Advanced level 2026 technical North West regional mock mathematics 1-3 guide

SECTION A

1. Complex Numbers and Loci

a) z² – 3z + (3 – i) = 0 M1M1A1A1

Δ² = -3 – 4i → Δ = ±(1 + 2i)

∴ z₁ = (3 + 1 + 2i) / 2 = 2 + i;   z₂ = (3 – 1 – 2i) / 2 = 1 – i

S = {2 + i, 1 – i}

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b) Let z = x + iy M1M1M1A1

∴ f(z) = (x² – y² – 3x + 3) + i(2xy – 3y – 1)

For f(z) to be purely imaginary, Re[f(z)] = 0

→ x² – y² – 3x + 3 = 0 … (E)

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c) (E): x² – y² – 3x + 3 = 0 M1M1A0.5, 0.5

→ (x – 3/2)² / (√3/2)² – y² / (√3/2)² = -1

(E) is a rectangular hyperbola symmetrical about the Oy-axis.

d) Center of (E): Ω(3/2, 0) 1

e) Vertices: S(3/2, √3/2);   S'(3/2, -√3/2) 1.5

Oblique asymptotes: { y = x – 3/2 | y = -x + 3/2 } 1.5

2. Function Analysis: f(x) = (x + 1)e^x+1

a) D_f = ]-∞, +∞[ 1

b) lim_x→-∞ f(x) = 0;   lim_x→+∞ f(x) = +∞ 1

c) f'(x) = (x + 2)e^x+1 0.5

For turning points, f'(x) = 0 → x = -2

f(-2) = -1/e ≈ -0.367 0.5

Table of Variations: 1.5
x: -∞ … -2 … +∞
f'(x): (-) 0 (+)
f(x): Decreases to -1/e, then Increases.

d) Intercepts: x-axis: (-1,0); y-axis: (0,e) 1, 1

e) f”(x) = (x + 3)e^x+1 0.5

f”(x) = 0 → x = -3. Point of inflection at (-3, -2/e²) 1

3. Differential Equations

3a) (E): g”(x) + g'(x) – 6g(x) = 0

i. g(x) = Ae^2x + Be^-3x 1, 1, 1

ii. g(0) = 3 → A + B = 3

g'(0) = -4 → 2A – 3B = -4

∴ A = 1; B = 2. Result: g(x) = e^2x + 2e^-3x 1, 1, 1