Correct answer: 1, 3. Factorising gives (x − 1)(x − 3) = 0, so x = 1 or x = 3.
Background: The x-intercepts are the points where the graph crosses the x-axis (y = 0).
Did you know? For x² + bx + c, find two numbers with product c and sum b: here −1 and −3.
Memory trick: x-intercepts = roots.
Correct answer: 12. Substituting x = −3 gives 9 + 6 − (m + 3) = 0, so 15 − m − 3 = 0 and m = 12.
Background: A zero of a polynomial is a value of x that makes the polynomial equal to 0.
Did you know? Check: (−3)² − 2(−3) − 15 = 9 + 6 − 15 = 0.
Memory trick: Zero → plug in and solve.
Correct answer: 169 cm². Side = ∛2197 = 13 cm, so the area of one face = 13² = 169 cm².
Background: Volume of a cube = side³, and area of one face = side².
Did you know? 13³ = 2197 and 13² = 169.
Memory trick: Cube root for the side, square for the face.
Correct answer: None of these. Multiplying by x(x − 3) gives 2(x − 3) + 3x = x(x − 3), so x² − 8x + 6 = 0 and x = 4 ± √10, which is not among the options.
Background: The restrictions are x ≠ 0 and x ≠ 3, because those values make a denominator zero.
Did you know? The roots 4 + √10 and 4 − √10 are about 7.16 and 0.84.
Memory trick: Multiply through by the LCM x(x − 3) first.
Correct answer: −2. 4x − 10x = 15 − 3, so −6x = 12 and x = −2.
Background: Put the x terms on one side and the numbers on the other.
Did you know? Check: 4(−2) + 3 = −5 and 10(−2) + 15 = −5.
Memory trick: x terms left, numbers right.
Correct answer: 14. 14 × 14 × 14 = 2744, so the cube root is 14.
Background: 2744 ends in 4, so its cube root must also end in 4.
Did you know? Perfect cubes to remember: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Memory trick: Last digit 4 → cube root ends in 4.
Correct answer: 2⁶. (2³)² = 2^(3 × 2) = 2⁶ = 64.
Background: Power rule: (aᵐ)ⁿ = aᵐⁿ.
Did you know? 2⁶ = 64, not 32.
Memory trick: Power of a power → multiply the powers.
Correct answer: 0, 8. Equal roots need b² − 4ac = 0, so k² − 8k = 0, i.e. k(k − 8) = 0, giving k = 0 or k = 8.
Background: A quadratic has equal roots when its discriminant is zero.
Did you know? For k = 0 the equation becomes 2x² = 0.
Memory trick: Equal roots → D = 0.
Correct answer: 15, 20, 25. 15² + 20² = 225 + 400 = 625 = 25², so these sides satisfy the Pythagorean theorem.
Background: The longest side is the hypotenuse.
Did you know? 15, 20, 25 is the 3-4-5 triangle multiplied by 5.
Memory trick: 3-4-5 × 5 = 15-20-25.
Correct answer: √5. Distance = √(x² + y²) = √(2² + 1²) = √5.
Background: The distance formula comes from the Pythagorean theorem.
Did you know? √5 is about 2.24.
Memory trick: d = √(x² + y²).
Correct answer: 16. 48 = 2⁴ × 3 and 80 = 2⁴ × 5, so the HCF is 2⁴ = 16.
Background: The HCF is found by multiplying the common prime factors with their lowest powers.
Did you know? For two numbers, HCF × LCM = product of the numbers. Here LCM = 240 and 16 × 240 = 48 × 80.
Memory trick: HCF = common primes, lowest power.
Correct answer: 48. x² − y² = (x + y)(x − y) = 16 × 3 = 48.
Background: This uses the difference-of-squares identity, x² − y² = (x + y)(x − y), so you do not need to find x and y.
Did you know? The same identity helps with quick mental sums such as 99² − 1² = 100 × 98.
Memory trick: Sum × Difference = 16 × 3 = 48.
Correct answer: x + y = 0. For (a, −a), x + y = a + (−a) = 0, so every such point lies on x + y = 0.
Background: For the point (a, −a), the x- and y-coordinates are opposites, so their sum is always zero.
Did you know? Examples are (1, −1), (2, −2), and (3, −3). All lie on a line through the origin with slope −1.
Memory trick: (a, −a): a + (−a) = 0.
Correct answer: 6050. The nth term is 2n × n² = 2n³, so the sum is 2(1³ + 2³ + … + 10³) = 2 × 55² = 2 × 3025 = 6050.
Background: Each term has the form (2n) × n² = 2n³, so the series is 2(1³ + 2³ + … + 10³).
Did you know? The sum of the first n cubes equals the square of the sum of the first n numbers: [n(n + 1) ÷ 2]².
Memory trick: Sum of cubes = (sum of numbers)², then multiply by 2.
Correct answer: 14 m. Circumference = 22,000 m ÷ 500 = 44 m, and diameter = 44 × 7 ÷ 22 = 14 m.
Background: The distance covered in one revolution is the wheel’s circumference, so circumference = total distance ÷ number of revolutions.
Did you know? The value 22/7 is a convenient approximation of π (3.14159…).
Memory trick: Distance ÷ revolutions = circumference; diameter = circumference × 7/22.
Correct answer: 150.24 cm. New total = 20 × 150 + 155 = 3155 cm, and new average = 3155 ÷ 21 ≈ 150.24 cm.
Background: The new average = (old total + new value) ÷ (new number of students).
Did you know? Adding a value just above the old average raises the average only slightly, which is why the answer is close to 150.
Memory trick: Total first, then add the new value, then divide.
Correct answer: 21. With x 20-cent coins and 54 − x 50-cent coins: 0.20x + 0.50(54 − x) = 20.70, so x = 21. Check: 21 × 0.20 + 33 × 0.50 = 20.70.
Background: Problems with two unknown quantities need two equations: one for the count (x + y = 54) and one for the total value (0.20x + 0.50y = 20.70).
Did you know? You can always check an answer by substituting it back: 21 × 0.20 + 33 × 0.50 = 20.70.
Memory trick: One equation for count, one for value.
Correct answer: 50. The first four multiples of 5 are 5, 10, 15, 20, and 5 + 10 + 15 + 20 = 50.
Background: The first four multiples of 5 are 5, 10, 15, and 20. Their sum is 50.
Did you know? A shortcut formula: the sum of the first n multiples of k is k × n(n + 1) ÷ 2, so 5 × 4 × 5 ÷ 2 = 50.
Memory trick: 5 × (1 + 2 + 3 + 4) = 5 × 10 = 50.
Correct answer: 2,496. 2,400 × 1.3 = 3,120; then 3,120 × 0.8 = 2,496.
Correct answer: 1/19. Denominators follow 2n + 1, so the 9th term has denominator 19.
Correct answer: 27. The sequence is squares of odd numbers: 1, 9, 25, 49, 81, 121, so 27 should be 25.
Correct answer: Rs. 0.04. 2.5 hours = 150 minutes; 6 ÷ 150 = Rs. 0.04.
Correct answer: 78. Sum of 1 to 12 = 12 × 13 ÷ 2 = 78.
Correct answer: On the x-axis. Points with y = 0 lie on the x-axis.
Correct answer: 0.4. 2 ÷ 5 = 0.4.