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    1214 - Year - BACHELOR OF SCIENCE (B.SC.) FIRST SEMESTER OLD COURSE Subject - MAT 101 - MATHEMATICS PAPER-I - (Algebra and Trigonometry)

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    Ramnaresh
    1. The document is a mathematics exam paper covering topics in algebra and trigonometry across 4 units. 2. The first unit covers proofs involving trigonometric identities, separating trig functions into real and imaginary parts, and relating trig functions. 3. The second unit covers matrix operations including finding matrices to put a given matrix in normal form and using matrices to solve systems of linear equations. 4. The third unit covers solving polynomial equations including proving properties of polynomial roots and using specific methods to solve cubic and quartic equations. 5. The fourth unit covers mathematical induction, properties of fractions, greatest common divisors, and simultaneously solving systems of equations.

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    1214 - Year - BACHELOR OF SCIENCE (B.SC.) FIRST SEMESTER OLD COURSE Subject - MAT 101 - MATHEMATICS PAPER-I - (Algebra and Trigonometry)

    Uploaded by

    Ramnaresh
    18 views2 pages
    1. The document is a mathematics exam paper covering topics in algebra and trigonometry across 4 units. 2. The first unit covers proofs involving trigonometric identities, separating trig functions into real and imaginary parts, and relating trig functions. 3. The second unit covers matrix operations including finding matrices to put a given matrix in normal form and using matrices to solve systems of linear equations. 4. The third unit covers solving polynomial equations including proving properties of polynomial roots and using specific methods to solve cubic and quartic equations. 5. The fourth unit covers mathematical induction, properties of fractions, greatest common divisors, and simultaneously solving systems of equations.

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    1214 - Year - BACHELOR OF SCIENCE (B.Sc.) FIRST SEMESTER OLD COURSE Subject - MAT 101 - MATHEMATICS PAPER-I - (Algebra and Trigonometry)

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    1. The document is a mathematics exam paper covering topics in algebra and trigonometry across 4 units. 2. The first unit covers proofs involving trigonometric identities, separating trig functions into real and imaginary parts, and relating trig functions. 3. The second unit covers matrix operations including finding matrices to put a given matrix in normal form and using matrices to solve systems of linear equations. 4. The third unit covers solving polynomial equations including proving properties of polynomial roots and using specific methods to solve cubic and quartic equations. 5. The fourth unit covers mathematical induction, properties of fractions, greatest common divisors, and simultaneously solving systems of equations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    18 views2 pages

    1214 - Year - BACHELOR OF SCIENCE (B.SC.) FIRST SEMESTER OLD COURSE Subject - MAT 101 - MATHEMATICS PAPER-I - (Algebra and Trigonometry)

    Uploaded by

    Ramnaresh
    1. The document is a mathematics exam paper covering topics in algebra and trigonometry across 4 units. 2. The first unit covers proofs involving trigonometric identities, separating trig functions into real and imaginary parts, and relating trig functions. 3. The second unit covers matrix operations including finding matrices to put a given matrix in normal form and using matrices to solve systems of linear equations. 4. The third unit covers solving polynomial equations including proving properties of polynomial roots and using specific methods to solve cubic and quartic equations. 5. The fourth unit covers mathematical induction, properties of fractions, greatest common divisors, and simultaneously solving systems of equations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    Bachelor of Science (B.Sc.

    ) First Semester Old Course


    MAT 101 - Mathematics Paper – I – (Algebra and Trigonometry)

    P. Pages : 2 GUG/W/18/1214
    Time : Three Hours *0927* Max. Marks : 60
    _____________________________________________________________________
    Notes : 1. Solve all five questions.
    2. Each question carries equal marks.

    UNIT – I

    1. a) 1 + sin  + i cos  6
    Prove that = sin  + i cos 
    1 + sin  − i cos 
    Hence prove that
    5 5
         
    1 + sin + i cos  + 1 + sin − i cos  = 0
     5 5  5 5

    b) Prove that 6
    5 3 3 5
    sin 6 = 6 cos  sin  − 20 cos  sin  + 6 cos  sin 

    OR

    c) Separate into real and imaginary parts of tan h (x + iy) . 6

    d) If sin ( + i) = cos  + isin  . Prove that cos2  =  sin  . 6

    UNIT – II

    2. a) 1 1 2  6
    For the matrix A = 1 2 3  , find non singular matrices P and Q such that PAQ is in
    0 −1 −1
    the normal form.

    b) Solve the linear equations 2x + 3y − z = 0, x − y + 2z = 5, 3x + y − z = 1 by matrix method. 6

    OR

    c) Show that if B be an invertible matrix of the same order as A, then the matrices A and 6
    B−1  A  B have the same characteristic roots.

    d) 1 4 6
    Find the characteristic equation of the matrix A =   and verify Cayley – Hamilton
    2 3
    theorem for the matrix. Hence or otherwise express A5 − 4A4 − 7 A3 + 11A2 − A −10I
    as linear polynomial in A.

    GUG/W/18/1214 1 P.T.O
    UNIT – III
    3. a) Prove that every equation of degree n has n roots and no more. 6

    b) Solve the following equation 6x 3 − 11x 2 − 3x + 2 = 0 , the roots being in harmonic 6


    progression.
    OR
    c) Show that the same transformation removes both 2nd and 4th terms of the equation 6
    x 4 + 16x 3 + 83x 2 + 152x + 84 = 0 .

    d) Solve by Cardon’s method x 3 − 21x = 344 . 6

    UNIT – IV
    4. a) Use the first principle of induction to prove the following 6
    1
    12 + 22 + 33 + .... + n 2 = n(n + 1) (2n + 1),  n  N
    6

    b) a c 6
    Let and be fractions in lowest terms so that (a, b) = (c,d) = 1 .
    b d
    Prove that if their sum is an integer, then b = d .
    OR
    c) Find the gcd of 275 and 200 and express it in the form 275m + 200n. 6

    d) Find positive integers a and b satisfying the equations (a, b) = 10 and [a, b] = 100 6
    simultaneously. Find all solutions.

    5. Solve any six.

    a) State De Moivre’s Theorem. 2

    b) Separate sin (x + iy) into real and imaginary parts. 2

    c) Define transpose of a matrix. 2

    d) Show that if  is the eigenvalue of a non singular matrix A, then  −1 is the 2


    eigenvalue of A −1 .

    e) Find the nature of the roots of the equation 3x 4 + 12x 2 + 5x − 4 = 0 . 2

    f) Determine the values of  2 for the cubic equation x3 + px 2 + qx + r = 0 whose roots 2


    are , ,  .

    g) Prove that (a,a + k) | k for all integers a, k not both zero. 2

    h) Evaluate (n, n + 1) , where n  N . 2

    *************

    GUG/W/18/1214 2

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