Equations and matrices– CA Foundation, CPT notes, PDF

This article is about Equations and matrices – Business Mathematics and Logical Reasoning & Statistics for CA foundation CPT students. we also provide PDF file at the end.

Equations and matrices

What we will study in this chapter: Equations and matrices

LEARNING OBJECTIVES

After studying this chapter, you will be able to:

Understand the concept of equations and its various degrees – linear, simultaneous, quadratic and cubic equations.

Know how to solve the differential equations using different methods of solutions.

UNIT OVERVIEW

Unit overview

2.1.1 INTRODUCTION:

The equation is defined to be a mathematical statement of equality. If equality is true for a certain value of the variable involved, the equation is often called a conditional equation and equality sign ‘=’ is used; while if the equality is true for all values of the variable involved, the equation is called an identity.

For Example: \frac { x+2 }{ 3 } +\frac { x+3 }{ 2 } =3   holds true only for

So it is a conditional. On the other hand,

is an identity since it holds for all values of the variable x.

Determination of value of the variable which satisfies an equation is called solution of the equation or root of the equation. An equation in which highest power of the variable is 1 is called a Linear (or a simple) equation. This is also called the equation of degree 1. Two or more linear equations involving two or more variables are called Simultaneous Linear Equations. An equation of degree 2 (highest Power of the variable is 2) is called Quadratic equation and the equation of degree 3 is called Cubic Equation.

For Example: is a Linear equation.

is a Quadratic equation.

is a Cubic equation.

are jointly called Simultaneous equations.

2.1.2 SIMPLE EQUATION

A simple equation in one unknown x is in the form

Where a, b are known constants and

Note: A simple equation has only one root.

Example:

Solution: By transposing the variables in one side and the constants in other side we have

UNIT-I : EXERCISE (A)

Choose the most appropriate option (a) (b) (c) or (d).

ILLUSTRATIONS:

UNIT-I : EXERCISE

(B) Choose the most appropriate option (a) (b) (c) or (d).

2.1.3 SIMULTANEOUS LINEAR EQUATIONS IN TWO UNKNOWNS

The general form of a linear equations in two unknowns x and y is where a, b are non-zero coefficients and c is a constant. Two such equations and form a pair of simultaneous equations in x and y. A value for each unknown which satisfies simultaneously both the equations will give the roots of the equations.

2.1.4 METHOD OF SOLUTION

Elimination Method: In this method two given linear equations are reduced to a linear equation in one unknown by eliminating one of the unknowns and then solving for the other unknown.

Example 1: Solve: and

Solution:

By making

and by making

Adding. Substituting this values of x in  we find;

Cross Multiplication Method: Let two equations be:

We write the coefficients of x, y and constant terms and two more columns by repeating the coefficients of x and y as follows:

1 2 3 4 b1 c1 a1 b1 b2 c2 a2 b2

and the result is given by:

so the solution is :

Example 2: Solve

Solution:

Method of elimination: By  we get

Adding (ii) & (iii) we get

or

Putting we get

or,

So

Method of cross-multiplication:

or

or

or

2.1.5 METHOD OF SOLVING SIMULTANEOUS LINEAR

EQUATION WITH THREE VARIABLES

Example 1: Solve for x, y and z:

Solution: (a) Method of elimination

By (i) × 2 we get

By [the variable z is thus eliminated]

By

By

By

Putting

Putting

or

or

or

So is the required solution.

(b) Method of cross multiplication

We write the equations as follows:

By cross multiplication

Substituting above values for x and y in equation (iii) i.e., we have

[the variable z is thus eliminated]

or

or

Now

Thus

Example 2: Solve for x y and z:

Solution: We put and get

By

By

or

By

By

By

Putting

or By

or By

Putting

or

Thus is the solution.

Example 3: Solve for x, y and z

Solution: We can write as

By we get

or

By

By

Required solution is

UNIT-I : EXERCISE (C)

Choose the most appropriate option (a), (b), (c) or (d).

UNIT-I : EXERCISE (D)

Choose the most appropriate option (a), (b), (c) or (d) as the solution to the given set of equations:

2.1.6 PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS

ILLUSTRATIONS:

UNIT-I : EXERCISE (E)

Choose the most appropriate option (a), (b), (c) or (d).

2.1.7 QUADRATIC EQUATION

An equation of the form where x is a variable and a, b, c are constants with a  is called a quadratic equation or equation of the second degree.

When  the equation is called a pure quadratic equation; when the equation is called an affected quadratic.

Examples:

1. i)
2. ii)

iii)

The value of the variable say x is called the root of the equation. A quadratic equation has got two roots.

