Linear Equations
Linear equations in one variable: ax + b = 0 → x = −b/a. In two variables: lines with slope-intercept form y = mx + c and simultaneous linear equations solved by substitution, elimination, or matrix methods for higher dimensions.
Tips: For simultaneous equations, use elimination with coefficients to cancel variables; for systems with parameters, look for consistency (determinant ≠ 0).
Solved examples
Example 1: Solve
3x − 7 = 2. Solution: 3x = 9 → x = 3.Example 2 (simultaneous): Solve
2x + 3y = 13 and 4x − y = 5. Eliminate y: multiply 2nd by 3 → 12x − 3y = 15. Add to first → 14x = 28 → x = 2. Then 2×2 + 3y = 13 → y = 3.Quadratic Equations
Standard form ax^2 + bx + c = 0. Roots via formula x = (−b ± √(b^2 − 4ac)) / (2a). Discriminant Δ = b^2 − 4ac determines nature of roots (Δ>0 two reals; Δ=0 one repeated; Δ<0 complex).
Vieta's formulas: If roots r1, r2 then
r1 + r2 = −b/a and r1 r2 = c/a.Tricks: Use factoring when integer roots suspected; complete the square for vertex form; use substitution for symmetric equations.
Solved examples
Example 3: Solve
x^2 − 5x + 6 = 0. Factor: (x−2)(x−3)=0 → x=2,3.Example 4: Find sum and product of roots of
2x^2 − 7x + 3 = 0. Sum = 7/2, product = 3/2.Example 5 (parameter): For what k does
x^2 + 2(k−1)x + k = 0 have equal roots? Δ=4(k−1)^2 − 4k = 0 → (k−1)^2 − k =0 → k^2 −3k +1 =0 → k = (3 ± √5)/2.Inequalities
Inequalities: linear (ax + b < 0), quadratic (ax^2 + bx + c ≥ 0), and absolute value. Key rules: multiplying both sides by negative reverses inequality; solution sets often intervals; for quadratics use sign-analysis with roots as critical points.
Absolute value:
|x|, |x|>a → x < −a or x > a.Solved examples
Example 6: Solve
2x − 5 > 1 → 2x > 6 → x > 3.Example 7: Solve
x^2 − 5x + 6 < 0. Roots 2 and 3 → quadratic opens up → negative between roots → 2 < x < 3.Example 8 (modulus): Solve
|x − 2| ≤ 3 → −3 ≤ x − 2 ≤ 3 → −1 ≤ x ≤ 5.Functions
Function f maps domain → range. Key concepts: injective (one-to-one), surjective (onto), bijective (both). Composition (f∘g)(x) = f(g(x)). Inverse f^{-1} exists iff f is bijection. Common special functions: modulus, floor/ceiling, piecewise linear, polynomials.
Domain & Range: Determine domain by excluding values causing division by zero or negative inside even root; range via algebraic manipulation or monotonicity.
Solved examples
Example 9: If
f(x)=2x+3, inverse f^{-1}(y)=(y−3)/2.Example 10: For
f(x)=x^2 defined on ℝ, inverse doesn't exist globally; restrict domain x≥0 then inverse is √x.Advanced Techniques & Shortcuts
- Symmetric equations: For equations symmetric in x and y, use substitution s=x+y, p=xy and reduce degree.
- Root bounding: Use rational root theorem for integer roots, Sturm/Descartes' rule for sign changes if needed.
- Inequality tricks: AM-GM for non-negative variables, Cauchy-Schwarz for sums/products, rearrangement for ordered sums.
- Functions: Check monotonicity by derivative (if allowed) or sign of increments for integer problems.
50 Practice MCQs — Algebra (Answers highlighted, solved examples included above)
Questions cover linear/quadratic solving, parameter conditions, inequalities, function domain/range, composition, and conceptual traps.
Q1. Solve:
5x − 20 = 0. x = ?Q2. Roots of
x^2 − 7x + 10 = 0 are:Q3. For what k does
x^2 + kx + 9 have real roots?Q4. Solve inequality:
−2 < x − 3 ≤ 4.Q5. If f(x)=3x−1 and g(x)=x^2, (f∘g)(2) = ?
Q6. If sum of roots of
ax^2 + bx + c = 0 is 6 and product is 8, equation (monic) is:Q7. Solve:
2x + 3 = 3x − 1.Q8. If roots of
x^2 − sx + p = 0 are r1 and r2, and r1^2 + r2^2 = 25, and r1 + r2 = 7, find p.Q9. Solve |2x − 1| = 5.
Q10. If f(x)=x^2+1, find f^{-1}(2) (assuming principal branch).
Q11. For which x is
1/(x−2) defined?Q12. If polynomial p(x) has root 3, then (x−3) is a:
Q13. If sum of roots of
x^2 − 4x + c = 0 is 4 and roots are equal, c = ?Q14. Solve inequality:
x/(x−1) > 0.Q15. If f(x)=ax + b and f(1)=3, f(2)=5 → find a and b.
Q16. Quadratic: minimum value of
x^2 − 4x + 5 is:Q17. If roots of
x^2 − 10x + 21 are α and β, compute α^2 + β^2.Q18. Solve:
(x−1)(x−2) = 0.Q19. For which x is √(x−3) defined (real)?
Q20. If f(x)=x+1 and g(x)=2x, find (g∘f)(3).
Q21. If discriminant of
x^2 + kx + 16 equals 0, k = ?Q22. Solve inequality:
x^2 − 1 < 0.Q23. If f(x)=x^3, is f bijective on ℝ?
Q24. If polynomial p(x) = x^2 + 2x + 1, factorization is:
Q25. If r is root of
2x − 4 = 0, r = ?Q26. If f(x)=|x|, is it differentiable at 0?
Q27. For which k does
x^2 + kx + 1 have no real roots?Q28. If f(x)=2x and f^{-1}(x) = ?
Q29. If roots of
x^2 + x − 6 are p and q, p−q = ?Q30. Solve:
3x − 2 < 1.Q31. If f(x)=x^2−2x+1, what is vertex?
Q32. If polynomial divisible by (x−2), remainder theorem says p(2)=?
Q33. If f(x)=1/x, domain excludes:
Q34. If quadratic has roots 4 and −1, equation is:
Q35. If f(x)=x^2 and g(x)=x+1, solve f(x)=g(x).
Q36. If p(x)=x^2+bx+c has integer roots and product 12, possible integer factors include:
Q37. Solve |x+2| > 1.
Q38. If f(x)=2x+3 and y=7, x = ?
Q39. If quadratic ax^2+bx+c has Δ<0, number of real roots =
Q40. If f(x)=x^2 and domain restricted to x≥0, is inverse defined?
Q41. Solve: x^2 − 2x − 3 = 0.
Q42. If f(x)=ax^2 and f(2)=8 → a = ?
Q43. If r1 + r2 = 0 for quadratic, it implies b = ? (for monic x^2+bx+c)
Q44. If function f is odd, then f(−x) = ?
Q45. If polynomial has degree 3, maximum number of real roots =
Q46. If quadratic x^2 + 4x + 4 = 0, roots equal to:
Q47. If f(x)=x/(x+1), find f(1).
Q48. If roots of ax^2+bx+c are real and positive, which must hold?
Q49. If f(x)=x^2−1, f(2)=?
Q50. Best general strategy for algebra MCQs in CAT is:
Practice & Test-day Tips
- Practice factoring by patterns and use Vieta for quick root sums/products.
- For inequalities, draw a number line and mark test intervals.
- For functions, check domain first, then explore monotonicity for inverses.
- When parameter appears, use discriminant/consistency conditions to find required ranges.