Advanced level 2026 laval mock further mathematics 2
Advanced level 2026 laval mock further mathematics 2
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Given the differential equation $\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 12(x + e^x)$ has general solution $y = g(x) + f(x)$ where $f(x)$ is a particular integral and satisfying the above differential equation.
a) find $f(x)$ (8 marks)
b) the complementary function $g(x)$ and hence the general solution of $y = g(x) + f(x)$.
i) Given that $I_n = \int_0^1 x^n \sqrt{1 – x^2} dx$
a) Show that $I_{n+2} = (\frac{n+1}{n+4})I_n$
b) hence evaluate $I_4$ (8 marks)
ii) A curve is defined by $y^2 = \frac{x^3}{a}$ where $a$ is a constant, for $0 \leq x \leq \frac{7a}{3}$. Show that the arc length of the curve is given by $\frac{13a}{3}$ (4 marks)
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Given the points $A(1, 4, 3), B(5, 3, 1)$ and $C(-4, -2, 3)$, find
a) $\vec{AB} \times \vec{BC}$
b) the Cartesian equation of the plane ABC
c) the distance from the plane ABC from the origin
d) the area of the triangle ABC (8 marks)
i) Given that $z^3 = (1 + i\sqrt{3})^8 (1 – i)^5$
Find the three roots of the above equation, giving the answers in the form $k\sqrt{2}e^{i\theta}$, where $-\pi \leq \theta \leq \pi, k \in \mathbb{Z}$ (4 marks)
ii) The general point $P(x, y)$ which is represent by the complex number $z = x + iy$ in the $z$ plane, lies on the locus of $|z| = 1$
A transformation from the $z$ plane to the $w$ plane is defined by $w = \frac{z + 2}{z + 1}, z \in \mathbb{C}$
and maps the point $P(x, y)$ onto the point $Q(u, v)$
Find, in Cartesian form, the equation of the locus of the point $Q$ in the $w$ plane (4 marks)
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let $G$ be a group defined under multiplication on the set $\{x, x^2, x^3, x^4\}, x \in \mathbb{R}$ such that $x^5 = x$
a) Draw the group table for $G$
b) find the identity element and show that $G$ is a commutative group
c) Show that the set $H = \{1, 3, 7, 9\}$ forms a group under multiplication modulo 10 ($\times_{10}$)
d) show that $G$ is isomorphic to $H$, that $G \cong H$ (12 marks)
i) Find the value of $\lambda$ for which the vectors $v_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, v_2 = \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix}, v_3 = \begin{pmatrix} \lambda \\ 0 \\ -3 \end{pmatrix}$ are linearly independent (4 marks)
ii) Show that the linear transformation T represented by the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 0 & -2 \\ 3 & -2 & -7 \end{pmatrix}$ maps the whole space unto the plane $x – 2y + z = 0$ (4 marks)
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a) Find the equation of the normal at the point $P(4, 1)$ to the rectangular hyperbola $xy = 4$. This normal meets the hyperbola again at the point $Q$, find the length of PQ.
0775 Further Maths 2/1A @Laval Mock 2026
