Osmania University (OU) 2010-1st Year B.E Electronics & Communication Engineering main , mathematics - i - Question Paper
Time: 3 Hours] [Max. Marks: 75
B.E. I YEAR MAIN EXAMINATION, JUNE 2010 MATHEMATICS - I
Note: Answer all questions from Part A.
Answer any five questions from Part B.
[MARKS 25]
PART- A
1. Are the vectors (1, 0, 1), (0, 1, 1), (2, 2, 4) linearly dependent ? If so, find the relation between them. [MARKS 3]
2. Find the sum of the eigen values of the matrix
12 3 1
2 3 5 6
12 4 3
.0 0 2 1.
[MARKS 2]
3. Test the series (-1)n-1(1/n2n) for absolute convergence. [MARKS 3]
4. Discuss the convergence of the series . [MARKS 2]
5. Expand f(x) = tan x about x = 0 upto the term containing x3. [MARKS 3]
6. Find the radius of curvature of x2 + y2 = 4 at (1, 2). [MARKS 2]
X2 + y2
7. Determine limo , if it exists. [MARKS 2]
' ' x y
8. If z = eax+by f(ax-by), then pprove that bX + a|y = 2abz. [MARKS 3]
9. Evaluate // xy dxdy over the first quadrant of the circle x2 + y2 = 4. [MARKS 3]
10. Find the directional derivative of f(x,y) = x2 + y2 at (1, 1) in the direction of 2i-4j.
10. Find the [MARKS 2]
PART - B [MARKS 50]
112 0-2 0 003
11. a) Using Caley-Hamilton theorem, find the inverse of matrix
[MARKS 5]
b) Reduce the quadratic form Q = x2 + 3y2 + 3z2 - 2yz to canonical form [MARKS 5]
12. a) Discuss the convergence of the geometric series rn, r is any real number. [MARKS 5]
b) Test the series (V(n4+1) - V(n4-1)) for convergence. [MARKS 5]
13. a) State and prove Cauchys mean value theorem. [MARKS 5]
b) Find the envelop of | cos k + y sin k = 1, k is a parameter. [MARKS 5]
14. a) Sketch the graph of the curve x2 - xy + 1 = 0. [MARKS 5]
b) Find the local minimum and local maximum values of the function f(x, y) = x3 + y3 - 3xy. [MARKS 5]
15. a) State and prove Greens theorem in a plane. [MARKS 5]
b) If a is a constant vector and f = xi + yj + zk, find curl (a * f). [MARKS 5]
2 1 -3 -6 3-3 1 2 1 1 1 2
16. a) Reduce the matrix [MARKS 5]
b) Discuss the convergence of the series ((n + 1)4 /(nn+1))xn, x>0. [MARKS 5]
17. a) If z = f(x, y), x = e2u+e-2v, y = e-2u+e2v, then show that Of - Of = 2(x - y).
7 ' J du dv dx dy
[MARKS 5]
b) Evaluate ffs F.n ds, where F = x2i + 3y2k and S is the position of the plane x + y + z = 1 in the first octant. [MARKS 5]
to normal form and hence find its rank.
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