MM 1131.9: Mathematics- I - Model Paper 1- S1 BCA Kerala university

Time:    3 Hours                                                                                                                                Max. Marks: 80

SECTION –A

1.      Express hyperbolic secant function as exponential function.

2.      What is the geometrical interpretation of Rolle’s mean value theorem?

3.      Define linear differential equation.

5.      Determine whether y= sin 2x is a solution to the initial value problem y^”+ 4y = 0 ; y(0) = 0 and y’(0)=1

6.      What do you mean by exact differential equation?

7.      What is a complementary function?

8.      Define Laplace transform.

9.      Find an integrating factor of y’ – 3y =6

10.  State the linearity of Laplace transform.

                                                                                             

SECTION –B

11.  State Leibniz theorem.

12.  Give an example of an exact differential equation.

13.  Find the differential equation of the family of curves y=ae^2x + be^-2x where a and b are arbitrary constants.

14.  What is first shifting property of Laplace transform? Give one example.

16.  Solve (D³ -3D +2)y = 0

18.  Find the maximum value of 2 cos x + x for 0<x< pi 

20.  L{cos 2x cos 4x}

22.  L{e^ax}

                                                                       

                                                            

SECTION –C

23.  State Lagrange’s mean value theorem with an example.

24.  Compute the Laplace transform of cosh at and sinh at

25.  Find the derivative of x^{sin x} + (log x)^x

26.  Find the differential equation of all circles of radius r

27.  Solve (D²- 2D +5)y = e^2x sin x

28.  State Rolle’s theorem. Discuss the validity of Rolle’s theorem for

            f(x) =   1- x² , for x < =0

                        cos x for x>0 on [-1,pi/2 ]

                                                            SECTION- D

32.  If cos^-1(y/b) = n log (x/n) prove that

 (1) x²y₂ + xy₁+x²y=0

(2)x²y_n+2 + (2x+1)xy_n+1+2x²y_n=0

33. Solve         (1) (x²y – 2xy²) dx = (x³  - 3x²y)dy

                        (2) dy/dx = [x(2log x +1)]/in y + y cos y)

34. Solve the non homogeneous differential equation (D² -4D+4)y = 8(x² + e^2x+ sin 2x)

                35. Find the inverse Laplace

                                                (1) L^-1 [s / ((s-2)^2 + 9)]

                                                (2) L^-1 { 1/ ((s+1)(s^2 + 1) )]

                                                                                                                                 (2*15=30)