Second year BSC degree - Kerala university

Second year BSC degree examination march/april 2005

Part 11 – algebra  and calculus

1.      Prove that the least divisor of a composite number is a prime number

2.      Find the number and sum of the divisors of 7128

3.      If n is ay natural number prove that n (n+1)(n+2) is divisible by 6

4.      Show that the remainder when 2⁴⁶ is divided by 47 is 1

5.      Show that the 8^th power of any number is of the form 17m or 17m+1 or 17m-1

6.      Solve 4x⁴-8x³+7x²=2x-2 = 0 given that 1+I is a root

7.      Solve x⁴+2x³-25x²-26x+120=0 given that the product of two of its roots is 8

8.      Find the sum of the fourth powers of the roots of x³-x-1=0

9.      Find the number and position of the real roots of the equation x⁵-5x+1=0

10.  Find the value of integral x+4 dx as limit of sums from 0 to 2

11.  Find the length of the arc of the parabola y²=4ax measured from the vertex to one extremity of the latus rectum

12.  Find the integration, the volume of a right circular cone of base radius r and height h

13.  Prove that square root of ½ and square root of pie are equal.

14.  Compute the inverse of the matrix

15.  Find the rank of the matrix

16.  Test for consistency and solve

x+y+z=3

x+2y+3z=4

x+4y+9z=6

17.  Using Cayley Hamilton theorem show that A³-6A²+11A-61=0 where and hence find A^-1