(Pages 1402
P.T.O.
MAR IVANIOS COLLEGE (AUTONOMOUS)
THIRUVANANTHAPURAM
Reg. No. :.………………… Name :………………….
Second Semester B.Voc. Degree Examination, June 2016
First Degree Programme under CSS
General Course VI: (for Software Development)
AUSD263: Mathematics I
Time: 3 Hours Max. Marks: 80
SECTION A
Answer ALL questions in one or two sentences. Each question carries 1 mark.
1. What is the derivative of x
2. What is the derivative of
3. Write an example for a differential equation of first order and first degree
4. Define Laplace transform
5. Give an example of a prime number and a composite number.
6. Define greatest common divisor of two numbers.
7. Write the conjugate of i6.
8. Define a complex number
9. What is the formula for solving a quadratic equation
10. cos(ix)
(10 × 1 10 Marks)
SECTION B
Answer any EIGHT questions. Each question carries 2 marks.
11. Differentiate (x2 with respect to x.
12. Differentiate eax log (sin bx).
13. Prove that 1 s.
14. Solve d2y dx2 2dy dx 5y 0.
1402
2
15. Find the smallest number with 20 divisors.
16. Find the g.c.d of 2210 and 493.
17. Prove that the number of primes is infinite.
18. Write the conjugate of 6 10i and 3 8i.
19. Define hyperbolic functions.
20. Write sin and cos in Euler's form.
21. Find the Fourier cosine series for the function x in 0 x
22. Prove that sin i sin hx.
× 2 16 Marks)
SECTION C
Answer any SIX questions. Each question carries 4 marks.
23. If y prove th t x2) yn+2 (2n x yn+1 n2 yn 0.
24. Find the differential coefficient of (sin x)cos x (cos x)sin x.
25. Solve d2y dx2 6dy dx 9y 2e-3x sin 2x.
26. Prove that L cos at] (s2 a2) s2+ a2)2
27. Prove that n13 n is divisible by 2730.
28. If n is odd, show that n (n2 is divisible by 24.
29. St te De Moivre's theorem nd prove using Euler's formula.
30. Find all the cube roots of 1 i .
31. Find the Fourier Series for the fun tion in nd dedu e th t
1 1 32 52 …… 8.
× 4 24 Marks)
SECTION D
Answer any TWO questions. Each question carries 15 marks.
32. St te Ferm theorem and prove the following.
i). 12! 25 (mod 13)
ii). 18 0 mod
iii). n5 n 0(mod30)
33. i). Solve x8 x5 x3 1 0.
ii). Solve x9 x5 x4 1 0.
1402
3
34. Find
i). L-1 [3s 7 s2 2s
ii). L-1 [3s 1 (s2
iii). L-1
35. Find the Fourier series expansion for the function, where x in . Deduce that ……
× 15 30 Marks)