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    Bachelor of Science (B.SC.) Semester-I Examination Mathematics (Algebra and Trigonometry) Optional Paper-1

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    Swapnil Kamdi
    This document contains: 1) A mathematics exam for Bachelor of Science (B.Sc.) Semester I with questions covering algebra, trigonometry, and other topics. 2) The exam is divided into 4 units and contains 5 questions with multiple parts covering concepts such as matrices, equations, groups, logarithms, and trigonometric functions. 3) Students are instructed to attempt all questions which are equally weighted and to fully solve either the question or its listed alternative.

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    Bachelor of Science (B.SC.) Semester-I Examination Mathematics (Algebra and Trigonometry) Optional Paper-1

    Uploaded by

    Swapnil Kamdi
    29 views3 pages
    This document contains: 1) A mathematics exam for Bachelor of Science (B.Sc.) Semester I with questions covering algebra, trigonometry, and other topics. 2) The exam is divided into 4 units and contains 5 questions with multiple parts covering concepts such as matrices, equations, groups, logarithms, and trigonometric functions. 3) Students are instructed to attempt all questions which are equally weighted and to fully solve either the question or its listed alternative.

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    2008

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    This document contains: 1) A mathematics exam for Bachelor of Science (B.Sc.) Semester I with questions covering algebra, trigonometry, and other topics. 2) The exam is divided into 4 units and contains 5 questions with multiple parts covering concepts such as matrices, equations, groups, logarithms, and trigonometric functions. 3) Students are instructed to attempt all questions which are equally weighted and to fully solve either the question or its listed alternative.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    29 views3 pages

    Bachelor of Science (B.SC.) Semester-I Examination Mathematics (Algebra and Trigonometry) Optional Paper-1

    Uploaded by

    Swapnil Kamdi
    This document contains: 1) A mathematics exam for Bachelor of Science (B.Sc.) Semester I with questions covering algebra, trigonometry, and other topics. 2) The exam is divided into 4 units and contains 5 questions with multiple parts covering concepts such as matrices, equations, groups, logarithms, and trigonometric functions. 3) Students are instructed to attempt all questions which are equally weighted and to fully solve either the question or its listed alternative.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    NIR/KW/18/2008

    Bachelor of Science (B.Sc.) Semester—I Examination

    MATHEMATICS (ALGEBRA AND TRIGONOMETRY)


    Optional Paper—1

    Time : Three Hours] [Maximum Marks : 60

    N.B. :— (1) Solve all the five questions.


    (2) All questions carry equal marks.

    (3) Question Nos. 1 to 4 have an alternative, solve each question in full or its alternative
    in full.
    UNIT—I
    1. (A) Find non-singular matrices P and Q such that PAQ is in the normal form. Also find rank
    of A where :

    1 1 1 
    1 − 1 − 1
    A=   6
    3 1 1 

    (B) Find the value of λ, so that following equations :


    x – y + z = 1, x + 2y + 4z = λ, 2x + 4y + 8z = λ 2
    may have solution and solve them completely in each case. 6
    OR
    (C) Find eigen values of the matrix :

    3 1 4
    0 2 6
     
    0 0 5

    and eigen vectors corresponding to the greatest eigen value of the matrix. 6

     0 0 1
     3 1 0
    (D) Verify Cayley-Hamilton theorem for the matrix A =   6
    − 2 1 4

    UNIT—II
    2. (A) Solve the equation
    x4 – 2x3 – 17x2 + 18x + 72 = 0,
    given that sum of the two roots being equal to the sum of the other two. 6

    BKR—11055 1 (Contd.)
    (B) Find the condition that the roots of the equation x3 + ax2 – bx + c = 0 are in arithmetic
    progression. Hence solve the equation x3 – 9x2 + 23x – 15x = 0. 6
    OR
    (C) Solve x3 – 27x + 54 = 0 by Cardon’s method. 6
    (D) Solve the reciprocal equation 2x4 – 5x3 + 4x2 – 5x + 2 = 0. 6
    UNIT—III
    3. (A) Solve the equation
    x5 – x3 + x2 – 1 = 0
    using DeMoivre’s theorem. 6
    (B) Prove that :

     nπ 
    (i) (1+ i 3 ) n
    (
    + 1− i 3 ) n
    = 2n+1 . cos  
     3 
    (ii) Separate sinh (α – iβ ) into real and imaginary parts. 6
    OR
    (C) If sin (θ + iφ) = cos α + i sin α ,
    Prove that :
    (i) cos2 θ = ± sin α
    (ii) sinh2 φ = ± sin α . 6
    (D) Prove that :

     a + ib  b
    (i) log   = 2 i tan–1  
     a − ib  a 

    π  1 1 1  1 1  1  1 1 
    (ii) =  +  −  +  +  +  − ....... 6
    4  2 3  3  23 33  5  25 35 

    UNIT—IV
    4. (A) Prove that the set G = {0, 1, 2, 3, 4, 5, 6} is a finite abelian group of order 7 with respect
    to addition modulo 7. 6
    (B) Prove that a necessary and sufficient condition for a non-empty subset H of a group (G, o )
    to be a subgroup is that

    a, b ∈ H ⇒ a o b−1 ∈ H 6

    OR
    (C) Prove that intersection of two subgroups of a group G is a subgroup of G. Give an example
    to prove that union of two subgroups of G is not necessarily a subgroup of G. 6
    (D) Show that the four permutations I = (a), (a b), (c d), (a b) . (c d) on four symbols
    a, b, c, d form a finite abelian group with respect to the multiplication of permutations.
    6
    BKR—11055 2 NIR/KW/18/2008
    Question—V
    5 0 5
    0 2 0
    5. (A) Find the sum and product of eigen values of the matrix  . 1½
    4 0 3

    (B) Find K, if the system of equations x + 4y + 2z = 0, 4x + 9y + z = 0, Kx + 3y + Kz = 0


    has non-trivial solution. 1½
    (C) Use Descartes’s rule of signs to discuss about real roots (positive and negative) and imaginary
    roots of the equation x4 + 2x3 – 16x2 – 22x + 7 = 0. 1½
    (D) If α, β and γ are the roots of the equation x3 + 9x2 – 7x + 5 = 0, find Σα 2 . 1½
    (E) Find general value of Log (–7 i). 1½
    (F) Prove that :
    cos (ix) = cosh x and
    sin (ix) = i sinh x. 1½
    (G) Verify whether the set of integers I with binary operation ‘o ’ defined by a o b = a − b ∀ a , b ∈ I
    is a group. 1½

    1 2 3   1 2 3
    (H) If A = 1 3 2  and B =  2 1 3 , find AB and BA. 1½
       

    BKR—11055 3 NIR/KW/18/2008

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