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    Maths

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    MALLAREDDY ENGINEERING COLLEGE (AUTONOMOUS)

    I B.Tech. I Semester (MR-24) IMidQuestionBank 2024-25(Subjective)

    Subject: Matrices &Calculus (D0B01) Branch:Common to all


    Name of the Faculty: Dr.K. Malleswari
    Marks BT
    Q. Questions Level CO
    No.
    Module-I
    2 1 3 5
    4 2 1 3 
    a) Find the rank of the matrix A=  by reducing into echelon form. 5 3 1
    8 4 7 13 
    1.  
    8 4  3  1
    b) Define Hermitian, skew-hermitian and unitary matrix.
    2 3  1  1 
    1  1  2  4 
    Reducing the matrix   into normal form and hence find its rank. 5 3 1
    3 1 3  2
    2.  
    6 3 0  7
    a)Discuss for what values of 𝑎, 𝑏 the simultaneous equations
    3. 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 = 10, 𝑥 + 2𝑦 + 𝑎𝑧 = 𝑏 have (i)no solution 5 6 1
    (ii) a unique solution (iii) an infinite number of solutions.
    1 1 −1 1
    b) Find the value of k such that the rank of A = [ 1 − 1 𝑘 − 1 ] is 2
    3 1 0 1
    Find for what values of λ the equations 5 4 1
    4. 𝑥 + 𝑦 + 𝑧 = 1, 𝑥 + 2𝑦 + 4𝑧 = 𝜆, 𝑥 + 4𝑦 + 10𝑧 = 𝜆2 have a solution. And solve
    them completely in each case.
    5.  1  3 3  1 5 3 1
    1 1  1 0 
    Find the inverse of A =  using Gauss-Jordan Method.
     2  5 2  3
     
     1 1 0 1
    Solve the system of equations
    6. 𝑥 + 2𝑦 + (2 + 𝑘)𝑧 = 0, 2𝑥 + (2 + 𝑘)𝑦 + 4𝑧 = 0,7𝑥 + 13𝑦 + (18 + k)𝑧 = 0 for 5 3 1
    all values ofk.
    7. a)Show that the only real number 𝜆 for which the system 5 3 1
    𝑥 + 2𝑦 + 3 = 𝜆𝑥, 3𝑥 + 𝑦 + 2𝑧 = 𝜆𝑦, 2𝑥 + 3𝑦 + 𝑧 = 𝜆𝑧has non-zero solution is 6
    and solve them, when 𝜆 = 6.
    8. Solve the following system of equations by LU-Decomposition method 5 3 1
    2𝑥 + 𝑦 − 𝑧 = 3, 𝑥 − 2𝑦 − 2𝑧 = 1, 𝑥 + 2𝑦 − 3𝑧 = 9

    Module-II
    Determine the characteristic roots and corresponding characteristic vectors of the
    1. 8 −6 2 5 5 2
    matrix𝐴 = [−6 7 −4]
    2 −4 3

    a) If  is the eigen value of A corresponding eigen vector X, then prove that  n is


    2. the eigen value of An corresponding to the eigen vector X 5 2 2
    b) Prove that the eigen values of 𝐴−1 are the reciprocals of the eigen values of A.
    1 2 −1
    3. Verify Cayley-Hamilton theorem for 𝐴 = [2 1 −2]. Hence find A-1and A4. 5 4 2
    2 −2 1
    4. Determine the diagonal matrix orthogonally similar to the following symmetric
    3 −1 1 5 5 2
    matrix 𝐴 = [−1 5 −1]
    1 −1 3
    5. Reduce the quadratic form 3𝑥 2 + 5𝑦 2 + 3𝑧 2 − 2𝑦𝑧 + 2𝑧𝑥 − 2𝑥𝑦 to the canonical 5 5 2
    form by orthogonal reduction.
    6 −2 2
    6. Determine the eigen values and eigen vectors of the matrix 𝐴 = [−2 3 −1] 5 5 2
    2 −1 3
    2 1 1
    7. If 𝐴 = [0 1 0] find the value of the matrix 5 4 2
    1 1 2
    𝐴8 − 5𝐴7 + 7𝐴6 − 3𝐴5 + 𝐴4 − 5𝐴3 + 8𝐴2 − 2𝐴 + 𝐼.
    1 0 −1 5 5 2
    8. Diagonalize the matrix 𝐴 = [1 2 1 ]
    2 2 3

    Module-III

    1. 𝑠𝑖𝑛𝑥 5 4 3
    Verify the Rolle’s theorem for f(x) = in [0, 𝜋].
    𝑒𝑥
    2. Verify the Lagrange’s mean value theorem for the function f(x) =x2 – 2x + 4 on 4 3
    [1, 5] 5
    3. Verify cauchy’s mean value theorem for f(x) = sinx, g(x) = cosx in [𝑎, 𝑏] 5 4 3
    4. Expand x4 – 3x3 + 2x2 – x + 1 in power of (x-3)
    5 4 3

    Signature of the faculty Signature of HOD

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