📑 Topics on this page (Jump to Topic):
📌 1. Key Identities
📘 Formulas:
- (a ± b)² = a² ± 2ab + b²
- (a + b)² − (a − b)² = 4ab
- (a ± b)³ = a³ ± b³ ± 3ab(a ± b)
- a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
- If a + b + c = 0 then a³ + b³ + c³ = 3abc
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
📌 2. x + 1/x Type
| If x + 1/x = k | then |
|---|---|
| x² + 1/x² | k² − 2 |
| x³ + 1/x³ | k³ − 3k |
| x⁴ + 1/x⁴ | (k² − 2)² − 2 |
| If x − 1/x = k, x² + 1/x² | k² + 2 |
💡 Shortcut Trick: x + 1/x = 2 → x = 1, so xⁿ + 1/xⁿ = 2; x + 1/x = −2 → x = −1.
📌 3. Equations & Remainder Theorem
📘 Formulas:
- ax² + bx + c = 0: sum of roots = −b/a, product = c/a
- Roots = [−b ± √(b² − 4ac)]/2a; D > 0 real distinct, = 0 equal, < 0 imaginary
- Remainder when p(x) is divided by (x − a) = p(a)
- Unique solution if a₁/a₂ ≠ b₁/b₂
⚠️ Exam Traps:
- For x² + 1/x², subtract 2 with (x + 1/x) and add 2 with (x − 1/x) — do not swap.
📌 4. Graphs of Linear Equations
The graph of ax + by + c = 0 is always a straight line. Every point (x, y) on the line satisfies the equation.
📘 Formulas:
- x-intercept (put y = 0) = −c/a; y-intercept (put x = 0) = −c/b
- Intercept form: x/p + y/q = 1 (p, q = x- and y-intercepts)
- Slope m = −a/b; in y = mx + c, m = slope and c = y-intercept
- Slope through two points = (y₂ − y₁)/(x₂ − x₁)
- Area of the triangle formed by the line and the axes = ½ × |p × q| = c²/(2|ab|)
- x = k: line parallel to the y-axis; y = k: parallel to the x-axis; x = 0 is the y-axis itself and y = 0 the x-axis
| Condition (a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0) | Graph | Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines intersect at one point | Unique solution |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel lines | No solution |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident lines | Infinitely many solutions |
⚠️ Exam Traps:
- In 3x + 4y = 12 the x-intercept is 4 (12/3), not 3 — find it by putting y = 0.
- The point of intersection of two lines = the simultaneous solution of the two equations.
📌 5. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. If x + 1/x = 5, then x² + 1/x² = ? Practice
Correct answer: (B)
Solution: (x + 1/x)² − 2 = 25 − 2 = 23.
Solution: (x + 1/x)² − 2 = 25 − 2 = 23.
Q2. If x + 1/x = 6, then x³ + 1/x³ = ? Practice
Correct answer: (C)
Solution: k³ − 3k = 216 − 18 = 198.
Solution: k³ − 3k = 216 − 18 = 198.
Q3. If x − 1/x = 6, then x² + 1/x² = ? Practice
Correct answer: (D)
Solution: (x − 1/x)² + 2 = 36 + 2 = 38.
Solution: (x − 1/x)² + 2 = 36 + 2 = 38.
Q4. If a = 8, b = 3, c = -11, then a³ + b³ + c³ = ? Practice
Correct answer: (A)
Solution: a + b + c = 0 → a³ + b³ + c³ = 3abc = 3×8×3×(-11) = -792.
Solution: a + b + c = 0 → a³ + b³ + c³ = 3abc = 3×8×3×(-11) = -792.
Q5. If a + b = 15 and ab = 19, then a² + b² = ? Practice
Correct answer: (B)
Solution: (a + b)² − 2ab = 225 − 38 = 187.
Solution: (a + b)² − 2ab = 225 − 38 = 187.
Q6. The remainder when p(x) = 3x³ − 4x − 8 is divided by (x − 2) is: Practice
Correct answer: (C)
Solution: Remainder = p(2) = 8.
Solution: Remainder = p(2) = 8.
Q7. The sum and product of the roots of x² − 8x + 15 = 0 are respectively: Practice
Correct answer: (D)
Solution: Sum = −b/a = 8, product = c/a = 15 (roots 3, 5).
Solution: Sum = −b/a = 8, product = c/a = 15 (roots 3, 5).
Q8. If 5x + 3y = 69 and 5x − 3y = 21, then x + y = ? Practice
Correct answer: (A)
Solution: Solving gives x = 9, y = 8 → x + y = 17.
Solution: Solving gives x = 9, y = 8 → x + y = 17.
Q9. If x + 1/x = 4, then x⁴ + 1/x⁴ = ? Practice
Correct answer: (B)
Solution: x² + 1/x² = 14; x⁴ + 1/x⁴ = 14² − 2 = 194.
