Every question, cut by section

Simplify $\dfrac{x^2 + 2x - 15}{x^2 - 25}$, stating any values of $x$ for which it is undefined.

$g(x) = x^3 + x - 9$. A candidate writes: '$g(1) = -7$, $g(2) = 1$, so there's a root between 1 and 2.' The values and the conclusion are both correct. What is this answer missing for full marks?

Which of these is a correct way to write $\cos 2A$?

A deer population in a nature reserve is modelled by $N = 340(1.06)^t$, where $N$ is the number of deer and $t$ is the number of years since the reserve was established. What was the population when the reserve was established?

What is $\frac{d}{dx}[2x^4\sin x]$?

$y = e^{2x-6} + 1$. What is the equation of its asymptote, and its $y$-intercept to 3 s.f.?

For $a\cos\theta + b\sin\theta$ with $a > 0$ and $b < 0$, which form keeps $R > 0$ and $0 < \alpha < 90°$?

Which response to 'Prove that $(\sin\theta+\cos\theta)^2 \equiv 1+\sin 2\theta$' is certain to earn full marks?

Which of these correctly rewrites $4\,\text{cosec}\,\theta$ as a single fraction in terms of $\sin\theta$?

Given that $\tan\theta = 3$, find the exact value of $\sec^2\theta$.

The curve $y = f(x)$ passes through the point $(3, -5)$. What are the coordinates of the corresponding point on $y = f(x + 2)$?

Given $y = 3x + 4$, what is $x$ in terms of $y$?

$f(x) = x^3 - 2x^2 - 5x + 6$. What is the remainder when $f(x)$ is divided by $(x-1)$, and how do you know without doing the division?

The iteration $x_{n+1} = \sqrt[3]{5x_n + 3}$ is applied repeatedly, and the values of $x_n$ settle down to a limit $L$ — they stop changing at the number of decimal places being used. What must be true about $L$?

Which of these integrals is of the form $\int \frac{f'(x)}{f(x)}\,dx$?

What is $\frac{d}{dx}\left[6(2.5)^x\right]$?

What is $\frac{d}{dx}\left[\frac{x^2}{2x+1}\right]$?

Solve $e^{2x-1} = 5$, giving your answer to 3 significant figures.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Express $3\cos\theta + 4\sin\theta$ in the form $R\cos(\theta - \alpha)$ and hence solve $3\cos\theta + 4\sin\theta = 2$ for $0° \leq \theta < 360°$.

Which response is certain to earn full marks under the printed instruction above?

Using $\cos 2A \equiv 1-2\sin^2 A$ with $A=\frac{\theta}{2}$, which expression equals $1-\cos\theta$?

Before it's derived: which do you expect $\sin 2A$ to equal, in terms of $\sin A$ and $\cos A$?

Which restricted domain of $y = \sin\theta$ makes it one-one, which is exactly what is needed before $\arcsin$ can be defined as its inverse?

The curve $y = f(x)$ passes through the point $(3, -5)$. What are the coordinates of the corresponding point on $y = -f(x)$?

For which value of $x$ is the expression $\dfrac{1}{2x - 5}$ undefined?

A cubic $f(x)$ is divided by a quadratic $g(x)$, giving $f(x) \equiv g(x)Q(x) + R(x)$. What must be true of $R(x)$?

$f(x) = \tan x$ ($x$ in radians). $f(1) \approx 1.5574$ and $f(2) \approx -2.1850$ — a sign change. Does the sign-change test prove that $\tan x = 0$ has a root between $x = 1$ and $x = 2$?

Find $\int \frac{\cos 3x}{2+\sin 3x}\,dx$.

A population of a native fish species in a lake, monitored after an invasive species was introduced, is modelled by $P=2000(0.9)^t$, where $P$ is the number of fish and $t$ is the number of years since monitoring began. Find the exact rate at which the population is decreasing when $t=10$, to 3 significant figures.

What is $\frac{d}{dx}\left[e^{5x^2}\right]$?

Solve $\ln(3x+2) = 1.5$, giving your answer to 3 significant figures.

In part (a) you meant to write α = 53.1° but transposed the digits and wrote 51.3°. Part (b) says 'hence' state the maximum value of $g(\theta)$ and where it occurs. What is the best thing to do?

Using the compound-angle formula $\cos(A - B) \equiv \cos A \cos B + \sin A \sin B$, what is $\cos(\theta - 60°)$ written in terms of $\cos\theta$ and $\sin\theta$?

The compound-angle formula for tangent is $\tan(A-B) \equiv \dfrac{\tan A - \tan B}{1+\tan A\tan B}$. What sign sits in the denominator, and is that the same sign as in the numerator?

