B.A. / B.Sc. Maths Semester-4th Previous Year Old Question Paper Jammu University Under Graduation

B.A. / B.Sc. IV - Semester Examination
BS-IVS/5
26966
MATHEMATICS
Course No. : 401

Time Allowed: 3 hours                                                                        Maximum Marks: 80

Note: Do any five questions, selecting one from each unit.

UNIT - I

1. a) If A is a non-empty subset of R that is bounded below, show that A has a greatest lower bound in R. Find g.l.b. of {π+1, π+1/2, π+1/3,.....}

b) When do we say that sequence {x_n} of real numbers has the limit L? Show that the limit of a convergent sequence of real numbers is unique.

2. a) i) If a sequence {x_n} of real numbers is convergent, show that {x_n} is bounded.

ii) Show that a non- decreasing sequence which is bounded above is convergent.

b) Define Cauchy sequence of real numbers and show that a Cauchy sequence of real numbers is convergent.

b) State D’Alembert’s ratio test and Raabe’s test and show by giving an example that Raabe’s test is stronger than D’Alembert’s ratio test.

UNIT – III

5. a) Prove that ƒ(x) = xⁿ, x ∈ R (n is a positive integer) is continuous at each x and deduce that easy polynomial function is continuous at each point of R. Give an example to show that not easy continuous function is differentiable.

b) State Rolle’s theorem and find a suitable point c of Rolle’s theorem for the function,
ƒ(x) = (x-a) (b-x), x ∈ [a,b]. Show that between two consecutive zeroes of ƒ'(x) there lies a zero of ƒ(x).

6. a) State Mean value theorem & use it to show that if ƒ'(x) = 0 ∀ x ∈ (a, b), then ƒ(x) is constant on [a, b]. What happens, if you replace ƒ'(x) = 0 ∀ x ∈ (a, b) by ƒ'(x) = 0 ∀ x ∈ (a,b)? 

b) State Taylor’s theorem with Lagrange’s form of the remainder and obtain Maclaurin's infinite series for ƒ(x) = log (1 + x).

UNIT – IV

7. a) Prove that if θ is real and n is rational, then (cos θ + isin θ)ⁿ = cos nθ + isin nθ.

b) Determine all roots of x⁹ – 1 = 0 by De - Moivre’s theorem and find out which of these roots satisfy x³ – 1 = 0.