IIT-JEE 1982

There are ______ \(\pi\) bonds in a nitrogen molecule.

The compound with no dipole moment is

Among the following, the molecule that is linear is

____ hybrid orbitals of nitrogen atom are involved in the formation of ammonium ion.

Helium atom is two times heavier than a hydrogen molecule. At 298 K, the average kinetic energy of a helium atom is

At room temparature, ammonia gas at 1 atm pressure and hydrogen chloride gas at P atm pressure are allowed to effuse through identical pin holes from opposite ends of a glass tube of one metre length and of uniform cross-section. Ammonium c...

The energy released when an electron is added to a neutral gaseous atom is called ______ of the atom.

The element with the highest first ionization potential is

The oxidation number of carbon in CH2O is

Write down the balanced equations for the reaction when:
An alkaline solution of potassium ferricyanide is reacted with hydrogen peroxide.

MgCl2.6H2O on heating give anhydrous MgCl2

How will you prepare bleaching powder from slaked lime.

Hydroxylamine reduces iron (III) according to the equation:
2NH2OH + 4Fe3+ \(\to\) N2O(g) \(\uparrow\) + H2O + 4Fe2+ + 4H+
Iron (II) thus produced is estimated by titration with a standard permanganate solution. The reaction is :
$$MnO_...

When there are two electrons in the same orbital, they have _____ spins.

The mass of a hydrogen atom is ______ kg.

Isotopes of an element differ in the number of ______ in their nuclei.

The ion that is isoelectronic with CO is

Calculate the wavelength in Angstrom of the photon that is emitted when an electron in the Bohr orbit, n = 2 returns to the orbit, n = 1 in the hydrogen atom. The ionization potential of the ground state hydrogen atom is 2.17 \(\times\) 10-...

The outer electronic configuration of the ground state chromium atom is 3d4 4s2

If \(a{x^2} + {b \over x} \ge c\) for all positive \(x\) where \(a>0\) and \(b>0\) show that \(27a{b^2} \ge 4{c^3}\).

If \(f(x)\) and \(g(x)\) are differentiable function for \(0 \le x \le 1\) such that \(f(0)=2\), \(g(0)=0\), \(f(1)=6\); \(g(1)=2\), then show that there exist \(c\) satisfying \(0 < c < 1\) and \(f'(c)=2g'(c)\).

The area bounded by the curves \(y=f(x)\), the \(x\)-axis and the ordinates \(x=1\) and \(x=b\) is \((b-1)\) sin \((3b+4)\). Then \(f(x)\) is

For any real \(t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}\) is a point on the
hyperbola \({x^2} - {y^2} = 1\). Show that the area bounded by this hyperbola and the lines joining its centre to the points c...

If A and B are points in the plane such that PA/PB = k (constant) for all P on a given circle, then the value of k cannot be equal to ..........................................

If \(z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5},\) then

The inequality |z-4| < |z-2| represents the region given by

Show that \(\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.}\)

Find the value of \(\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx}\)

If \(y = f\left( {{{2x - 1} \over {{x^2} + 1}}} \right)\) and \(f'\left( x \right) = \sin {x^2}\), then \({{dy} \over {dx}} = ..........\)

Let \(f\) be a twice differentiable function such that
\(f''\left( x \right) = - f\left( x \right),\) and $$f'\left( x \right) = g\left( x \right),h\left( x \right) = {\left[ {f\left( x \right)} \right]^2} + {\left[ {g\left( x \right)} \r...

The sum of the coefficients of the plynomial \({\left( {1 + x - 3{x^2}} \right)^{2163}}\) is ...............

The larger of \({99^{50}} + {100^{50}}\) and \({101^{50}}\) is ..............

Prove that \({7^{2n}} + \left( {{2^{3n - 3}}} \right)\left( {3n - 1} \right)\) is divisible by 25 for any natural number \(n\).

\(A\) is point on the parabola \({y^2} = 4ax\). The normal at \(A\) cuts the parabola again at point \(B\). If \(AB\) subtends a right angle at the vertex of the parabola. Find the slope of \(AB\).

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the numbers of words which have at least one letter repeated are

In a certain test, \({a_i}\) students gave wrong answers to atleast i questions, where i = 1, 2,..., k. No student gave more than k wrong answers. The total number of wrong answers given is.....................................

The value of the expression \(\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}}\) is equal to

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from amongst the remaining. The number of p...

\(A\) and \(B\) are two candidates seeking admission in \(IIT.\) The probability that \(A\) is selected is \(0.5\) and the probability that both \(A\) and \(B\) are selected is atmost \(0.3\). Is it possible that the probability of \(B\) g...

If \(A\) and \(B\) are two events such that \(P\left( A \right) > 0,\) and \(P\left( B \right) \ne 1,\) then \(P\left( {{{\overline A } \over {\overline B }}} \right)\) is equal to

A vertical pole stands at a point \(Q\) on a horizontal ground. \(A\) and \(B\) are points on the ground, \(d\) meters apart. The pole subtends angles \(\alpha\) and \(\beta\) at \(A\) and \(B\) respectively. \(AB\) subtends an angle $$\g...

The number of real solutions of the equation \({\left| x \right|^2} - 3\left| x \right| + 2 = 0\) is

\(mn\) squares of equal size are arranged to from a rectangle of dimension \(m\) by \(n\), where \(m\) and \(n\) are natural numbers. Two squares will be called ' neighbours ' if they have exactly one common side. A natural number is writte...

Show that the equation \({e^{\sin x}} - {e^{ - \sin x}} - 4 = 0\) has no real solution.

Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built a...

The largest interval for which \({x^{12}} - {x^9} + {x^4} - x + 1 > 0\) is

The coeffcient of \({x^{99}}\) in the polynomial (x -1) (x - 2)...(x - 100) is ..............

If \(2 + i\sqrt 3\) is root of the equation \({x^2} + px + q = 0\), where p and q are real, then (p, q) = (..........,....................).

If p, q, r are any real numbers, then

If \(x,\,y\) and \(z\) are \(pth\), \(qth\) and \(rth\) terms respectively of an A.P. and also of a G.P., then \({x^{y - z}}\,{y^{z - x}}\,{z^{x - y}}\) is equal to :