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Intersection - Linear Algebra for Arts and Sciences - Exam, Exams of Linear Algebra

Jaypee University of Engineering & TechnologyLinear Algebra

This is the Past Exam of Linear Algebra for Arts and Sciences which includes Linear Combination, Rank, Basis, Column, Dimension, Vectors, Basis, Rank, Condition, Subspace etc. Key important points are: Intersection, System, No Intersection, Point, Plane, Line, Tomato Plants, Insure Rapid, Plant Growth, Fertilizer

Typology: Exams

2012/2013

Uploaded on 02/27/2013

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FINAL EXAM - December 2010 Linear Algebra 201-105-RE Page 1
1. Solve each of the following systems, or explain why the system has no solution.
a)
251852
32
17124
zyx
zy
zyx
b)
11181242
41074
4922
921333
wzyx
wzyx
wzyx
wzyx
2. Given the system
27)19(82
762
9103
2zkyx
zyx
zyx
Find all values of k for which the system is:
a) consistent b) inconsistent.
3. a) Find the intersection of the planes -3x +16y +17z = -39 and x -6y -7z = 15.
b) Which of the following best describes this intersection? Circle your selection.
i) No intersection ii) Point iii) Line iv) Plane v)
3
4. A greenhouse is used to raise plants for winter use. Among these are tomato plants, zucchini
plants and cucumber plants. To insure rapid plant growth, the plants are given two types of
fertilizer each week, Fertilizer A and Fertilizer B. Tomato plants require 9 units of Fertilizer A
and 3 units of Fertilizer B, zucchini plants require 8 units of Fertilizer A and 4 units of Fertilizer
B, cucumber plants require 6 units of Fertilizer A and 6 units of Fertilizer B. If only 252 units of
Fertilizer A and 216 units of Fertilizer B are on hand, how many of each type of plant will get a
proper dose of each fertilizer this week?
a) Define all the necessary variables x, y, and z, and set up the system of equations
required to solve the problem. DO NOT SOLVE.
b) Given that the parametric solution for the problem is
),399,602( ttt
, find all particular
solutions that are realistic.
5. Given
610
226
A
,
59
49
B
,
00
91
84
C
Find the following or explain why they don't exist:
a)
b)
CAt
3
c)
41 )(5 BB
6. Let
A
and
B
be
44
matrices such that
6)det(A
and
2)det(B
. Find:
a)
])det[( t
AB
b)
)det(BAB
c)
)3det( 1
B
d)
])det[( 21
BA
7. Let a simple economy consist of two industries: Paper and Wood. The production of $1 of
paper requires 40¢ of paper and 70¢ of wood. The production of $1 of wood requires 10¢ of
paper and 20¢ of wood. There is external demand for $8200 of paper and $4100 of wood.
a) What dollar amount of paper and wood should be produced to meet the demand?
b) Determine whether or not each industry is profitable. Why or why not?
c) Is the economy productive? Why or why not?
pf3
pf4

Partial preview of the text

Download Intersection - Linear Algebra for Arts and Sciences - Exam and more Exams Linear Algebra in PDF only on Docsity!

  1. Solve each of the following systems, or explain why the system has no solution.

a) 2 5 18 25

x y z

y z

x y z b)

2 4 12 18 11

x y z w

x y z w

x y z w

x y z w

  1. Given the system

2 8 ( 19 ) 27

x y k^2 z

x y z

x y z

Find all values of k for which the system is: a) consistent b) inconsistent.

  1. a) Find the intersection of the planes -3 x +16 y +17 z = -39 and x -6 y -7 z = 15. b) Which of the following best describes this intersection? Circle your selection. i) No intersection ii) Point iii) Line iv) Plane v)^3
  2. A greenhouse is used to raise plants for winter use. Among these are tomato plants, zucchini plants and cucumber plants. To insure rapid plant growth, the plants are given two types of fertilizer each week, Fertilizer A and Fertilizer B. Tomato plants require 9 units of Fertilizer A and 3 units of Fertilizer B, zucchini plants require 8 units of Fertilizer A and 4 units of Fertilizer B, cucumber plants require 6 units of Fertilizer A and 6 units of Fertilizer B. If only 252 units of Fertilizer A and 216 units of Fertilizer B are on hand, how many of each type of plant will get a proper dose of each fertilizer this week? a) Define all the necessary variables x , y , and z , and set up the system of equations required to solve the problem. DO NOT SOLVE. b) Given that the parametric solution for the problem is ( 2 t 60 , 99 3 t , t ), find all particular solutions that are realistic.
  3. Given 0 1 6

