Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 31
9709 Mathematics November 2025 Question Paper 31
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Find the exact value of $\displaystyle\int_1^2 \ln 3x\,dx$. Give your answer in the form $a + \ln b$, where $a$ and $b$ are integers.
Show that the equation $\log_4(2x+1) = 2\log_4(3x-1) - 2$ can be written as a quadratic equation in $x$.
Hence solve the equation $\log_4(2x+1) = 2\log_4(3x-1) - 2$.
Express $3\sqrt2\sin(x+45\degree) + \cos x$ in the form $R\cos(x-\alpha)$, where $R > 0$ and $0\degree < \alpha < 90\degree$.
Hence solve the equation $3\sqrt2\sin(3\theta+45\degree) + \cos 3\theta = -4$ for $0\degree < \theta < 180\degree$.
The diagram shows the graph of $y = e^{\sin 2x}\cos 4x$ for $0 \leqslant x \leqslant \tfrac14\pi$, and its maximum point $M$. Find the $x$-coordinate of $M$.
The shaded region on the Argand diagram shows points representing complex numbers $z$ defined by two inequalities. The shaded region is bounded by a circle and a line parallel to the imaginary axis. The boundaries of the region are included in the shaded region.
Find two inequalities in terms of $z$ that define the shaded region.
Calculate the least value of $\arg z$ for points in this region.
Solve the quadratic equation $(2+i)w^2 + 4w + 2 - i = 0$. Give your answers in the form $x + iy$, where $x$ and $y$ are real.
The parametric equations of a curve are $x = t^2 - \ln(2t+1)$, $y = \dfrac{t}{2t+1}$. Obtain a simplified expression for $\dfrac{dy}{dx}$ in terms of $t$.
The variables $x$ and $y$ satisfy the differential equation $(x^2+1)\dfrac{dy}{dx} = kxe^{2y}$, where $k$ is a constant. It is given that $y = 0$ when $x = 0$ and that $y = -\tfrac12$ when $x = 1$. Solve the differential equation and find the exact value of $y$ when $x = \sqrt3$.
By sketching a suitable pair of graphs, show that the equation $\sec 2x = -e^x$ has only one root in the interval $0 < x < \tfrac12\pi$.
Show by calculation that this root lies between 0.9 and 1.
Show that if a sequence of values given by the iterative formula $x_{n+1} = \tfrac12\cos^{-1}(-e^{-x_n})$ converges, then it converges to the root of the equation in part (a).
Use the iterative formula given in part (c) to calculate $x$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
Let $f(x) = \dfrac{x^3 + 2x - 11}{(3+x)(2+x^2)}$.
Express $f(x)$ in partial fractions.
Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.
With respect to the origin $O$, the points $A$, $B$, $C$, $D$ have position vectors $\overrightarrow{OA} = \begin{pmatrix}1\\5\\3\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}0\\4\\1\end{pmatrix}$, $\overrightarrow{OC} = \begin{pmatrix}1\\-3\\1\end{pmatrix}$, $\overrightarrow{OD} = \begin{pmatrix}3\\-5\\4\end{pmatrix}$. The line $m$ passes through $A$ and $B$.
Find a vector equation for $m$.
Find the position vector of the point of intersection of $m$ and the line passing through the points $C$ and $D$.
Find the position vector of the foot of the perpendicular from $C$ to $m$.