HSC Science (Computer Science) 12th Board ExamMaharashtra State Board
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Solution - The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices. - HSC Science (Computer Science) 12th Board Exam - Mathematics and Statistics

Question

The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices.

Solution

Let Rs.’x’, Rs.’y’ and Rs.’z’ be the cost of one dozen pencils, one dozen pens and one dozen erasers.
Thus, the system of equations are:

`{:(4x+3y+2z=60),(2x+4y+6z=90),(6x+2y+3z=70):}`

Let us write the above equations in the matrix form as:

`[[4,3,2],[2,4,6],[6,2,3]][[x],[y],[z]]=[[60],[90],[70]] " i.e "AX=B`

`"Using "R_z->R_2-1/2R_1 and R_3->R_3-3/2R_1`

`[[4,3,2],[0,5/2,5],[0,-5/2,0]][[x],[y],[z]]=[[60],[60],[-20]]`

`"Using "R_3->R_3+R_2`

`[[4,3,2],[0,5/2,5],[0,0,0]][[x],[y],[z]]=[[60],[60],[40]`

As matrix A is reduced to its upper triangular form we can write

4x + 3y + 2z = 60..........(i)

5/2y + 5z = 60..........(ii)

5z = 40
z = 8.....(iii)

Substituting (iii) in (ii) we get,


5/2y + 5(8) = 60

y = 8.....(iv)

Substituting (iii) and (iv) in (i) we get,
4x + 3 (8) + 2 (8) = 60
x = 5

∴ Cost of one dozen pencils, one dozen pens and one dozen erasers is Rs. 5, Rs. 8 and Rs. 8 respectively

Is there an error in this question or solution?

APPEARS IN

 2015-2016 (March) (with solutions)
Question 2.2.3 | 4 marks
 2015-2016 (July) (with solutions)
Question 3.2.2 | 4 marks

Reference Material

Solution for question: The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each item per dozen by using matrices. concept: null - Elementary Operation (Transformation) of a Matrix. For the courses HSC Science (Computer Science), HSC Science (General) , HSC Arts, HSC Science (Electronics)
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