EXERCISE - A
1) The sum of three numbers is 6, if we multiply the third number by 2 and add the first number to the result, we get 7, by adding second and third number to three times the first number, we get 12, using Matrices find the numbers. 3,1,2
2) An amount of Rs 5000 is put into 3 investments at the rate of interest of 6%, 7%, 8% per annum respectively. The total annual income is 358. If the combined income from the first two investments is Rs 70 more than the income from the third, find the amount of each investment by Matrix method. 1000,2200,1800
3). A mixture is to be made of three foods A, B, C. The three foods A, B, C contain nutrients P, Q, R as shown below
Ounces per pound of Nutrients
Food P Q R
A 1 2 5
B 3 1 1
C 4 2 1
How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R ? 1,1,1
4) A transport company uses 3 types of trucks A, B , C to transport three types of vehicles M,N, P. The capacity of each truck in terms of 3 types of vehicle is given as follows:
M N P
A 1 3 2
B 2 2 3
C 3 2 2
Using Matrix method, find the number of trucks of each type required to transport 85, 105, 110 vehicles of M,N, P types respectively. 15,20,10
5) In an engineering workshop there are 10 machines for drilling, 8 machines for turning and 7 machines for grinding. Three types of brackets are made. Type I required 0 minutes for drilling, 5 minutes for turning and 4 minutes for grinding. The corresponding times for type II and III brackets are 3, 3, 2 and 3, 2, 2, minutes respectively. How many brackets of each types should be produced per hour so that all the machines remains fully occupied during an hour ? Solve by using Matrix method . 5,55,145
6) Given the following National Income model :
C= a+ bY (a> 0, 0 < b < 1)
I= d + eY (d> 0, 0< e < 1
Y= I + C
Solve the indigenous variables C, I and Y using Matrix method. (a- ac+ bd)/(1- b - c), (d+ ac - bd)/(1- b - c), (a+ d)/(1- b - c)
7) The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1, By adding second and third number to five times the first number, we get 6 p. Find the three numbers by using metrix . 1,-1?2
8) An amount of Rs 10000 is put into 3 investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using Matrix . 2000,3000,5000
9) A company produces three products everyday. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using Matrix method. 11,15,19
10) The price of 3 commodities P, Q and R are Rs x, y, z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q, B purchases 3 units of Q and sells 2 units of P 1 units of R. C purchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earns Rs 6000, Rs 5000 and Rs 13000 respectively. If selling the unit is positive earnings and buying the units is negative earnings , find the price per unit three commodities by using metric method. 3000, 1000, 2000
11) Two factories decided to award their employees for three values of (a) adaptable to new techniques, (b) careful and alert in difficult situations and (c) keeping calm in tense situations at the rate of Rs x, y, z per person respectively. The first factory decided to honour respectively 2,4 and 3 employees with a total prize money of Rs 29000. The second factory decided to honour respectively 5,2 and 3 employees with the prize money of Rs 30500. If the three prizes per person together cost Rs 9500, then
i) represent the above situation by Matrix equation and form linear equations using matrix multiplication.
ii) solve these equations using matrices. 2500,3000,4000
12) A total amount of Rs 7000 depreciated in three different saving bank accounts with annual interest rates 5%, 8% and 17/2% respectively. The total annual interest from these 3 account is Rs 550. Equal amounts have been deposited in the 5%, 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices. 1125,1125,4750
13) A shopkeeper has 3 varieties of pens A, B and C. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeen purchased 4 pens A veriety, 3 pens of B variety and 2 penc of C variety for Rs 60. While Shikha purchased 6 pens of A variety, 2 pens of B variety and 3 pens of C variety for Rs 70. using Matrix method find the cost of each pen. 5,8,8
14) Purvi has invested a part of his investment in 10% bond A and a part in 15% bond B. His interest interest during first year is Rs 4000. if he invests 20% more in 10% Bond A and 10% more in 15% bond B his income during 2nd year increases by Rs 500. Find his initial investments and new investment in bonds A and B using Matrix method. 10000 in A, 20000 in B, new Rs 12000 in A Rs 22000 in B
15) To control a crop disease it is necessary to use 8 units of chemical A, 14 units of chemical B and 13 units of chemical C. One barrel of spray P contains one units of A, 2 units of B and 3 units of C. One barrel of spray Q contains 2 units of A, 3 units of B and 2 units of C. One barrel of spray R contains 1 unit of A, 2 units of B and 2 units of C. Using Matrix method, find how many barrel of each spray be used to just meet the requirement ? P: 1 barrel ; Q: 2 barrel; R: 3 barrel
16) A firm produces two products And B passing through two machines X and Y before completion. X can produce either 10 units of A or 15 units of B per hour. Y can produce 15 units of either product per hour. Find daily production of A and B if time available is 12 hours on machine X and 10 hours on Y per day using Matrix inversion method. 60,90
17) The equilibrium condition for three related markets is given by
11p₁ - p₂ - p₃ = 31
- p₁ + 6p₂ - 2p₃ = 26
- p₁ - 2p₂ + 7p₃ = 24
Using Matrix inversion method, find the equilibrium prices of each market. 4,7,6
EXERCISE - C
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