Learn Extracted exam questions A-Level Mathematics 9709 Mathematics June 2025 Question Paper 32
9709 Mathematics June 2025 Question Paper 32
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Solve the equation $\dfrac{e^x + 2e^{-x}}{e^x - 3} = 4$. Give your answer correct to 3 decimal places.
Expand $(6-x)(1-2x)^{-\frac{3}{2}}$ in ascending powers of $x$, up to and including the term in $x^2$, simplifying the coefficients.
State the set of values of $x$ for which the expansion is valid.
On an Argand diagram shade the region whose points represent complex numbers $z$ which satisfy both the inequalities $|z - 3i| \leqslant 2$ and $\tfrac{1}{4}\pi \leqslant \arg(z - 1 - 2i) \leqslant \tfrac{3}{4}\pi$.
Solve the equation $3\cot x - 4\cot 2x = 3$ for $0\degree \leqslant x \leqslant 180\degree$.
The square roots of $-1 - 4\sqrt{5}\,i$ can be expressed in the Cartesian form $x + iy$, where $x$ and $y$ are real and exact. By first forming a quartic equation in $x$ or $y$, find the square roots of $-1 - 4\sqrt{5}\,i$ in exact Cartesian form.
By sketching a suitable pair of graphs, show that the equation $|x - 2| = 2\sin\tfrac{1}{2}x$ has only one root in the interval $0 < x < \pi$.
Show by calculation that this root lies between 1 and 1.5.
Use the iterative formula $x_{n+1} = 2 - 2\sin\tfrac{1}{2}x_n$ with an initial value of 1.03 to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Express $7\sin\theta + 24\cos\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \tfrac{1}{2}\pi$. Give the value of $\alpha$ correct to 4 decimal places.
Hence solve the equation $7\sin\tfrac{1}{3}x + 24\cos\tfrac{1}{3}x = 24.5$ for $0 < x < \pi$.
The variables $x$ and $\theta$ satisfy the differential equation $\sin 2\theta\,\dfrac{dx}{d\theta} = (4x + 3)\cos 2\theta$, and $x = 0$ when $\theta = \tfrac{1}{12}\pi$. Solve the differential equation and obtain an expression for $x$ in terms of $\theta$.
With respect to the origin $O$, the points $A$, $B$ and $C$ have position vectors given by $\overrightarrow{OA} = \begin{pmatrix}1\\-4\\2\end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix}-2\\1\\3\end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix}2\\3\\5\end{pmatrix}$.
Find a vector equation for the line through $A$ and $B$.
Using a scalar product, find the exact value of $\cos BAC$.
Hence find the exact area of triangle $ABC$.
Find the quotient and remainder when $x^2$ is divided by $1 + 4x^2$.
Find the exact value of $\displaystyle\int_0^{0.5} x\tan^{-1}(2x)\,dx$.
The diagram shows the graph of $y = 5\sin 2x\cos^2 x$ for $0 \leqslant x \leqslant \tfrac{1}{2}\pi$ and its maximum point $M$.
Find the exact $x$-coordinate of $M$.
By using the substitution $u = \cos x$, find the area of the region bounded by the curve, the $x$-axis between $x = 0$ and $x = \tfrac{1}{4}\pi$, and the line $x = \tfrac{1}{4}\pi$.