An isotone of $${}_{32}^{76}Ge$$ is

Correct set of four quantum numbers for the valence (outermost) electron of rubidium (Z = 37) is

Which electronic level would allow the hydrogen atom to absorb a photon but not to emit a photon?

The increasing order (lowest first) for the values of e/m (charges/mass) for electron (e), proton (p), neutron (n) and alpha particle ($$\alpha$$) is:

2.68 $$\times$$ 10^-3 moles of a solution containing an ion Aⁿ⁺ require 1.61 $$\times$$ 10^-3 moles of $$MnO_4^-$$ for the oxidation of An⁺ to $$AO_3^-$$ in acid medium. What is the value of n?

When alpha particles are sent through a thin metal foil, most of them go straight through the foil because

Many elements have non-integral atomic masses because:

The electron energy in hydrogen atom is given by E = (-21.7 $$\times$$ 10^-12)/n² ergs. Calculate the energy required to remove an electron completely from the n = 2 orbit. What is the longest wavelength (in cm) of light that can be used to cause this transition?

On hybridization of one s and one p orbitals we get:

C_p - C_v for an ideal gas is _______

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

The total energy of one mole of an ideal monatomic gas at 27^o is _______ calories.

The hydration energy of Mg⁺⁺ is larger than that of:

The IUPAC name of the compound having the formula is :

When 16.8 g of white solid X were heated, 4.4 g of acid gas A, that turned lime water milky was driven off together with 1.8 g of a gas B which condensed to a colourless liquid.

The solid that remained, Y, dissolved in water to give an alkaline solution, which with excess barium chloride solution gave a white precipitate Z. The precipitate efferversced with acid giving off carbon dioxide. Identify A, B and Y and write down the equation for the thermal decomposition of X.

Find the area of the region bounded by the $$x$$-axis and the curves defined by $$$y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot x,{\pi \over 6} \le x \le {{3\pi } \over 2}$$$

Given a function $$f(x)$$ such that
(i) it is integrable over every interval on the real line and
(ii) $$f(t+x)=f(x),$$ for every $$x$$ and a real $$t$$, then show that
the integral $$\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx$$ is independent of a.

If $$M$$ and $$N$$ are any two events, the probability that exactly one of them occurs is

A box contains $$24$$ identical balls of which $$12$$ are white and $$12$$ are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $$4$$th time on the $$7$$th draw is

Three identical dice are rolled. The probability that the same number will appear on each of them is

The points with position vectors $$a+b,$$ $$a-b,$$ and $$a+kb$$ are collinear for all real values of $$k.$$

$$A, B, C$$ and $$D,$$ are four points in a plane with position vectors $$a, b, c$$ and $$d$$ respectively such that $$$\left( {\overrightarrow a - \overrightarrow d } \right)\left( {\overrightarrow b - \overrightarrow c } \right) = \left( {\overrightarrow b - \overrightarrow d } \right)\left( {\overrightarrow c - \overrightarrow a } \right) = 0$$$

The point $$D,$$ then, is the ................ of the triangle $$ABC.$$

In a certain city only two newspapers $$A$$ and $$B$$ are published, it is known that $$25$$% of the city population reads $$A$$ and $$20$$% reads $$B$$ while $$8$$% reads both $$A$$ and $$B$$. It is also known that $$30$$% of those who read $$A$$ but not $$B$$ look into advertisements and $$40$$% of those who read $$B$$ but not $$A$$ look into advertisements while $$50$$% of those who read both $$A$$ and $$B$$ look into advertisements. What is the percentage of the population that reads an advertisement?

If 1, $${{a_1}}$$, $${{a_2}}$$......,$${a_{n - 1}}$$ are the n roots of unity, then show that (1- $${{a_1}}$$) (1- $${{a_2}}$$) (1- $${{a_3}}$$) ....$$(1 - \,a{ - _{n - 1}}) = n$$

Evaluate the following $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $$

$$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)$$ is equal to

There exists a value of $$\theta $$ between 0 and $$2\pi $$ that satisfies the equation $$\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$$

If the complex numbers, $${Z_1},{Z_2}$$ and $${Z_3}$$ represent the vertics of an equilateral triangle such that
$$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$$ then $${Z_1} + {Z_2} + {Z_3} = 0.$$

Find the values of $$x \in \left( { - \pi , + \pi } \right)$$ which satisfy the equation $${g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \right| + \left| {{{\cos }^3}x} \right| + ...)}} = {4^3}$$

If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct.

