CSIR NET Mathematical Sciences - June 2010 - Memory Based Questions

I put some questions of CSIR NET Maths June 2010, based on memory. If you know any questions other than these write as a comment here. May many get benefit.

Paper I

1. If A has m elements and B has n elements then number of functions f: A -> B is ……….
2. Upto an isomorphism, no. of groups of order 33 is ..........
3. No. of generators of a cyclic group of order 12 is .........
4. No. of subfields of a finite field of order 3^10 is .........
5. If (xn) is a Cauchy sequence of real numbers then the sequence (cos xn) is ........
6. Variance of 12, 14, 15, 16 and 18 is 5 then var of 2004, 2008, 2010, 2012 and 2015 is .........
7. Value of matrix  A^ 4 if  A^3 - A^2 + A + I = 0 is ......
8. Singularities of 1/(z - z^3) is ..........
9. If X, Y, Z are subsets of U and X n Z = empty then complement of (XnYnZ) is ..........
10. Z[x] is ……… (PID, not PID, Euclidean domain, …) 

Paper II

1. 22 identical biscuits are to be distributed among 5 children, each at least 3 and at the most 7. How many ways it can be done?
2. x =  5 (mod 35) and x = 10 (mod 29). Find the smallest positive integer satisfying both.
3. Solve the integral equation u(x) = sec^2 x + q INTEGRAL 0 to 1 u(t) dt, where q not = 1.
4. Let D be a unit disc at 0. Show that there does not exist an analytic function f : D -> C which is bijective.
5. Let H be a proper subgroup of a finite group G. Show that G is never equal to the union of all conjugates of H.
6. Show that an entire function is constant iff it has a removable singularity at infinity.
7. Show that f(x) = (x^2 + 1) ^ (1/2) is uniformly continuous on (0, oo ).
8. Find the maximum value of f(x, y, z) = 2x + 4y - 4z subject to the condition x^2 + y^2 - z^2 = 1.
9. f (x) = k x^3          1< x < 2, k > 0

                    0                otherwise
          Find k for which f(x) is a p.d.f and find median of X.
10. Verify Green's theorem    y'' + f(x) = 0, y(0) = y(1) = 0.