Advanced level 2026 west regional mock further mathematics 1

Here is the extracted text from the sixth image, which contains advanced mathematics questions:

1. $\frac{4}{(x + 1)(x^2 + 2x + 3)}, x \neq -1$, expressed in partial fractions where $A, B$ and $C$ are real constants is

   • A. $\frac{A}{x+1} + \frac{Bx+C}{x^2+2x+3}$

   • B. $\frac{A}{x+1} + \frac{B}{x^2+2x+3}$

   • C. $\frac{A}{x+1} + \frac{Bx}{x^2+2x+3}$

   • D. $\frac{A}{x+1} + \frac{B}{x-1} + \frac{C}{x+3}$

2. The particular integral of the differential equation $\frac{d^2y}{dx^2} – 8\frac{dy}{dx} + 12y = e^{2x}$ is

   • A. $\lambda e^{2x}$

   • B. $\lambda x^2 e^{2x}$

   • C. $\lambda x e^{2x}$

   • D. $2\lambda e^{2x}$

3. The maximum value of $f(x) = \frac{3}{2 \cosh x + 1}$ is

   • A. 4

   • B. 2

   • C. 1

   • D. 0

4. If $\mathbf{a} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ and $\mathbf{b} = 3\mathbf{i} – 2\mathbf{j} + \mathbf{k}$, then $\mathbf{a} \times \mathbf{b} =$

   • A. $8\mathbf{i} + 8\mathbf{j} – 8\mathbf{k}$

   • B. $8\mathbf{i} – 8\mathbf{j} – 8\mathbf{k}$

   • C. $8\mathbf{i} – 8\mathbf{j} + 8\mathbf{k}$

   • D. $-8\mathbf{i} + 8\mathbf{j} – 8\mathbf{k}$

5. The Maclaurin series expansion of $f(x) = \ln \left[ \frac{1-3x}{(1+3x)^2} \right]$ is valid for

   • A. $-\frac{1}{3} < x < \frac{1}{3}$

   • B. $-\frac{1}{3} \le x < \frac{1}{3}$

   • C. $-\frac{1}{3} < x \le \frac{1}{3}$

   • D. $-\frac{1}{3} \le x \le \frac{1}{3}$

6. The equation of the locus of points which lie on the half line with equation $\arg z = \frac{\pi}{4}, z \neq 0$ after it has been transformed by the complex function $\omega = \frac{1}{z}$ is

   • A. $\arg \omega = 0$

   • B. $\arg \omega = \frac{\pi}{2}$

   • C. $\arg \omega = -\frac{\pi}{4}$

   • D. $\arg \omega = -\frac{\pi}{2}$

7. For every natural number $n$, $n(n^2 – 1)$ is divisible by

   • A. 4

   • B. 6

   • C. 10

   • D. 15

8. Given that $F(x) = \int_{0}^{x} f(t) dt$ where $f$ is defined on $[-2, 3]$ as shown on the graph below. On which interval is $F$ decreasing?

   (Image shows a parabolic graph $f(x)$ crossing the x-axis at $x=2$)

   • A. $[-2, -1]$

   • B. $[-2, 2]$

   • C. $[2, 3]$

   • D. $[-2, 1]$

9. The root mean square value of the function $f(t) = \sin t$ on the interval $[0, \frac{\pi}{2}]$ is

   • A. $\sqrt{2}$

   • B. $\frac{\sqrt{2}}{2}$

   • C. $\frac{1}{2}$

   • D. 2

10. The eccentricity of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ if its latus rectum is equal to one half of its minor axis is

    (The question text cuts off slightly at “is”)

Would you like me to help you solve the cross product in question 4 or find the divisibility proof for question 7?