IIT-JEE 1993

Pick out the isoelectronic structures from the following
I. \(CH_3^+\) II. \(H_3O^+\)
III. \(NH_3\)
IV. \(CH_3^-\)

The dipole moment of KCl is 3.336 \(\times\) 10-29 Coulomb meters which indicates that it is a highly polar molecule. The interatomic distance between K+ and Cl- in this molecule is 2.6 \(\times\) 10-10. Calculate the dipole moment of KCl m...

The dipole moment of CH3F is greater than of CH3Cl

A first order reaction A \(\to\) B, requires activation energy of 70 kJ mol-1. When 20% solution of A was kept at 25oC for 20 minutes, 25% decomposition tooks place. What will be the percentage decomposition in the same time in a 30% soluti...

The standard reduction potential for the half-cell
\(NO_3^-\) + 2H+ (aq) + e \(\to\) NO2 (g) + H2O is 0.78 V
(i) Calculate the reduction potential in 8 M H+
(ii) What will be the reduction potential of the half-cell in a neutral solution? A...

Chromium metal can be plated out from an acidic solution containing CrO3 according to the following equation
CrO3 (aq) + 6H+ (aq) + 6e- \(\to\) Cr(s) + 3H2O
Calculate (i) how many grams of chromium will be plated out by 24,000 coulombs and...

A gas bulb of 1 litre capacity contains 2.0 \(\times\) 1021 molecules of nitrogen exerting a pressure of 7.57 \(\times\) 103 Nm-2. Calculate the root mean square (r.m.s) speed and the temparature of the gas molecules. If the ratio of the m...

In the van der Waal's equation (P + \({{{n^2}a} \over {{V^2}}}\))(V - nb) = nRT the constant 'a' reflects the actual volume of the gas molecules.

Give briefly the isolation of magnesium from sea water by the Dow process. Give equations for the steps involved.

The basic nature of the hydroxides of group 13 (Gr. III B) decreases progressively down the group.

The decreasing order of electron affinity of F, Cl, Br is F > Cl > Br

Ca2+ has a small ionic radius than K+ because it has ______.

What weight of the non-volatile solute, urea(NH2 - CO - NH2) needs to be dissolved in 100g of water, in order to decrease the vapour pressure of water by 25%? What will be the molality of the solution?

Upon mixing 45.0 ml. of 0.25 M lead nitrate solution with 25.0 ml. of 0.10M chromic sulphate solution, precipitation of lead sulphate takes place. How many moles of lead sulphate are formed? Also, calculate the molar concentrations of the s...

In a given electric field, \(\beta-particles\) are deflected more than \(\alpha-particles\) in spite of \(\alpha-particles\) having larger charge.

Wave functions of electrons in atoms and molecules are called ______.

What transition in the hydrogen spectrum would have the same wavelength as the Balmer transition n = 4 to n = 2 of He+ spectrum?

Estimate the difference in energy between 1st and 2nd Bohr orbit for a hydrogen atom. At what minimum atomic number, a transition from n = 2 to n = 1 energy level would result in the emission of X-rays with $$\lambda = 3.0 \times {10^{ - 8...

The 2px, 2py and 2pz orbitals of atom have identical shapes but differ in their _____.

The light radiations with discrete quantities of energy are called ______.

Let \(f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr {2x - 3} & {1 \le x \le 3} \cr } } \right.\)
Find all possible re...

Find the equation of the normal to the curve
\(y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)\) at \(x=0\)

If \(f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right.\) then:

Find the coordinates of the point at which the circles \({x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36\) touch each other. Also find equations common tangests touching the circles in the distinct ...

The locus of the centre of a circle, which touches externally the circle \({x^2} + {y^2} - 6x - 6y + 14 = 0\) and also touches the y-axis, is given by the equation:

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle \({x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0\) cuts the members of the family are concurrent at a point. Find the coo...

