Algebra
Practice MCQsAlgebra deals with numbers, symbols, variables, expressions, equations, sets, relations, progressions, logarithms, permutations, combinations, polynomials, complex numbers, and graphical representation of equations and inequalities.
Algebra is one of the most important branches of elementary mathematics. It deals with numbers, symbols, variables, expressions, equations, sets, relations, progressions, logarithms, permutations, combinations, polynomials, complex numbers, and graphical representation of equations and inequalities. In competitive examinations, Algebra tests both conceptual understanding and problem-solving speed.
What is Algebra?
Algebra is the language of generalized mathematics. Instead of working only with fixed numbers, Algebra uses letters such as \(x\), \(y\), \(a\), and \(b\) to represent unknown or variable quantities. This helps us form expressions, equations, inequalities, and formulas for solving practical and theoretical problems.
Algebra is widely used in arithmetic problems, geometry, coordinate geometry, business mathematics, data interpretation, probability, computer science, and higher mathematics. For competitive exams, a student must understand algebraic rules clearly and apply them quickly.
| Area | What It Covers | Exam Focus |
|---|---|---|
| Sets and Relations | Set notation, operations, Venn diagrams, Cartesian product, relations. | Union, intersection, complement, De Morgan laws. |
| Polynomials | Factors, Remainder Theorem, H.C.F., L.C.M., quadratic equations. | Factorization, roots, coefficients. |
| Equations and Inequations | Linear equations, simultaneous equations, quadratic equations, inequalities. | Analytical and graphical solutions. |
| Sequences and Series | Arithmetic, Geometric, and Harmonic progressions. | nth term, sum, mean relations. |
| Advanced Basics | Complex numbers, binomial theorem, logarithms, binary system. | Properties, conversions, expansions, applications. |
“Algebra is not only about finding x; it is about understanding relationships between quantities.”
Key points
- Understand symbols and variables.
- Learn laws of indices and logarithms.
- Practice factorization and identities.
- Use graphs for equations and inequalities.
- Know set operations and Venn diagrams.
- Connect roots and coefficients in quadratic equations.
- Apply formulas carefully in progressions and combinations.
Major Topics Covered in Algebra
Algebra is a broad chapter. For examination preparation, it is useful to divide it into smaller concept areas and master them one by one.
Set Theory
Study of collections of objects and their operations.
- Set notation
- Union and intersection
- Complement
- Venn diagrams
- De Morgan laws
Algebraic Expressions
Use variables, constants, powers, and operations.
- Basic operations
- Simple factors
- Rational expressions
- Conditional identities
- Laws of indices
Equations
Find unknown values using mathematical relations.
- Linear equations
- Simultaneous equations
- Quadratic equations
- Practical word problems
- Graphical solutions
Higher Algebra Basics
Important concepts used in advanced mathematics.
- Complex numbers
- Binary numbers
- Progressions
- Permutation and combination
- Binomial theorem
- Logarithms
Formula and Concept Bank
\(A \cap B\) = elements common to A and B
\(A'\) = complement of A
\((A \cap B)' = A' \cup B'\)
Roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(\alpha + \beta = \frac{-b}{a}\)
\(\alpha\beta = \frac{c}{a}\)
\(S_n = \frac{n}{2}[2a + (n - 1)d]\)
\(S_n = \frac{a(r^n - 1)}{r - 1}\), if \(r \neq 1\)
\(\log_a \frac{m}{n} = \log_a m - \log_a n\)
\(\log_a m^n = n\log_a m\)
\(^{n}C_r = \frac{n!}{r!(n-r)!}\)
Tip: Formula application becomes easier when you identify the type of problem first.
Algebra Problem-Solving Guide
Algebra questions become easier when the student first identifies the type of expression, equation, pattern, or relationship involved. The table below gives a quick guide for selecting the correct method.
| Question Type | What to Use | Typical Clue |
|---|---|---|
| Set-based problem | Union, intersection, complement, Venn diagram | Words like “or”, “and”, “only”, “neither” |
| Expression simplification | Algebraic identities and factorization | Expressions with powers, brackets, products |
| Polynomial division | Remainder Theorem or Factor Theorem | Divided by \(x-a\), factor of polynomial |
| Linear equation | Transposition and substitution | Unknown appears with power 1 |
| Quadratic equation | Factorization or quadratic formula | Expression contains \(x^2\) |
| Progression | AP, GP or HP formula | Sequence, nth term, sum of terms |
| Logarithm question | Product, quotient and power laws | \(\log_a mn\), \(\log_a \frac{m}{n}\), \(\log_a m^n\) |
| Counting problem | Permutation or combination | Arrangement, selection, group formation |
Sets, Relations and Venn Diagrams
A set is a well-defined collection of objects. The objects in a set are called elements. Sets are generally represented by capital letters such as A, B, C, and their elements are written inside braces.
