MTH501 Midterm Past Paper 9

MTH501 Linear Algebra · Mid Term Examination - November 2004 · Time Allowed: 90 Minutes · Instructions

make an appropriate assumption and use it to solve the problem.

**WARNING: Please note that Virtual University takes serious note of

unfair means. Anyone found involved in cheating will get an `F` grade in this course.

Questions: 11

Q1. Consider the system of Linear equations

x1 4x2� 5x3 0 2x1� x2 8x3 9 x1 x2� 8x3 2 Then determinant of the Matrix of Coefficients of the above system is (a) 91 (b) 123 (c) 141 (d) 81

Total

Marks : 1

A and B2

A B

T 9

18

0

45

9

Q2. Marks : 2

2 0 20

2 5 12 1 2 x y1� 2

2 3 x� y0 2

Then the values of x and y are x� 5 y

3x such that

(a) (b) (c)

(d)

x = - 2 and y = 4 x = 2 and y = - 6 x = 0 and y = 0

x = 2 and y = -4

Q3. Which statement about the set S is false where S = {(1, 1, 3), (2, 3, 7),(2, 2, 6)}

Marks : 1

(a)

(b)

(c) (d) The set S contain an element which is solution of the equation 5x – yz = 0 The Set S is linearly independent. The set S contain two elements which are multiple of each other. The Set S is linearly dependent.

Q4. Marks : 2

Consider a linear transformation T such that 1 6

Then 5

T25 15

9

(a)

(b) 1

5

30

(c)

12

(d) is

Q5. How many subspaces R2 have?

Marks : 1

(a) (b)

(c)

(d) only two: {0} and R2 Only four: {0} x- axis and y -axis and R2 Infinitely many. None of the above.

Q6. Marks : 8

An oil refinery produces low-sulfur and high-sulfur fuel. Each ton of low-sulfur requires 5 minutes in the blending plant and 4 minutes in the refining plant; each ton of high sulfur fuel requires 4 minutes in the blending plant and 2 minutes in the refining plant. If the blending plant is available for 3 hours and the refining plant is available for 2 hours, how many tons of each type of fuel should be manufactured so that the plants are fully utilized?

Solution:

The data given in the question can be formed into system of linear equations as Low-sulfur High-sulfur Blending plant Refining Plant As we are given in the question that we have blending plant and refining plant available for 3 and 2 hours respectively. Let x tons of low-sulfur and y tons of high sulfur be the amount should be manufactured so that plants are fully utilized. Then from the above data we must have the system,

5x + 4y = 180 4x + 2y = 120 Augmented matrix for the above system 5 4 1801 4 2 120 ∼4 2 60

2 120 by R1� R2

1 2 60 0� 6�120 So we have x + 2y = 60 and -6y = - 120, thus we have y= 20 tons and x = 20tons are the required manufactured tons of each low-sulfur and high-sulfur so that we can utilize both plants for the given time.