Every question, cut by section

A proof by contradiction correctly reaches the equation "2(j² − m) = 1", where j and m are integers, and stops there with the words "this has no solutions." Based on real examiner-report guidance on exactly this question type, does this earn full marks?

A curve is given parametrically by $x=x(t)$, $y=y(t)$. The area under the curve between $x=a$ and $x=b$ (with $a<b$) is found by evaluating which integral?

A is a point not on line l, and X is the general point on l — so X = c + t·d for a fixed point c on l, a fixed direction vector d, and a variable parameter t. Which equation, once solved for t, gives the point on l closest to A?

An implicit equation contains the isolated term $5x^2y$. Differentiated correctly with respect to $x$, what does this term become?

$\displaystyle\int \frac{5}{(x+2)^2}\,dx=$?

A real examiner report on a parametric-conversion question states that some candidates "assumed the general form of the answer, substituted for x in terms of t and compared this to the expression for y" — and that this strategy was "often unsuccessful." What is this approach actually doing wrong?

A "show that" part asks you to show that $\dfrac{dh}{dt} = \dfrac{4-h}{50}$. Partway through your own derivation you reach $\dfrac{dh}{dt} = \dfrac{4-h}{45}$ — close to, but not exactly, the printed answer. What is the safest thing to do?

$\int_1^4 f(x)\,dx$ is to be evaluated using the substitution $u=2x-1$. What are the correct limits for the resulting integral in u?

The worked chain earlier in this lesson found $(9-6x)^{-\frac12} \approx \frac13+\frac19x+\frac{1}{18}x^2+\frac{5}{162}x^3$. Substituting $x=\frac{1}{10}$ gives an approximation to $\frac{1}{\sqrt{8.4}}$.

A question asks for this approximation "as an exact fraction." Which final line is safest?

While finding a volume of revolution from parametric equations, you reach $y^2 = \cos^2\theta - 2\sin\theta\cos\theta + \sin^2\theta$ partway through squaring $y=\cos\theta-\sin\theta$. What is the correct next simplification?

A student shows two lines do not intersect, then argues: "the direction vectors $(1,0,1)$ and $(0,1,1)$ are not perpendicular, since their scalar product is $1$, not $0$ — so the lines are not skew." What is wrong with this argument?

Triangle ABC has vertex B. To find the angle ABC using the scalar product formula, which pair of vectors should be dotted together?

A candidate assumes √2 = p/q for integers p and q, does all the algebra correctly, and reaches "p and q are both even." What sentence does a real examiner report say is needed here — and most often missing — for this to count as a finished proof?

Why do the x-limits of a parametric area integral have to be converted into t-values before you integrate with respect to $t$, rather than just used as they are?

Find the unit vector in the direction of $\mathbf{a} = (6,-8,0)$.

An implicit equation contains the isolated term $3y^2$. Differentiated correctly with respect to $x$, what does this term become?

A denominator is $(x+1)(2x-3)^2$. How many unknown constants does its full partial-fraction decomposition need, and why?

A real examiner report on a different parametric-conversion question states: "A few candidates thought it appropriate to use calculus and scored no marks." Why does differentiating x=f(t) and y=g(t) and forming dy/dx not answer a "convert to cartesian form" question?

A curve has parametric equations $x = 2t + 1$, $y = t^2 - 3t$. Which is the correct way to find a single cartesian equation connecting $x$ and $y$?

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

A question gives the expansion of $(1+2x)^{-1}$ and says: 'by substituting x = 0.01, find an approximation to $\frac{1}{1.02}$.' A student substitutes into the series only, and writes down the resulting decimal as the final answer with no further comment.

What, specifically, is put at risk by stopping there?

Which of these binomial expansions is an INFINITE series that only converges — only equals the expression it expands — for $x$ inside some range?

The region between the curve $y=f(x)$, the x-axis, $x=0$ and $x=b$ is rotated 360° about the x-axis. Which expression gives the volume of the solid formed?

Two lines in three dimensions are shown not to intersect — solving their equations simultaneously gives no consistent solution. Is it correct to now conclude the lines are skew?

A real WMA14 mark scheme's own guidance on grading the opening line of a proof by contradiction states: "No need for explicit statement of assumption - accept if just a suitable equation is set up" (Jun 2022, Q9). What does this actually tell you about how the opening method mark is awarded?

A curve has parametric equations $x=5-t$, $y=t^2$, for $t\ge0$. As $t$ increases from 0, what happens to $x$?

Find the distance between $P=(1,-2,4)$ and $Q=(5,1,-8)$.

An implicit equation contains the isolated term $7x^3+4$. Differentiated correctly with respect to $x$, what does this become?