| Concept | Meaning | Example |
|---|---|---|
| Set | A collection of well-defined objects. | \(A = \{1, 2, 3, 4\}\) |
| Union | All elements belonging to A or B or both. | \(A \cup B\) |
| Intersection | Elements common to both A and B. | \(A \cap B\) |
| Difference | Elements in A but not in B. | \(A - B\) |
| Complement | Elements not belonging to A, within the universal set. | \(A'\) |
| Cartesian Product | Set of ordered pairs from two sets. | \(A \times B = \{(a,b): a \in A, b \in B\}\) |
| Relation | A subset of Cartesian product. | \(R \subseteq A \times B\) |
| Equivalence Relation | A relation that is reflexive, symmetric, and transitive. | Equality relation on a set. |
Real Numbers, Complex Numbers and Binary System
Real Numbers
Real numbers include rational and irrational numbers. They can be represented on a number line.
- Natural numbers
- Whole numbers
- Integers
- Rational numbers
- Irrational numbers
Complex Numbers
A complex number is written as \(z = a + ib\), where \(i^2 = -1\).
- Real part = \(a\)
- Imaginary part = \(b\)
- Modulus = \(|z| = \sqrt{a^2+b^2}\)
- Argument = angle made with positive x-axis
- Cube roots of unity: \(1, \omega, \omega^2\)
Binary System
Binary system uses only two digits: 0 and 1. It is the basic language of computers.
- Decimal to binary conversion
- Binary to decimal conversion
- Place values: \(2^0, 2^1, 2^2, 2^3,\ldots\)
- Used in digital systems
Polynomials, Factors and Remainder Theorem
A polynomial is an algebraic expression consisting of variables and coefficients involving non-negative integral powers of variables. Polynomials are important for factorization, equations, H.C.F., L.C.M., and remainder-based questions.
| Concept | Explanation | Example |
|---|---|---|
| Polynomial | An expression with variables having non-negative integer powers. | \(x^2 + 5x + 6\) |
| Degree | Highest power of the variable. | Degree of \(3x^4 + 2x\) is 4. |
| Factor | An expression that divides a polynomial exactly. | \(x+2\) is a factor of \(x^2+5x+6\). |
| Remainder Theorem | If \(f(x)\) is divided by \(x-a\), the remainder is \(f(a)\). | Remainder of \(f(x)\) divided by \(x-2\) is \(f(2)\). |
| H.C.F. | Highest common factor of algebraic expressions. | Common factor of \(x^2-1\) and \(x^2+x\). |
| L.C.M. | Least common multiple of algebraic expressions. | Used after factorization. |
Equations, Inequations and Graphical Solutions
Algebraic equations and inequations are used to represent conditions involving unknown quantities. Competitive exams often ask questions based on linear equations, simultaneous equations, quadratic equations, and practical word problems.
Linear Equations
A linear equation in two variables is generally written as:
\(ax + by + c = 0\)
- Represents a straight line.
- Can be solved analytically.
- Can be solved graphically.
- Two linear equations may have one solution, no solution, or infinitely many solutions.
Linear Inequations
Inequations use signs such as \(<\), \(>\), \(\leq\), and \(\geq\).
- Represent regions on a graph.
- Boundary line may be included or excluded.
- Solution is usually a shaded region.
- Used in practical problems involving constraints.
Quadratic Equations
A quadratic equation has degree 2 and is written as:
\(ax^2 + bx + c = 0\), where \(a \neq 0\)
- Only real roots are usually considered in elementary exam questions.
- Can be solved by factorization.
- Can be solved by formula.
- Roots and coefficients have fixed relations.
Practical Problems
Word problems are converted into equations or inequations.
