PTU B.Tech CSE 1st Semester May 2019 75362 MATHEMATICS I Question Papers

Roll No.

Total No. of Pages : 03

Total No. of Questions : 18

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B.Tech. (CSE/IT) (2018 Batch) (Sem.-1)

MATHEMATICS-I

Subject Code : BTAM-104-18

M.Code: 75362

Time: 3 Hrs.

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Max. Marks : 60

INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION - B & C have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.

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4. Select atleast TWO questions EACH from SECTION - B & C.

SECTION-A

1. Can Rolle's theorem be applied to the function f (x) = 2 + (x - 1)^2/3, x ? [0, 2].
2. Define beta function.
3. Evaluate lim _x?0 (x cos x-sin x) / (x² sin x)

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4. Find the values of x, y, z to satisfy the relation
   

   x+3	2y+x
   0	z-1

   =

   3	2a
   -7	4a-6

   2a

5. Find adjoint of
6. Define basis of vector spaces.
7. Give the statement of rank nullity theorem.

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8. Give any two properties of Eigen values.
9. Define symmetric matrix with an example.
10. Find sum and product of latent roots of the matrix

    2	1
    2	3

SECTION-B

11. a) Expand f (x) = sin^-1x by Maclaurin's theorem.
    

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    b) Evaluate lim _x?a (x^a-a^x) / (x^x-a^a)
12. a) Evaluate the integral ?₀¹ dx / (1-x⁴) in terms of gamma function.
    b) Find maxima of f (x, y) = 2 (x² - y²) – x⁴ + y⁴.
13. a) Prove that

    1+a	1	1
    1	1+b	1
    1	1	1+c

    = abc (1/a + 1/b + 1/c) + ab + bc + ca + a + b + c + 1
    b) Solve the equations x +y + z = 1, x +2y+3z = 6, x + 3y + 4z = 6 using Cramer's rule.

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14. a) Are the vectors (2, 1, 1), (2, 0, -1), (4, 2, 1) linearly dependent?
    b) Find the rank of the matrix :

    5	3	7
    3	2	6
    7	2	10

SECTION-C

15. Show that the matrix

    2	0	-1
    5	1	0
    0	1	3

    satisfies the equation A³ – 6A² + 11A – I = 0.
16. Let T: R³ ? R² be the linear transformation defined by T

    x

    y

    z

    =

    x+y

    x-z

    then find the matrix representation of T w.r.t. the ordered basis X = {(1, 0, 1), (1, 1, 0), (0, 1, 1)} in R³ and Y = {(1, 0), (0, 1)} in R².

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17. a) Is the matrix

    4	2	1
    6	3	4
    2	1	0

    orthogonal ?
    b) Write the matrix

    1	2	3
    4	5	6
    7	-8	9

    as the sum of symmetric and skew symmetric matrices.
18. Reduce the matrix

    -1	1	2	-2
    1	-1	-2	2
    2	-2	-4	4
    -1	1	2	-2

    to the diagonal form. FirstRanker.com

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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