Solution: x² + 1/x² = 14; x⁴ + 1/x⁴ = 14² − 2 = 194.
Q10. If a + b = 5 and ab = 3, then a³ + b³ = ? Practice
Correct answer: (C)
Solution: (a + b)³ − 3ab(a + b) = 125 − 45 = 80.
Solution: (a + b)³ − 3ab(a + b) = 125 − 45 = 80.
Q11. If x + 1/x = 4, then x² + 1/x² = ? Practice
Correct answer: (D)
Solution: (x + 1/x)² − 2 = 16 − 2 = 14.
Solution: (x + 1/x)² − 2 = 16 − 2 = 14.
Q12. If x + 1/x = 7, then x³ + 1/x³ = ? Practice
Correct answer: (A)
Solution: k³ − 3k = 343 − 21 = 322.
Solution: k³ − 3k = 343 − 21 = 322.
Q13. If x − 1/x = 7, then x² + 1/x² = ? Practice
Correct answer: (B)
Solution: (x − 1/x)² + 2 = 49 + 2 = 51.
Solution: (x − 1/x)² + 2 = 49 + 2 = 51.
Q14. If a = 5, b = 4, c = -9, then a³ + b³ + c³ = ? Practice
Correct answer: (C)
Solution: a + b + c = 0 → a³ + b³ + c³ = 3abc = 3×5×4×(-9) = -540.
Solution: a + b + c = 0 → a³ + b³ + c³ = 3abc = 3×5×4×(-9) = -540.
Q15. If a + b = 12 and ab = 16, then a² + b² = ? Practice
Correct answer: (D)
Solution: (a + b)² − 2ab = 144 − 32 = 112.
Solution: (a + b)² − 2ab = 144 − 32 = 112.
Q16. The remainder when p(x) = x³ + 5x + 1 is divided by (x + 2) is: Practice
Correct answer: (A)
Solution: Remainder = p(-2) = -17.
Solution: Remainder = p(-2) = -17.
Q17. The sum and product of the roots of x² − 3x + 2 = 0 are respectively: Practice
Correct answer: (B)
Solution: Sum = −b/a = 3, product = c/a = 2 (roots 1, 2).
Solution: Sum = −b/a = 3, product = c/a = 2 (roots 1, 2).
Q18. If 5x + 4y = 21 and 3x − 3y = −9, then x + y = ? Practice
Correct answer: (C)
Solution: Solving gives x = 1, y = 4 → x + y = 5.
Solution: Solving gives x = 1, y = 4 → x + y = 5.
Q19. If x + 1/x = 5, then x⁴ + 1/x⁴ = ? Practice
Correct answer: (D)
Solution: x² + 1/x² = 23; x⁴ + 1/x⁴ = 23² − 2 = 527.
Solution: x² + 1/x² = 23; x⁴ + 1/x⁴ = 23² − 2 = 527.
Q20. If a + b = 7 and ab = 2, then a³ + b³ = ? Practice
Correct answer: (A)
Solution: (a + b)³ − 3ab(a + b) = 343 − 42 = 301.
Solution: (a + b)³ − 3ab(a + b) = 343 − 42 = 301.
Q21. What is the area of the triangle formed by the line 3x + 4y = 12 and the coordinate axes? Practice
Correct answer: (C)
Solution: Intercepts: x = 4, y = 3 → area = ½ × 4 × 3 = 6 sq units.
Solution: Intercepts: x = 4, y = 3 → area = ½ × 4 × 3 = 6 sq units.
Q22. At which point does the line 2x − 3y + 6 = 0 cut the y-axis? Practice
Correct answer: (A)
Solution: Put x = 0: −3y + 6 = 0 → y = 2 → (0, 2). (−3, 0) is the x-intercept.
Solution: Put x = 0: −3y + 6 = 0 → y = 2 → (0, 2). (−3, 0) is the x-intercept.
Q23. What is the slope of the line 4x + 2y = 7? Practice
Correct answer: (D)
Solution: m = −a/b = −4/2 = −2.
Solution: m = −a/b = −4/2 = −2.
Q24. For what value of k will 2x + 3y = 5 and 4x + ky = 7 have no solution (parallel lines)? Practice
Correct answer: (B)
Solution: For parallel lines 2/4 = 3/k → k = 6; the c-ratio 5/7 ≠ 1/2, so there is no solution.
Solution: For parallel lines 2/4 = 3/k → k = 6; the c-ratio 5/7 ≠ 1/2, so there is no solution.
Q25. At which point do the graphs of x + y = 4 and x − y = 2 intersect? Practice
Correct answer: (D)
Solution: Adding: 2x = 6 → x = 3, then y = 1 → (3, 1).
Solution: Adding: 2x = 6 → x = 3, then y = 1 → (3, 1).