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Quant02. Equations

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Revision Notes for CPT …

Revision Notes for CPT
Chapter: Equations
Subject: Quantitative Aptitude

Published in: Education
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• 1. CPTSuccess
Revision Notes – Quantitative Aptitude
Chapter 2. Equations
• 2. Chapter 2. Equations
• An equation is defined to be a mathematical statement of equality.
• 3. Simple equation
• 4. A simple equation in one unknown x is in the form
• 5. ax + b = 0, where a, b are known as constants and a ≠ 0
• 6. A simple equation has only one root
• 7.  Simultaneous linear equations in two unknown
The general form of a linear equation in two unknowns x and y is ax + by + c = 0 where a and b are non-zero coefficients.
Two equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 form a pair of simultaneous equations in x and y. A value for each unknown which satisfies both equations at the same time gives the roots / solution of the equation.
•
Revision Notes – Quantitative Aptitude
CPTSuccess
www.cptsuccess.com
• 8. Chapter 2. Equations
• Methods to Solve Simultaneous linear equations
Elimination method: In this method one unknown is eliminated, thus reducing two linear equations to a linear equation in one unknown. This unknown is solved and its value substituted in the equation to find the other unknown.
Revision Notes – Quantitative Aptitude
CPTSuccess
www.cptsuccess.com
• 9. Chapter 2. Equations
• Methods to Solve Simultaneous linear equations
Cross-multiplication method
For two equation, a1x + b1y + c1 = 0, and a2x + b2y + c2 = 0
Coefficients of x and y and constant term are arranged as:

which gives: x / (b1 c2 – b2 c1) = y / (c1 a2 – c2 a1) = 1 / (a1 b2 – a2 b1)
Hence, x = (b1 c2 – b2 c1) / (a1 b2 – a2 b1)
y = (c1 a2 – c2 a1) / (a1 b2 – a2 b1)
Equations in three variables can also be solved using the above two methods
Revision Notes – Quantitative Aptitude
CPTSuccess
www.cptsuccess.com
• 10. Chapter 2. Equations
An equation in the form ax2 + bx + c = 0, where x is a variable and a, b and c are constants with a ≠ 0 is called a quadratic equation. When b = 0 the equation is called a pure quadratic equation and when b ≠ 0 the equation is called an affected quadratic.

• Roots of a Quadratic Equation
x = [- b ±  (b2 – 4ac)] / 2a
Sum of roots = - b / a = - (coefficient of x / coefficient of x2)
Product of roots = c / a = (constant term / coefficient of x2)
x2 – (sum of roots) x + (product of roots) = 0
Revision Notes – Quantitative Aptitude
CPTSuccess
www.cptsuccess.com
• 11. Chapter 2. Equations
• Roots of a Quadratic Equation
b2 – 4ac is known as the discriminant in the equation as it discriminates the nature of roots of the equation

If b2 – 4ac = 0, the roots are real and equal
If b2 – 4ac > 0, the roots are real and distinct (unequal)
If b2 – 4ac < 0, the roots are imaginary
If b2 – 4ac is a perfect square the roots are real rational and distinct
If b2 – 4ac > 0 but not a perfect square the roots are real irrational and distinct

Other properties
Irrational roots occur in pairs. If p+ q is one root, then the other root p - q
If a = c then one root is reciprocal to the other
If b = 0 the roots are equal but of opposite signs
Revision Notes – Quantitative Aptitude
CPTSuccess
www.cptsuccess.com
• 12. Chapter 2. Equations
• Application of Equations to Coordinate Geometry

Distance of a point P (x, y) from Origin (0, 0) is  (x2 + y2)
Distance between two points P (x1, y1) and Q (x2, y2) is  [(x1 – x2)2 + (y1 – y2)2]
Equation of a straight line is written as y = mx + c, where m is the slope and c is the constant
The Slope of the line is given by, m = (y2 – y1) / (x2 – x1)