Algebra Notes for CUET, CLAT & IPMAT 2026
Algebra Notes for CUET, CLAT & IPMAT 2026
Algebra is one of the most important topics in Quantitative Aptitude for exams like CUET, CLAT, and IPMAT. Questions from Algebra are frequently asked in the form of simplification, equations, identities, functions, inequalities, logarithms, and word problems. A strong command over Algebra improves calculation speed, logical thinking, and problem-solving ability.
This article covers all major Algebra concepts in a simple, exam-oriented, and SEO-friendly format for aspirants preparing for competitive exams.
What is Algebra?
Algebra is a branch of Mathematics in which letters and symbols are used to represent numbers and quantities in formulas and equations.
Example:
x + 5 = 12
Here, (x) is an unknown variable.
Important Terms in Algebra
| Term | Meaning |
|---|---|
| Variable | Symbol representing a number |
| Constant | Fixed numerical value |
| Coefficient | Number multiplied with a variable |
| Expression | Combination of variables and numbers |
| Equation | Two expressions connected by “=” |
| Polynomial | Algebraic expression with powers |
Example:
[
3x^2 + 5x + 7
]
Variable = (x)
Coefficient of (x^2) = 3
Constant = 7
Types of Algebra Questions Asked in Exams
Algebraic Identities
Linear Equations
Quadratic Equations
Polynomials
Factorization
Inequalities
Functions
Logarithms
Surds & Indices
Word Problems Based on Algebra
1. Algebraic Identities
Algebraic identities are fixed formulas used for simplification and solving problems quickly.
Important Identities
Square Identities
(a+b)^2=a^2+b^2+2ab
(a-b)^2=a^2+b^2-2ab
Cube Identities
(a+b)^3=a^3+b^3+3ab(a+b)
(a-b)^3=a^3-b^3-3ab(a-b)
Shortcut Tips
Use identities for fast calculations.
Frequently asked in simplification questions.
Useful in quadratic equations and factorization.
Example
Find:
99^2
Using identity:
(a-b)^2
(100-1)^2
10000 + 1 - 200 = 9801
2. Linear Equations
A linear equation has the highest power of variable equal to 1.
General form:
ax+b=0
Formula
x = -b/a
Example
Solve:
3x + 5 = 20
3x = 15
x = 5
Types of Linear Equations
1. One Variable Equation
Example:
2x + 7 = 13
2. Two Variable Equation
Example:
2x + 3y = 12
These are solved using:
Elimination Method
Substitution Method
3. Quadratic Equations
A quadratic equation has degree 2.
General form:
ax^2+bx+c=0
Nature of Roots
Discriminant
D = b^2 - 4ac
| Condition | Nature of Roots |
|---|---|
| D > 0 | Real and distinct |
| D = 0 | Real and equal |
| D < 0 | Imaginary |
Example
Solve:
x^2 - 5x + 6 = 0
Factorization:
(x-2)(x-3)=0
x=2,\ 3
4. Polynomials
A polynomial is an algebraic expression consisting of variables and coefficients.
Example:
2x^3 + 5x^2 - 7x + 9
Types of Polynomials
| Type | Degree |
|---|---|
| Constant | 0 |
| Linear | 1 |
| Quadratic | 2 |
| Cubic | 3 |
Important Polynomial Theorems
Remainder Theorem
If polynomial (f(x)) is divided by ((x-a)), remainder is (f(a)).
Factor Theorem
If (f(a)=0), then ((x-a)) is a factor.
5. Factorization
Factorization means expressing an algebraic expression as product of factors.
Common Methods
1. Common Factor Method
6x + 12 = 6(x+2)
2. Identity Method
a^2-b^2=(a+b)(a-b)
3. Splitting Middle Term
x^2+5x+6
= x^2+2x+3x+6
= x(x+2)+3(x+2)
= (x+2)(x+3)
6. Inequalities
An inequality compares two algebraic expressions.
Symbols used:
(>)
(<)
Rules of Inequalities
Adding same number on both sides keeps inequality unchanged.
Multiplying by negative number reverses sign.
Example:
-2x > 10
Divide by -2:
x < -5
7. Functions
A function relates one quantity to another.
General form:
f(x)=x^2+3
Here:
Input = (x)
Output = (f(x))
Types of Functions
| Function | Example |
|---|---|
| Constant | (f(x)=5) |
| Linear | (f(x)=2x+1) |
| Quadratic | (f(x)=x^2) |
Important Concepts
Domain
All possible values of (x).
Range
All possible output values.
Important Log Rules
Product Rule
log(ab)=log a + log b
Division Rule
log a/b = log a - log b
Power Rule
log(a^n)=nlog a
Example
log 100 = 2
10^2 = 100
9. Surds and Indices
Laws of Indices
Product Rule
[
a^m + a^n = a^{m+n}
Division Rule
frac{a^m} / {a^n}=a^{m-n}
Power Rule
[
(a^m)^n=a^{mn}
]
Surds
Irrational roots are called surds.
Example:
sqrt{2}
sqrt{3}
10. Algebraic Word Problems
Word problems are common in CUET and IPMAT.
Steps to Solve
Assume variable.
Frame equation.
Solve equation.
Verify answer.
Example
A number increased by 5 becomes 20. Find the number.
