Miscellaneous problems

Miscellaneous problems

1. Find the rank of the matrix 

2. Find the rank of the matrix A = 

3. Find the rank of the matrix A = 

4. Examine the consistency of the system of equations: x + y + z = 7 , x + 2 y + 3z = 18, y + 2z = 6 .

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

6. Find k if the equations x + y + z = 1, 3x − y − z = 4, x + 5y + 5z = k are inconsistent.

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

8. The cost of 2kg. of wheat and 1kg. of sugar is ₹100. The cost of 1kg. of wheat and 1kg. of rice is ₹80. The cost of 3kg. of wheat, 2kg. of sugar and 1kg of rice is ₹220. Find the cost of each per kg., using Cramer’s rule.

9. A salesman has the following record of sales during three months for three items A,B and C, which have different rates of commission.

Find out the rate of commission on the items A,B and C by using Cramer’s rule.

10. The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 60% of those who already subscribe will subscribe again while 25% of those who do not now subscribe will subscribe. On the last letter it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

Answers:

1. ρ(A)= 2

2. ρ(A) = 3

3. ρ(A) = 3

4. The given system is inconsistent and has no solution.

5. k = 8.

6. The equations are inconsistent when k assumes any real value other than 0.

7. x=1,  y = 2 , and z = 2

8. Cost of wheat is ₹30/kg, Cost of sugar is ₹40/kg and Cost of rice is ₹50/kg.

9. The rates of commission for A,B and C are ₹2, ₹4 and ₹11 respectively

10. 39%