JEE Advanced 2003 Question Paper with Solutions

If n th 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.

The value of $'a'$ so that the volume of parallelopiped formed by $\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k$ and $a\widehat i + \widehat k$ becomes minimum is

A cube has a side of length 1.2 ✕ 10 -2 m. Calculate its volume.

If W 1 , W 2 and W 3 represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between W 1 , W 2 and W 3 .

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 $\overrightarrow w ,$ $\overrightarrow w$ and $\overrightarrow u$ respectively and $\overrightarrow x ,\overrightarrow y ,\overrightarrow z ,$ are unit vectors along the bisectors of the angles $\alpha ,\,\,\beta ,\,\,\gamma$ respectively. Prove that $\,\left[ {\overrightarrow x \times \overrightarrow y \,\,\overrightarrow y \times \overrightarrow z \,\,\overrightarrow z \times \overrightarrow x } \right] = {1 \over {16}}{\left[ {\overrightarrow u \,\,\overrightarrow v \,\,\overrightarrow w } \right]^2}\,{\sec ^2}{\alpha \over 2}{\sec ^2}{\beta \over 2}{\sec ^2}{\gamma \over 2}.$

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

Using VSEPR theory deduce the structures of PCl 5 and BrF 5

Calculate the molarity of water if it's density is 1000 kg/m 3

Which has maximum number of atoms?

Mixture X = 0.02 mol of [Co(NH 3 ) 5 SO 4 ]Br and 0.02 mol of [Co(NH 3 ) 5 Br]SO 4 was prepared in 2 litre of solution. 1 litre of mixture X + excess AgNO 3 → Y. 1 litre of mixture X + excess BaCl 2 → Z No. of moles of Y and Z are

Prove that there exists no complex number z such that | z | < 1 3 a n d ∑ r = 1 n a r z r = 1 where | a r | < 2 .

Prove that 2 k ( n 0 ) ( n k ) − 2 k − 1 ( n 2 ) ( n 1 ) ( n − 1 k − 1 ) + 2 k − 2 ( n − 2 k − 2 ) − . . . . . ( − 1 ) k ( n k ) ( n − k 0 ) = ( n k ) ⋅

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 2 ( 1 + 1 − ( 2 I n n ) 2 ) $

If z 1 and z 2 are two complex numbers such that | z 1 | < 1 < | z 2 | then prove that | 1 − z 1 z 2 ― z 1 − z 2 | < 1 .

If x 2 + ( a − b ) x + ( 1 − a − b ) = 0 where a , b ∈ R then find the values of a for which equation has unequal real roots for all values of b .

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.

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 P ( 1 ) = 0 and d p ( x ) d x > P ( x ) for all x ≥ 1 then prove that P ( x ) > 0 for all x > 1 .

If f is an even function then prove that ∫ 0 π / 2 f ( cos ⁡ 2 x ) cos ⁡ x d x = 2 ∫ 0 π / 4 f ( sin ⁡ 2 x ) cos ⁡ x d x .

$A$ 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 $B$ hits the target and $C$ does not.

Two students use the same stock solution of ZnSO 4 and solution of CuSO 4 . The emf of one cell is 0.03 V higher than other. The conc. of CuSO 4 in the cell with higher emf value is 0.5 M. Find out the conc. of CuSO 4 in the other cell (2.203 RT/F = 0.06)

1 gm of charcoal adsorbs 100 ml 0.5 M CH 3 COOH to form a monolayer, and thereby the molarity of CH 3 COOH reduces to 0.49. Calculate the surface area of the charcoal adsorbed by each molecule of acetic acid. Surface area of charcoal = 3.01 × 10 2 m 2 /gm.