📑 Topics on this page (Jump to Topic):
📌 1. BODMAS Rule
| Letter | Meaning |
|---|---|
| V | Vinculum (bar) — first of all |
| B | Brackets: ( ) → { } → [ ] |
| O | Of and powers |
| D, M | Division and multiplication — left to right |
| A, S | Addition and subtraction — left to right |
✏️ Example: 48 ÷ 4 × 3 = 12 × 3 = 36. "of" comes before division: 48 ÷ 4 of 3 = 48 ÷ 12 = 4.
⚠️ Exam Traps:
- ÷ and × have equal priority — whichever comes first is done first.
📌 2. Laws of Indices
📘 Formulas:
- am × an = am+n
- am ÷ an = am−n
- (am)n = amn
- a0 = 1 (a ≠ 0), a−n = 1/an
- ap/q = q√(ap)
- (ab)n = anbn; if ax = ay then x = y
⚠️ Exam Traps:
- (am)n ≠ amn: (2³)² = 64 but 23² = 2⁹ = 512.
📌 3. Surds & Roots
📘 Formulas:
- √a × √b = √(ab); √a/√b = √(a/b)
- Rationalisation: 1/(√a + √b) = (√a − √b)/(a − b)
- √(a + 2√b) = √x + √y where x + y = a, xy = b
- √(x√(x√(x…∞))) = x; √(x + √(x + …)) = n where n(n − 1) = x
| Number | Square | Cube |
|---|---|---|
| 11 | 121 | 1331 |
| 12 | 144 | 1728 |
| 13 | 169 | 2197 |
| 14 | 196 | 2744 |
| 15 | 225 | 3375 |
| 16 | 256 | 4096 |
| 17 | 289 | 4913 |
| 18 | 324 | 5832 |
| 19 | 361 | 6859 |
| 25 | 625 | 15625 |
💡 Shortcut Trick: A perfect square never ends in 2, 3, 7 or 8. Estimate roots: √50 ≈ 7.07.
📌 4. Quick Simplification with Identities
📘 Formulas:
- (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²
- a² − b² = (a + b)(a − b)
- a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²)
- (a³ + b³)/(a² − ab + b²) = a + b
✏️ Example: (997)² = (1000 − 3)² = 1000000 − 6000 + 9 = 994009.
Recurring decimals
- 0.ab̅ (both recurring) = ab/99; 0.ab̅ (only b recurring) = (ab − a)/90
📌 5. Practice MCQs
📝 Practice MCQs (prepared by SF Prep — not PYQs)
Q1. The value of 135 ÷ 9 × 3 − 5 is: Practice
Correct answer: (B)
Solution: Left to right: 135 ÷ 9 = 15; × 3 = 45; − 5 = 40.
Solution: Left to right: 135 ÷ 9 = 15; × 3 = 45; − 5 = 40.
Q2. If 23 × 27 ÷ 26 = 2x, then x = ? Practice
Correct answer: (C)
Solution: Add/subtract powers: 3 + 7 − 6 = 4.
Solution: Add/subtract powers: 3 + 7 − 6 = 4.
Q3. The value of (16)3/4 is: Practice
Correct answer: (D)
Solution: 16 = 24 → (24)3/4 = 23 = 8.
Solution: 16 = 24 → (24)3/4 = 23 = 8.
Q4. The value of (1001)² is: Practice
Correct answer: (A)
Solution: (1000 + 1)² = 1000000 + 2000 + 1 = 1002001.
Solution: (1000 + 1)² = 1000000 + 2000 + 1 = 1002001.
Q5. On rationalising (√13 + √11) / (√13 − √11), we get: Practice
Correct answer: (B)
Solution: Multiply numerator and denominator by (√13 + √11): (√13 + √11)²/(13 − 11) = (24 + 2√143)/2.
Solution: Multiply numerator and denominator by (√13 + √11): (√13 + √11)²/(13 − 11) = (24 + 2√143)/2.
Q6. 120 ÷ 5 of 4 + 4 = ? Practice
Correct answer: (C)
Solution: 'of' first: 5 of 4 = 20; 120 ÷ 20 = 6; + 4 = 10.
