Algebra Questions for CDS

Algebraic Identities

These are the building blocks of CDS algebra. Most simplification questions are solved by recognising which identity to apply rather than expanding everything manually. The standard identities appear directly in questions disguised as numerical or expression problems. If you see numbers like 998 or 1001, think (a+b) or (a-b) identities immediately.

• (a+b)^2 = a^2 + 2ab + b^2
• (a-b)^2 = a^2 - 2ab + b^2
• (a+b)(a-b) = a^2 - b^2
• (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
• (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
• a^3 + b^3 = (a+b)(a^2 - ab + b^2) and a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Linear Equations in One and Two Variables

CDS regularly tests word-problem-based linear equations — ages, money, mixtures, and work problems. For two-variable systems, you must choose between substitution and elimination. Elimination is faster in exam conditions. Watch out for inconsistent systems (no solution) and dependent systems (infinite solutions) — conceptual MCQs on these appear occasionally.

• For ax + b = 0, solution is x = -b/a
• For two equations, use elimination: multiply equations to make one coefficient equal, then subtract
• If a1/a2 = b1/b2 not equal to c1/c2, the system is inconsistent — no solution
• If a1/a2 = b1/b2 = c1/c2, infinite solutions (dependent lines)
• If a1/a2 not equal to b1/b2, unique solution — lines intersect once

Quadratic Equations

Quadratic questions in CDS usually involve finding roots, the sum and product of roots, or forming a new equation from given roots. The discriminant tells you the nature of roots — memorise the three cases. Questions also appear as: given sum and product of roots, find the equation. Think of it backwards from Vieta's formulas.

• Standard form: ax^2 + bx + c = 0
• Roots by formula: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
• Sum of roots = -b/a, Product of roots = c/a (Vieta's formulas)
• If D = b^2 - 4ac > 0: two real distinct roots
• If D = 0: two equal real roots (roots are repeated)
• If D < 0: no real roots (complex/imaginary)

Polynomials and Remainder/Factor Theorem

CDS asks direct applications of the Remainder and Factor theorems. Remainder theorem: when polynomial p(x) is divided by (x - a), remainder is p(a). Factor theorem: if p(a) = 0, then (x - a) is a factor. These questions are quick once you remember to just substitute the value directly instead of doing long division.

• Remainder when p(x) is divided by (x - a) = p(a)
• If p(a) = 0 then (x - a) is a factor of p(x)
• For divisor (ax - b), substitute x = b/a into p(x)
• Degree of remainder is always less than degree of divisor
• A polynomial of degree n has at most n real roots

Special Algebraic Results and Surds

CDS frequently tests expressions involving x + 1/x type problems and rationalisation of surds. If you know x + 1/x = k, you can chain-derive x^2 + 1/x^2, x^3 + 1/x^3 etc. by squaring or cubing. For surds, multiply numerator and denominator by the conjugate to rationalise. These questions look hard but follow a fixed pattern.

• If x + 1/x = k then x^2 + 1/x^2 = k^2 - 2
• If x + 1/x = k then x^3 + 1/x^3 = k^3 - 3k
• Rationalise 1/(a + sqrt(b)) by multiplying by (a - sqrt(b))/(a - sqrt(b))
• sqrt(a) x sqrt(b) = sqrt(ab) but sqrt(a) + sqrt(b) is NOT sqrt(a+b) — common error
• Order of surds: compare by squaring both sides when needed

Inequalities

Linear inequalities appear in CDS as straightforward solve-and-shade problems. The critical rule is: when you multiply or divide both sides by a negative number, flip the inequality sign. This is the most common trap. Number line representation may be tested conceptually.

• Add or subtract any number from both sides without changing the sign
• Multiply or divide by positive number — sign stays same
• Multiply or divide by NEGATIVE number — sign FLIPS
• For combined inequalities like a < x < b, treat each part separately
• Absolute value inequality: |x| < a means -a < x < a

• Forgetting to flip the inequality sign when multiplying or dividing by a negative number — this single mistake causes wrong answers in inequality MCQs
• Confusing sum of roots (-b/a) with product of roots (c/a) in quadratic Vieta formulas — note the negative sign is on the SUM not the product
• Writing sqrt(a+b) = sqrt(a) + sqrt(b) — this is WRONG and surds questions are designed to catch this exact error
• In remainder theorem, using the wrong substitution value: for (x + a) the substitution is x = -a, not +a
• In cube identity questions, using (a+b)^3 pattern but forgetting the middle terms 3a^2b and 3ab^2 — always verify with a simple number before applying

• SPOON for Vieta: S = Sum of roots = -b/a (S starts negative), P = Product = c/a (P is positive with no negative sign)
• DIP for discriminant conditions: D positive = two different roots, D is zero = identical roots, D is negative = no real roots — think DIP into the number line
• For x + 1/x chain: Square it and minus 2, Cube it and minus 3 times itself — Square minus 2, Cube minus 3x
• Remainder theorem shortcut: PLUG and CHUG — just plug the zero of the divisor into the polynomial and chug out the answer, no long division

• In CDS, identity-based simplification questions (especially using a^2 - b^2 and a^3 +/- b^3) appear almost every paper — spend 30 seconds max by spotting the pattern, not expanding
• Quadratic roots questions often come in pairs: nature of roots (use discriminant) and value of expression using roots (use Vieta) — expect 2-3 such questions per paper
• Word problems on linear equations (age, money, mixture) are framed in simple English — read twice, assign variables clearly, and write equations before solving to avoid setup errors
• Surds rationalisation questions are usually 1-step — identify conjugate, multiply, simplify. If your answer has a surd in denominator, you have not finished
• Timing guide: identity questions should take under 45 seconds, quadratic root questions under 90 seconds, word problem linear equations under 2 minutes. If a question exceeds this, mark and move on