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Applications of Matrices and Determinants Model Question Paper

12th Standard

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Business Maths

Time : 01:30:00 Hrs
Total Marks : 50
    5 x 1 = 5
  1. If A = (1 2 3), then the rank of AAT is ________.

    (a)

    0

    (b)

    2

    (c)

    3

    (d)

    1

  2. The rank of the unit matrix of order n is ________.

    (a)

    n −1

    (b)

    n

    (c)

    n +1

    (d)

    n2

  3. If A =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AAT is ________.

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  4. Which of the following is not an elementary transformation?

    (a)

    \({ R }_{ i }\leftrightarrow { R }_{ j }\)

    (b)

    \({ R }_{ i }\rightarrow { 2R }_{ i }+{ 2C }_{ j }\)

    (c)

    \({ R }_{ i }\rightarrow { 2R }_{ i }-{ 4R }_{ j}\)

    (d)

    \({ C }_{ i }\rightarrow { C }_{ i }+{ 5C }_{ j }\)

  5. The system of linear equations x + y + z = 2, 2x + y − z = 3, 3x + 2y + k = 4 has unique solution, if k is not equal to _______.

    (a)

    4

    (b)

    0

    (c)

    -4

    (d)

    1

  6. 5 x 1 = 5
  7. For any 2 x 2 matrix, if A (adj A = \(\left| \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right| \) then IAI is _______

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    10

  8. If A is a matrix of order 3 and IAI = 8 then |adj AI|=_______

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    64

  9. The system of equation x + y + z = 2, 3x - y + 2z = 6 and 3x + y - z = -18 has________solution

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    unique

  10. The system of linear equations x + y + Z = 2, 2x + Y - z = 3, 3x + 2y + kz = 4 has a unique solution if k is_______

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    Not equal to 0

  11. A set of values of the variable x1,x2,...xn satisfying all the equations simultaneously is called__________ of the system

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    Solution

  12. 6 x 2 = 12
  13. Find the rank of each of the following matrices.
    \(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

  14. Show that the equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 are consistent and solve them by rank method.

  15. Solve the following equations by using Cramer’s rule
     2x + 3y = 7; 3x + 5y = 9

  16. In a market survey three commodities A, B and C were considered. In finding out the index number some fixed weights were assigned to the three varieties in each of the commodities. The table below provides the information regarding the consumption of three commodities according to the three varieties and also the total weight received by the commodity

    Commodity Variety Variety Total weight
    I II III
    A 1 2 3 11
    B 2 4 5 21
    C 3 5 6 27

    Find the weights assigned to the three varieties by using Cramer’s Rule.

  17. A total of Rs. 8,500 was invested in three interest earning accounts. The interest rates were 2%, 3% and 6% if the total simple interest for one year was Rs. 380 and the amount, invested at 6% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account? (use Cramer’s rule).

  18. Find the rank of the matrix A =\(\left( \begin{matrix} -2 & 1 & 3 \\ 0 & 1 & 1 \\ 1 & 3 & 4 \end{matrix}\begin{matrix} 4 \\ 2 \\ 7 \end{matrix} \right) \)

  19. 6 x 3 = 18
  20. Find the rank of the matrix \(\begin{pmatrix} 1 & 5 \\ 3 & 9 \end{pmatrix}\)

  21. Find the rank of the matrix \(\left( \begin{matrix} 1 & 2 & -1 \\ 2 & 4 & 1 \\ 3 & 6 & 3 \end{matrix}\begin{matrix} 3 \\ -2 \\ -7 \end{matrix} \right) \)

  22. Show that the equations 3x − 2y = 6, 6x − 4y = 10 are inconsistent

  23. Investigate for what values of ‘a’ and ‘b’ the following system of equations x + y + z = 6,x + 2y + 3z = 10, x + 2y + az = b have
    (i) no solution
    (ii) a unique solution
    (iii) an infinite number of solutions.

  24. Consider the matrix of transition probabilities of a product available in the market in two brands A and B.
    \(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.9 } & \overset { B }{ 0.1 } \\ 0.3 & 0.7 \end{matrix} \right) \)
    Determine the market share of each brand in equilibrium position.

  25. If \(A=\left( \begin{matrix} 2 & 4 \\ 4 & 3 \end{matrix} \right) ,X=\left( \begin{matrix} n \\ 1 \end{matrix} \right) B=\left( \begin{matrix} 8 \\ 11 \end{matrix} \right) \) and AX = B then find n.

  26. 2 x 5 = 10
  27. Solve the equations 2x + 3y = 7, 3x + 5y = 9 by Cramer’s rule.

  28. Solve by Cramer’s rule x + y + z = 4, 2x − y + 3z = 1, 3x + 2y − z = 1

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