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    Matrices & Determinants

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    Matrices & Determinants

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    Tuff

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    (Matrices & Determinants)

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    5 views3 pages

    Matrices & Determinants

    Uploaded by

    sirikimurthy
    Tuff

    Copyright:

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

    VISWABHARATI IIT OLYMPIAD PROGRAMME

    TEST – I
    CLASS : VIII OLYMPIAD 1 SUB : MATHS
    I. Single Correct Answer :

    1. For each real number ‘x’ such that , let A (x) be the matrix( ) [ ] and
    .Then,
    A) A(z) = A(x) + A(y) B) A(z) = A(x) [A(y)]–1
    C) A(z) = A(x) A(y) D) A(z) = A(x) – A(y)
    cos  sin    1 0
    A  B 
    2. If  sin   cos  ,  1 1  , C  ABAT , then A T Cn A equals to n  I  
     n 1  1  n  0 1   1 0
           
    A)  1 0 B) 0 1  C) 1 n  D)  n 1 
    0 1 1
    3. A is an involutary matrix given by A   4 3 4  , then the inverse of A/2 will be
     3 3 4 
    A 1 A
    2
    A) 2 A B) 2 C) 2 D) A
    a b 
    4. A  and MA = A2m, m  N for some matrix M, then which one of the following is correct?
     b a 
     a 2m b2m  m 1 0
    M   2m  M   a 2  b2   
    A) b a 2m  B) 0 1 
    1 0  m 1  a b
    M   a m  bm    M   a 2  b2   
    C) 0 1  D)  b a  .
     x 0 2
    5. If A, B, C are three square matrices of third order such that A   0 y 0  , det  B   22.32 ,
     0 0 z 

     
    det C   22 where x, y, z  I  and det adj  adj  ABC    232  316  74 then the digit in the unit

    place in the value of det adjA is

    A) 4 B) 6 C) 1 D) 8

    II. Multi Correct Answer :

    6. If the sum of two idempotent matrices is idempotent then


    A) B)
    C) D)

    1 0 0 
    7. Let A  0 1 1  and aA1  bA2  cA  dI where G. C. D of a , d is one then
    0 2 4 

    A) a  6 B) b  1 C) c  6 D) d  1
    8. Let A and B be the different matrices satisfying A3  B3 and A2 B  B2 A . Then
    det  A2  B 2  is equal to

    A) 1 B) 2 C) 0 D) 4
     x if i  j , x  R
    9. Let A  aij  be a matrix of order 3 where aij  1 if | i  j | 1 then which of the following
     0 otherwise

    hold(s) good?
    (A) for x= 2 , A is a diagonal matrix
    (B) A is a symmetric matrix
    (C) For x = 2 , det A has the value equal to 6 (D) Let f(x) = det A, then the function f(x) has both
    local maxima and local minima
    10. A  M n (c) (where M is a matrix of order n x n ) is non-singular matrix such that

    3ABA1  A  2A1BA then


    A) A & B both are identity matrices
    AB 0
    B)
    ABA1  AB1A  0
    C)
    D) A + B is not a singular matrix
    11. Let A,B,C be square matrices of same order and I is the unit matrix of same order such that
    A  B  C  AB  BC  CA . Consider the following statements
    (i) ABC  AC  CA (ii) BCA  BA  AB (iii) CAB  CB  BC
    (A) (i) and (ii) are equivalent (B) (ii) and (iii) are equivalent
    (C) (i) and (iii) are equivalent (D) (i) , (ii) and (iii) are equivalent
    12. If the sum of two idempotent matrices is idempotent then

    A) AB  BA  0 B) AB  BA  I C) AB  BA  0 D) AB  BA

    1 0 0 
    13. Let A  0 1 1  and aA1  bA2  cA  dI where G. C. D of a , d is one then
    0 2 4 

    A) a  6 B) b  1 C) c  6 D) d  1
    III. Numerical
     
    14. If    0,  , and k  4sec2   9 cos ec 2 , then the number of solutions of the system of equations
     2
    3x  y  4z  3, x  2 y  3z  2,6x  5 y  kz  3 is
    15. Let a,b,c be distinct positive integers such that ab  bc  ca  107 then the minimum value of

     a  b3  c3  3abc  is
    1 3
    6
     1  1
     r.3r
    1  3 2   2r 
    16. let A  3 ,B  3 and Cr    be given matrices. If
      5 
     1 1   0  r  1 3r 
    1 2 

     tr   AB  
    50
    cr  3  a.3b where tr (A) denotes trace of matrix A , then a  b is
    r

    r 1

    17. if the matrix ( ) is an orthogonal matrix then the value of 3a  6b is

    18. A possible value of ‘K’ for which the system of equations


    2x  3 y  6z  5t  3, y  4z  t  1, 4 x  5 y  8z  9t  k in x, y, z has infinite number of solutions.
    Key :

    1. C 7. ABC 13. ABC


    2. C 8. C 14. 1
    3. A 9. BD 15. 9
    4. D 10. BC 16. 2
    5. D 11. ABCD 17. 5
    6. AB 12. AC 18. 7

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