Advanced level 2026 west regional mock pure mathematics 2

(i) Express $\frac{5x – 2}{(x + 1)(2x – 3)}, x \in \mathbb{R}, x \neq -1, \frac{3}{2}$ in partial fractions. (4 marks)

(ii) If the roots of the quadratic equation $3x^2 + kx + 12 = 0$ are $\alpha$ and $\beta$, find the values of the real constant $k$ for which $\alpha^2 + \beta^2 = 10$. (5 marks)

2. The polynomial $f(x) = 2x^3 + ax^2 + bx – 15$ has factor $(x – 3)$.

   $f(x)$ leaves a remainder $-35$ when divided by $(x + 2)$. Find

   a. the values of the constants $a$ and $b$.

   b. The values of $x$ for which $f(x) = 0$. (8 marks)

(i) Given that $y = e^{2x} \ln 3x$, find $\frac{dy}{dx}$ when $x = 1$. (3 marks)

(ii) Evaluate $\int_{3}^{8} \frac{1}{x\sqrt{x + 1}} dx$ using the substitution $u^2 = x + 1$. (5 marks)

(i) The equations of two circles $S_1$ and $S_2$ are given by:

$S_1 : x^2 + y^2 – 4x – 6y – 3 = 0$ and $S_2 : x^2 + y^2 + 8x + 10y = -5$

Show that $S_1$ and $S_2$ touch each other externally. (6 marks)

(ii) Given the periodic function $f$ with period 4, where $f(x) = \begin{cases} 3x – 1, & 0 \le x \le 1 \\ -x + 5, & 1 < x \le 4 \end{cases}$

a. Show that $f$ is not continuous at $x = 1$. (2 marks)

b. Find $f(9)$. (2 marks)

c. Sketch the graph of $f(x)$ in the interval $-2 \le x \le 6$. (3 marks)

(i) Evaluate $\sum_{r=1}^{\infty} 5(\frac{2}{3})^r$ (4 marks)

(ii) The variables $x$ and $y$ are related by the relationship $y = \log_{10}(a + bx)$, where $a$ and $b$ are constants.

Approximate values of $x$ are tabulated below.

$x$ | 1 | 2 | 3 | 4 | 5 | 6

$y$ | 0.857 | 0.924 | 0.982 | 1.033 | 1.079 | 1.21

By drawing a linear graph relating $10^y$ and $x$, estimate to one decimal place

a. the values of $a$ and $b$.

b. The value of $y$ when $x = 1.5$. (9 marks)

(i) Given the lines $L_1 : \frac{x – 2}{1} = \frac{y – 1}{1} = \frac{z}{2}$ and $L_2 : \mathbf{r} = 2\mathbf{i} + 2\mathbf{j} + mk + \mu(\mathbf{i} + 2\mathbf{j} + \mathbf{k})$. Find the value of $m$ for which $L_1$ and $L_2$ intersect.

(ii) Prove by mathematical induction that $\sum_{r=1}^{n} \frac{r}{2^r} = 2 – \frac{n + 2}{2^n}, n \ge 1, n \in \mathbb{N}$. (5,5 marks)

7. Given that $f(\theta) = \frac{2 \sin 2\theta – 3 \cos 2\theta + 3}{\sin \theta}, \theta \neq n\pi$ where $n$ is an integer,

MTPG MOCK/ 2026/ /0765-0770/02 2/3