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its digits reversed. For example, the value of reverseDigit(12345) is 54321; the value of

reverseDigit(5600) is 65; the value of reverseDigit(7008) is 8007; and the value of reverseDigit(-

532) is -235.

served took a random sample of 100 customers and found that the mean waiting time

was 7.2 minutes. Assuming that the population standard deviation is known to be

15 minutes, find the 90% confidence interval estimate of the mean waiting time for

all the bank’s customers.

short proof or a counterexample.

i) The relation ∼ defined by R by x ∼ y if x ≥ y is an equivalence relation.

ii) If S1 and S2 are finite non-empty subsets of a vector space V such that [S1] = [S2], then

S1 and S2 have the same number of elements.

iii) For any square matrix A, ρ(A) = det(A)

iv) The determinant of any unitary matrix is 1.

v) If the characteristic polynomials of two matrices are equal, their minimal polynomials are

also equal.

vi) If the determinant of a matrix is 0, the matrix is not diagonalisable.

vii) Any set of mutually orthogonal vectors is linearly independent.

viii) Any two real quadratic forms of the same rank are equivalent over R.

ix) There is no system of linear equations over R that has exactly two solutions.

x) If a square matrix A satisfies the equation A2 = A, then 0 and 1 are the eigenvalues of

A.

f (x) ={2e^(-2x); 0 is greater than equal to x and 0; 0 is smaller than x

find

i) ) P (| X − μ | >1

ii) Use Chebyshev’s inequality to obtain an upper bound on P[| X − μ | >1] and

compare with the result in (i).