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Practice Problems-1 (PG Mathematics)

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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 roots of $x^2-5|x|+6$ is ____

14. Find all the values for $x$ such that $\sum \frac{x^n}{n}$ converges.

15. Give an example of a finite non commutative division ring.

16. For which n, the unitary group $U_n$ is cyclic?

17. Find $\int_2^5 [x^2] dx$, where [•] is integral part function.

18. Find True or False... 
● Subset of every measurable set is measurable.
● Every Libstcitz function is uniformly continuous
● Every positive measure set has a non measurable subset
● The product of all elements in a group is identity.
● If a matrix is diagonalizable, then the eigen values are distinct.
● Every open set in $\mathbb(R)$ is the union of a finite disjoint collection of open intervals.
● Two random variables are uncorrelated, then they are independent.

19. Find the number of homomorphisms from $A_5$ to $S_4$.

20. Let G be the multiplicative group of 4th roots of unity. Then the number of 2-sylow subgroups of G is ____. (Try to find the same in $V_4$.

Comments

  1. UnknownFebruary 13, 2020 at 5:30 PM

    3) 16
    4) 1
    2) infinity
    1) -1
    15) Q8
    16)2,4,p^alpha,2p^alpha where p^alpha is odd prime
    19) 1 since A5 is simple
    12)12
    8)2
    9) C
    10) S4
    11)3

    ReplyDelete
  2. UnknownFebruary 13, 2020 at 5:31 PM

    18) b ) False eg .1/x

    ReplyDelete
  3. UnknownFebruary 13, 2020 at 5:32 PM

    18) e) False

    ReplyDelete
  4. UnknownFebruary 13, 2020 at 5:42 PM

    10) V4 not S4

    ReplyDelete

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