Algebra forms the bridge between arithmetic and higher mathematics, introducing students to the powerful idea of using letters (variables) to represent unknown quantities. For UPTET Paper II, algebra questions test both your conceptual understanding and your ability to solve problems involving expressions, equations and identities at the Class 6–8 level.
This topic typically contributes 3–5 questions in the Mathematics section. Questions range from simplifying expressions and applying identities to solving linear equations and factorising polynomials. Mastery here also supports your pedagogical understanding—knowing where students commonly struggle helps you teach algebra effectively.
The scope covers four interconnected areas: forming and simplifying algebraic expressions, memorising and applying standard identities, solving linear equations in one and two variables, and factorising expressions using various methods.
Key Concepts
**Variable and Constant**: A variable (x, y, a) can take different values; a constant (5, −3, π) has a fixed value. An algebraic expression combines both using operations.
**Terms, Coefficients and Like Terms**: In 3x² + 5x − 7, there are three terms. The coefficient of x² is 3. Like terms have identical variable parts (e.g., 4xy and −2xy) and can be combined.
**Degree of a Polynomial**: The highest sum of exponents in any term. For 2x³y + 5x²y² − y, degrees of terms are 4, 4 and 1 respectively, so polynomial degree is 4.
**Identity vs Equation**: An identity holds true for all values of variables (e.g., (a + b)² = a² + 2ab + b²). An equation is true only for specific values (e.g., 2x + 3 = 7 is true only when x = 2).
**Linear Equation in One Variable**: Has the form ax + b = 0 (a ≠ 0) with exactly one solution.
**Linear Equation in Two Variables**: Has the form ax + by + c = 0. Represents a straight line; infinite solutions exist as ordered pairs (x, y).
**Factorisation**: Expressing an expression as a product of its factors—reverse of expansion.
Formulas / Key Facts
**Standard Algebraic Identities (must memorise)**
| Identity | Expanded Form | |----------|---------------| | (a + b)² | a² + 2ab + b² | | (a − b)² | a² − 2ab + b² | | (a + b)(a − b) | a² − b² | | (x + a)(x + b) | x² + (a + b)x + ab | | (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca | | (a + b)³ | a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) | | (a − b)³ | a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a − b) | | a³ + b³ | (a + b)(a² − ab + b²) | | a³ − b³ | (a − b)(a² + ab + b²) |
**Solving Linear Equations**
Isolate the variable by performing inverse operations on both sides.
For two-variable equations, use substitution or elimination when a pair of equations is given.
**Sign errors when removing brackets**: Students write −(a − b) = −a − b instead of the correct −a + b. Fix: Multiply the minus sign with every term inside carefully.
**Confusing (a + b)² with a² + b²**: The term 2ab is often forgotten. Fix: Always recall that squaring a binomial produces three terms, not two.
**Adding unlike terms**: Writing 3x + 2y = 5xy is incorrect because x and y are different variables. Fix: Only like terms (same variable and power) can be combined.
**Incorrect splitting of middle term**: When factorising x² + 5x + 6, students may try 2 and 3 but write (x + 2)(x − 3). Fix: Check that both factors give positive middle term—signs must match the original expression.
**Forgetting to verify solutions**: After solving an equation, substituting the answer back catches arithmetic errors. Fix: Make verification a habit, especially in exams.
**Applying identities mechanically without checking structure**: Using a³ + b³ formula when the expression is a³ − b³. Fix: Carefully observe signs before selecting the identity.
Quick Reference
**(a + b)² = a² + 2ab + b²** and **(a − b)² = a² − 2ab + b²** — never forget the middle term.
**a² − b² = (a + b)(a − b)** — fastest way to factorise difference of squares.
**Linear equation solution**: Transpose, simplify, isolate the variable.
**Factorising quadratics**: Find two numbers with required product and sum, then split the middle term.
**Degree of polynomial** = highest power of the variable (or sum of powers in multi-variable terms).
**Verification** takes 10 seconds and saves marks—always substitute your answer back.
The sum of two numbers is 25 and their difference is 7. If the larger number is x and the smaller number is y, which pair of equations correctly represents this situation?
Q5 · Algebra (Class 6–8) · MEDIUM
Using the identity (a + b)² = a² + 2ab + b², find the value of 103².