Algebra Formulas For SSC CGL 2024- (With Solved Previous Year Questions).

Algebra holds immense significance in the SSC CGL exam, constituting a substantial portion of the quantitative aptitude section. To succeed in this segment, a solid understanding of algebraic concepts and formulas is essential. This document provides a comprehensive compilation of important algebra formulas for the SSC CGL 2024 exam. Each formula is accompanied by solved examples, allowing candidates to grasp their application effectively. By familiarizing oneself with these formulas, test-takers can enhance their problem-solving skills and increase their chances of scoring well in the algebraic portion of the SSC CGL exam.

Algebra is a branch of Mathematics that deals with symbols and the rules for manipulating those symbols. It includes the study of symbols and the rules for manipulating these symbols to represent numbers and quantities in equations and formulas. Algebraic techniques are used to solve equations, analyze patterns, and describe relationships between quantities.

Algebra Formula Definition

An algebraic formula is a mathematical expression that relates two or more variables. It typically consists of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Formulas in Algebra can be used to describe patterns, make prediction, and solve problems.

For example: (a + b)² = a² + 2ab + b²

Components of Algebra Formulas:

Variables: Variables are symbols that represent unknown quantities or values that can change. Commonly used variables include x, y, z although any symbol can be used as a variable.
Constants: Constants are fixed values that do not change. They are typically represented by numbers in algebraic formulas.
Operators: Operators are symbols that denote mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.

Application of Algebra Formulas:

Physics: Algebraic formulas are used to describe physical laws and principles, such as Newton’s second law of motion (F = ma) and the equations of motion.
Finance: Formulas in algebra are used in finance to calculate interest rate, loan payments, and investments returns.
Engineering: Algebraic formulas are essential in engineering for solving problems related to mechanics, electrical circuits, fluid dynamics and more.

Algebraic Identities for SSC CGL 2024:

In algebra, an identity is a mathematical equation that holds true for all values of the variables involved. It’s essentially a statement of equality that remains true regardless of the values of the variables. Here are some commonly used algebraic identities:

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a + b) (a – b) = a² – b²
(x + a)(x + b) = x ²+ x(a + b) +ab
a² + b² = (a + b)² – 2ab
a² + b² = (a – b)² + 2ab

Important Algebra Formulas for Squares: (For Class- 9, 10 and SSC CGL)

Here are some common algebra formulas related to squares:

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b² = (a + b)(a – b)
a² + b² = (a – b)² + 2ab
(a + b)² + (a – b)² = 2(a² + b²)
(a + b)² – (a – b)² = 4ab
(a² + b²)² = a⁴ +2a²b² + b⁴
(a + b + c)² = a² + b² + c ²+ 2(ab + bc + ca)
(a + b – c)² = a² + b² + c² + 2(ab – bc – ca)
(a – b + c)² = a² + b² + c² + 2(-ab – bc + ca)
(-a + b + c)² = a² + b² + c² + 2(-ab + bc – ca)

Important Algebra Formulas for Cubes : (For Class- 9, 10 and SSC CGL)

Here are some common algebra formulas related to cubes.

(a + b)³ = a³ + b³ + 3ab(a + b)
(a + b)³ = a³ + b³+ 3a²b + 3ab²
(a – b)³ = a³ – b³ – 3ab (a – b)
(a – b)³ = a³ – b³ – 3a²b + 3ab²
a³ + b³ = (a + b)(a² – ab + b²)
a³ – b³ = (a – b)(a² + ab + b²)
a³ + b³ = (a + b){(a + b)²-3ab}
a³ – b³ = (a- b){(a – b)² + 3ab}
(a + b)³ + (a – b)³ = 2a (a² + 3b²)
(a + b)³ – (a – b)³ = 2b (b² + 3a²)
(a + b + c)³ = a³ + b³ + c³ + 3(a + b)(b + c)(c + a)
a³ + b³ + c³ – 3abc = (a + b + c)(a² + b² + c² – ab – bc – ca)
a³ + b³ + c³ – 3abc =½ (a + b + c){(a – b)² + (b – c)² + (c – a)²}

