### NBHM Problem...

Let 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."

### Practice Problems-1 (PG Mathematics)

Those who are preparing for competitive examinations in mathematics may utilize...
All the best...
1. Find the radius of convergence $\Sigma \frac{(-1)^n}{n}(z-i)^n$

2. Find the limit $x log x$ as $x$ tend to 0.

3. Find the number of generators in $(Z_{60}, \oplus)$.

4. Let $J$ be the square matrix with all entries 1. Then the rank of $J$ is _____

5. Let V be the space of all polynomials of degree n, with complex coefficients. Then the dim V over $\mathbb{R}$ is _____

6. Give an example of a Lebesgue integrable but not Riemann integrable function.

7. If $Y=2X+3$, find the correlation coefficient between $X$ and $Y$.

8. Find the number of nonisomorphic abelian groups of order 60.

9. Let $G$ be a group of order 15. Then G is $A$. nonabelian $B$. Simple $C$. Cyclic $D$. None of these

10. Give an example of a group in which every subgroup is normal
11. In a cyclic group of order 36, no. Of subgroups of order 6 is _____
12. In $(Z_{12}, \oplus)$, order of 5 is ____
13. The number of real…

### Practice problems -3(PG TRB)

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).

### Problems on Field Extensions and Galois theory

Let $F$ be a field.

Define Splitting field - Finite extension - Algebraic extension - Simple extension - Normal extension - Separable extension - Galois group - Solvable group

1. Differentiate Algebraic integer and Algebraic number.

Justify the following Statements:
1. The splitting field of a polynomial over $F$ need not be a finite extension. [TRB]

2. An algebraic extension is always a finite extension. [TRB]

3. $\mathbb{R}$ is an algebraic extension of $\mathbb{Q}$.

4. Any finite extension of a field is a simple extension. [TRB]

5. Any finite extension of a field of characteristic zero is a simple extension. [TRB]

6. Normal extension of a field is always a finite extension. [TRB]

7. Every algebraic integer is algebraic number.
8. Every group with order less than 60 is solvable.
9. $S_n$ is solvable for $n$=
(4/6/60/30)
10. Which of the following group os not solvable? ($S_3/S_5/Z_4/Quaternion group$)