Advanced level 2026 BAEBOC pure mathematics 2

a) Given that $g(x) = (x-2)(x+3)Q(x) + ax+b$ is a polynomial function. When $g(x)$ is divided by $(x + 3)$, the remainder is 8, and when $g(x)$ is divided by $(x – 2)$ the remainder is 3. Find the remainder when $g(x)$ is divided by $(x – 2)(x + 3)$. (7marks)

b) Show that the quadratic equation $x^2 + x(2p + q) + pq = 0$ has real roots for all real values of $p$ and $q$. (5marks)

i) The first term of an AP is 17, and the sum of the first 16 terms is -16. Find the sixteenth term and the common difference of the progression. (6marks)

ii) Using the binomial expansion, show that $(\sqrt{7} + 2)^4 + (\sqrt{7} – 2)^4 = 466$ (5marks)

Given that $f(x) = \frac{7}{(3x-1)(x+2)}$,

a) Express $f(x)$ in partial fractions (3marks)

b) State the domain of $f(x)$ (2mark)

c) State the equations of the asymptotes to the curve $y = f(x)$ (3marks)

d) Determine the turning point of the function $y = f(x)$ (2marks)

e) Sketch the curve of $y = f(x)$, showing the behavior of the curve as it approaches the asymptotes and the point where the curve crosses the coordinate axes. (3marks)

i) Given that $y = \frac{\cos x + \sin x}{\cos x – \sin x}$, show that $\frac{d^2y}{dx^2} = 2y\frac{dy}{dx}$ (5marks)

ii) Evaluate $\int_{0}^{\frac{\pi}{3}} \sin 2x \cos x dx$ (4marks)

i) Given that $z$ is a complex number, find the Cartesian equation of the loci $\left\{ z : \text{arg}(z – i) = \frac{\pi}{4} \right\}$. (2marks)

ii) Show that $\frac{\cos 3A – \cos A}{\sin 3A + \sin A} = -\tan A$ (3marks)

iii) Show that the matrix $M$ is invertible where $M = \begin{pmatrix} 2 & 1 & 0 \\ 1 & -1 & 1 \\ 5 & 1 & 0 \end{pmatrix}$ hence find $M^{-1}$, the inverse of $M$ (5marks)

i) Obtain the centres of the circles, $s_1 : x^2 + y^2 – 2x + 4y – 5 = 0$ and $s_2 : x^2 + y^2 – 10x – 8y + 2 = 0$.

Hence, find the locus of the point which moves so that it is always equidistant from the centres of $S_1$ and $S_2$. (6marks)

ii) The lines $l_1$ and $l_2$ have equations

$l_1 : r_1 = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -2 \\ 3 \end{pmatrix}, l_2 : r_2 = \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$ respectively. Show that the lines $l_1$ and $l_2$ are skew. (5marks)

Define a partial order relation. Let $B$ be a set and $P(A)$, the power set of $A$. Define a relation $R$ on $P(A)$ by $ARB \iff A \subseteq B$, where $A, B \in P(A)$. Show that $R$ is a partial order relation. (8marks)