qm problem set 1

7/31/2019 QM Problem Set 1

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INDIAN INSTITUTE OF MANAGEMENT CALCUTTA

Qualifying Mathematics (OM 100) Course Problem Set

- Page 1

Prof.Bodhibrata Nag

Q.1. Let A & B be sets. What does it mean if we say that A is NOT equal to B?

(a) For every x in A, there is some y in B, such that x is not equal to y (b) There is some element x

of A which is not in B, and there is some element y in B which is not in A. (c) For every x in A and

every y in B, x is not equal to y (d) There is some x in A and some y in B such that x is not equal to

y (e) Either there is some element x of A which is not in B, or there is some element y in B which is

not in A, or both

A.(e)

Q.2. Prove that C)CA()CB()CBA( =IUIUII

Q.3. If A={1,2,3}, B={3,4,5,6}, C={1,2,7,5}, find

(a) BAU (b) CBI (c) BBU (d) A-B (e) B-A (f) ( ) CBA UU (g) ( )CBA UI (h) ( ) CBA II (i) A-(B-C) (j) ( )CBA U (k) A x BA.(a) {1,2,3,4,5,6} (b) {5} (c) {3,4,5,6} (d) {1,2} (e) {4,5,6} (f) {1,2,3,4,5,6,7}

(g) {1,2,3} (h) (i) {1,2} (j) {1,2,3,4,6} (k) {(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5), (2,6), (3,3),(3,4), (3,5),(3,6)}

Q.4. If A={1,2,3}, B={1,2,3,4}, C={3,4,5,6,7}, U={1,2,3,4,5,6,7,8}, find C,B,A .

A. A ={4,5,6,7,8}, B ={5,6,7,8},C ={1,2,8}

Q.5. An analysis of an insurance companys new policyholders revealed the following figures:

796 bought automobile insurance

402 bought life insurance

667 bought home insurance

347 bought automobile and life insurance

580 bought automobile and home insurance

291 bought life and home insurance

263 bought automobile, life and home insurance

What is the number of new policy holders?

A.910

Q.6. In a survey of 75 college students, it was found that of the three news magazines A,B and C:23 read A

18 read B

14 read C

10 read A and B

9 read A and C

8 read B and C

5 read all three

(a) how many read none of these three magazines? A.42

(b) how many read A alone? A.9

(c) how many read B alone? A.5

(d) how many read C alone? A.2


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Prof.Bodhibrata Nag

(e) how many read neither A nor B? A.44

(f) how many read A or B or both? A.31

Q.7. Let A be a 3 x 4 matrix and let B be a 3 x 4 matrix. Which of the four operations A x B, B x

A, A +B, A-B make sense?

(a)none of the operations make sense (b) all the operations make sense (c)A x B makes sense (d) A

x B and B x A make sense (e) A+B and A-B make sense

A.(e)

Q.8. For matrices

=

31-0

1-30

001

A ,

=

1-10

110

001

P ,

=

1-10

110

002

2

1Q , find (a)PA (b) AQ (c)

PQ (d) (PA)Q (e) P(AQ) (f) Q

T

A

T

P

T

A.(a)

=

4-40

220

001

PA (b)

=

2-10

210

001

AQ (c) PQ=I (d),(e),(f)

400

020

001

Q.9. Given

=

213

201A ,

=

01

32

01

B , compute (a)AB (b)BA .

A.(a)

=

3703AB (b)

=

201

10311

201

BA

Q.10. Calculate the given matrix products:

(a)

101

010

101

2-04

12-1-

21-1

A.

202

02-0

31-3

(b)

2-04

12-1-

21-1

101

010

101

A.

01-5

12-1-

01-5

(c)

210

233

1-13

120

21-1A.

676

100

(d)

1-52

13-1-

1-2-2

001-

01-0

1-00

A.

122-

1-31

15-2-


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Prof.Bodhibrata Nag

(e)

1120

321-1

1-1

32

11-

11

A.

213-1

9742

2-1-31-

4311

Q.11. Solve Ax=b by row reducing the augmented matrix, where

=

413

3-1-2

111

A ,

=

3

2

1

x

x

x

x and

=

5

1

0

b .

