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

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

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