IIT-JEE 2003

Using VSEPR theory deduce the structures of PCl5 and BrF5

Two students use the same stock solution of ZnSO4 and solution of CuSO4. The emf of one cell is 0.03 V higher than other. The conc. of CuSO4 in the cell with higher emf value is 0.5 M. Find out the conc. of CuSO4 in the other cell (2.203 RT...

Calculate the molarity of water if it's density is 1000 kg/m3

Wavelength of high energy transition of H-atoms is 91.2 nm. Calculate the corresponding wavelength of He atoms.

1 gm of charcoal adsorbs 100 ml 0.5 M CH3COOH to form a monolayer, and thereby the molarity of CH3COOH reduces to 0.49. Calculate the surface area of the charcoal adsorbed by each molecule of acetic acid. Surface area of charcoal = 3.01 $$\...

(i) Find the equation of the plane passing through the points \((2, 1, 0), (5, 0, 1)\) and \((4, 1, 1).\)
(ii) If \(P\) is the point \((2, 1, 6)\) then find the point \(Q\) such that \(PQ\) is perpendicular to the plane in (i) and the mid ...

Using the relation \(2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0\) or otherwise,
prove that \(\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]\)

If the function \(f:\left[ {0,4} \right] \to R\) is differentiable then show that
(i)\(\,\,\,\,\,\) For \(a, b\)\(\,\,\)$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left(...

Find a point on the curve \({x^2} + 2{y^2} = 6\) whose distance from
the line \(x+y=7\), is minimum.

If \(P(1)=0\) and \({{dp\left( x \right)} \over {dx}} > P\left( x \right)\) for all \(x \ge 1\) then prove that
\(P(x)>0\) for all \(x>1\).

For the circle \({x^2}\, + \,{y^2} = {r^2}\), find the value of r for which the area enclosed by the tangents drawn from the point P (6, 8) to the circle and the chord of contact is maximum.

Prove that there exists no complex number z such that \(\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1\) where \(\left| {{a_r}} \right| < 2\).

If \({z_1}\) and \({z_2}\) are two complex numbers such that \(\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,\) then prove that \(\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1\).

If \(f\) is an even function then prove that
\(\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.}\)

A right circular cone with radius \(R\) and height \(H\) contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant \(= k > 0\). Find the time after which the come is empty.

Prove that
$${2^k}\left( {\matrix{
n \cr
0 \cr

} } \right)\left( {\matrix{
n \cr
k \cr

} } \right) - {2^{^{k - 1}\left( {\matrix{
n \cr
2 \cr

} } \right)}}\left( {\matrix{
n \cr
1 \cr

} } \ri...

Normals are drawn from the point \(P\) with slopes \({m_1}\), \({m_2}\), \({m_3}\) to the parabola \({y^2} = 4x\). If locus of \(P\) with \({m_1}\) \({m_2}\)\(= \alpha\) is a part of the parabola itself then find \(\alpha\).

\(A\) is targeting to \(B, B\) and \(C\) are targeting to \(A.\) Probability of hitting the target by \(A,B\) and \(C\) are \({2 \over 3},{1 \over 2}\) and \({1 \over 3}\) respectively. If \(A\) is hit then find the probability that \(B\) ...

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is \(p.\) If he fails in one of the exams then the probability of his passing in the next exam is \({p \over 2}\) otherwi...

If \({I_n}\) is the area of \(n\) sided regular polygon inscribed in a circle of unit radius and \({O_n}\) be the area of the polygon circumscribing the given circle, prove that
$$${I_n} = {{{O_n}} \over 2}\left( {1 + \sqrt {1 - {{\left( {...

If \({x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0\) where \(a,\,b\, \in \,R\) then find the values of a for which equation has unequal real roots for all values of \(b\).

If a, b, c are in A.P., \({a^2}\), \({b^2}\), \({c^2}\) are in H.P., then prove that either a = b = c or a, b, \({ - {c \over 2}}\) form a G.P.

If \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,\) are three non-coplanar unit vectors and \(\alpha ,\beta ,\gamma\) are the angles between \(\overrightarrow u\) and \(\overrightarrow v\) and \(\overrightarrow w ,\) $$\ove...

If nth divisions of the main scale coincide with (n+1)th divisions
of vernier scale. Given one main scale division is equal to 'a' units. Find the
least count of the vernier.