A.x1=-1, x2=2,x3=-1

Q.12. Solve the system of linear equations by reducing the augmented matrix to row reduced

form:

(a) 2x+4y-z+w=1, x+2y+3w=4, x+y+3z=2, 3y+2z+w=-1

A.x=2,y=-1,z=1/3,w=4/3

(b) x+y+z+w=9,x+y+2z=11, x+2z-w=8,y+3z-w=11

A.x=2,y=1,z=4,w=2

(c) x+y+z+4w=0, x-y-2z=0, x+2y+8z-w=0, y+3z+w=0

A.x=y=z=w=0

(d) x+z=2,y+w=4,z+v=6,x+w=8,y+v=0

A.x=0,y=-4,z=2,w=8,v=4

Q.13. Solve the following underdetermined system by reducing its augmented matrix to row

reduced form. State the solution in terms of a parameter t.

(a) x-z=2,y+z=7 A.x=2+t,y=7-t,z=t

(b) x+y-2z=2, -3x+y+6z=7 A.x=-5/4+2t,y=13/4,z=t

Q.14. Compute the following determinants:

(a)

1234

1-123

11-12

1-11-1

(b)

4-102

1-303

0124

0231

(c)

13101

32202

002-00

1411-2

31-22-1

A.(a) 27 (b) 120 (c) 32

Q.15. Find the solution to the following equations by using matrix inverse:

(a) 2x+y+3z=3, 2y+z=2, x+y+2z=1

(b) 2x+y-z=2, x+2y+3z=1, 2x+3y+4z=1


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(c) 2x-y+3z=33, y=-2, 2x+y+z=11

(d) x+z=2, -x+2z=1, y+3z=3

A.(a) x=6,y=3,z=-4 (b)x=-4, y=7, z=-3 (c) x=2,y=-2,z=9 (d) x=z=1,y=0

Q.16. Find the inversion of the following matrices using the principle of adjoint:

(a)

425

210

301

(b)

3201

2412

0113

1230

(c)

8234

3100

2-010

1001

(d)

1005

0102

0013

5231

(e)

400

221

112

(f)

214

011

023

(g)

7210

1431-1

822-1

2100

A.(a)

1-25

21110-

36-0

15

1(b)

2817-25

17-2813-5-

213-820

55-205

55

1(c)

12-3-4-

3-10912

24-2-8-

1-238

4

1

(d)

1210-15-5

10-336-2

15-6-283

5231

37

1(f)

153-

062-

04-2

2

1(g)

1-11-1

22-21-

43-35-

1210-1116

Q.17. Find the inverse of each matrix A and solve the equation AX=B by use of A-1

:

(a)

=

=

11

7B,

31

12A A.

=

=

3

2X,

21-

1-3

5

1A 1

(b)

=

=

11

8

3

B,

232

4-103

1-31

A A.

=

=

2

1

2

X,

1311-

1414-

2-9-32

A 1

(c)

=

=17

166

B,

432

273121

A A.

=

=

3

11

X,

115-

128-3-5-22

A 1

(d)

=

=

12

35

15

B,

2-2-1-

1115

321

A A.

=

=

3

2

2

X,

101

1419

31-2-20

A 1

(e)

=

=

5

5

3

B,

11-0

1-01

01-2

A A.

=

=

2

3

3

X,

1-2-1

2-2-1

1-1-1

A 1


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Prof.Bodhibrata Nag

Q.18. Solve the following equations by Cramers Rule:

(a) 2x-z=1, 2x+4y-z=1, x-8y-3z=-2 A. x=1,y=0,z=1

(b) 2x+y+z=8, 3x-y-2z=-6, x-y+3z=7 A. x=1,y=3,z=3

(c) 2x+3y-z=9,3x-2y-z=1,2x-y+2z=4 A. x=2,y=2,z=1

Q.19. The frozen sea fish market is essentially made up of two competing producers- Big Fish

and Tasty Ones. Suppose that 65% of the customers buying the Big Fish product one month will

remain loyal the next month, while 25% of those buying the Tasty Ones product one month switch

to Big Fish the next month. Set up the brand switching matrix describing the loyalty transitions

from one product to the other. Supposing that the original market share of Big Fish is 90%,

determine its share two months hence. What is the equilibrium value of market share of Big Fish?

A.

0.750.35

0.2565.0, 49%, 42%.

