Test 2: Real Analysis

Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom

Total Marks: 75

1) The value of limn→∞1n(1+21/2+31/3+…..+n1/n)=… is given by:

a) ∞
b) 0
c) 1
d) limit does not exist

2) If x≥0, the series 12⋅x24+1⋅3⋅52⋅4⋅6⋅x48+1⋅3⋅5⋅7⋅92⋅4⋅6⋅8⋅10⋅x612+… converges for:

a) x∈[0,1]
b) x∈[0,1)
c) x∈[1,∞)
d) x∈(1,∞)

3) Find the values of p for which the series 1(log2)p+1(log3)p+1(log4)p+… converges:

a) p∈[0,1)
b) p∈(1,∞)
c) p∈(0,∞)
d) The series is divergent for all values of p

4) The series 12x+1⋅32⋅4x2+1⋅3⋅52⋅4⋅6x3+…..,x>0 converges for:

a) x∈(0,1]
b) x∈(0,1)
c) x∈[1,∞)
d) x∈(1,∞)

5) The series 1+α⋅β1⋅γx+α(α+1)β(β+1)1⋅2⋅γ⋅(γ+1)x2+α(α+1)(α+2)β(β+1)(β+2)1⋅2⋅3⋅γ⋅(γ+1)⋅(γ+2)x3+…, x≠1 converges for:

a) x<1
b) x>1
c) x∈R
d) The series does not converge for any values of x

6) The series ∑1⋅3⋅5⋅…⋅⋅(2n−I)2⋅4⋅6…⋅2n⋅x2n2n converges for:

a) x2<1
b) x2≤1
c) x2>1
d) The series does not converge for any values of x

7) For Riemann integrability, condition of continuity is

a) necessary
b) sufficient
c) necessary and sufficent
d) None of these

8) Let fn(x)=(sinnx)/n,0≤x≤1. Find m∈N such that |fn(x)−0|<1/10 for n>m for all x∈[0,1].

a) 1
b) 10
c) 11
d) 12

9) The integral ∫π/20sinxlogsinx converges to:

a) log(6/e)
b) log(4/e)
c) log(3/e)
d) log(2/e)

10) The integral ∫π/20sinxxpdx converges for:

a) p<2
b) p<1
c) p<−1
d) p>1

11) If α≥0,β≥0, then show that ∫∞0xβdx1+xαsin2x converges for:

a) α>2(β+1)
b) α<2(β+1)
c) β>2(α+1)
d) β><(α+1)

12) If F(x)=(1/x2)×∫x4{4t2−3F′(t)}dt, then what is F′(4)? a) 32/19
b) 64/3
c) 64/19
d) 16/3

13) The value of limx→0xex2∫x0et2dt is given by:

a) 0 b) 1
c) does not exist
d) -1

14) The maximum value of (1/x2)2x2 is given by:

a) ∞
b) e1/e
c) e2/e
d) e4/e

15) The the dimensions of a right circular cone of maximum volume which can be circumscribed about a sphere of radius a are given by:

a) radius= a√2, height = 4a
b) radius= a/√2, height = 2a
c) radius= a√2, height = 8a
d) radius= a/√2, height = 4√2a