Advanced level 2026 south west regional mock mathematics with statistics 2
Advanced level 2026 south west regional mock mathematics with statistics 2
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 $(2\alpha + a\beta)$ and $(2\beta + a\beta)$.
(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.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| $y$ | 0.250 | 0.133 | 0.091 | 0.069 | 0.056 | 0.047 |
It was observed that $x$ and $y$ satisfy a relation of the form:
where $\alpha$ and $\beta$ are real constants.
(a) Show that $\frac{1}{y} = \left(\frac{\alpha}{\beta}\right)x + \beta$
(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
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
(3 marks)
(b) Find the equation of the tangent to the curve at the point $P(2, e^2)$. (3 marks)
Source Information:
© TRU/RPI-Sc/SWAMT/PMS/0770/P2/MOCK 2026 | pg. 2/3
