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A-Level Maths Practice Questions

Advanced practice questions covering Pure, Statistics & Mechanics. Each question includes detailed step-by-step solutions.

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Question 1
Pure Maths AS Level

Differentiate: y = 3x³ + 2x² - 5x + 7

Solution:
  1. Apply the power rule: d/dx(xⁿ) = nxⁿ⁻¹
  2. Differentiate each term:
    dy/dx = 3(3x²) + 2(2x) - 5(1) + 0
    dy/dx = 9x² + 4x - 5
Answer: dy/dx = 9x² + 4x - 5
Question 2
Pure Maths A2 Level

Integrate: ∫(4x³ - 6x + 2) dx

Solution:
  1. Apply the power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
  2. Integrate each term:
    ∫4x³ dx = 4x⁴/4 = x⁴
    ∫-6x dx = -6x²/2 = -3x²
    ∫2 dx = 2x
  3. Combine and add constant: x⁴ - 3x² + 2x + C
Answer: x⁴ - 3x² + 2x + C
Question 3
Pure Maths AS Level

Solve: 2ˣ = 32

Solution:
  1. Express 32 as a power of 2: 32 = 2⁵
  2. Therefore: 2ˣ = 2⁵
  3. Since the bases are equal, the exponents must be equal: x = 5
Answer: x = 5
Question 4
Statistics A2 Level

The probability of event A is 0.6 and event B is 0.4.

If A and B are independent, find P(A ∩ B).

Solution:
  1. For independent events: P(A ∩ B) = P(A) × P(B)
  2. Given: P(A) = 0.6, P(B) = 0.4
  3. Calculate: P(A ∩ B) = 0.6 × 0.4 = 0.24
Answer: 0.24
Question 5
Mechanics A2 Level

A particle has mass 5 kg and accelerates at 3 m/s².

Calculate the force acting on the particle.

Solution:
  1. Use Newton's Second Law: F = ma
  2. Where m = 5 kg, a = 3 m/s²
  3. Substitute: F = 5 × 3 = 15 N
Answer: 15 N
Question 6
Pure Maths Challenge

Find the equation of the tangent to the curve y = x² + 3x - 2 at the point (1, 2).

Solution:
  1. Find the gradient by differentiating: dy/dx = 2x + 3
  2. At x = 1: m = 2(1) + 3 = 5
  3. Use point-slope form: y - 2 = 5(x - 1)
  4. Simplify: y = 5x - 3
Answer: y = 5x - 3
Question 7
Statistics AS Level

Find the standard deviation of: 4, 8, 12, 16, 20

Solution:
  1. Find the mean: (4+8+12+16+20)/5 = 12
  2. Find deviations squared: 64, 16, 0, 16, 64
  3. Find variance: (64+16+0+16+64)/5 = 32
  4. Standard deviation: √32 = 5.66 (2 d.p.)
Answer: 5.66
Question 8
Mechanics Challenge

A projectile is launched at 20 m/s at 30° to the horizontal.

Calculate the maximum height reached. (g = 10 m/s²)

Solution:
  1. Vertical component: v_y = 20 sin(30°) = 10 m/s
  2. At max height, v = 0. Use: v² = u² - 2gh
  3. 0 = 10² - 2(10)h
  4. h = 100/20 = 5 m
Answer: 5 m
Question 9
Pure Maths A2 Level

Solve: log₂(x) + log₂(x-3) = 2

Solution:
  1. Use log rules: log₂(x(x-3)) = 2
  2. Convert to exponential: x(x-3) = 2² = 4
  3. Expand: x² - 3x = 4
  4. Rearrange: x² - 3x - 4 = 0
  5. Factorise: (x-4)(x+1) = 0
  6. Solutions: x = 4 or x = -1 (reject negative)
Answer: x = 4
Question 10
Statistics A2 Level

X ~ N(50, 16). Find P(X > 54).

Solution:
  1. Mean μ = 50, Variance σ² = 16, so σ = 4
  2. Standardise: Z = (54-50)/4 = 1
  3. Find P(Z > 1) from tables: P(Z > 1) = 1 - 0.8413 = 0.1587
Answer: 0.1587