Some more algebraic formulas

(a + b)(b + c)(c + a) = (a + b + c)(ab + bc + ca) – abc
(a + b)(b + c)(c + a) = a²b + a²c + b²a + b²c + c²a + c²b + 2abc
a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
a – b = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)

Algebra Formulas including Natural Numbers

Algebraic formulas involving natural numbers can include equations and expressions such as:

Addition: a + b = c
Subtraction: a – b = c
Multiplication: a × b = c
Division: a/b = c, where b ≠ 0.
Exponentiation: aⁿ = c, where n is a natural number.
Square of a number: a² = c
Cube of a number: a³ = c
Square root: √a = c, where c is the non-negative square root of a.
Cube root: ∛a = c, where c is the real root of a

Algebra Formulas (Law of Exponents)

The law of exponents in algebra are rules that govern the manipulation and simplification of expressions involving exponents. Here are the basic laws:

Product Rule: a^m × aⁿ = a^{m + n}
Quotient Rule: a^m÷ aⁿ = a^{m – n}
Power Rule: (a^m )ⁿ = a^mn
Zero Exponent Rule: a⁰ = 1, (Where a ≠ 0)
Negative Exponent Rule: a^-1 = 1/a¹, (where a ≠ 0)

Algebra Formulas (Quadratic Formula)

The Quadratic Formula is a formula used to find the solutions (roots) of a quadratic equation of the form ax + bx + c = 0, where a, b, and c are constants and x is the variable. The formulas is:

Some more important algebraic formulas are

If x + 1∕x = K, then x²+ 1∕x² = K² -2
If x² + 1∕x² =T, then x⁴ + 1/x⁴ = T² – 2
If x – 1/x = K, then x² + 1/x² = K² + 2
If x + 1∕x = P, then x³+ 1/x³ = P³ -3P
If x – 1/x = T, then x³+ 1∕x³ = T³+ 3T

Previous Year Questions

1. If a²+b²+c² = 21 and a + b + c = 7, then (ab + bc + ca) is equal to:

Answer- Given, a²+b²+c² = 21, a + b + c = 7

So, (a + b + c)² = 49

=a² + b² + c² + 2(ab + bc + ca) = 49

=21 + 2(ab + bc + ca) = 49

=2(ab + bc + ca) = 49 – 21 = 28

=ab + bc + ca = 28/2

Therefore, ab + bc + ca = 14

2. If a – b = 5 and ab = 2, then a³ – b³ is equal to:

Answer- Given, a – b = 5, ab =2

We know that, a² + b² = (a – b)² + 2ab =5² + 2 × 2 = 25 + 4 = 29

And, a³ – b³ = (a – b)(a² + b² + ab) = 5 × (29 + 2) = 5 × 31 = 155

3. If a + b + c = 9 and ab + bc + ca = -22, then what is the value of a³ + b³ + c³ – 3abc?

Answer- Given, a + b + c =9 and ab + bc + ca = -22

We know that, a³ + b³ + c³ – 3abc = (a + b + c){(a + b + c)² -3 (ab + bc + ca)}

=9 {(9)² – 3(-22)} = 9 {81 + 66} = 9 × 147 = 1323

4. If x + 1/x = 5, then x³ + 1/x³ is equal to:

Answer- x +1/x = 5

Cube both sides-

=x³ + 1/x³ = (5)³ – 3 × 5 = 125 – 15 = 110

5. If x + 3y + 2 = 0 then the value of x³ + 27y³ + 8 – 18xy is:

Answer- Put, x = 1 and y = -1

Therefore, 1 – 27 + 8 + 18 = 0.

FAQs

Q.1 What is algebra?

Answer– Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.

Q.2 Is algebra important for SSC CGL?

Answer- Algebra questions for SSC CGL are an important aspect for the aspirants who are planning to appear for the upcoming examination.

Q.3 What are algebraic identities?

Answer- Algebraic identities are equations in algebra where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation.

Q.4 Who is the father of algebra?

Answer- Muhammad ibn Musa al-Khwarizmi.

Algebra Formulas for SSC CGL 2024- (With Solved Previous Year Questions).

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