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#### Applications of Matrices and Determinants Two Marks Questions

12th Standard EM

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Time : 00:45:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Solve the following system of equations by rank method
x+y+z=9,2x+5y+7z=52,2x−y−z =0

2. For what values of the parameterl , will the following equations fail to have unique solution: 3x−y+λz=1,2x+y+z=2,x+2y−lz = −1 by rank method.

3. An amount of Rs.5,000/- is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is Rs.358/-. If the income from first two investments is `70/- more than the income from the third, then find the amount of investment in each bond by rank method.

4. Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?

5. Find k if the equations 2x+3y−z=5,3x−y+4z=2,x+7y−6z=k are consistent.

6. Solve the equations x+2y+z=7,2x−y+2z=4,x+y−2z = −1 by using Cramer’s rule

7. Find the rank of the following matrices
$\left( \begin{matrix} 2 & -1 & 1 \\ 3 & 1 & -5 \\ 1 & 1 & 1 \end{matrix} \right)$

8. Find the rank of the following matrices
$\left( \begin{matrix} 3 & 1 & -5 \\ 1 & -2 & 1 \\ 1 & 5 & -7 \end{matrix}\begin{matrix} -1 \\ -5 \\ 2 \end{matrix} \right)$

9. Solve the following equation by using Cramer’s rule
5x + 3y = 17; 3x + 7y = 31

10. Solve the following equation by using Cramer’s rule
x + y + z = 6, 2x + 3y− z =5, 6x−2y− 3z = −7

11. Find the rank of the matrix $\left[ \begin{matrix} 7 & -1 \\ 2 & 1 \end{matrix} \right]$

12. Solve: x + 2y = 3 and 2x + 4y = 6 using rank method.

13. Show that the equations x +y + Z = 6, x.+ 2y + 3z = 14 and x + 4y + 7z = 30 are consistent

14. Solve: 2x + 3y = 4 and 4x + 6y = 8 using Cramer's rule.

15. Two newspapers A and B are published in a city . Their market shares are 15% for A and 85% for B of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year