For real $$x$$, the function $$\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}$$ will assume all real values provided

If $$\,{a^2} + {b^2} + {c^2} = 1$$, then ab + bc + ca lies in the interval

The equation $$x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}$$ has

If the product of the roots of the equation $$\,{x^2} - 3\,k\,x + 2\,{e^{2lnk}} - 1 = 0\,\,\,\,is\,7$$, then the roots are real for k = .................................

If $$p$$ be a natural number then prove that $${p^{n + 1}} + {\left( {p + 1} \right)^{2n - 1}}$$ is divisible by $${p^2} + p + 1$$ for every positive integer $$n$$.

Given $${s_n} = 1 + q + {q^2} + ...... + {q^2};$$
$${S_n} = 1 + {{q + 1} \over 2} + {\left( {{{q + 1} \over 2}} \right)^2} + ........ + {\left( {{{q + 1} \over 2}} \right)^n}\,\,\,,q \ne 1$$
Prove that $${}^{n + 1}{C_1} + {}^{n + 1}{C_2}{s_1} + {}^{n + 1}{C_3}{s_2} + ..... + {}^{n + 1}{C_n}{s_n} = {2^n}{S_n}$$

The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .......................

If $$n$$ is a natural number such that
$$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}$$ and $${p_1},{p_2},\,\,......,\,{p_k}$$ are distinct primes, then show that $$In$$ $$n \ge k$$ $$in$$ 2

If $$a,\,b$$ and $$c$$ are in A.P., then the straight line $$ax + by + c = 0$$ will always pass through a fixed point whose coordinates are ...............

Two equal sides of an isosceles triangle are given by the equations $$7x - y + 3 = 0$$ and $$x + y - 3 = 0$$ and its thirds side passes through the point $$(1, -10)$$. Determine the equation of the third side.

If $$a > 0,\,b > 0$$ and $$\,c > 0,$$ prove that $$\,c > 0,$$ prove that $$\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9$$

The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle. The radius of this circle is ........................................

The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............

The abscissa of the two points A and B are the roots of the equation $${x^2}\, + \,2ax\, - {b^2} = 0$$ and their ordinates are the roots of the equation $${x^2}\, + \,2px\, - {q^2} = 0$$. Find the equation and the radius of the circle with AB as diameter.

The locus of the mid-point of a chord of the circle $${x^2} + {y^2} = 4$$ which subtends a right angle at the origin is

If $$\alpha $$ be a repeated root of a quadratic equation $$f(x)=0$$ and $$A(x), B(x)$$ and $$C(x)$$ be polynomials of degree $$3$$, $$4$$ and $$5$$ respectively,
then show that $$\left| {\matrix{ {A\left( x \right)} & {B\left( x \right)} & {C\left( x \right)} \cr {A\left( \alpha \right)} & {B\left( \alpha \right)} & {C\left( \alpha \right)} \cr {A'\left( \alpha \right)} & {B'\left( \alpha \right)} & {C'\left( \alpha \right)} \cr } } \right|$$ is
divisible by $$f(x)$$, where prime denotes the derivatives.

With usual notation, if in a triangle $$ABC$$;
$${{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}$$ then prove that $${{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}$$.

For a triangle $$ABC$$ it is given that $$\cos A + \cos B + \cos C = {3 \over 2}$$. Prove that the triangle is equilateral.

The numerical value of $$\tan \left\{ {2{{\tan }^{ - 1}}\left( {{1 \over 5}} \right) - {\pi \over 4}} \right\}$$ is equal to __________

For $$0 < a < x,$$ the minimum value of the function $$lo{g_a}x + {\log _x}a$$ is $$2$$.

Evaluate the following $$\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} $$

Four person K, L, M, N are initially at the four corners of a square of side d. Each person now moves with a uniform speed v in such a way that K always moves directly towards L, L directly towards M, M directly towards N, and N directly towards K. The four person will meet at a time _____________.

A projectile fired from the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.

A simple pendulum with a bob of mass m swings with an angular amplitude of $$40^\circ $$. When its angular displacement is $$20^\circ $$, the tension in the string is greater than mg cos$$20^\circ $$.

A block of mass 1 kg lies on a horizontal surface in a truck. The coefficient of static friction between the block and the surface is 0.6. If the acceleration of the truck is 5 m/s², the frictional force acting on the block is _________ newtons.

A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in the time t is proportional to