f'(0) = g(0).

The three planes do not nave a common point.

The determinant for k ≠ 3

The probability that the system of equations has a solution is 1.

15) A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F respectively. The line PQ is given by the equation + y - 6 = 0 and the point D is (3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ.

The equation of circle C is

16) A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F respectively. The line PQ is given by the equation + y - 6 = 0 and the point D is (3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ.

Points E and F are given by

17) A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F respectively. The line PQ is given by the equation + y - 6 = 0 and the point D is (3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ.

Equations of the sides QR, RP are

If f(-10) = 2, then f''(-10,) =

The area of the region bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b, where -∞< a < b < -2, is

The number of elements in the set A ∩ B ∩ C is

Let z be any point in A ∩ B ∩ C. The |z+1-i|² + |z-5-i|² lies between

Let z be any point in A ∩ B ∩ C and let w be any point satisfying |w-2-i | < 3. Then, |z|-|w|+3 lies between

Resistance of a metal increases with increase in temperature.

An object moving around the Earth under the influence of Earth'\'s gravitational force is in a state of 'free-fall'.

By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.

38) A small spherical monoatomic ideal gas bubble (λ = 5/3) is trapped inside a liquid of density ρ_l(see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T₀, the height of the liquid is H and the atmospheric pressure is P₀ (Neglect surface tension).

As the bubble moves upwards, besides the buoyancy force the following forces are acting on it

39) A small spherical monoatomic ideal gas bubble (λ = 5/3) is trapped inside a liquid of density ρ_l(see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T₀, the height of the liquid is H and the atmospheric pressure is P₀ (Neglect surface tension).

When the gas bubble is at height y from the bottom, its temperature is

40) A small spherical monoatomic ideal gas bubble (λ = 5/3) is trapped inside a liquid of density ρ_l(see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains n moles of gas. The temperature of the gas when the bubble is at the bottom is T₀, the height of the liquid is H and the atmospheric pressure is P₀ (Neglect surface tension).

The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant)

The quantum number of n of the state finally populated in He⁺ ions is

The wavelength of light emitted in the visible region by He⁺ ions after collisions with H atoms is

The ratio of the kinetic energy of the n = 2 electron for the H atom to that of He⁺ ion is

44) A small block of mass M moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from 60° to 30° at point B. The block is initially at rest at A. Assume that collisions between the block and the incline are totally inelastic (g = 10 m/s²)
Figure :

The speed of the block at point B immediately after it strikes the second incline is

45) A small block of mass M moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from 60° to 30° at point B. The block is initially at rest at A. Assume that collisions between the block and the incline are totally inelastic (g = 10 m/s²)
Figure :

The speed of the block at point C, immediately before it leaves the second incline is

46) A small block of mass M moves on a frictionless surface of an inclined plane, as shown in figure. The angle of the incline suddenly changes from 60° to 30° at point B. The block is initially at rest at A. Assume that collisions between the block and the incline are totally inelastic (g = 10 m/s²)
Figure :

If collision between the block and the incline is completely elastic, then the vertical (upward) component of the velocity of the block at point B, immediately after it strikes the second incline is

61) In the following reaction sequence, products I, J and L are formed. K represents a reagent.

The structure of the product I is

62) In the following reaction sequence, products I, J and L are formed. K represents a reagent.

The structures of compounds J and K, respectively, are

63) In the following reaction sequence, products I, J and L are formed. K represents a reagent.

The structure of product L is

Among the following, the correct statement is

Among the following, the correct statement is

White phosphorus on reaction with NaOH gives PH₃ as one of the products. This is a

The freezing point of the solution M is

The vapour pressure of the solution M is

Water is added to the solution M such that the mole fraction of water in the solution becomes 0.9. The boiling point of this solution is