Cu is +0.34 V. Calculate the reduction potential at pH = 14 for the above coup

The ionisation constant of $$NH_4^+$$ in water is 5.6 $$\times$$ 10-10 at 25oC. The rate constant for the reaction of $$

For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$

Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the lines $$x=0,$$ $$y=0,$$ and

Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential

For the three events $$A, B,$$ and $$C,P$$ (exactly one of the events $$A$$ or $$B$$ occurs) $$=P$$ (exactly one of the

In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, $$3$$ in the front an

A nonzero vector $$\overrightarrow a $$ is parallel to the line of intersection of the plane determined by the vectors $

If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$

The position vectors of the vertices $$A, B$$ and $$C$$ of a tetrahedron $$ABCD$$ are $$\widehat i + \widehat j + \wideh

If for nonzero $$x$$, $$af(x)+$$ $$bf\left( {{1 \over x}} \right) = {1 \over x} - 5$$ where $$a \ne b,$$ then $$\int_1^

The angle between a pair of tangents drawn from a point P to the circle $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,

$${\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,$$ is true if and only if

The value of the expression $$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left(

For positive integers $${n_1},\,{n_2}$$ the value of the expression $${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 +

Find all non-zero complex numbers Z satisfying $$\overline Z = i{Z^2}$$.

Find all values of $$\theta $$ in the interval $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ satisfying the equ

Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_

Using mathematical induction prove that for every integer $$n \ge 1,\,\,\left( {{3^{2n}} - 1} \right)$$ is divisible by

For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3

The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are i

A rectangle $$PQRS$$ has its side $$PQ$$ parallel to the line $$y = mx$$ and vertices $$P, Q$$ and $$S$$ on the lines $$

The intercept on the line y = x by the circle $${x^2} + {y^2} - 2x = 0$$ is AB. Equation of the circle with AB as a dia

General value of $$\theta $$ satisfying the equation $${\tan ^2}\theta + \sec \,2\,\theta = 1$$ is _________.

A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angle

Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $$\left( {{{1\, + \,

An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one dire

Points $$A, B$$ and $$C$$ lie on the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$A, B$$ and $$C$$, taken

From a point $$A$$ common tangents are drawn to the circle $${x^2} + {y^2} = {a^2}/2$$ and parabola $${y^2} = 4ax$$. Fin

If $$x{e^{xy}} = y + {\sin ^2}x,$$ then at $$x = 0,{{dy} \over {dx}} = ..............$$

In a triangle $$ABC$$, $$a:b:c=4:5:6$$. The ratio of the radius of the circumcircle to that of the incircle is .........

A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the

Determine the points of maxima and minima of the function $$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x &gt

Let $$f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0

Evaluate $$\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx} $$.

Two guns, situated on the top of a hill of height 10 m, fire one shot each with the same speed $$5\sqrt 3 $$ m s-1 at so