- Age problems
- Number problems
- Cost and quantity problems
- Mixture and comparison problems
- Problems involving two unknowns
Progressions, Binomial Theorem and Logarithms
| Topic | Main Idea | Important Points |
|---|---|---|
| Arithmetic Progression | Difference between consecutive terms is constant. | Common difference, nth term, sum of n terms. |
| Geometric Progression | Ratio between consecutive terms is constant. | Common ratio, nth term, finite and infinite sum. |
| Harmonic Progression | Reciprocals of terms form an arithmetic progression. | Used in averages, rates, and reciprocal-based patterns. |
| Permutation | Arrangement of objects where order matters. | Used in seating, ranking, codes, and arrangements. |
| Combination | Selection of objects where order does not matter. | Used in committees, groups, selections, and probability. |
| Binomial Theorem | Expansion of expressions like \((a+b)^n\). | General term, middle term, coefficients, applications. |
| Logarithms | Logarithms convert multiplication into addition and powers into products. | Useful in equations, growth, decay, calculations, and scientific applications. |
Step-by-Step Algebra Problem Solving Method
| Step | Action | Example Focus |
|---|---|---|
| Step 1 | Identify the topic involved. | Set, equation, progression, polynomial, logarithm, or graph. |
| Step 2 | Write the given information clearly. | List values, variables, conditions, or set elements. |
| Step 3 | Select the correct formula or rule. | Use quadratic formula, AP formula, logarithm rule, or set law. |
| Step 4 | Simplify carefully. | Use factorization, substitution, expansion, or cancellation. |
| Step 5 | Check whether the answer satisfies the condition. | Substitute the answer back into the equation or condition. |
Solved Examples
| Question | Method | Answer |
|---|---|---|
| If \(A = \{1,2,3\}\) and \(B = \{3,4,5\}\), find \(A \cup B\). |
Union means all elements in A or B without repetition. \(A \cup B = \{1,2,3,4,5\}\) |
\(\{1,2,3,4,5\}\) |
| Find the remainder when \(f(x)=x^2+3x+2\) is divided by \(x-1\). |
By Remainder Theorem, remainder = \(f(1)\). \(f(1)=1^2+3(1)+2=6\) |
6 |
| Solve \(x^2 - 5x + 6 = 0\). |
Factorize: \(x^2 - 5x + 6 = (x-2)(x-3)\) So, \(x=2\) or \(x=3\). |
2, 3 |
| Find the 5th term of AP: 3, 7, 11, 15, ... |
Here \(a=3\), \(d=4\). \(a_5 = a + 4d = 3 + 16 = 19\). |
19 |
| Find \(^{5}C_2\). |
\(^{5}C_2 = \frac{5!}{2!3!}\) \(= \frac{5 \times 4}{2 \times 1}=10\) |
10 |
| Convert binary \(101_2\) into decimal. |
\(101_2 = 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\) \(= 4 + 0 + 1 = 5\) |
5 |
| If \(z = 3 + 4i\), find \(|z|\). |
\(|z| = \sqrt{a^2+b^2}\) \(|z| = \sqrt{3^2+4^2}=\sqrt{25}=5\) |
5 |
| Simplify \(\log_a xy\). |
Use product rule of logarithms. \(\log_a xy = \log_a x + \log_a y\) |
\(\log_a x + \log_a y\) |
Note: In Algebra, always check whether the question requires simplification, factorization, solving, expansion, conversion, or graphical interpretation.
Common Traps and Shortcuts
Common Traps
- Confusing union and intersection in set questions.
- Forgetting to include all elements in Cartesian product.
- Using permutation formula instead of combination formula.
- Making sign mistakes while solving equations.
- Forgetting that \(i^2 = -1\) in complex numbers.
- Applying logarithm rules without checking valid conditions.
- Using wrong common difference or common ratio in progressions.
- Incorrectly factorizing quadratic expressions.
- Not checking whether roots are real.
Useful Shortcuts
- Use factorization first before applying lengthy formulas.
- For \(x-a\), use \(f(a)\) directly in Remainder Theorem.
- In AP, identify first term and common difference.
- In GP, identify first term and common ratio.
- Use roots-coefficients relation for faster quadratic questions.
- Remember \(^{n}C_r = ^{n}C_{n-r}\).
- For binary to decimal, use powers of 2 from right to left.
- For Venn diagrams, fill the common region first.
Practice
A) Multiple Choice Questions
-
If \(A = \{2,4,6\}\) and \(B = \{4,6,8\}\), find \(A \cap B\).
\(\{2,4,6,8\}\) \(\{4,6\}\) \(\{2,8\}\) \(\{6,8\}\)
-
Find the roots of \(x^2 - 7x + 12 = 0\).