Let number be (x)
x+5=20
x=15
Important Algebra Formulas for Exams
Identities
(a+b)^2=a^2+b^2+2ab
(a-b)^2=a^2+b^2-2ab
a^2-b^2=(a+b)(a-b)
Laws of Exponents
a^m + a^n = a^{m+n}
{a^m} - {a^n}=a^{m-n}
Logarithm Rules
log(ab)=log a + log b
log{a}/ log{b}=log a - log b
Most Important Topics for CUET
Linear Equations
Quadratic Equations
Simplification
Algebraic Identities
Polynomials
Most Important Topics for CLAT
Basic Algebra
Simplification
Number-based equations
Logical algebraic reasoning
Most Important Topics for IPMAT
Functions
Quadratic Equations
Logarithms
Inequalities
Advanced Algebra
Preparation Strategy for Algebra
For Beginners
Learn formulas first.
Practice identities daily.
Solve easy equations.
For Intermediate Students
Practice mixed questions.
Learn shortcuts.
Improve calculation speed.
For Advanced Students
Solve previous year papers.
Practice timed mocks.
Focus on difficult algebraic manipulation.
Common Mistakes in Algebra
Sign errors (+/- mistakes)
Wrong factorization
Calculation mistakes
Forgetting formulas
Incorrect simplification
Best Tips to Score High in Algebra
Memorize identities thoroughly.
Practice at least 20 questions daily.
Revise formulas regularly.
Focus on accuracy first, then speed.
Attempt easy questions first in exam.
Previous Year Trend Analysis
| Exam | Difficulty Level |
|---|---|
| CUET | Easy to Moderate |
| CLAT | Moderate |
| IPMAT | Moderate to Difficult |
Quadratic equations and simplification are among the most repeated topics.
Top 20 Algebra MCQs for CUET, CLAT & IPMAT (With Answers)
1. If (x + 3 = 10), then the value of (x) is:
A) 5
B) 6
C) 7
D) 8
Answer:
C) 7
2. Simplify:
[
( a+b )^2
]
(a+b)^2=a^2+b^2+2ab
A) (a^2+b^2)
B) (a^2+b^2-2ab)
C) (a^2+b^2+2ab)
D) (2a+2b)
Answer:
C) (a^2+b^2+2ab)
3. Solve:
[
2x - 5 = 9
]
A) 5
B) 6
C) 7
D) 8
Answer:
C) 7
4. The roots of the equation:
[
x^2 - 9 = 0
]
are:
A) (3,-3)
B) (9,-9)
C) (0,9)
D) (1,-1)
Answer:
A) (3,-3)
5. Find the value of:
[
\log_{10}1000
]
A) 1
B) 2
C) 3
D) 4
Answer:
C) 3
6. Simplify:
[
a^3 \times a^2
]
A) (a^5)
B) (a^6)
C) (a)
D) (2a^3)
Answer:
A) (a^5)
7. If:
[
x^2 - 5x + 6 = 0
]
then the roots are:
A) 1, 6
B) 2, 3
C) 3, 6
D) 1, 2
Answer:
B) 2, 3
8. Factorize:
[
a^2-b^2
]
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A) ((a+b)^2)
B) ((a-b)^2)
C) ((a+b)(a-b))
D) (a(a-b))
Answer:
C) ((a+b)(a-b))
9. Solve:
[
3x+7=22
]
A) 3
B) 4
C) 5
D) 6
Answer:
C) 5
10. The degree of polynomial:
[
5x^4+3x^2+7
]
is:
A) 2
B) 3
C) 4
D) 5
Answer:
C) 4
11. Simplify:
[
(a-b)^2
]
(a-b)^2=a^2+b^2-2ab
A) (a^2+b^2+2ab)
B) (a^2+b^2-2ab)
C) (a^2-b^2)
D) (2ab)
Answer:
B) (a^2+b^2-2ab)
12. If:
[
2x=18
]
then (x) equals:
A) 6
B) 7
C) 8
D) 9
Answer:
D) 9
13. Simplify:
[
\frac{a^5}{a^2}
]
A) (a^7)
B) (a^3)
C) (a^2)
D) (5a^2)
Answer:
B) (a^3)
14. Find the discriminant of:
[
x^2-4x+3=0
]
A) 2
B) 3
C) 4
D) 5
Answer:
C) 4
15. If:
[
f(x)=x^2+2
]
then (f(3)) is:
A) 9
B) 10
C) 11
D) 12
Answer:
C) 11
16. Solve:
[
x^2=49
]
A) (\pm5)
B) (\pm6)
C) (\pm7)
D) (\pm8)
Answer:
C) (\pm7)
17. Simplify:
[
\sqrt{81}
]
A) 7
B) 8
C) 9
D) 10
Answer:
C) 9
18. The coefficient of (x^2) in:
[
7x^2+5x+1
]
is:
A) 1
B) 5
C) 7
D) 2
Answer:
C) 7
19. Solve:
[
5x-15=0
]
A) 2
B) 3
C) 4
D) 5
Answer:
B) 3
20. Simplify:
[
(2+3)^2
]
A) 15
B) 20
C) 25
D) 30
Answer:
C) 25
Conclusion
Algebra is a scoring and highly important section for CUET, CLAT, and IPMAT preparation. With proper understanding of identities, equations, logarithms, functions, and inequalities, students can solve questions quickly and accurately.
Consistent practice, formula revision, and solving previous year questions are the keys to mastering Algebra. Build strong fundamentals and focus on speed + accuracy to maximize your score in competitive exams.
Algebra Notes for CUET
Algebra Notes for CLAT
Algebra Notes for IPMAT
Algebra Formulas PDF
Algebra Short Notes
Quadratic Equations for CUET
Algebra Questions for IPMAT
Algebra Tricks and Shortcuts
Algebra for Competitive Exams
CUET Quantitative Aptitude Algebra Notes
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