Matrices And Determinants - Important Question Paper

11th Standard

    Reg.No. :
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Business Mathematics

Time : 01:00:00 Hrs
Total Marks : 50

    Part A

    10 x 1 = 10
  1. The value of x if \(\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0\) is

    (a)

    0, -1

    (b)

    0, 1

    (c)

    -1, 1

    (d)

    -1, -1

  2. The value of the determinant \({\begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & c \end{vmatrix}}^{2}\)is

    (a)

    abc

    (b)

    0

    (c)

    a2b2c2

    (d)

    -abc

  3. The number of Hawkins-Simon conditions for the viability of an input - output analysis is

    (a)

    1

    (b)

    3

    (c)

    4

    (d)

    2

  4. Which of the following matrix has no inverse

    (a)

    \(\begin{pmatrix} -1 & 1 \\ 1 &-4 \end{pmatrix}\)

    (b)

    \(\begin{pmatrix} 2 & -1 \\ -4 &2 \end{pmatrix}\)

    (c)

    \(\begin{pmatrix} cos\ a & sin\ a \\ -sin\ a & cos\ a \end{pmatrix}\)

    (d)

    \(\begin{pmatrix} sin\ a & cos\ a \\ -cos\ a & sin\ a \end{pmatrix}\)

  5. The Inverse of matrix of\(\begin{pmatrix} 3 & 1 \\ 5 & 2\end{pmatrix}\) is

    (a)

    \(\begin{pmatrix} 2 & -1 \\-5 & 3 \end{pmatrix}\)

    (b)

    \(\begin{pmatrix} -2 & 5 \\1 & -3 \end{pmatrix}\)

    (c)

    \(\begin{pmatrix} 3 & -1 \\-5 & -3 \end{pmatrix}\)

    (d)

    \(\begin{pmatrix} -3 & 5 \\1 & -2 \end{pmatrix}\)

  6. If A \(=\begin{pmatrix} -1 & 2 \\ 1 & -4 \end{pmatrix}\) then A (adj A) is 

    (a)

    \(\begin{pmatrix} -4 & -2 \\ -1 & -1 \end{pmatrix}\)

    (b)

    \(\begin{pmatrix} 4 & -2 \\ -1 & 1 \end{pmatrix}\)

    (c)

    \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

    (d)

    \(\begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}\)

  7. If A is an invertible matrix of order 2, then det (A-1) be equal to

    (a)

    det (A)

    (b)

    \({{1}\over{det(A)}}\)

    (c)

    1

    (d)

    0

  8. If A is a square matrix of order 3 and IAI = 3 then | adj A| is equal to

    (a)

    81

    (b)

    27

    (c)

    3

    (d)

    9

  9. If \(\begin{vmatrix} x & 2 \\ 8 &5 \end{vmatrix}=0\) then the value of x is

    (a)

    \({{-5}\over{6}}\)

    (b)

    \({{5}\over{6}}\)

    (c)

    \({{-16}\over{5}}\)

    (d)

    \({{16}\over{5}}\)

  10. If \(\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5\) then value of \(\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}\) is

    (a)

    -5

    (b)

    -125

    (c)

    -25

    (d)

    0

  11. Part B

    6 x 2 = 12
  12. Suppose the inter-industry flow of the product of two sectors X and Yare given as under.

    Production Sector Consumption Sector Domestic demand Gross output
      X Y    
    X 15 10 10 35
    Y 20 30 15 65

    Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

  13. Solve: \(\begin{vmatrix}2& x&3\\4&1&6\\1&2&7 \end{vmatrix}=0\)

  14. Two commodities A and B are produced such that 0.4 tonne of A and 0.7 tonne of B are required to produce a tonne of A. Similarly 0.1 tonne of A and 0.7 tonne of B are needed to produce a tonne of B. Write down the technology matrix. If 6.8 tonnes of A and 10.2 tonnes of B are required, find the gross production of both of them.

  15. If \(A=\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}\) then show that |2A| = 4 |A|.

  16. Using the property of determinant, evaluate \(\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.\)

  17. Using the property of determinants show that \(\begin{vmatrix} x &a &x+a \\ y & b &y+b \\z & c & z+c \end{vmatrix}=0.\)

  18. Part C

    6 x 3 = 18
  19. If\(A=\begin{bmatrix}1&3&3\\1&4&3\\1&3&4 \end{bmatrix}\)then verify that A (adj A) = |A| I and also find A-1.

  20. Find the inverse of \(\begin{bmatrix}-1 & 5 \\-3 & 2 \end{bmatrix}.\)

  21. Solve: 2x+ 5y = 1 and 3x + 2y = 7 using matrix method.

  22. Show that \(\begin{vmatrix}x+a &b&c \\a &x+b&c\\a&b&x+c \end{vmatrix}=x^2(x+a+b+c)\)

  23. Using the properties of determinants, show that \(\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right| \)=0

  24. Find the adjoint of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right] \)

  25. Part D

    2 x 5 = 10
  26. If \(A=\left[ \begin{matrix} 1 & tan\quad x \\ -tan\quad x & \quad \quad \quad 1 \end{matrix} \right] \), then show that ATA-1=\(\left[ \begin{matrix} cos\quad 2x & -sin2x \\ sin\quad 2x & cos2x \end{matrix} \right] .\)

  27. Solve by matrix inversion method: 3x - y + 2z = 13 ; 2x + Y - z = 3 ; x + 3y - 5z = - 8.

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