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A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

Concept: Independent Events

What is space wave propagation?

Concept: Propagation of Electromagnetic Waves

If A = `([cos alpha, sin alpha],[-sinalpha, cos alpha])` , find α satisfying 0 < α < `pi/r`when `A+A^T=sqrt2I_2` where A^{T} is transpose of A.

Concept: Operations on Matrices > Addition of Matrices

Solve the equations x + y = 4 and 2x - y = 5 using the method of reduction.

Concept: Introduction of Matrices

If A = `[(1,2,3), (2,k,2), (5,7,3)]` is a singular matrix then find the value of 'k'.

Concept: Introduction of Matrices

If A = `[(7,1), (2,5)]` and B = `[(1,2), (3,-1)]` then verify that |AB| = |A| |B|.

Concept: Introduction of Matrices

Express the truth of each of the following statements by Venn diagram:

(a) Some hardworking students are obedient.

(b) No circles are polygons.

(c) All teachers are scholars and scholars are teachers.

Concept: Venn Diagrams

If the function *f* : R → R be defined by *f*(*x*) = 2*x* − 3 and *g* : R → R by *g*(*x*) = *x*^{3} + 5, then find the value of (fog)^{−1} (*x*).

Concept: Composition of Functions and Invertible Function

An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive the next 30 years is known to be 2/3 . Find the probability that in the next 30 years at most 3 men will survive.

Concept: Conditional Probability

Suppose that 80% of all families own a television set. If 5 families are interviewed at random, find the probability that

a. three families own a television set.

b. at least two families own a television set.

Concept: Conditional Probability

Obtain an expression for torque acting on a rotating body with constant angular acceleration. Hence state the dimensions and SI unit of torque.

Concept: Torque and Angular Momentum

The distance between two bodies is doubled. How is the magnitude of the gravitational force between them affected?

Concept: Centre of Gravity

Define emissive power and coefficient of emmision of a body.

Concept: Temperature and Heat

Two copper spheres of radii 6 cm and 12 cm respectively are suspended in an evacuated enclosure. Each of them are at a temperature 15°C above the surroundings. The ratio of their rate of loss of heat is.................

- 2:1
- 1:4
- 1:8
- 8:1

Concept: Temperature and Heat

The dimensions of emissive power are

Concept: Temperature and Heat

A pinhole is made in a hollow sphere of radius 5 cm whose inner wall is at temperature 727^{o}C. Find the power radiated per unit area. [Stefan’s constant σ = 5.7 x 10^{-8} J/m^{2} s K4 , emissivity (e) = 0.2]

Concept: Temperature and Heat

Compute the temperature at which the r.m.s. speed of nitrogen molecules is 832 m/s. [Universal gas constant, R = 8320 J/k mole K, molecular weight of nitrogen = 28.]

Concept: Temperature and Heat

The light from the Sun is found to have a maximum intensity near the wavelength of 470 nm. Assuming the surface of the Sun as a black body, the temperature of the Sun is .................

[Wien's constant b = 2 .898 x l0^{- 3}mK]

(a) 5800 K

(b) 6050 K

(c) 6166 K

(d) 6500 K

Concept: Temperature and Heat

A metal ball cools from 64 ^{°}C to 50 ^{°}C in 10 minutes and to 42 ^{°}C in next 10 minutes. The ratio of rates of fall of temperature during the two intervals is _______.

Concept: Temperature and Heat

The susceptibility of magnesium at 300 K is 2.4 x 10^{-5}. At what temperature will the susceptibility increase to 3.6 x 10^{-5}?

Concept: Temperature and Heat