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#### Applications of Matrices and Determinants Book Back Questions

12th Standard EM

Reg.No. :
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Time : 01:00:00 Hrs
Total Marks : 30
5 x 1 = 5
1. 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

2. Cramer’s rule is applicable only to get an unique solution when

(a)

${ \triangle }_{ z }\neq 0$

(b)

${ \triangle }_{ x }\neq 0$

(c)

${ \triangle }_\neq 0$

(d)

${ \triangle }_{ y }\neq 0$

3. if $\frac { { a }_{ 1 } }{ x } +\frac { { b }_{ 1 } }{ y } ={ c }_{ 1 },\frac { { a }_{ 2 } }{ x } +\frac { { b }_{ 2 } }{ y } ={ c }_{ 2 },{ \triangle }_{ 1= }\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } \end{vmatrix};\quad { \triangle }_{ 2 }=\begin{vmatrix} { b }_{ 1 } & { c }_{ 1 } \\ { b }_{ 2 } & { c }_{ 2 } \end{vmatrix}{ \triangle }_{ 3 }=\begin{vmatrix} { c }_{ 1 } & { a }_{ 1 } \\ { c }_{ 2 } & a_{ 2 } \end{vmatrix}$ then (x,y) is

(a)

$\left( \frac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } } \frac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } } \right)$

(b)

$\left( \frac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } } \frac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } } \right)$

(c)

$\left( \frac { { \triangle }_{ 1 } }{ { \triangle }_{ 2 } } \frac { { \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right)$

(d)

$\left( \frac { { -\triangle }_{ 1 } }{ { \triangle }_{ 2 } } \frac { {- \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right)$

4. $\left| { A }_{ n\times n } \right|$=3 $\left| adjA \right|$ =243 then the value n is

(a)

4

(b)

5

(c)

6

(d)

7

5. Rank of a null matrix is

(a)

0

(b)

-1

(c)

$\infty$

(d)

1

6. 3 x 2 = 6
7. Solve the following equation by using Cramer’s rule
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

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

9. Solve the following equation by using Cramer’s rule
x + 4y + 3z =2,2x−6y + 6z=−3, 5x− 2y + 3z =−5

10. 3 x 3 = 9
11. Parithi is either sad (S) or happy (H) each day. If he is happy in one day, he is sad on the next day by four times out of five. If he is sad on one day, he is happy on the next day by two times out of three. Over a long run, what are the chances that Parithi is happy on any given day?

12. Akash bats according to the following traits. If he makes a hit (S), there is a 25% chance that he will make a hit his next time at bat. If he fails to hit (F), there is a 35% chance that he will make a hit his next time at bat. Find the transition probability matrix for the data and determine Akash’s long- range batting average.

13. 80% of students who do maths work during one study period, will do the maths work at the next study period. 30% of students who do english work during one study period, will do the english work at the next study period. Initially there were 60 students do maths work and 40 students do english work.
Calculate,
(i) The transition probability matrix
(ii) The number of students who do maths work, english work for the next subsequent 2 study periods.

14. 2 x 5 = 10
15. Solve by Cramer’s rule x+y+z=4,2x−y+3z=1,3x+2y−z = 1

16. An automobile company uses three types of Steel S1, S2 and S3 for providing three different types of Cars C1, C2 and C3. Steel requirement R (in tonnes) for each type of car and total available steel of all the three types are summarized in the following table.

 Types of Steel Types of Car Total Steel available C1 C2 C3 S1 3 2 1 28 S2 1 1 2 13 S3 2 2 2 14

Determine the number of Cars of each type which can be produced by Cramer’s rule.