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# Quant02. Equations

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Revision Notes for CPT
Chapter: Equations
Subject: Quantitative Aptitude

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### Quant02. Equations

1. 1. CPTSuccess<br />Revision Notes – Quantitative Aptitude<br />Chapter 2. Equations<br />
2. 2. Chapter 2. Equations<br /><ul><li> An equation is defined to be a mathematical statement of equality.
3. 3. Simple equation
4. 4. A simple equation in one unknown x is in the form
5. 5. ax + b = 0, where a, b are known as constants and a ≠ 0
6. 6. A simple equation has only one root
7. 7.  Simultaneous linear equations in two unknown</li></ul>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. <br />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.<br /><ul><li> </li></ul>©<br />Revision Notes – Quantitative Aptitude<br />CPTSuccess<br />www.cptsuccess.com<br />
8. 8. Chapter 2. Equations<br /><ul><li>Methods to Solve Simultaneous linear equations</li></ul>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.<br />©<br />Revision Notes – Quantitative Aptitude<br />CPTSuccess<br />www.cptsuccess.com<br />
9. 9. Chapter 2. Equations<br /><ul><li>Methods to Solve Simultaneous linear equations</li></ul>Cross-multiplication method<br />For two equation, a1x + b1y + c1 = 0, and a2x + b2y + c2 = 0 <br />Coefficients of x and y and constant term are arranged as:<br />  <br />which gives: x / (b1 c2 – b2 c1) = y / (c1 a2 – c2 a1) = 1 / (a1 b2 – a2 b1) <br />Hence, x = (b1 c2 – b2 c1) / (a1 b2 – a2 b1)<br /> y = (c1 a2 – c2 a1) / (a1 b2 – a2 b1) <br /> Equations in three variables can also be solved using the above two methods<br />©<br />Revision Notes – Quantitative Aptitude<br />CPTSuccess<br />www.cptsuccess.com<br />
10. 10. Chapter 2. Equations<br /><ul><li>Quadratic Equations</li></ul>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. <br /> <br /><ul><li>Roots of a Quadratic Equation</li></ul>x = [- b ±  (b2 – 4ac)] / 2a<br />Sum of roots = - b / a = - (coefficient of x / coefficient of x2)<br />Product of roots = c / a = (constant term / coefficient of x2)<br />Construction of a Quadratic Equation<br />x2 – (sum of roots) x + (product of roots) = 0<br />©<br />Revision Notes – Quantitative Aptitude<br />CPTSuccess<br />www.cptsuccess.com<br />
11. 11. Chapter 2. Equations<br /><ul><li>Roots of a Quadratic Equation </li></ul>b2 – 4ac is known as the discriminant in the equation as it discriminates the nature of roots of the equation<br /> <br />If b2 – 4ac = 0, the roots are real and equal<br />If b2 – 4ac > 0, the roots are real and distinct (unequal)<br />If b2 – 4ac < 0, the roots are imaginary<br />If b2 – 4ac is a perfect square the roots are real rational and distinct<br />If b2 – 4ac > 0 but not a perfect square the roots are real irrational and distinct<br /> <br />Other properties<br />Irrational roots occur in pairs. If p+ q is one root, then the other root p - q<br />If a = c then one root is reciprocal to the other<br />If b = 0 the roots are equal but of opposite signs<br />©<br />Revision Notes – Quantitative Aptitude<br />CPTSuccess<br />www.cptsuccess.com<br />
12. 12. Chapter 2. Equations<br /><ul><li>Application of Equations to Coordinate Geometry</li></ul> <br />Distance of a point P (x, y) from Origin (0, 0) is  (x2 + y2)<br />Distance between two points P (x1, y1) and Q (x2, y2) is  [(x1 – x2)2 + (y1 – y2)2]<br />Equation of a straight line is written as y = mx + c, where m is the slope and c is the constant<br />The Slope of the line is given by, m = (y2 – y1) / (x2 – x1)<br />©<br />Revision Notes – Quantitative Aptitude<br />CPTSuccess<br />www.cptsuccess.com<br />
13. 13. CPTSuccess<br />