IIT-JEE 1997

Which one of the following compounds has sp2 hybridization?

The critical temperature of water is higher than that of O2 because the H2O molecule has

Which contains both polar and non-polar bonds?

A compound of vanadium has a magnetic moment of 1.73 BM. Work out the electronic configuration of the vanadium ion in the compound.

Calculate the equilibrium constant for the reaction
Fe2+ + Ce4+ \(\leftrightharpoons\) Fe3+ + Ce3+
(given \(E_{C{e^{4 + }}/C{e^{3 + }}}^o\) = 1.44 V; \(E_{F{e^{3 + }}/F{e^{2 + }}}^o\) = 0.68 V)

How many grams of silver could be plated out on a serving tray by electrolysis of a solution containing silver in +1 oxidation state for a period of 8.0 hours at a current of 8.46 amperes? What is the area of the tray if the thickness of th...

Give reactions for the oxidation of hydrogen peroxide with potassium permanganate in acidic medium.

To a 25ml H2O2 solution, excess of acidified solution of potassium iodide was added. The iodine liberated required 20 ml of 0.3 N sodium thiosulphate solution. Calculate the volume strength of H2O2 solution.

Element A burns in nitrogen to give an ionic compound B. Compound B reacts with water to give C and D. A solution of C becomes 'milky' on bubbling carbon dioxide. Identify A, B, C and D.

Arrange the following sulphates of Alkaline earth metals in order of decreasing thermal stability : BeSO4, MgSO4, CaSO4, SrSO4

Give reasons of the following:
The crystalline salts of alkaline earth metals contain more water of crystallisation than the corresponding alkali metal salts.

For a d-electron the orbital angular momentum is

If \(f\left( x \right) = {x \over {\sin x}}\) and \(g\left( x \right) = {x \over {\tan x}}\), where \(0 < x \le 1\), then in this interval

Let \(a+b=4\), where \(a<2,\) and let \(g(x)\) be a differentiable function.
If \({{dg} \over {dx}} > 0\) for all \(x\), prove that \(\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} }\)
increases as \((b-a)\) increases.

Let \(f(x)= Maximum\) \(\,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},\) where \(0 \le x \le 1.\)
Determine the area of the region bounded by the curves
\(y = f\left( x \right),\) \(x\)-axes, \(x=0\) a...

If \(g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,}\) then \(g\left( {x + \pi } \right)\) equals

Let C be any circle with centre \(\,\left( {0\, , \sqrt {2} } \right)\). Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)

The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle \({x^2} + {y^2} = 1\) pass through the point........................

For each natural number k, let \({C_k}\) denote the circle with radius k centimetres and centre at the origin. On the circle \({C_k}\), a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on $${C_k...

Let \({z_1}\) and \({z_2}\) be roots of the equation \({z^2} + pz + q = 0\,\) , where the coefficients p and q may be complex numbers. Let A and B represent \({z_1}\) and \({z_2}\) in the complex plane. If \(\angle AOB = \alpha \ne 0\,\) a...

Let \({d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.\) If \(\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \right) - F\left( 1 \right)\)
then one of the possible values of \(k\) is ............

Determine the value of \(\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.\)

The value of \(\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx}\) is ...............

Let \(u(x)\) and \(v(x)\) satisfy the differential equation \({{du} \over {dx}} + p\left( x \right)u = f\left( x \right)\) and \({{dv} \over {dx}} + p\left( x \right)v = g\left( x \right),\) where \(p(x) f(x)\) and \(g(x)\) are continuous f...

A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x2 + 2y2 = 6 are at right angles.

The sum of the rational terms in the expansion of \({\left( {\sqrt 2 + {3^{1/5}}} \right)^{10}}\) is ...............

Let \(0 < {A_i} < n\) for \(i = 1,\,2....,\,n.\) Use mathematical induction to prove that
\($\sin {A_1} + \sin {A_2}....... + \sin {A_n} \le n\,\sin \,\,\left( {{{{A_1} + {A_2} + ...... + {A_n}} \over n}} \right)\)$
where \(\ge 1\) is a n...

If \(p\) and \(q\) are chosen randomly from the set \(\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},\) with replacement, determine the probability that the roots of the equation \({x^2} + px + q = 0\) are real.

Let \(S\) be a square of unit area. Consider any quadrilateral which has one vertex on each side of \(S\). If \(a,\,b,\,c\) and \(d\) denote the lengths of the sides of the quadrilateral, prove that $$2 \le {a^2} + {b^2} + {c^2} + {d^2} \le...

The sum of all the real roots of the equation \({\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0\) is ............................

Let \(p\) and \(q\) be roots of the equation \({x^2} - 2x + A = 0\) and let \(r\) and \(s\) be the roots of the equation \({x^2} - 18x + B = 0.\) If \(p < q < r < s\) are in arithmetic progression, then \(A = \,..........\) and $$B = \,.......

The real roots of the equation \(\,{\cos ^7}x + {\sin ^4}x = 1\) in the interval \(\left( { - \pi ,\pi } \right)\) are ...., ...., and ______.

Prove that \(\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},}\) where \(n \ge 3\) is an integer.

Prove that the values of the function \({{\sin x\cos 3x} \over {\sin 3x\cos x}}\) do not lie between \({1 \over 3}\) and 3 for any real \(x.\)

If \(A,B\) and \(C\) are vectors such that \(\left| B \right| = \left| C \right|.\) Prove that
\(\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.\)

Let \(OA=a,\) \(OB=10a+2b\) and \(OC=b\) where \(O,A\) and \(C\) are non-collinear points. Let \(p\) denote the area of the quadrilateral \(OABC,\) and let \(q\) denote the area of the parallelogram with \(OA\) and \(OC\) as adjacent sides....

The equation of state for real gas is given by \(\left( {P + {a \over {{V^2}}}} \right)\left( {V - b} \right) = RT\). The dimention of the constant a is ___________.