The equation of the locus of the mid-points of the circle \(4{x^2} + 4{y^2} - 12x + 4y + 1 = 0\) that subtend an angle of \(2\pi /3\) at its centre is.................................

\(ABCD\) is a rhombus. Its diagonals \(AC\) and \(BD\) intersect at the point \(M\) and satisfy \(BD\) = 2\(AC\). If the points \(D\) and \(M\) represent the complex numbers \(1 + i\) and \(2 - i\) respectively, then A represents the comp[l...

Evaluate \(\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.\)

The value of \(\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}}\) is

The value of \(\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi }\) is ..............

Prove that \(\sum\limits_{r = 1}^k {{{\left( { - 3} \right)}^{r - 1}}\,\,{}^{3n}{C_{2r - 1}} = 0,}\) where \(k = \left( {3n} \right)/2\) and \(n\) is an even positive integer.

Using mathematical induction, prove that
$${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right...

\(E\) and \(F\) are two independent events. The probability that both \(E\) and \(F\) happen is \(1/12\) and the probability that neither \(E\) nor \(F\) happens is \(1/2.\) Then,

Numbers are selected at random, one at a time, from the two- digit numbers \(00, 01, 02 ......, 99\) with replacement. An event \(E\) occurs if only if the product of the two digits of a selected number is \(18\). If four numbers are select...

An unbiased die with faces marked \(1,2,3,4,5\) and \(6\) is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than \(2\) and the maximum face value is not greater than \(5,\) is th...

If in a triangle \(ABC\), \({{2\cos A} \over a} + {{\cos B} \over b} + {{2\cos C} \over c} = {a \over {bc}} + {b \over {ca}},\) then the value of the angle \(A\) is .................... degrees.

An observer at \(O\) notices that the angle of elevation of the top of a tower is \({30^ \circ }\). The line joining \(O\) to the base of the tower makes an angle of \({\tan ^{ - 1}}\left( {1/\sqrt 2 } \right)\) with the North and is inclin...

For \(0 < \phi < \pi /2,\) if
\(x =\)\(\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty\) then

A line through \(A (-5, -4)\) meets the line \(x + 3y + 2 = 0,\) \(2x + y + 4 = 0\) and \(x - y - 5 = 0\) at the points \(B, C\) and \(D\) respectively. If $${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)...

The vertices of a triangle are \(A\left( { - 1, - 7} \right)B\left( {5,\,1} \right)\) and \(C\left( {1,\,4} \right).\) The equation of the bisector of the angle \(\angle ABC\) is ............... .

Tagent at a point \({P_1}\) {other than \((0, 0)\)} on the curve \(y = {x^3}\) meets the curve again at \({P_2}\). The tangent at \({P_2}\) meets the curve at \({P_3}\), and so on. Show that the abscissae of $${P_1},\,{P_2},{P_3}......{P_n}...

Number of solutions of the equation \(\tan x + \sec x = 2\cos x\,\) lying in the interval \(\left[ {0,2\pi } \right]\) is:

If \(A > 0,B > 0\,\) and \(A + B = \pi /3,\) then the maximum value of tan A tan B is _______.

Determine the smallest positive value of number \(x\) (in degrees) for which
\($\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).\)$

If \(K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),\) then the numerical value of K is ______.

In a triangle \(ABC, D\) and \(E\) are points on \(BC\) and \(AC\) respectively, such that \(BD=2DC\) and \(AE=3EC.\) Let \(P\) be the point of intersection of \(AD\) and \(BE.\) Find \(BP/PE\) using vector methods.

Let \(\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k\) and \(\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k\) be three vectors. A vector in the plane of \({\overrightarrow b }\) and $${\ove...

Let \(a, b, c\) be distinct non-negative numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) lie in a plane, then \(c\) is

A uniform rod of length L and density \(\rho\) is being pulled along a smooth floor with a horizontal acceleration \(\alpha\) (see Fig.). The magnitude of the stress at the transverse cross-secion through the mid-point of the rod is _____...