Dr Babasaheb Ambedkar Marathwada University 2011-1st Year M.C.A Computer Aplications Mathematics, first semester - Question Paper
Q.1: Attempt any 2 of the subsequent (12)
(a) explain the convergence of the series
……(x>0)
(b) Examine the convergence of the series
…………
(c) explain the convergence of the infinite series
Q.2: Attempt any 2 of the subsequent (12)
(a) Test for the consistency and solve if possible
x+2y+z=3
2x+3y+2z=5
3x-5y+5z=2
3x+9y-z=4
(b) obtain the Eigen values and Eigen vectors for the subsequent matrix
(c) obtain the rank of the matrix by reducing it to its normal form
Q.3: Attempt any 2 of the subsequent (12)
(a) Let T:U?V be described by T(x1,x2)=(x1,0) show that T is a linear map.
(b) obtain the matrix representation of every of the mapping T on R2
Relative to the usual basis {(1,0), (0,1)}
(i) T(x, y)=(2y, 3x-y)
(ii) T(x,y)=(3x-4y, x+5y)
(c) Determine whether the vectors (1, -2, 3), (5, 6, -1) and (3, 2, 1) are linearly dependent? If so obtain the relation ranging from them.
Q.4: Attempt any 2 of the subsequent (12)
(a) Show that the vectors and are orthogonal. Apply Gram-Schmidt process to orthogonalize these vectors.
(b) obtain 2 vectors of norm one that are orthogonal to 3 vectors .
(c) Let V2 be the vector space with inner product If show that f and g are orthogonal.
Q.5: Attempt any 2 of the subsequent (12)
(a) State whether the quadratic form
Is (i) Positive definite (ii) negative definite (iii) semi positive definite (iv) semi negative definite?
(b) describe trace, index and signature.
obtain the matrix of the quadratic form
.
(c) decrease the quadratic form to the canonical form
Government college of Engineering, Aurangbad
(An autonomous Institute of Government of Maharashtra)
FY MCA Examination End semester Examination Dec 2011
MCA 104: MATHEMATICS
Time: Three Hours Max. Marks: 60
"Verify the course code and check whether you have got the correct question paper".
N.B.
1. All questions are compulsory.
2. Figures to the right indicate full marks.
3. Assume suitable data if necessary and state it clearly.
4. Use of non programmable calculator is allowed.
Q.1: Attempt any two of the following (12)
(a) Discuss the convergence of the series
x 1 x3 13 x5 1 3 5 x7 , _.
- +--+ +----+......(X>0)
1 23 245 2467 v '
(b) Examine the convergence of the series 1 + ............
1.2.3 2.3.4 3.4.5
(c) Discuss the convergence of the infinite series ,2
7T
Q.2: Attempt any two of the following (12)
(a) Test for the consistency and solve if possible x+2y+z=3
2x+3y+2z=5
3x-5y+5z=2
3x+9y-z=4
(b) Find the Eigen values and Eigen vectors for the following matrix
r 1 2-23] 2 5-46 -1 -3 2 -2 .2 4-16.
Q.3: Attempt any two of the following
(a) Let T:UV be defined by T(xi,x2)=(xi,0) show that T is a linear map.
(b) Find the matrix representation of each of the mapping T on R2
Relative to the usual basis {(1,0), (0,1)}
(i) T(x, y)=(2y, 3x-y)
(ii) T(x,y)=(3x-4y, x+5y)
(c) Determine whether the vectors (1, -2, 3), (5, 6, -1) and (3, 2, 1) are linearly dependent? If so find the relation between them.
Q.4: Attempt any two of the following (12)
(a) Show that the vectors are orthogonal. Apply Gram-Schmidt process to orthogonalize these vectors.
(b) Find two vectors of norm 1 that are orthogonal to three vectors u = ( 2,1, 4, 0) , v =
(1, 1,2,2 ) ,w = (3 ,2 , 5,4) .
(c) Let V2 be the vector space with inner product (/,g) = JQ /(x) g(x) dx . If /(x) c 0 s ( 2nx) , g(x) = s in ( 2 nx) show that f and g are orthogonal.
(12)
Q.5: Attempt any two of the following
(a) State whether the quadratic form
5x + 3xf + 6x| + 2xx2 + 3x2x3 8xx3
Is (i) Positive definite (ii) negative definite (iii) semi positive definite (iv) semi negative definite?
(b) Define trace, index and signature.
Find the matrix of the quadratic form
3 xi + 4xf + 5 x I 5 Xi x2 5 x2 x 3 + 7 x 3 x
(c) Reduce the quadratic form to the canonical form
3x2 + 5 y2 + 3 z2 2 yz + 2zx 2 xy
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