Given that $\cos\theta = -\frac{2}{5}$, find the exact value of $\sec\theta$.

The curve $y = f(x)$ has a minimum turning point at $(2, -4)$. What is the corresponding point on $y = |f(x)|$?

$f(x) = (x - 5)^2 + 2$, $x \geq 5$. Find $f^{-1}(x)$ and state its domain.

Simplify $\dfrac{2x^2+x-6}{4x^2-9}$.

$g(x) = x^3 + x - 9$. A candidate writes: '$g(1) = -7$, $g(2) = 1$, so there's a root between 1 and 2.' Which single addition earns full marks?

Find $\int 8x(x^2+3)^3\,dx$.

A tank is being drained, and the volume of water remaining is modelled by $W=500(0.75)^t$, where $W$ is in litres and $t$ is in minutes. A student correctly finds $\frac{dW}{dt}\big|_{t=2} \approx -80.9$, and writes their final answer as simply "$80.9$". Which of the following best describes this answer?

Given that $x = y^3 + y$, what is $\frac{dy}{dx}$?

A set of data pairs $(x, y)$ is believed to fit either $y = ax^n$ or $y = kb^x$. A graph of $\log_{10} y$ against $x$ (unlogged) turns out to be a straight line. Which model is this consistent with, and why?

A real WMA13 question of this exact type (Oct 2020, Q7(a)) reads: 'Express cos x + 4 sin x in the form R cos(x − α) where R > 0 and 0 < α < π/2. Give the exact value of R and give the value of α, in radians, to 3 decimal places.' Its real mark scheme awards B1 for R = √17 with the note 'do not allow decimals for this mark', and states for α = awrt 1.326 that 'the degree equivalent α = awrt 75.96° is A0' — the same angle, in the wrong unit.

Applying that same real mark scheme's rules: $2\cos\theta + 3\sin\theta$ is written as $R\cos(\theta-\alpha)$, with $R>0$ and $0 < \alpha < \frac{\pi}{2}$, and the question asks for the EXACT value of $R$ and the value of $\alpha$ in RADIANS to 3 decimal places. Which pair earns full marks?

If $5\cos\theta + 12\sin\theta$ is written in the form $R\cos(\theta - \alpha)$ with $R > 0$, what is the value of $R$?

An exam question instructs: 'Prove that $\cos x\cos 2x + \sin x\sin 2x \equiv \cos x$.' What determines whether this earns full marks?

For $0° \leq \theta < 360°$, at which values does $y = \text{cosec}\,\theta$ have vertical asymptotes?

The curve $y = f(x)$ crosses the x-axis at exactly one point, $(6, 0)$, and nowhere else. Where does $y = f(|x|)$ cross the x-axis?

$f(x) = 3x + 2$ and $g(x) = x^2 - 1$, both with domain $x \in \mathbb{R}$. What is $gf(x)$?

$f(x) = x^3+4x^2-3x-18$ and $f(2)=0$, so $(x-2)$ is a factor. Dividing gives $f(x) \equiv (x-2)(x^2+ax+b)$. What are $a$ and $b$?

A question asks for $x_n$ values 'to 4 decimal places'. A candidate's working shows $x_1 = 2.571$. What happens to the accuracy mark for $x_1$?

Given that $\cos 4x \equiv 1 - 2\sin^2 2x$.

Find $\int \sin^2 2x\,dx$.

A perfume's scent intensity is modelled by $S=5+45(0.8)^t$ (on a 0–100 scale), $t$ hours after application, approaching a faint background level of $5$ as $t\to\infty$. Which of the following correctly explains why $S=3$ has no solution for $t\geq0$?

A candidate is asked to find $\frac{dy}{dx}$ for $y = \frac{x^2+1}{3x-2}$ (3 marks: M1 A1 A1) and writes only the final line "$\frac{dy}{dx} = \frac{6x^2-4x-6}{(3x-2)^2}$" — no working shown. The correct answer is $\frac{3x^2-4x-3}{(3x-2)^2}$.

Under the general Pearson marking guidance quoted in this lesson, how many of the 3 marks can be awarded?

A graph of $\log_{10} y$ against $\log_{10} x$, for a relationship $y = ax^n$, is a straight line through $(0, 2)$ and $(2, 6)$. Find $n$ and $a$.

$f(x) = e^x$ for $x \in \mathbb{R}$ has range $f(x) > 0$. Given that $f$ has an inverse function $f^{-1}$, what are the domain and range of $f^{-1}$?