First Semester BCA Degree Examination BCA 102 Mathematics (2007 Admission onwards) mg university

Time : 3 hrs Marks: 7

                                                Part               

     (Answer any TEN questions, each question carries 3 marks)

1. Define a)conjugate of a matrix b) hermitian matrix
2. Prove that 1+a 1 1
1 1+b 1 = abc(1+1/a +1/b+1/c)
1 1 1+c
3. What is a singular matrix ? Give an example.
4. Derive the partial differential equation
z = ( x + a )( y + b)
5. Find d 2 y / d x2 when x=a(t-sint) , y=a(1+cost)
6. Differentiate e tan¯ 1 x with respect to cos ¯ ¹x
7. If y=a cos(logx)+b sin (logx), prove that x2 y2+x y1+y =0
8. State the Drichlet’s conditions of the Fourier series
9. Find the Laplaces transform of sin 2 3t
10. State the convolution theorem
11. Let A= 1 2 and f(x) = x2 – 3x+4 Find f(A)
2 2
12. If L{f(t)}= f(s), then prove that L{eatf(t) }=f(s-a)

                                             PART B

                                  ( Answer all questions )

13. Examine whether the following system of equations are consistent, if so, solve
2x – y + 2z =8 , 3x + 2y - 2z = -1 , 5x + 3y - 3z = 3 (9 marks)
OR
5 3 3
14. Find A-1 where A = 2 6 -3 and hence solve the equations
8 -3 -2
5x+3y + 3z = 48 ,2x + 6y -3z = 18 ,8x -3y – 2z = 21 (9 marks)
15. (a) Find the nth derivative log(9x2-4) (4 marks)
(b) If y=sin-1x, prove that (1-x2) y n+2 – (2n+1)xy n+1 – n2yn=0 (5 marks)
OR
16. (a) Find the nth derivative of (10x-21)/(2x-3)(2x+5) (4 marks)
(b) If y=[x+√(1+x2)]m, prove that (1+x2)y n+2+(2n+1)xy n+1+(n2-m2)yn=0
(5 marks)
17. Solve (a) ( x 2 - y 2-z 2)p+2xyz =2xz
(b) p tanx+q tany = tant (9marks)
OR
18. a)Form the partial differentiate equation
z = f (xy/z) (4 marks)
b)Solve ( y + z )p+(z+x)q = x+y (5 marks)
19. Find a fourier series to represent x – x 2 from x = - Π to x = Π (9 marks)
OR
20. Explain f(x) = x sinx, 0 < x < 2Π ,as a fourier series (9 marks)
21. (a) Find the Laplace transform of (1-et )/t (4 marks)
(b) Find the Inverse laplace transform of s/(s+a) 2 (5 marks)
OR
22. Using convolution theorem, find the inverse Laplace Transforms
( a )1/s (s2+4) (b) s2 / (s 2+ 4)2 (9 marks)