JEE Advanced 2009 Question Paper with Solutions

A metal rod AB of length 10x has its one end A in ice at 0 $^\circ$ C and the other end B in water at 100 $^\circ$ C. If a point P on the rod is maintained at 400 $^\circ$ C, then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is 540 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of $\lambda x$ from the ice end A, find the value of $\lambda$ . (Neglect any heat loss to the surrounding.)

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s 2 , find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest.

The mass M shown in the figure below oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. find the separation (in cm) between the successive nodes on the string.

Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure 8 N/m $^2$ . The radii of bubbles A and B are 2 cm and 4 cm, respectively. Surface tension of the soap-water used to make bubbles is 0.04 N/m. Find the ratio $n_B/n_A$ , where $n_A$ and $n_B$ are the number of moles of air in bubbles A and B, respectively. (Neglect the effect of gravity.)

In a constant volume calorimeter, 3.5 g of a gas with molecular weight 28 was burnt in excess oxygen at 298.0 K. The temperature of the calorimeter was found to increase from 298.0 K to 298.45 K due to the combustion process. Given that the heat capacity of the calorimeter is 2.5 kJ K $^{-1}$ , the numerical value for the enthalpy of combustion of the gas in kJ mol $^{-1}$ is ____________.

For $0 < \theta < {\pi \over 2},$ the solution (s) of $ ∑ m = 1 6 cos ⁡ e c ( θ + ( m − 1 ) π 4 ) cos ⁡ e c ( θ + m π 4 ) = 4 2 $ is (are)

If the sum of first $n$ terms of an A.P. is $c{n^2}$ , then the sum of squares of these $n$ terms is

The speed of the particle, that can take discrete values, is proportional to

Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?

Assume that two deuteron nuclei in the core of fusion reactor at temperature T are moving towards each other, each with kinetic energy 1.5 kT, when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature T required for them to reach a separation of 4 $\times$ 10 $^{-15}$ m is in the range

A steady current I goes through a wire loop PQR having shape of a right angle triangle wit6h PQ = 3, PR = 4x and QR = 5x. If the magnitude of the magnetic field at P due to this loop is $k\left( {{{{\mu _0}I} \over {48\pi x}}} \right)$ , find the value of $k$ .

A solid sphere of radius R has a charge Q distributed in its volume with a charge density $\rho = K{r^a}$ , where K and a are constants and r is the distance from its centre. If the electric field at $r = R/2$ is 1/8 times than at $r = R$ , find the value of $a$ .

Let $p(x)$ be a polynomial of degree $4$ having extremum at $x = 1,2$ and $\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \over {{x^2}}}} \right) = 2$ . Then the value of $p (2)$ is

For the function $ f ( x ) = x cos 1 x , x ≥ 1 , $