$\dfrac{11x^2-26x+1}{(3x+1)(x-2)^2}$ is decomposed and each term expanded separately, giving individual ranges of validity $|x|<\dfrac13$, $|x|<2$ and $|x|<2$. What is the range of validity of the COMBINED expansion (all three series added together)?

A rational function has denominator $(x+2)(3x-1)^2$. Which is the correct partial-fraction form to decompose it into?

A curve has parametric equations $x=t+2$, $y=t^2$ for $-1 \leq t \leq 3$. A student is asked to find the RANGE of the resulting cartesian function. Based on real examiner-report guidance on exactly this confusion, which answer below makes the documented mistake?

Which of the following is separable — that is, can be rearranged so every y-term (with dy) sits on one side and every x-term (with dx) sits on the other?

For $y = uv$, where $u$ and $v$ are both functions of $x$, which is the product rule?

To apply the general binomial series to $(5+2x)^{-1}$, what has to be done first?

$y = x + \frac{3}{x}$. What is $y^2$?

Two lines have direction vectors that are not written identically to each other. Does this, on its own, prove the lines are not parallel?

VERIDIAN-original scenario, in the style of a real question of this shape (Jan 2024, Q8): "Prove, by contradiction, that the curve $y = x^3 + 3x + 1$ has no stationary points." A candidate correctly assumes a stationary point exists, sets $\frac{dy}{dx} = 0$, and reaches $3x^2 + 3 = 0$, i.e. $x^2 = -1$.

Which closing sentence earns full marks, based on real examiner-report guidance on exactly this question type?

A curve has parametric equations $x=t^2+3$, $y=2t+1$. Which integral gives the area under the curve between the t-values $t=1$ and $t=3$?

Given $\mathbf{a}=(2,-1,3)$ and $\mathbf{b}=(-1,4,2)$, find $3\mathbf{a}-2\mathbf{b}$.

A curve is given parametrically by $x=t^2+1$, $y=2t^3$. Find an equation of the tangent to the curve at $t=2$.

An equation contains both $x$ and $y$, and $y$ is being treated as an (unknown) function of $x$. Differentiated with respect to $x$, why does a term like $y^3$ produce $3y^2\dfrac{dy}{dx}$, and not just $3y^2$?

Without carrying out any working: which of these is the correct antiderivative for $\displaystyle\int \frac{5}{2x+3}\,dx$?

A parameter $t$ is restricted to $0 \leq t \leq 6$, and $y = t^2 - 6t + 2$. A student substitutes only $t=0$ and $t=6$, gets $y=2$ both times, and concludes the range of $y$ is just the single value $2$. Based on real examiner-report guidance on exactly this question type, is this a safe method?

A quantity V is changing with time t. V is also known as a function of another quantity r, which is itself changing with time. Which equation correctly connects the rate of change of V with time to the rate of change of r with time?

If $y = f(u)$ where $u = g(x)$, which is the chain rule?

An expansion of $(1+ax)^n$ is used to approximate $\sqrt{k}$ for some target number $k$, by substituting a chosen value of $x$ into the series. Besides doing that substitution, what else has to be checked before a final answer is safe to write down?

A curve is given parametrically by $x=\theta^2$, $y=g(\theta)$. In the integral $V=\pi\int y^2\,dx$, what does $dx$ become once everything is written in terms of θ?

$A=(1,0,2)$, $B=(3,4,1)$, $C=(2,1,5)$. Find the angle $ABC$, to 1 decimal place.

A proof that √2 is irrational assumes √2 = p/q for integers p, q ≠ 0, but never states that p/q is in its lowest terms. What does the real record show about this specific omission?

A curve has parametric equations $x=t^2-4$, $y=t$, for $t\ge0$. What t-values correspond to the x-limits $x=0$ and $x=5$?

$A$ and $B$ have position vectors $\mathbf{a}=(3,0,-2)$ and $\mathbf{b}=(-1,5,4)$. Find $\overrightarrow{AB}$.

A curve is given parametrically by $x=t^3$, $y=t^2$. Correctly differentiated, $\dfrac{dy}{dx}=\dfrac{2}{3t}$. At $t=2$, a candidate 'simplifies' this — the same style of reciprocal slip documented on a real question of this type — to $\dfrac{3t}{2}$, then substitutes. What gradient do they get, and what is the actual correct gradient at $t=2$?

A question gives an implicit equation in $x$ and $y$, and asks for the equation of the tangent at a specific, named point $P$ — not the origin. You correctly differentiate to find $\dfrac{dy}{dx}$ in terms of $x$ and $y$. What has to happen next for the gradient to be correct?

$\dfrac{3x^2+x-2}{(x-1)(x+3)}$ — before this can be split as simply $\dfrac{A}{x-1}+\dfrac{B}{x+3}$, what has to happen first, and why?