Advanced level 2026 south west regional mock mathematics with mechanics 2

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1. (i) $\alpha$ and $\beta$ are the roots of a quadratic equation, where $-\alpha – \beta = 5$ and $\alpha^2 + \beta^2 = 7$.

   Find the quadratic equation, with integer coefficients, whose roots are $(2\alpha + \alpha\beta)$ and $(2\beta + \alpha\beta)$. (6 marks)

   (ii) The coefficient of $x^3$ in the expansion of $(3 + ax)^6$ is 160, find the value of the constant $a$.

   Hence, find the coefficient of $x^3$ in the expansion of $(3 + ax)^6(1 – 2x)$. (6 marks)

2. The table below shows the corresponding values of $x$ and $y$ obtained from an experiment.

It was observed that $x$ and $y$ satisfy a relation of the form

where $\alpha$ and $\beta$ are real constants.

a) Show that $\frac{1}{y} = (\frac{\alpha}{\beta})x + \beta$

b) By drawing a suitable straight line graph, find the values of $\alpha$ and $\beta$, to 1 decimal place. (8 marks)

3. (i) The polynomial $p(x)$ is given by $p(x) = x^3 + kx^2 – x + 12$. Given that $p(x)$ leaves remainders of $R$ and $8R$ when divided by $(x – 1)$ and $(x – 4)$ respectively,

   (a) Find the values of $R$ and $k$. (5 marks)

   (b) Show that $(x + 4)$ is a factor of $p(x)$. (2 marks)

   (c) Hence, show that the equation $p(x) = 0$ has only one real root. (2 marks)

   (ii) Find the solution set of the inequality $|x – 1| > 2x + 1$. (3 marks)

4. (i) The function $f$ is periodic with period 4, where

   $$f(x) = \begin{cases} x^2 – 4, & \text{if } 0 \le x < 2 \\ -2x + k, & \text{if } 2 \le x < 4 \end{cases}$$

   Given that $f$ is continuous at the point $x = 2$, find

   (a) the value of the constant $k$, (2 marks)

   (b) $f(13)$ and $f(-41)$. (2 marks)

   (c) Investigate the parity of $f$ in the interval $0 \le x < 2$. (2 marks)

   (d) Sketch the graph of $f(x)$ in the interval $-4 \le x < 8$. (2 marks)

   (ii) Prove by contradiction, or otherwise, that $\frac{1}{\sqrt{2}}$ is irrational. (4 marks)

5. (i) A curve is given by $(x^2 – 3) \ln y + 6x = 14$

   (a) Show that $\frac{dy}{dx} = \frac{y(2x \ln y + 6)}{3 – x^2}$ (3 marks)

   (b) Find the equation of the tangent to the curve at the point $P(2, e^2)$. (3 marks)

   (ii) Solve the differential equation $(1 + x^2) \frac{dy}{dx} = 2xy$, given that $y = 1$ when $x = 0$. (5 marks)

Would you like me to work through the solution for the quadratic roots in question 1(i) or the differential equation in 5(ii)?