Solution: 'of' first: 5 of 4 = 20; 120 ÷ 20 = 6; + 4 = 10.
Q7. The value of (2/5)−3 is: Practice
Correct answer: (D)
Solution: A negative power inverts the fraction: (5/2)3 = 125/8.
Solution: A negative power inverts the fraction: (5/2)3 = 125/8.
Q8. (76² − 11²) ÷ (76 − 11) = ? Practice
Correct answer: (A)
Solution: a² − b² = (a + b)(a − b) → answer = a + b = 76 + 11 = 87.
Solution: a² − b² = (a + b)(a − b) → answer = a + b = 76 + 11 = 87.
Q9. The value of (90³ − 34³) ÷ (90² + 90×34 + 34²) is: Practice
Correct answer: (B)
Solution: a³ − b³ = (a − b) × (the given bracket) → answer = 90 − 34 = 56.
Solution: a³ − b³ = (a − b) × (the given bracket) → answer = 90 − 34 = 56.
Q10. The value of √(132 + √(132 + √(132 + …∞))) is: Practice
Correct answer: (C)
Solution: Let it be y: y² = 132 + y → y² − y − 132 = 0 → (y − 12)(y + 11) = 0 → y = 12.
Solution: Let it be y: y² = 132 + y → y² − y − 132 = 0 → (y − 12)(y + 11) = 0 → y = 12.
Q11. The fraction form of 0.2666… (i.e. 0.26̅) is: Practice
Correct answer: (D)
Solution: 0.ab̅ = (ab − a)/90 = (26 − 2)/90 = 24/90 = 4/15.
Solution: 0.ab̅ = (ab − a)/90 = (26 − 2)/90 = 24/90 = 4/15.
Q12. The value of √8281 is: Practice
Correct answer: (A)
Solution: 8281 ends in 1 → root ends in 1, 9; it is near 90² = 8100 → 91.
Solution: 8281 ends in 1 → root ends in 1, 9; it is near 90² = 8100 → 91.
Q13. If 22x − 3 = 512, then x = ? Practice
Correct answer: (B)
Solution: 512 = 29 → 2x − 3 = 9 → x = 6.
Solution: 512 = 29 → 2x − 3 = 9 → x = 6.
Q14. The value of 288 ÷ 9 × 4 − 6 is: Practice
Correct answer: (C)
Solution: Left to right: 288 ÷ 9 = 32; × 4 = 128; − 6 = 122.
Solution: Left to right: 288 ÷ 9 = 32; × 4 = 128; − 6 = 122.
Q15. If 24 × 26 ÷ 26 = 2x, then x = ? Practice
Correct answer: (D)
Solution: Add/subtract powers: 4 + 6 − 6 = 4.
Solution: Add/subtract powers: 4 + 6 − 6 = 4.
Q16. The value of (97)² is: Practice
Correct answer: (A)
Solution: (100 − 3)² = 10000 − 600 + 9 = 9409.
Solution: (100 − 3)² = 10000 − 600 + 9 = 9409.
Q17. On rationalising (√3 + √2) / (√3 − √2), we get: Practice
Correct answer: (B)
Solution: Multiply numerator and denominator by (√3 + √2): (√3 + √2)²/(3 − 2) = (5 + 2√6)/1.
Solution: Multiply numerator and denominator by (√3 + √2): (√3 + √2)²/(3 − 2) = (5 + 2√6)/1.
Q18. 72 ÷ 6 of 4 + 3 = ? Practice
Correct answer: (C)
Solution: 'of' first: 6 of 4 = 24; 72 ÷ 24 = 3; + 3 = 6.
Solution: 'of' first: 6 of 4 = 24; 72 ÷ 24 = 3; + 3 = 6.
Q19. The value of (2/3)−2 is: Practice
Correct answer: (D)
Solution: A negative power inverts the fraction: (3/2)2 = 9/4.
Solution: A negative power inverts the fraction: (3/2)2 = 9/4.
Q20. (66² − 51²) ÷ (66 − 51) = ? Practice
Correct answer: (A)
Solution: a² − b² = (a + b)(a − b) → answer = a + b = 66 + 51 = 117.
Solution: a² − b² = (a + b)(a − b) → answer = a + b = 66 + 51 = 117.