Q. 1.    (a) If three normals drawn from point A to the parabola  cut its axis at the points whose distances from the vertex of the parabola are in A.P., then prove that A lies on the curve given by

  . 

            (b) Show that the area of a triangle inscribed in an ellipse bears a constant ratio to the area of the triangle formed by joining the points on the auxillary circle corresponding to the vertices of the first triangle.

 Q. 2.    (a) A function f:R®  R satisfies the equation f(X+Y) = f (X)f(Y)   ,   ^{" x, y}  and  f(X) ¹ 0  for any X R . If the function is differentiable at     X=0 , such that f'(0)= 2 , then prove that  f' (X) = 2f(X)  for all x in R. Hence determine f (X) .

             (b) If the tangent at a point P₁ (other than (0, 0)) on the curve  meets the curve again at a point P₂, and the tangent at the point at P₂ meets the curve again at P₃, and so on, then prove that the abscissae of the points P₁, P₂, P₃, …, P_n are in G.P.

 Chemistry

 Q. 1.    (a) An LPG cylinder contains 10 kg of liquefied butane at 2 atm and 298 K. The total kinetic energy of gaseous butane in the cylinder was found to be 450 J. When the cylinder was completely used up, (i.e. no further butane came out of the cylinder), the total kinetic energy of gaseous butane was found to be 2100 J. Calculate the density of liquefied butane. Consider atmospheric pressure to be 1 atm =  Pa. 

(b) A certain organic compound (A) decomposes by two parallel first order mechanisms giving B and C, as follows:

The rate constants of the two reaction paths are k_B and k_C. Given that k_B : k_C = 1 : 9, and , calculate the ratio of A to C if an experiment is started with pure A and is allowed to run for one hour.

 Q. 2.    A one litre solution is prepared by mixing 50 mL of (M/10)  and

30 mL (M/5) HCl followed by the addition of water. 100 mL of this solution requires 40 mL of a hydrazine solution for “complete” neutralization. 50 mL of the same hydrazine solution requires 60 mL of a  solution for complete oxidation in acidic medium. If 100 mL of this  solution is made alkaline and excess BaCl₂ is added, what will be the mass of the precipitate obtained? 
(Atomic mass – Ba = 137, Cr = 52, K = 39)

Physics

 Q. 1.    (a) A solid sphere of mass M and radius R is set into motion on a rough horizontal surface with a linear C.M. velocity v_o along the x-axis, and with an angular speed  in the anticlockwise direction, as shown. Show that the sphere would stop rotating in the anticlockwise direction at some instant, and find the linear velocity of the sphere at that instant.

            (b) Consider the same sphere as in the question above. A thread is tied to the sphere at height R above the surface. At what angle q with respect to the horizontal and with what force F must the thread be pulled so that the sphere neither rotates about its center of mass, nor moves forward linearly? The coefficient of friction between the sphere and the horizontal surface is m.

Q. 2.    (a) The resistivity of a semiconductor can be modified by adding different amounts of impurities. A rod of semiconducting material of length L and cross-sectional area A lies along the x-axis between x = 0 and x = L. The material obeys Ohm’s law and its resistivity varies along the length of the rod according to the relation . The emf at end x = 0 is V_o and at end x = L is zero. Find

1.      the resistance of the rod

2.      the current in the rod, and

3.      the electric potential at x.

(b) Consider two thin circular rings A and B of identical radius R. A carries positive charge of linear density l, and B carries the same quantity of negative charge. Both these rings are placed on the x-y plane, at different instances, with their centres lying at (0,0). A small particle of negligible mass and carrying a positive charge q is initially placed at (0, R) in both the cases and imparted a velocity of –  and –  respectively, for A and B, such that it just passes through the ring and continues in its track indefinitely. Find the ratio of  to .

 Q. 3.    An enemy cargo ship moving due east of the point B with a uniform speed of 10 m/s is at point C when it is spotted by a frigate which is at point A due south of B and moving due north with a uniform speed of 15 m/s, as shown in the figure. The distances of the cargo ship and the frigate from B are 6.53 km and 12.60 km respectively. At the instant when the cargo ship is spotted, an air missile is fired by the frigate with a velocity of 400 m/s with respect to itself. The direction of this velocity makes an angle of 30^o with respect to the line joining A and B. Find out the angle above the horizontal at which the missile has to be fired so as to strike the cargo ship and the time of flight of the missile.

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