Skip to main content

Practice problems -3(PG Mathematics)

1. Find lim $\sum_{k=m}^{\infty}a_k$ as m tends to
$\infty$ , if the series $\sum_{k=1}^{\infty}a_k$ is convergent.

2. Every k- cell is _____

3. Give an example of a non analytic function for which CR equations are satisfied.

4. Prove that if u and v are harmonic functions need not to imply f=u+iv is analytic.

5. Verify the analyticity of the functions $|z|^2$ and $z \overline{z}$.

6. The Petersen graph is __________ (Eulerian/ Hamiltonian / Both/ Neither)

7. The radius of curvature at any point on the Helix $x=acos \theta$, $y=asin \theta$ and $z=a \theta tan \alpha$ is __________

8. Any countable set in real line has measure _____.

9. The function $f(z)=|z|^2$, is ____ (Analyic everywhere/Analytic nowhere/ Analytic at z=0 /Noneofthese)

10. Any __________ separable space is separable (Subspace/ Open subspace / Closed subspace).
Labels:

Comments

Post a Comment

Popular posts from this blog

Countable and Uncountable

Here we collect the various examples given in books and options asked in CSIR, GATE, TRB, JAM examinations about countable sets.  Find which of the following sets are countable?  1. The set of rational numbers $\mathbb{Q}$  2. The set of natural numbers $\mathbb{N}$ 3. The set of real numbers $\mathbb{R}$ 4. The set of prime numbers $\mathbb{P}$ 5. The set of complex numbers with unit modulus. 6. The set of algebraic numbers. 7. The set of transcendental numbers. 8. The set of all polynomials with integer coefficients. 9. The set of all polynomials with rational coefficients. 10. The set of all polynomials over $\mathbb{R}$ with rational roots. 11. The set of all monic polynomials over $\mathbb{R}$ with rational roots. 12. The product set $\mathbb{N} \times \mathbb{N}$ 13. The product set $\mathbb{N} \times \mathbb{R}$ 14. $\wp(\mathbb{N})$, The set of all subsets of $\mathbb{N}$  15. The set of all finite subsets of $\mathbb{N}$  16. The set of all functions from   $\mathbb{N}$ to $\
Post a Comment
Read more

NBHM Problem...

Let T  be a nilpotent linear operator on the vector space $\mathbb{ R}^5$  (i.e., $ T^k=0$  for some k). Let $d_i$  denote the dimension of the kernel of $ T^i$ . Which of the following can possibly occur as a value of $ (d_1,d_2,d_3)$ ?  1. (1,2,3) 2. (2,3,5) , 3. (2,2,4) , 4. (2,4,5) For the excellent proof using Jordan Canonical form visit  https://nbhmcsirgate.theindianmathematician.com/2020/04/nbhm-2020-part-c-question-26-solution.html#.XqCM3QR0R10.whatsapp Here we present a proof (incomplete in preciousness) on the fact "A linear transformation is completely determined by its behaviour on a basis."
Post a Comment
Read more