Q.20. Three major suppliers dominate the compact disk market, their products being MAG5,

D5MR and 5MEM. Computer users regularly purchase these disks in packs of 20 on a quarterly

basis. Of those users purchasing MAG5 one quarter, 90% will purchase the same product the next

quarter, the remainder evenly switching between D5MR and 5MEM; 70% of D5MRs customers

remain loyal, 20% switching to 5MEM; of those customers purchasing 5MEM one quarter, an

average 10% will switch to MAG5, and another 10% will switch to D5MR the next quarter. Set up

a brand switching matrix describing this consumer behaviour.

A.

0.80.20.05

0.10.70.05

0.10.19.0

Q.21. A small country produces three forms of energy- electricity, oil and coal. Production of 1

unit of electricity requires 0.2 units of electricity, 0.2 units of oil and 0.1 units of coal. Production

of 1 unit of oil requires 0.4 units of electricity, 0.4 units of oil and 0.2 units of coal. Production of 1

unit of coal requires 0.1 units of electricity, 0.1 units of oil and 0.3 units of coal. What gross

production is necessary to meet an external demand for 100 units of electricity, 200 units of oil and

100 units of coal.

A. 440 units electricity, 540 units oil and 360 units coal.

Q.22. If one is solving three linear equations involving two unknowns, what happens?(a)There will always be infinitely many solutions (b)Usually there will be infinitely many solutions,

but occasionally there will be one or no solutions (c) Usually there will be no solution, but

occasionally there will be one or more solutions (d)There will always be exactly one solution (e)

There will never be a solution

A.(c)

Q.23. Determine whether the following systems are consistent, inconsistent or dependent. If

consistent, determine the solution:

(a) x+y=2, x-y=4 A.consistent, x=3,y=-1

(b) y=2x+8, x-y/2=-4 A.dependent

(c) y=-2x-5, y=x/2+5 A.consistent, x=-4,y=3


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(d) x/2+y/6=1,x/3+y/3=1 A.consistent,x=y=3/2

(e) 12x+8y=5,-9x-6y=5 A.inconsistent

(f) y=-2x-3,y=x/2+5 A.consistent, x=-16/5,y=17/5

Q.24. In an election, the three candidates for office received 20%, 27% and 53% of the vote,

respectively. The winning candidate had a majority of 732 votes over the total of her opponents.

How many votes did each get? Set up a system of linear equations and solve by the Gauss-Jordan

method.

A.2440,3294,6466

Q.25. In the following exercises, the augmented matrix for a system of linear equations has been

changed to row echelon form. Decide in each case whether the corresponding system is consistent

or inconsistent and, if it has any solutions, find all of them.

(a)

100

210

531

(b)

0000

4100

12-2-1

(c)

0000

10002100

4-23-1

A.(a) inconsistent (b) consistent, solution is 9+2t,t,4 for any real number t

(c) inconsistent

Q.26. Use a Gauss Jordan reduction of the appropriate matrix to find all the solutions to the given

system of linear equations, if any exist.

(a) x+y+z=0, x-y+z=0, x-3y+z=0 A.(-t,0,t) for t (b) x-y+z=0, 2x+3y-5z=0, 3x+2y-4z=0 A. (2t,7t,5t) for t (c) x+y=w, w+x=y, w+z=2x, x-y=z A. (0,t,-t,t) for t (d) x-2y-2z=-1, 2x+3y-z=1,x+y-z=0 A. (1,0,1)

(e) x-y-z=3, 2x-2y+3z=5 A.

+5

1,,

5

514t

tfor t

(f) x+2y-2z=0, 3x-y+3z=1, 4x+y+z=2 A.inconsistent

(g) u+v-w-x=0,u-v+w-x=0, u+v-w+x=0 A.(0,t,t,0) for t

(h) x-3y+z=1, 2x+y-z=2, x+4y-2z=1, 5x-8y+2z=5 A.

+ ttt ,

7

3,

7

21 for t

Q.27. Write the augmented matrix for the system of equations:

x-y+z=1, x+2y-z=2, 2x-3y+3z=1

Reduce the matrix to its reduced row echelon form and solve the system, if it has any solution.