How to find out the roots of a quadratic equation:

or

or

or

or

or

Sum and Product of the Roots:

Let one root be and the other root be

Now

Thus sum of roots

Net

So the product of the roots

2.1.8 HOW TO CONSTRUCT A QUADRATIC EQUATION

For the equation we have

or

or

or

2.1.9 NATURE OF THE ROOTS

1. i) If the roots are real and equal;
2. ii) If then the roots are real and unequal (or distinct);

iii)  If  then the roots are imaginary;

1. iv) Ifperfect squarethe roots are real, rational and unequal (distinct);
2. v) If but not a perfect square the rots are real, irrational and unequal.

Since discriminates the roots is called the discriminate in the equationas it actually discriminates between the roots.

Note:   (a)        Irrational roots occur in conjugate pairs that is if is a root then             is the other root of the same equation.

(b)        If one root is reciprocal to the other root then their product is 1 and so

(c)        If one root is equal to other root but opposite in sign then.

their sum = 0 and so i.e.

Example 1:  Solve

Solution:  1st method :

or

or

or

or

2nd method (By formula)

Here a = 1, b = –5 , c = 6 (comparing the equation with

Example 2:  Examine the nature of the roots of the following equations.

1. i) ii)

iii)                   iv)

Solution: (i)      a = 1, b = –8, c = 16

The roots are real and equal.

(ii)

and a perfect square

The roots are real, rational and unequal

(iii)

The roots are imaginary and unequal

(iv)

The roots are real and unequal. Since is not a perfect square the roots are realirrational and unequal.

ILLUSTRATIONS:

1. If and be the roots of find the equation whose roots are

SOLUTION:    Now sum of the roots of the required equation

2

Product of the roots of the required equation

Hence the required equation is

or

2. If be the roots of find the value of

SOLUTION:

3. Solve

SOLUTION:

or

or ( taking or )

or

or

orTherefore

4. Solve

Solution:

or

or

or           Taking

or

or

or

or

i.e.

i.e.

5. Solve i.e.

SOLUTION   

or

or

orwhen

or

or

or

or

or

For i.e,

For i.e,

6. If one root of the equation is from the equation given that the roots are irrational

SOLUTION    other roots is   sum of two roots

Product of roots

Required equation is : – (sum of roots )x + (Product of roots) = 0

or

7. If are the two of the equation from the equation

whose roots are

SOLUTION :   As are the roots of the equation

Now

Required equation is

or

8. If the roots of the equation

are equal show that

SOLUTION:    Since the roots of the given equation are equal the discriminant must be

zeroie.

or

or

or

or

UNIT-I : EXERCISE (F)

Choose the most appropriate option (a) (b) (c) or (d).

UNIT-I : EXERCISE (G)

Choose the most appropriate option (a) (b) (c) or (d).

2.1.10 PROBLEMS ON QUADRATIC EQUATION

UNIT-I : EXERCISE (H)

Choose the most appropriate option (a) (b) (c) or (d).

2.1.11 SOLUTION OF CUBIC EQUATION

On trial basis putting if some value of x stratifies the equation then we get a factor. This is a trial and error method. With this factor to factorise the LHS and then other get values of x.

ILLUSTRATIONS:

UNIT-I : EXERCISE (I)

Choose the most appropriate option (a), (b), (c) or (d)

SUMMARY

• A simple equation in one unknown xis in the form

Where a, b are known constants and

• The general form of a linear equations in two unknowns x and y is wherea, b are non-zero coefficients and c is a constant. Two such equations and form a pair of simultaneous equations in x and y. A value for each unknown which satisfies simultaneously both the equations will give the roots of theequations.
• Elimination Method: In this method two given linear equations are reduced to a linear

equation in one unknown by eliminating one of the unknowns and then solving for the other unknown.

• Cross Multiplication Method:Let two equations be:
• An equation of the form where xis a variable and a, b, c are constantswith  is called a quadratic equation or equation of the second degree.

When b=0 the equation is called a pure quadratic equation; when b = 0 the equation is

called an affected quadratic.

• The roots of a quadratic equation:
• The Sum and Product of the Roots of quadratic equation
• sum of roots
• product of the roots
• To construct a quadratic equation for the equation we have
• (Sum of the roots ) x + Product of the roots = 0
• Nature of the roots
• If the roots are real and equal;
• if then the roots are real and unequal (or distinct);
• If then the roots are imaginary;
• If is a perfect square the roots are real, rational and unequal (distinct);
• If but not a perfect square the rots are real, irrational and unequal.

Since discriminates the roots is called the discriminant in theequationas it actually discriminates between the roots.

ANSWERS

UNIT –I

Exercise (A)

Exercise (B)

Exercise (C)

Exercise (D)

Exercise (E)

Exercise (F)

Exercise (G)

Exercise (H)

Exercise (I)

ADDITIONAL QUESTION BANK

ANSWERS

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