UCMA Algebra & Trigonometry B.Sc Question Paper : ideunom.ac.in

University : University of Madras
Degree : B.Sc
Department :Mathematics
Subject :UCMA Algebra & Trigonometry
Document type : Question Paper
Website : ideunom.ac.in

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IDEUNOM UCMA Algebra & Trigonometry Model Paper

 U/ID 32351/UCMA
Time : Three hours
Maximum : 100 marks

Related : University of Madras PAH Dynamics B.Sc Question Paper : www.pdfquestion.in/6852.html

PART A — (10 ´ 3 = 30 marks)

OCTOBER 2011

Answer any TEN questions.
Each question carries 3 marks.
1. If a ,a , ,a n 1 2 L are the roots of the equation find the value of ( ) 1 1 +a ( ) ( ) n 1 +a 1 +a 2 L .
2 11 38 39 0 x3 – x2 + x – = is 2 – 3i solve the equation.
3. Find the equation whose roots are the roots of the equation x3 + x + = multiplied by 3.
5. Find approximately the value of q in radians
6. Prove that ? sinh- = log? + +1 1 2 x x x .
7. Show that ? is an unitary matrix.
8. Find the rank of the matrix
9. Define order of an element in a group. Give the order of the identity element in any group
10. Define prime number and composite number. Give examples.
11. State Euler function.
12. Show ( ) p Log i i n .

PART B — (5 ´ 6 = 30 marks)
Answer any FIVE questions.
Each question carries 6 marks.
13. Solve the equation 4 24 23 18 0 x3 – x2 + x + = given that the roots are in Arithmetic progression
14. Show that e 1+ 2+3 +4 +5 +6 +L= .3e/2
15. Prove that 2 cos cos 6 6 cos 4 15 cos2 10 5 6q = q +
16. Show that the equations are consistent 2x-3y+7 = z
17. Prove that the subgroup of a cyclic group is cyclic.
18. Show that if x and y are both prime to the prime number n then n-1 – n-1 x y is divisible by n . Deduce that 12 12 x – y is divible by 1365 .
19. If i a ib a +ib = + prove that ( n ) b a2 + b2 = e- 4 +1 p .

PART C — (4 ´ 10 = 40 marks)
Answer any FOUR questions.
Each question carries 10 marks.
20. Solve 3 27 27 3 0 x6 + x5 – x4 + x2 – x – = 0.
21. Separate -1 (x + iy) tan into real and imaginary
22. Verify Cayley-Hamilton theorem for the matrix
23. State and prove Lagrange’s theorem.
24. Find the positive root of 3 0 x3 – x – = 0 correct to two places of decimals by Horner’s method.
25. Show that ( q ) q q q q sin sin sin cos + + +L= e .

OCTOBER 2012

U/ID 32351/UCMA
Time : Three hours
Maximum : 100 marks
PART A — (10 × 3 = 30 marks)
Answer any TEN questions.
Each question carries 3 marks.
1. Form the biquadratic equation two of whose roots are i and 3 .
2. Using descarte’s rule of signs find the nature of roots of the equation 3 1 0 x4 + x – = 0.
4. Write the expansion of tan nq .
5. Show that Tanq?
6. Show that the matrix ? ?
7. Using Cayley-Hamilton theorem find A-1 for the matrix is skew- Hermitian.?
8. If H and K are sub groups of a group G , show that H È K need not be a subgroup of G.
9. Test whether the number 1729 is prime or composite.
10. Show that (2 3) (2 3 )(mod5) 5 5 5 + º + .
11. Find the value of log (1 + i). log (1 + i)
12. Find the sum of the series

PART B : (5 × 6 = 30 marks)
Answer any FIVE questions.
Each question carries 6 marks :
13. Find the sum to infinity the series
14. If a , b , g are the roots of the equation 0 x3 + qx + r = , form the equation whose roots are a2+b2/ab
15. Show that cos7q 64cos q 112cos q 56cos q 7cosq = 7 – 5 + 3 – cos7q 64cos q 112cos q 56cos q 7cosq
16. Show that 1/2 is unitary
17. Show that a º b(modH) is a equivalence relation where H is a subgroup of G.
18. Show that 4 1 – + p = n i i e where n is an integer.

19. Sum the series.
sinh x + sinh(x + y)+ sinh(x + 2y)+ …. + to n term.
sinh x + sinh(x + y)+ sinh(x + 2y)+ … +

PART C : (4 × 10 = 40 marks)
Answer any FOUR questions.
Each question carries 10 marks :
20. Find an approximate root (correct to two decimals) between 1 and 2 for the equation 3 1 0 x3 – x + = using Horner’s method.
21. Increase the roots of the equation 6 11 7 8 4 0 x4 – x3 – x2 + x + = by 1 and hence solve it.
22. Show that q q (cos7q cos5q 3cos3q 3cosq ) 64 1cos sin3 4 = – – + . q q (cos7q cos5q 3cos3q 3cosq )64
23. Find characteristic roots and corresponding characteristic vectors for the matrix
24. State and prove Wilson’s theorem.
25. Separate into real and imaginary parts -1 (x + iy) tan .