A.(2,-1,-2)

Q.28. Use a Gauss-Jordan reduction of the appropriate matrix to find all of the solutions to the

following system of equations, if any exist:

-x-3y+z=1, -2x+y-z=2, -x+4y-2z=1, -5x-8y+2z=5

A.x=-1,y=0,z=0 is the unique solution

Q.29. Use a Gauss-Jordan reduction of the appropriate matrix to find all of the solutions to thefollowing system of equations, if any exist:


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Prof.Bodhibrata Nag

3x-3y+z+2w=1, -2x+y-z+3w=3, -x+4y-2z+w=-2, 6x-2y+2z+4w=8

A.x=-1, y=3/2, z=53/10, w=8/5 is the unique solution

Q.30. Given that the homogeneous system of equations,

(1-a)x+y-3z=0, ay+2z=0, (1+a)z=0

(a) does not have a unique solution, find all possible values of a; then find all the solutions of the

resulting homogeneous system of equations (b) for what values of a does the homogeneous system

of equations have a unique solution? What is the unique solution?

A.(a)a=0,1,-1; for a=0, x=-t,y=t,z=0, t being any real number; for a=-1, x=t/2, y=2t,z=t; for a=1,

x=t,y=z=0 (b) for all a different from a=0,1,-1, the system has the unique/trivial solution x=y=z=0.

Q.31. Find the value of Rs 1000 deposited in a bank at 10% interest for 8 years compounded (a)

annually (b) quarterly (c) continuously

A.(a) 2144(b) 2204 (c)2226

Q.32. A loan shark lends you Rs 100 at 2% compound interest per week rate. How much will you

owe after 3 years? What is the sharks profit on such a loan?

A. 2196,2096

Q.33. A rich uncle wants to make you a millionaire. How much money must he deposit in a trust

fund paying 8% compounded quarterly at the time of your birth to yield Rs 1,000,000 when you

retire at age 60?

A.8629

Q.34. A Rs 15,000 machine depreciates by 35% per year. Find its value after 4 years?

A. 2678

Q.35. A sum is invested at 12% interest compounded quarterly. How soon will it double in value?

A. 5.9 years

Q.36. Find the inverse of the given functions:

(a) )3x(5)x(h = A. 35

x)x(h 1 +=

(b) )3x(2)x(g = A.2

x3)x(g 1 =

(c) 4x)x(d3

+= A.31

4x)x(d =

(d) 3 4x)x(p += A. 4x)x(p 31 =

(e)3x

2)x(R

= A. 3

x

2)x(R 1 +=

Q.37. Determine the domain of the functions in interval notation:

(a)2x

x)x(G

= A. x2

(b) t115)t(G = A. (-,5/11]


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(c)3w

5)w(P

= A. (3, +)

Q.38. Determine whether the function given is even, odd or neither:

(a)1x

x2)x(g

2 = A.odd

(b) ( )22 x3x)x(f = A.neither(c) 2x2x)x(G = A.even

Q.39. Determine the equation of the line described in the form y=mx+b.

(a) The line passes through (2,1) and (-1,-8) A.y=3x-5

(b) The line passes through (2,1) and is parallel to the line 3x-7y=5 A.7

1x

7

3y +=

(c) The line passes through (12,-8) and is perpendicular to the line 12x+7y+10=0. A. 15x12

7y =

(d) The line is the perpendicular bisector of the line segment with end points (5,10) and (1,-2).

A. 5x3

1y +=

Q.40. Determine the functions f+g,f-g,fg and f/g and state their domains:

(a) ( )2224 3x)x(g,xx2)x(f +== A. ( ) ( ) ( )( ) 24682424 x9x12x11x2xfg,9x7x)x(gf,9x5x3)x(gf ++==++=+

( )( )

+=

:domainall,

3x

xx2x

gf

22

24

(b) ( )2x

2xxg,

x

3)x(f

+==

A. ( )( ) 0x:domain,x

2x4xgf

2

+=+ ; ( )( ) 0x:domain,

x

2x2xgf

2

=

( )( ) 0x:domain,x

6x3xfg

3

+= ; ( ) 2,0x:domain,

2x

x3x

gf

+=

(c)x

2x)x(g,4x

x)x(f +=+=

A. ( )x4x

8x6x2)x(gf

2

2

+++

=+ , domain: x0,-4; ( )( )x4x

8x6xgf

2 +

= , domain: x0,-4

( )( )4x

2xxfg

++

= , domain: x0,-4; ( )( )( )4x2x

xx

gf

2

++=

, domain: x0,4,-2

Q.41. Find gfo and fg o for the following functions:

(a) ( ) ( )

x

1xg,x5xf == A. ( )( )xgfo =

x

5, ( )( )xfg o =

x5

1


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