Practice problems -2(PG Mathematics)

1. The number of automorphisms from $\mathbb{C} \rightarrow \mathbb{C}$, that fixes $\mathbb{R}$ is ___

2. Let K be the splitting field of $x^2+1=0$ over $\mathbb{Q}$. Then [K:$\mathbb{Q}$] is ____.

3. Let f and g be two analytic functions with same real part. Then
●f=g
●f=g^2
●f-g is constant
●None of these

4. The maximum possible order of any element in $S_7$ is___

5. The number of primitive 8th roots of unity is___

6. The ratio between curvature and torsion is constant iff the curve is ____

7. If the radius of spherical curvature is constant then what can u say about the curve?

8. Let  $A_n$ denotes the area of the polygon in the complex plane with vertices as nth roots of unity. Then $lim\,A_n$ as n tends to $\infty$?

9. Number of generators in group of quaternions is____

10. The order of 5 Sylow subgroup in a group of order 125 is_____

11. The number of 5 slow subgroups in a group of order 15 is______

12. The radius of convergence of $\Sigma \frac{(-1)^n}{n}(z-2)^n$.

13.  Find all the values for $x$ such that $\sum \frac{(x-1)^n}{n}$ converges.

14. Find the number of homomorphisms from $Z_{10}$ to $Z_8$.

15. Classify the differential equation $u_{xx}+u_{yy}+u_{x}^2=0$.

16. Let $a_n$ be defined as $a_{n+1}=(a_n/4)+(3/4)$ and $a_1=2$. Find $lim\,a_n$.

17. Discuss the behaviour of $sin(1/x)$, $x^2sin(1/x)$ and $x sin(1/√x)$ at zero.

18. True or false.
●Cantor set has a measurable subset but not Borel.

●Every borel set is measurable.

●Every G-delta set is Borel.

●Cantor set has a non measurable subset.

19. If f is a Cantor Lebesgue function then the derivative of f is ____.

20. $lim \frac{x^2e^{nx}+cos(x)}{1+e^{nx}}$ as $n \rightarrow \infty$.