A ,

B ,

C

Find the following or explain why they don't exist: a) 2 B AC b) (^3) At C c) 5 ( B^1 B )^4

  1. Let A and B be 4 4 matrices such that (^) det( A ) 6 and (^) det( B ) 2. Find: a) det[( AB ) t ] b) (^) det( BAB ) c) det( 3 B^1 ) d)det[( BA^1 )^2 ]
  2. Let a simple economy consist of two industries: Paper and Wood. The production of $1 of paper requires 40¢ of paper and 70¢ of wood. The production of $1 of wood requires 10¢ of paper and 20¢ of wood. There is external demand for $8200 of paper and $4100 of wood. a) What dollar amount of paper and wood should be produced to meet the demand? b) Determine whether or not each industry is profitable. Why or why not? c) Is the economy productive? Why or why not?
  1. Given 5 5 2

A ,

z

y

x X , and 42

B , find:

a) M 31 b) det( A ) c) adj ( A ) d) A^1 using adj ( A ) e) The solution to AX B using A^1.

  1. Consider the system AX B , where

4

3

2

1 B x

x

x

x A X

a) Find det( A ). b) Use Cramer's rule to find x 3 only.

  1. Given two points P ( 3 , 1 , 4 , 0 ); Q ( 4 , 9 , 2 , 4 )and the line L 1 : x 5 4 t , y 3 2 t , z 2 t , w 2 2 t a) Find the vector PQ. b) Find the magnitude of PQ. c) Is the point P on the line L 1? Explain. d) Find a vector equation of a line L 2 that passes through the point Q and is parallel to L 1.
  2. Suppose A is a 4x7 matrix a) What is the maximum value for the rank of A? b) If dimension of Col( A ) is 2, what is the dimension of Null( A )? c) What is Dim(Col( A )) if the rank of A is 3? d) If the reduced form of A has 3 leading ones, what is the dimension of Col( At )?
  3. Let the set x t y z t wheretisanyrealnumber z

y

x S^32 , 0 , 3

a) Is 0

in S? Justify. b) Is S closed under scalar multiplication? Justify. c) Is S closed under addition? Justify. d) Is S a subspace of 3?

  1. Let , 1

u 1

u 2

u 3

u 4

u 5

ANSWERS:

  1. a) (5+4 t , -3-2 t , t ) b) inconsistent
  2. a) k 3 b) k 3
  3. a) ( x , y , z ) ( 3 5 t , 3 2 t , t ) b) iii) Line
  4. a) Let x = # of properly fertilized tomato plants y = # of properly fertilized zucchini plants 9 x + 8 y + 6 z = 252 z = # of properly fertilized cucumber plants 3 x + 4 y + 6 z = 216 b) 30 t 33 therefore solutions are: (0, 9, 30), (2, 6, 31), (4, 3, 32), (6, 0, 33)
  5. a) 19 19

4 74 b) 6 18

c) 0 5

  1. a) 12 b) 24 c) 2

81 d) 9

  1. a) $17 000 of paper and $20 000 of wood b) wood is profitable, paper not profitable c) economy is productive
  2. a) -11 b) -63 c) 45 15 6

d) 45 15 6

e) , 19 ) 3

,^17

(^31

  1. a) -34 b) 17
  1. a) ( 1 , 10 , 2 , 4 ) b) 11 c) P is not on the line d)( x , y , z , w ) ( 4 , 9 , 2 , 4 ) t ( 4 , 2 , 1 , 2 )
  2. a) 4 b) 5 c) 3 d) 3
  3. a) yes b) yes c) yes d) yes
  4. a) 3 k^1 b) i) L.D ii) L.I iii) L.I c) basis for Col(U) = { u  1 (^) , u  3 , u  4 },then u  (^) 2 3 u  1 , u  5 (^) 2 u  1 3 u  3 dim(Col(U)) = 3 d) Null(U) = span {( 3 , 1 , 0 , 0 , 0 ),( 2 , 0 , 3 , 0 , 1 )}, nullity (U) = 2
  5. a) Let x = # of cups made b) Maximize P 2 x 1. 5 y y = # of plates made c) 6 x 3 y 1200
  6. 5 x y 250 x 0 , y 0 d) Max P = $500 when 100 cups and 200 plates are made
  7. Min P = -14 at ( 2 , 0 , 1 , 1 , 0 , 0 )
  8. a) region with corners A(0, 0), B(0, 120), C(80, 120), D(150, 50), E(150, 0) b) Max P = 760 at (80, 120) c) 1- A, 2 - B, 3 - C (Max) (can be different; depends on steps from b) )