# B.Sc. (II YEAR) SEMESTER-IV MATHEMATICS MODEL QUESTION PAPER ADIKAVI NANNAYA UNIVERSITY

https://www.computersprofessor.com/2017/04/bsc-ii-year-semester-iv-mathematics.html

**MATHEMATICS MODEL PAPER**

**FOURTH SEMESTER – REAL ANALYSIS**

**COMMON FOR B.A & B.Sc**

**(w.e.f. 2015-16 admitted batch)**

**Time: 3 Hours Maximum Marks: 75**

**SECTION-A**

**Answer any FIVE questions. Each question carries FIVE marks. 5 x 5 = 25 Marks**

1. Prove that every convergent sequence
is bounded.

2. Show that the series

converges absolutely for all values of x.
3. Show that the series

converges
conditionally.
4. Examine the continuity of the
function defined by f(x) = |𝑥| + |𝑥 − 1| at x= 0, 1.

5. Verify Rolle’s
theorem in the interval [a, b] for the function f(x) = (𝑥 − 𝑎)(𝑥 – 𝑏)

^{𝑛}; m, n being +ve integers.
6. Prove that f(x) =

if x≠ 0 and
f(0) = 0 is continuous at x = 0 but not derivable at x = 0.
7. Evaluate

8. Prove that

**SECTION-B**

**Answer the all FIVE questions. Each carries TEN marks. 5 x 10 = 50 Marks**

9

**(a).**Prove that a monotone sequence is convergent if and only if it is bounded.**Or**

**(b).**Prove that the sequence

10

**(a).**State and prove Cauchy’s 𝑛^{𝑡ℎ}Root test**Or**

**(b)**Test for convergence

11

**(a).**If f is continuous on [a, b] then prove that f is bounded and attains its infimum and supremum.**Or**

**(b).**If f is continuous on [a, b] then prove that f is uniformly continuous on [a, b].

12

**(a).**State and prove Lagrange’s mean value theorem.**Or**

**(b)**Find c of Cauchy’s mean value theorem for

13

**(a).**If f:[a, b] ⟶R is continuous on [a, b] then prove that f is integrable on [a, b].**Or**

**(b).**State and prove Fundamental theorem of integral calculus.