Learn Extracted exam questions A-Level Mathematics 9709 Mathematics June 2025 Question Paper 31
9709 Mathematics June 2025 Question Paper 31
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Sketch the graph of $y = |2x-3|$.
Solve the inequality $3x - 1 < |2x-3|$.
It is given that $2\ln p + \ln(p-1) - \tfrac12\ln(q+1) = 3$. Find $q$ in terms of $p$.
Find the complex numbers $z$ for which $\dfrac{z+5i}{z-5}$ is real and $|z| = \sqrt{17}$. Give your answers in the form $z = x + iy$, where $x$ and $y$ are real.
The parametric equations of a curve are $x = e^{\tan t}$, $y = 3\tan^2 t$. Find the equation of the tangent to the curve at the point $(e, 3)$. Give your answer in the form $y = mx + c$, where $m$ and $c$ are exact.
The polynomial $3x^3 + pax^2 + 7a^2 x + qa^3$ is denoted by $f(x)$, where $p$, $q$ and $a$ are constants and $a \neq 0$. When $f(x)$ is divided by $(x+2a)$ the remainder is $-22a^3$. When $f(x)$ is divided by $(3x-a)$ the remainder is $-a^3$. Find the values of $p$ and $q$.
It is given that $z_1 = 3e^{\frac14\pi i}$, $z_2 = \tfrac32 e^{\frac56\pi i}$ and $\omega = 2e^{\frac12\pi i}$.
State the values of $\omega z_1$ and $\omega z_2$. Give your answers in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leqslant \pi$.
On a sketch of an Argand diagram with origin $O$, show the points $A$, $B$, $C$ and $D$ representing the complex numbers $z_1$, $z_2$, $\omega z_1$ and $\omega z_2$ respectively.
State the geometric effects of multiplying $z_1$ and $z_2$ by $\omega$.
Express $5\sin\!\left(x + \tfrac16\pi\right) - 4\cos x$ in the form $R\sin(x-\alpha)$, where $R > 0$ and $0 < \alpha < \tfrac12\pi$. State the exact value of $R$ and give the value of $\alpha$ correct to 3 decimal places.
Hence solve the equation $5\sin\!\left(2\theta + \tfrac16\pi\right) - 4\cos 2\theta = \sqrt7$ for $0 \leqslant \theta \leqslant \pi$. Give your answers correct to 2 decimal places.
With respect to the origin $O$, the points $A$ and $B$ have position vectors $2\mathbf{i} + 4\mathbf{k}$ and $5\mathbf{i} + \mathbf{j} + 6\mathbf{k}$ respectively. The line $l_1$ passes through the points $A$ and $B$.
Find a vector equation for the line $l_1$.
The line $l_2$ has equation $\mathbf{r} = 2\mathbf{i} + \mathbf{j} + 5\mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$. Show that $l_1$ and $l_2$ do not intersect.
Find the acute angle between the directions of $l_1$ and $l_2$.
The constant $a$ is such that $\displaystyle\int_1^a 6x\ln x\,dx = 4$.
Show that $a = \exp\!\left(\tfrac16\!\left(\tfrac{5}{a^2} + 3\right)\right)$, where $\exp(x)$ denotes $e^x$.
Verify by calculation that $a$ lies between 2 and 2.1.
Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Find the quotient and remainder when $x^3 + 5x^2 - 2x - 15$ is divided by $x^2 - 3$.
The variables $x$ and $y$ satisfy the differential equation $\dfrac{dy}{dx} = \dfrac{x^3 + 5x^2 - 2x - 15}{6y(x^2-3)}$. It is given that $y = 2$ when $x = 2$. Solve the differential equation to obtain an expression for $y^2$ in terms of $x$.
The diagram shows the curve $y = \cos x\sqrt{\sin 2x}$ for $0 \leqslant x \leqslant \tfrac12\pi$. The curve has a maximum point at $M$, where $x = a$.
Find the exact value of $a$.
The region enclosed between the $x$-axis and the curve is rotated through $2\pi$ radians about the $x$-axis. Find the exact volume of the solid generated.