Download MTE-04

MTE-04
Assignment Booklet
Bachelor’s Degree Programme
Elementary Algebra
School of Sciences
Indira Gandhi National Open University
New Delhi
2007
Dear Student,
Please read the section on assignments in the Programme Guide for Elective Courses that we sent you
after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous
evaluation, which would consist of one tutor-marked assignment for this course. This assignment is in
this booklet.
Instructions for Formating Your Assignments
Before attempting the assignment please read the following instructions carefully.
1) On top of the first page of your answer sheet, please write the details exactly in the following format :
ROLL NO:………………………….
NAME:………………………….
ADDRESS:………………………….
…………………………..
…………………………..
…………………………..
COURSE CODE:………………………….
COURSE TITLE:………………………….
ASSIGNMENT NO:………………………
STUDY CENTRE:………………………...
DATE:…………………….…….
PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND
TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3) Leave a 4 cm. margin on the left, top and bottom of your answer sheet.
4) Your answers should be precise.
5) While solving problems, clearly indicate which part of which question is being solved.
6) The assignment responses are to be submitted to your Study Centre Coordinator by November, 2007,
not later.
Please retain a copy of your answer sheets.
Wish you luck!
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ASSIGNMENT
(To be done after studying the course.)
Course Code : MTE-04
Assignment Code : MTE-04/TMA/2007
Total Marks : 100
1. Which of the following statements are true? Give reasons for your answers. (This means that if you
think a statement is false, give a short proof or an example that shows it is false. If it is true, give a
short proof for saying so. For instance, to show that ‘1, padma, blueis a set’ is true, you need to say
that this is true because it is a well-defined collection of 3 objects.)
i)
{MTE-04, –3, Indira Gandhi} is a set.
ii)
For any two sets A and B, A  Bc = A  B.
iii)
There is a unique z  C for which z  z 1 .
iv)
The least degree of the polynomial with real coefficients and with roots 2+i, 2i – 1 is 2.
v)
If a statement has a direct proof, then it cannot be proved by contradiction.
vi)
The equation x = 3 has the same geometric representation regardless of whether it is an
equation in one variable or two variables.
vii)
Any system of n linear equations in n – 1 variables has a solution.
viii) The CS inequality is a generalization of the triangle inequality.
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a)
(16)
Let m, n, x, y, z be positive real numbers, with x + y + z = 1. Prove that
x4
y4
z4
1



(mx  ny)(my  nx) (my  nz)(mz  ny) (mz  nx)(mx  nz) 3(m  n) 2
[Hint: Apply the AM  GM inequality to each term in the LHS, and then apply the CS
inequality.]
(8)
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b)
Given a, b > 0 such that ab = 18, check whether their sum is greatest when a = b.
a)
Consider the sets. A = {3x │ x Z}, B = {x  Q │ –5  x  10} and C = {factors of 20 in Z}.
Represent A  B, Bc  C and C \ A by the listing method and in a single Venn diagram.
(5)
b)
Give the geometric representation of the Cartesian product {–2, 1} × [–2, 1].
c)
Any subset of A × A is called a relation on the set A. A relation R on A is symmetric if
(a, b)  R  (b, a)  R  a, b  A. Give one example each, with justification, of
i)
(2)
(2)
a symmetric relation on N,
ii) a relation that is not symmetric on the set {2, 3, 5, 7}.
4. a) Obtain the geometric, polar and exponential representations of (i5 – 1)–1.
(3)
(5)
b) Use De Moivre’s theorem to obtain the 6th roots of i5 – 1. Also show them in an Argand diagram
(5)
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5. a) Solve the equation 9x4 – 18x3 – 31x2 + 8x + 12 = 0 by Ferrari’s method.
(8)
b) Solve the equation in (a) above, given that the sum of two of its roots is zero.
(6)
6. a) Obtain the solution set of the system x – 3y + 4z = 9, 4x + 3y + 2z = 7, y – 2x = 5 – 10z by
elimination.
(4)
b) Express the following problem situation as a linear system, and then solve it by substitution. Also
show the linear system geometrically.
A manufacturer produces two types of cupboards – deluxe and regular. Each deluxe cupboard
requires 12 worker hours to cut and assemble, and 5 worker hours to finish. Each regular
cupboard requires 8 worker hours to cut and assemble, and 3 hours to finish. On a daily basis,
the manufacturer has available 440 worker hours for cutting and assembling, and 175 worker
hours for finishing. How many cupboards of each type should be produced so that all the work
power is utilized?
(6)
7. a) In an electrical network it is possible to determine the amount of current in each branch in terms
of the resistances and the voltages. In the figure below, the
represents a battery
(measured in volts) that drives a charge and produces a current. The current will flow out of the
terminal of the battery represented by the longer vertical line, that is,
. The symbol
represents a resistor. The resistances are measured in ohms. The capital letters
represent nodes and the i’s represent the currents between the nodes. The currents are measured in
amperes. The arrows show the direction of the currents. If, however, one of the currents turns out
to be negative, this would mean that the current along that branch is in the opposite direction of
the arrow.
To determine the currents, Kirchhoff’s laws are used:
i)
At every node the sum of the incoming currents equals the sum of the outgoing currents.
ii) Around every closed loop the algebraic sum of the voltage must equal the algebraic sum of
the voltage drops. [The voltage drops E for each resistor are given by Ohm’s law: E = iR,
where i represents the current in amperes and R the resistance in ohms.]
Find the currents in the network given in the figure below using Cramer’s rule.
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b) Write the systems obtained in 6(a) and 6(b) in matrix form.
(2)
8. a) In the context of admission to IGNOU, give examples of the following:
i)
a one-way implication;
ii) the converse of (i) above;
iii) a two-way implication which is true;
iv) a statement involving both  and .
v) the contrapositive of (i) above.
(10)
b) Prove that if A and B are any two sets such that A  B, then A  B = B
i)
by direct method;
ii) by proving its contrapositive;
iii) by contradiction.
(10)
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