Cambridge AS & A Level. 2025. Mathematics (9709). May/June - Paper 3 (31) · Pure Mathematics 3

**(a)** Sketch the graph of $y = |2x-3|$.

**(b)** Solve the inequality $3x-1 \lt |2x-3|$.

It is given that $2\ln p + \ln(p-1) - \frac{1}{2}\ln(q+1) = 3$.

Find $q$ in terms of $p$.

Find the complex numbers $z$ for which $\dfrac{z+5\mathrm{i}}{z-5}$ is real and $|z| = \sqrt{17}$. Give your answers in the form $z = x+\mathrm{i}y$, where $x$ and $y$ are real.

The parametric equations of a curve are

$x = \mathrm{e}^{\tan t}, \quad y = 3\tan^2 t.$

Find the equation of the tangent to the curve at the point $(\mathrm{e}, 3)$. Give your answer in the form $y = mx+c$, where $m$ and $c$ are exact.

The polynomial $3x^3+pax^2+7a^2x+qa^3$ is denoted by $\mathrm{f}(x)$, where $p$, $q$ and $a$ are constants and $a \ne 0$.

When $\mathrm{f}(x)$ is divided by $(x+2a)$ the remainder is $-22a^3$. When $\mathrm{f}(x)$ is divided by $(3x-a)$ the remainder is $-a^3$.

Find the values of $p$ and $q$.

It is given that $z_1 = 3\mathrm{e}^{\frac{1}{4}\pi\mathrm{i}}$, $z_2 = \frac{3}{2}\mathrm{e}^{\frac{1}{6}\pi\mathrm{i}}$ and $\omega = 2\mathrm{e}^{\frac{1}{2}\pi\mathrm{i}}$.

**(a)** State the values of $\omega z_1$ and $\omega z_2$. Give your answers in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $r \gt 0$ and $-\pi \lt \theta \le \pi$.

It is given that $z_1 = 3\mathrm{e}^{\frac{1}{4}\pi\mathrm{i}}$, $z_2 = \frac{3}{2}\mathrm{e}^{\frac{1}{6}\pi\mathrm{i}}$ and $\omega = 2\mathrm{e}^{\frac{1}{2}\pi\mathrm{i}}$.

**(b)** 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.

(In part (a) you found $\omega z_1 = 6\mathrm{e}^{\frac{3}{4}\pi\mathrm{i}}$ and $\omega z_2 = 3\mathrm{e}^{\frac{2}{3}\pi\mathrm{i}}$.)

It is given that $z_1 = 3\mathrm{e}^{\frac{1}{4}\pi\mathrm{i}}$, $z_2 = \frac{3}{2}\mathrm{e}^{\frac{1}{6}\pi\mathrm{i}}$ and $\omega = 2\mathrm{e}^{\frac{1}{2}\pi\mathrm{i}}$.

**(c)** State the geometric effects of multiplying $z_1$ and $z_2$ by $\omega$.

**(a)** Express $5\sin\left(x+\frac{1}{6}\pi\right)-4\cos x$ in the form $R\sin(x-\alpha)$, where $R \gt 0$ and $0 \lt \alpha \lt \frac{1}{2}\pi$. State the exact value of $R$ and give the value of $\alpha$ correct to 3 decimal places.

**(b)** Hence solve the equation $5\sin\left(2\theta+\frac{1}{6}\pi\right)-4\cos 2\theta = \sqrt{7}$ for $0 \le \theta \le \pi$. Give your answers correct to 2 decimal places.

(In part (a) you found $5\sin\left(x+\frac{1}{6}\pi\right)-4\cos x = \sqrt{21}\sin(x-\alpha)$ with $\alpha = 0.333$.)

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 $B$.

**(a)** Find a vector equation for the line $l_1$.

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 $B$.

The line $l_2$ has equation $\mathbf{r} = 2\mathbf{i}+\mathbf{j}+5\mathbf{k}+\mu(\mathbf{i}+2\mathbf{j}+3\mathbf{k})$.

(In part (a) you found $l_1$: $\mathbf{r} = 2\mathbf{i}+4\mathbf{k}+\lambda(3\mathbf{i}+\mathbf{j}+2\mathbf{k})$.)

**(b)** Show that $l_1$ and $l_2$ do **not** intersect.

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 $B$.

The line $l_2$ has equation $\mathbf{r} = 2\mathbf{i}+\mathbf{j}+5\mathbf{k}+\mu(\mathbf{i}+2\mathbf{j}+3\mathbf{k})$.

(In part (a) you found $l_1$: $\mathbf{r} = 2\mathbf{i}+4\mathbf{k}+\lambda(3\mathbf{i}+\mathbf{j}+2\mathbf{k})$.)

**(c)** Find the acute angle between the directions of $l_1$ and $l_2$.

The constant $a$ is such that $\displaystyle\int_1^a 6x\ln x\,\mathrm{d}x = 4$.

**(a)** Show that $a = \exp\left(\frac{1}{6}\left(\frac{5}{a^2}+3\right)\right)$, where $\exp(x)$ denotes $\mathrm{e}^x$.

The constant $a$ is such that $\displaystyle\int_1^a 6x\ln x\,\mathrm{d}x = 4$.

(In part (a) it was shown that $a = \exp\left(\frac{1}{6}\left(\frac{5}{a^2}+3\right)\right)$.)

**(b)** Verify by calculation that $a$ lies between 2 and 2.1.

The constant $a$ is such that $\displaystyle\int_1^a 6x\ln x\,\mathrm{d}x = 4$.

(In part (a) it was shown that $a = \exp\left(\frac{1}{6}\left(\frac{5}{a^2}+3\right)\right)$.)

**(c)** 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.

**(a)** Find the quotient and remainder when $x^3+5x^2-2x-15$ is divided by $x^2-3$.

**(b)** The variables $x$ and $y$ satisfy the differential equation

$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{x^3+5x^2-2x-15}{6y(x^2-3)}.$

It is given that $y = 2$ when $x = 2$.

Solve the differential equation to obtain an expression for $y^2$ in terms of $x$.

(In part (a) you found $x^3+5x^2-2x-15 = (x+5)(x^2-3)+x$.)

The diagram shows the curve $y = \cos x\sqrt{\sin 2x}$ for $0 \le x \le \frac{1}{2}\pi$. The curve has a maximum point at $M$, where $x = a$.

**(a)** Find the exact value of $a$.