NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 94 - 100 of 101

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Question 94

Appeared in Year: 2008

Write in Short

Short Answer▾

Let be a simple closed contour in the complex plane described in the positive sense. Evaluate.

When

a) lies inside and

b) lies outside

Question 95

Appeared in Year: 2013

Describe in Detail

Essay▾

In each of the following find the points of continuity of f.

a)

b)

Explanation

Option (a) is discontinuous for all values of x on real line

Points of continuity

Option (b) is continous when

Points of continuity are

Points of continuity are

Case a)

b)

Case (a) (Explanation) Two case arise

Case I: is Rational

Now, we will check left hand limit and right hand limit at

Here is either rational or irrational

can be either , so R. …

… (61 more words) …

Question 96

Appeared in Year: 2012

Describe in Detail

Essay▾

Let . Fill in the blanks with the correct sign :

a) ________

b) ________

Explanation

(a) Let

Then satisfies the conditions of LMVT in consequently there exists some satisfying such that

… eq. (A)

(from eq. (A) )

Correct sign is “

Option (b) Let

So that

Since is continous in and derivable in .

So by LMVT, there exists some such that

… eq. (B)

Now and

(Using B)

Correct sign is “

… (6 more words) …

Question 97

Appeared in Year: 2007

Question

MCQ▾

Which of the following series are convergent?

Choices

Choice (4)Response

a.

b.

c.

d.

Both b. and c. are correct

Question 98

Appeared in Year: 2007

Describe in Detail

Essay▾

Find the interval of convergence of the series

Explanation

The given series is

Put

So the given series is

(By Cauchy՚s second theorem on limits)

The given series converges for

At , the series is

Which is convergent by Leibnitz test.

At , the series is

Which is not convergent by comparison test

Hence the interval of convergence is

… (2 more words) …

Question 99

Appeared in Year: 2011

Describe in Detail

Essay▾

Evaluate

Explanation

Let … eq. (1)

… eq. (2)

Subtracting eq. (2) from (1)

If [ First term, Common Ratio]

… (2 more words) …

Question 100

Appeared in Year: 2012

Question

True-False▾

Statements

  1. If … eq. (1)

    Where , then the polynomial has a root in the interval .

  2. If is continuous and differentiable in , where and if

    Then there exists such that

  3. The polynomial has only one real root.

Choices

Choice (4)Response

a.

None of the statements is correct.

b.

Both statement Ⅰ & statement Ⅱ are true.

c.

All the statements are correct.

d.

Only statement Ⅱ is true.