NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 1 - 9 of 101
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Question 1
Appeared in Year: 2006
Write in Short
Short Answer▾For what value of p does the following series converge?
Question 2
Appeared in Year: 2005
Describe in Detail
Essay▾Let is a differentiable function such that for all . For what value of will be the function be necessarily one-to-one?
Explanation
Let
Then
if … eq. (A)
If
Then [By LMVT]
[From (A) ]
then the function is not injective i.e. one-to-one
i.e..
so is we want f to be one-to-one
… (1 more words) …
Question 3
Appeared in Year: 2009
Describe in Detail
Essay▾Let and let f be defined by
If C is the straight-line segment joining ; compute
Explanation
Let P be the point in which represents the complex number .
is the line from
On so that
Question 4
Appeared in Year: 2006
Question
MCQ▾Pick out the function which are continous atleast at one point in the real line.
Choices
Choice (4) | Response | |
---|---|---|
a. | ||
b. | ||
c. | ||
d. | Both b. and c. are correct |
Question 5
Appeared in Year: 2008
Write in Short
Short Answer▾Write down an equation of degree four satisfied by all the complex fifth roots of unity.
Question 6
Appeared in Year: 2008
Describe in Detail
Essay▾Evaluate:
Explanation
This is exponential form form
Using the Result
If
Then
[using the result]
Question 7
Appeared in Year: 2014
Describe in Detail
Essay▾Find the sum of the following infinite series.
Explanation
Question 8
Appeared in Year: 2013
Describe in Detail
Essay▾Let . Find the sum of the infinite series.
Explanation
Let
Put
… eq. (A)
Now we know the exponential series
… eq. (1)
Replacing by in (1)
… eq. (2)
Adding eq. (1) & (2) and dividing by 2
From eq. (A)
… (3 more words) …
Question 9
Appeared in Year: 2005
Write in Short
Short Answer▾Let be two twice continuously differentiable functions on satisfying the Cauchy – Riemann equation. Let . Define . Express the complex derivative of i.e.. in terms of the partial derivatives of .