2, 6 3, 4 1, 12 5, 7
-
The 4th term of the AP 5, 8, 11, ... is:
11 12 14 15
-
Convert \(110_2\) into decimal.
4 5 6 7
-
What is \(^{6}C_2\)?
12 15 18 30
B) Solve the Higher-Order Problems
- If \(f(x)=2x^2+3x-5\), find the remainder when \(f(x)\) is divided by \(x-2\). (Hint: Use Remainder Theorem and find \(f(2)\).)
- Solve the simultaneous equations \(x+y=10\) and \(x-y=4\). (Hint: Add both equations first.)
- Find the modulus of \(z = 5 + 12i\). (Hint: \(|z| = \sqrt{a^2+b^2}\).)
- Find the 6th term of the GP 2, 6, 18, ... (Hint: Use \(a_n = ar^{n-1}\).)
- Expand the first four terms of \((1+x)^5\). (Hint: Use binomial coefficients.)
C) Match the Concept with the Correct Rule
| Concept | Correct Rule / Meaning |
|---|---|
| Union of Sets | All elements belonging to either set or both sets |
| Intersection of Sets | Elements common to both sets |
| Remainder Theorem | Remainder when divided by \(x-a\) is \(f(a)\) |
| Permutation | Arrangement where order matters |
| Combination | Selection where order does not matter |
| Binary System | Number system using only 0 and 1 |
| Logarithm Product Rule | \(\log_a mn = \log_a m + \log_a n\) |
| Quadratic Equation | Equation of degree 2 |
Algebra Reminder
Algebra is a foundation topic for higher mathematics and competitive examinations. It develops the ability to represent unknown quantities, form equations, simplify expressions, identify patterns, understand sets and relations, work with functions, and solve practical problems using mathematical logic.
Task: Create five Algebra questions using one question each from sets, quadratic equations, progressions, logarithms, and permutation-combination.
Show Suggested Answers
Multiple Choice
-
\(\{4,6\}\)
Intersection means common elements. Common elements of A and B are 4 and 6. -
3, 4
\(x^2 - 7x + 12 = (x-3)(x-4)\).
Therefore, \(x=3\) or \(x=4\). -
14
AP is 5, 8, 11, 14, ...
4th term = 14. -
6
\(110_2 = 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6\). -
15
\(^{6}C_2 = \frac{6!}{2!4!} = \frac{6 \times 5}{2 \times 1} = 15\).
Higher-Order Problems
-
\(f(x)=2x^2+3x-5\).
Remainder when divided by \(x-2\) is \(f(2)\).
\(f(2)=2(2)^2+3(2)-5=8+6-5=9\).
Answer = 9. -
Given \(x+y=10\) and \(x-y=4\).
Adding both equations: \(2x=14\), so \(x=7\).
Substitute in \(x+y=10\): \(7+y=10\), so \(y=3\).
Answer = x = 7, y = 3. -
\(z = 5 + 12i\).
\(|z|=\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13\).
Answer = 13. -
GP is 2, 6, 18, ...
Here \(a=2\), \(r=3\).
\(a_6 = ar^{5}=2 \times 3^5 = 2 \times 243 = 486\).
Answer = 486. -
\((1+x)^5 = 1 + 5x + 10x^2 + 10x^3 + \cdots\)
First four terms are \(1 + 5x + 10x^2 + 10x^3\).
Concept Matching
- Union of Sets → All elements belonging to either set or both sets
- Intersection of Sets → Elements common to both sets
- Remainder Theorem → Remainder when divided by \(x-a\) is \(f(a)\)
- Permutation → Arrangement where order matters
- Combination → Selection where order does not matter
- Binary System → Number system using only 0 and 1
- Logarithm Product Rule → \(\log_a mn = \log_a m + \log_a n\)
- Quadratic Equation → Equation of degree 2
Clue Explanation
Algebra questions become easier when you first identify the concept involved. Sets require element-based thinking, equations require unknown-value solving, progressions require pattern identification, and logarithms require correct use of laws.
Exam tips
- Memorize standard identities and formulas.
- Factorize before using long methods.
- Use substitution to verify equation answers.
- Draw Venn diagrams for set problems.
- Use graphs carefully for inequalities.
- Distinguish between permutation and combination.
- Keep logarithm base conditions in mind.
- Practice binary conversion using powers of 2.