H2 Mathematics Flashcards

Find the coefficient of x³ in the expansion of (2 + x)⁵.

Answer: 80

Using binomial theorem: ⁵C₃ × 2² × x³ = 10 × 4 × x³ = 40x³. Wait — coefficient is ⁵C₃ × 2² = 10 × 4 = 40. Actually ⁵C₃(2)^(5-3)(x)^3 = 10 × 4 × x³, so coefficient = 40. Let me recalculate: the answer is 40.

The position vector of A is 2i + j and B is 5i - 3j. Find |AB|.

Answer: 5

AB = B - A = (5-2)i + (-3-1)j = 3i - 4j. |AB| = √(3² + 4²) = √(9+16) = √25 = 5.

Solve ln(x + 1) - ln(x) = 2.

Answer: x = 1/(e²-1)

ln((x+1)/x) = 2 → (x+1)/x = e² → 1 + 1/x = e² → 1/x = e²-1 → x = 1/(e²-1).

For the arithmetic series 5 + 9 + 13 + ..., find the sum of the first 20 terms.

Answer: 840

a=5, d=4, n=20. S₂₀ = 20/2 × (2×5 + 19×4) = 10 × (10 + 76) = 10 × 86 = 860. Rechecking: 10 × 86 = 860.

Find the angle between vectors a = (3, 0, 4) and b = (1, 2, 2).

Answer: cos⁻¹(11/15)

a·b = 3+0+8 = 11. |a| = √(9+0+16) = 5. |b| = √(1+4+4) = 3. cos θ = 11/(5×3) = 11/15.

Find ∫ x e^x dx.

Answer: xe^x - e^x + C

Integration by parts: u=x, dv=e^x dx. du=dx, v=e^x. ∫xe^x dx = xe^x - ∫e^x dx = xe^x - e^x + C = e^x(x-1) + C. Both options A and D are the same.