The energy of the electron in the second and the third Bohr's orbits of the hydrogen atom is -5.42 $$\times$$ 10^-12 erg and -2.41 $$\times$$ 10^-12 erg respectively. Calculate the wavelength (in Å) of the emitted radiation when the electron drops from the third to second orbit.

If 0.50 mol of BaCl₂ is mixed with 0.20 mol of Na₃PO₄, the maximum amount of Ba₃(PO₄)₂ that can be formed is

1 mole of N₂H₄ loses 10 moles of electrons to form a new compound Y. assuming that all nitrogen appears in the new compound , what is the oxidation state of nitrogen in Y ? (there is no change in the oxidation state of hydrogen)

Balance the following Equations:

(i) Cu₂O + H⁺ + $$NO_3^ - \to $$ Cu²⁺ + NO + H₂O

(ii) K₄[Fe(CN)₆] + H₂SO₄ + H₂O $$\to$$ K₂SO₄ + FeSO₄ + (NH₄)₂SO₄ + CO

(iii) C2H5OH + I₂ + OH^- $$\to$$ CHI₃ + $$HCO_3^-$$ + I^- + 4H₂O

A 1.00 gm sample of H₂O₂ solution containing X percent H₂O₂ by weight requires X ml of a KMnO₄ solution for complete oxidation under acidic conditions. Calculate the normality of the KMnO₄ solution.

Rutherford's experiment on scattering of $$\alpha-particles$$ showed for the first time that the atom has

The correct order of second ionisation potential of carbon, nitrogen, oxygen and fluorine is

The angle between two convalent bonds is maximum in ________. (CH₄, H₂O, CO₂)

If a molecule MX₃ has zero dipole moment, the sigma bonding orbitals used by M (atomic number < 21) are

Equal weights of methane and oxygen are mixed in an empty container at 25^oC. The fraction of the total pressure exerted by oxygen is

The temperature at which a real gas obeys the ideal gas aws over a wide range of pressure is

The ratio of root mean square velocity to average velocity of a gas molecule at a particular temperature is

A solution of sodium metal in liquid ammonia is strongly reducing due to the presence of

Give reasons of the following:
Sodium carbonate is made by Solvay process but the same process is not extended to the manufacture of potassium carbonate.

The vapour pressure of pure benzene is 639.7 mm of mercury and the vapour of a solution of a solute in benzene at the same temperature is 631.9 mm of mercury. Calculate this molality of the solution.

Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$

For a biased die the probabilities for the different faces to turn up are given below :

This die tossed and you are told that either face $$1$$ or face $$2$$ has turned up. Then the probability that it is face $$1$$ is ...............

An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are $$0.4, 0.3, 0.2$$ and $$0.1$$ respectively. What is the probability that the gun hits the plane?

Let $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ be vectors of length $$3, 4, 5$$ respectively. Let $$\overrightarrow A $$ be perpendicular to $$\overrightarrow B + \overrightarrow C ,\overrightarrow B $$ to $$\overrightarrow C + \overrightarrow A $$ to $$\overrightarrow A + \overrightarrow B .$$ Then the length of vector $$\overrightarrow A + \overrightarrow B + \overrightarrow C $$ is ..........

Let $$\overrightarrow A ,\overrightarrow B $$ and $${\overrightarrow C }$$ be unit vectors suppose that $$\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,$$ and thatthe angle between $${\overrightarrow B }$$ and $${\overrightarrow C }$$ is $$\pi /6.$$ Then $$\overrightarrow A = \pm 2\left( {\overrightarrow B \times \overrightarrow C } \right).$$

The scalar $$\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)$$ equals :

Find the area bounded by the curve $${x^2} = 4y$$ and the straight

Suppose that the normals drawn at three different points on the parabola $${y^2} = 4x$$ pass through the point $$(h, k)$$. Show that $$h>2$$.

The general solution of the trigonometric equation sin x+cos x=1 is given by:

For complex number $${z_1} = {x_1} + i{y_1}$$ and $${z_2} = {x_2} + i{y_2},$$ we write $${z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.$$
Then for all complex numbers $$z\,\,with\,\,1 \cap z,$$ we have $${{1 - z} \over {1 + z}} \cap 0.$$

The complex numbers $$z = x + iy$$ which satisfy the equation $$\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1$$ lie on

Let the complex number $${{z_1}}$$, $${{z_2}}$$ and $${{z_3}}$$ be the vertices of an equilateral triangle. Let $${{z_0}}$$ be the circumcentre of the triangle. Then prove that $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2$$.

For every integer n > 1, the inequality $${(n!)^{1/n}} < {{n + 1} \over 2}$$ holds.

Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place the balls so that no box remains emply?

The area enclosed within the curve $$\left| x \right| + \left| y \right| = 1$$ is .................

Let A be the centre of the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$. Suppose that the tangents at the points B (1, 7) and D (4. - 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.

The equation $${{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1$$ represents

Each of the four inequalties given below defines a region in the $$xy$$ plane. One of these four regions does not have the following property. For any two points $$\left( {{x_1},{y_1}} \right)$$ and $$\left( {{x_2},{y_2}} \right)$$ in the region, the point $$\left( {{{{x_1} + {x_2}} \over 2},{{{y_1} + {y_2}} \over 2}} \right)$$ is also in the region. The inequality defining this region is

Suppose $${\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx} $$ is an identity in x, where C₀, C₁ ,....C_n are constants, and $${C_n} \ne 0$$ , then the value of n is _____.

Let $$y = {e^{x\,\sin \,{x^3}}} + {\left( {\tan x} \right)^x}$$. Find $${{dy} \over {dx}}$$

Let the angles $$A, B, C$$ of a triangle $$ABC$$ be in A.P. and let $$b:c = \sqrt 3 :\sqrt 2 $$. Find the angle $$A$$.

Let $$a, b, c$$ be positive real numbers Let
$$\theta = {\tan ^{ - 1}}\sqrt {{{a\left( {a + b + c} \right)} \over {bc}}} + {\tan ^{ - 1}}\sqrt {{{b\left( {a + b + c} \right)} \over {ca}}} $$ $$ + {\,\,\tan ^{ - 1}}\sqrt {{{c\left( {a + b + c} \right)} \over {ab}}} $$

Then $$\tan \theta = $$ ____________

Find the value of : $$\cos \left( {2{{\cos }^{ - 1}}x + {{\sin }^{ - 1}}x} \right)$$ at $$x = {1 \over 5}$$, where
$$0 \le {\cos ^{ - 1}}x \le \pi $$ and $$ - \pi /2 \le {\sin ^{ - 1}}x \le \pi /2$$.

Use the function $$f\left( x \right) = {x^{1/x}},x > 0$$. to determine the bigger of the two numbers $${e^\pi }$$ and $${\pi ^e}$$

Let $$x$$ and $$y$$ be two real variables such that $$x>0$$ and $$xy=1$$. Find the minimum value of $$x+y$$.

For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.

Evaluate $$\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.} $$

The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$

Let $$a, b, c$$ be non-zero real numbers such that
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has

A gas bubble, from an explosion under water, oscillates with a period T proportional to p^ad^bE^c. Where 'P' is the static pressure, 'd' is the density of water and 'E' is the total energy of the explosion. Find the value of a, b and c.

When a person walks on a rough surface, the frictional force exerted by the surface on the person is opposite to the direction of his motion.