IIT-JEE 2000

Write down the M.O. electron distribution of O2. Specify its bond order and magnetic property.

A hydrogenation reaction is carried out at 500 K. If the same reaction is carried out in the presence of a catalyst at the same rate, the temperature required is 400 K. Calculate the activation energy of the reaction if the catalyst lowers ...

Copper sulphate solution (250 mL) was electrolysed using platinum anode and a copper cathode. A constant current of 2mA was passed for 16 minutes. It was found that after electrolysis the absorbance of the solution was reduced to 50% of its...

The following electrochemical cell has been set up.
Pt(1) | Fe3+, Fe2+ (a = 1) | Ce4+, Ce3+ (a=1) | Pt(2)
Eo (Fe3+, Fe2+) = 0.77 V; Eo (Ce4+, Ce3+) = 1.61 V
If an ammeter is connected between the two platinum electrodes, predict the directi...

To 500 cm3 of water, 3.0 \(\times\) 10-3 kg of acetic acid is added. If 23% of acetic acid is dissociated, what will be the depression in freezing point? Kf and density of water are 1.86 K kg-1 mol-1 and 0.997 g cm-3, respectively

Calculate the energy required to excite one litre of hydrogen gas at 1 atm and 298 K to the first excited state of atomic hydrogen. The energy for the dissociation of H-H bond 436 kJ mol-1.

Suppose \(p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.\) If
\(\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|\) for all \(x \ge 0\), prove that
$$\left| {{a_1} + 2{a_2} + ........ + n...

For \(x>0,\) let \(f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.}\) Find the function
\(f\left( x \right) + f\left( {{1 \over x}} \right)\) and show that $$f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}....

If \({x^2} + {y^2} = 1\) then

Let \(ABC\) be an equilateral triangle inscribed in the circle \({x^2} + {y^2} = {a^2}\). Suppose perpendiculars from \(A, B, C\) to the major axis of the ellipse \(x.{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \((a>b)\) meets ...

A coin probability \(p\) of showing head when tossed. It is tossed \(n\) times. Let \({p_n}\) denote the probability that no two (or more) consecutive heads occur. Prove that \({p_1} = 1,\,\,{p_2} = 1 - {p^2}\) and $${p_n} = \left( {1 - p} ...

For any positive integer \(m\), \(n\) (with \(n \ge m\)), let \(\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}\)
Prove that $$\left( {\matrix{
n \cr
m \cr

} } \right) + \left( {\matrix{
{n - 1} \cr
m...

Let \(a,\,b,\,c\) be possitive real numbers such that \({b^2} - 4ac > 0\) and let \({\alpha _1} = c.\) Prove by induction that $${\alpha _{n + 1}} = {{a\alpha _n^2} \over {\left( {{b^2} - 2a\left( {{\alpha _1} + {\alpha _2} + ... + {\alpha ...

For every possitive integer \(n\), prove that
\(\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.\)
Hence or otherwise, prove that $$\left[ {\sqrt n + \sqrt {\left( {n + 1} \right)} } \right] = \left[ {\sqrt ...

Let \({C_1}\) and \({C_2}\) be respectively, the parabolas \({x^2} = y - 1\) and \({y^2} = x - 1\). Let \(P\) be any point on \({C_1}\) and \(Q\) be any point on \({C_2}\). Let \({P_1}\) and \({Q_1}\) be the reflections of \(P\) and \(Q\), ...

A coin has probability \(p\) of showing head when tossed. It is tossed \(n\) times. Let \({p_n}\) denote the probability that no two (or more) consecutive heads occur. Prove that \({p_1} = 1,{p_2} = 1 - {p^2}\) and $${p_n} = \left( {1 - p} ...

Let \(ABC\) be a triangle with incentre \(I\) and inradius \(r\). Let \(D,E,F\) be the feet of the perpendiculars from \(I\) to the sides \(BC\), \(CA\) and \(AB\) respectively. If \({r_1},{r_2}\) and \({r_3}\) are the radii of circles insc...

If \(\alpha ,\,\beta\) are the roots of \(a{x^2} + bx + c = 0\), \(\,\left( {a \ne 0} \right)\) and \(\alpha + \delta ,\,\,\beta + \delta\) are the roots of \(A{x^2} + Bx + c = 0,\) \(\left( {A \ne 0\,} \right)\,\) for some contant $...

The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

For points \(P\,\,\, = \left( {{x_1},\,{y_1}} \right)\) and \(Q\,\,\, = \left( {{x_2},\,{y_2}} \right)\) of the co-ordinate plane, a new distance \(d\left( {P,\,Q} \right)\) is defined by \(d\left( {P,\,Q} \right)\)$$ = \left( {{x_2},\,{y_2...

Let \(ABC\) and \(PQR\) be any two triangles in the same plane. Assume that the prependiculars from the points \(A, B, C\) to the sides \(QR, RP, PQ\) respectively are concurrent. Using vector methods or otherwise, prove that the prependicu...

In any triangle \(ABC,\) prove that
\($\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.\)$

A thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at $O$ as shown. The moment of inertia of the loop about the axis $XX'$ is: