Every question, cut by section

A statement claims "$f(n)$ is true for every positive integer $n$." How many values of $n$ for which $f(n)$ is false are needed to disprove it?

$f(x) = x^3 + x^2 - 2x + 12$. Which of these is a factor of $f(x)$?

An arithmetic series has first term $10$ and common difference $-2$. Find $S_{25}$.

What is the coefficient of $x^3$ in the expansion of $(5 - 2x)^6$?

Solve $\tan\theta = \sqrt{3}$ for $0° \leqslant \theta < 360°$. Which is the complete solution set?

A curve $y = f(x)$ crosses the x-axis exactly once inside the interval $[0, 4]$. Which of the following is guaranteed to give the correct total area enclosed between the curve and the x-axis on $[0, 4]$?

A "prove by exhaustion" question gives you a finite, correctly-listed set of cases and every one of them checks out. Based on real examiner reports on this exact question type, what else does the answer need to earn full marks?

The circle $C$ has equation $(x+5)^2 + (y-3)^2 = 16$. What are the coordinates of its centre?

In the standard proof that $S_n = \frac{1}{2}n[2a+(n-1)d]$, why is the series written a second time in reverse order before the two versions are added?

Pascal's triangle is used to expand $(2 + x)^7$. Which row gives the correct coefficients, and how many entries does it have?

Given that $\cos\theta = -\frac{3}{5}$ and $\theta$ is obtuse ($90° < \theta < 180°$), what is $\sin\theta$?

What is the exact value of $\displaystyle\int_{2}^{5} (4x-3)\,dx$?

$a$, $b$ and $c$ are positive integers. A question asks you to prove something about the product of $a$, $b$ and $c$. What number is being referred to?

The circle $C$ has equation $x^2 + y^2 - 10x + 6y + 18 = 0$. What is the radius of $C$?

A geometric sequence has first term $5$ and common ratio $3$. What is the value of $u_4$, the fourth term?

In the expansion of $(a+b)^n$, what does the coefficient $\binom{n}{r}$ actually count?

Solve $3\sin\theta\cos\theta = \sin\theta$ for $0° \leqslant \theta < 360°$.

A curve lies entirely below the x-axis for every x in $[0,3]$, and $\int_0^3 g(x)\,dx = -18$. What is the area of the region enclosed between the curve and the x-axis on this interval?

A question asks you to prove, by exhaustion, that a property holds for every pair of positive integers $(x, y)$ with $x + y = 11$ and $x < y$. How many cases must you list?

A circle has centre $(1, -2)$. The point $(4, 2)$ lies on the circle. What is the gradient of the tangent to the circle at $(4, 2)$?

A geometric series has first term $20$ and common ratio $r = -\frac{1}{3}$. Does it have a sum to infinity, and if so what is it?

Which expression equals ${}^7C_3$?

The spec's own worked example for this topic asks you to solve $\sin(x + \frac{\pi}{2}) = \frac{3}{4}$ for $0 < x < 2\pi$. A student's calculator gives $\arcsin(0.75) \approx 48.6°$, which they convert to $\approx 0.848$ radians and give as their only answer.

Which option correctly identifies every problem with this response?

Find the exact area of the finite region enclosed between the curves y = x² and y = 8 − x².

Positive integers $a$ and $b$ satisfy $b = a + 4$ and $a + b \leq 14$. Which is the correct, complete list of pairs $(a, b)$ to check in a proof by exhaustion covering every valid pair?

The circle $C$ has equation $x^2 + y^2 - 4x + 2y + k = 0$, where $k$ is a constant. $C$ has radius $3$. What is the value of $k$?

A geometric series has first term $3$ and common ratio $2$. What is $S_6$, the sum of the first six terms?

For which of these is the graph of $y = a^x$ a valid exponential curve, defined for every real $x$?

Why is $\tan\theta$ undefined at $\theta = 90°$ and $\theta = 270°$, but not at $\theta = 0°$ or $\theta = 180°$?

Before evaluating $\int_a^b f(x)\,dx$ and calling the result 'the area of the region bounded by the curve, the x-axis, and the lines x = a and x = b', what must be checked first?

You are asked to disprove the statement "$n^2 + 2n + 3$ is even for every positive integer $n$." You find that $n = 1$ gives $6$ (even) and $n = 2$ gives $11$ (odd). What should you write next?

The circle $C$ has equation $x^2 + y^2 + 10x - 4y - 7 = 0$. What are the coordinates of its centre?

Four geometric series have common ratios $r = -0.9$, $r = -1.5$, $r = -1$ and $r = 1$ respectively. Which one has a sum to infinity?

Given that $\log_a x = t$, what is $\log_a(x^2)$ in terms of $t$?

To find the stationary points of a curve $y = f(x)$, which equation must be solved?

The instructions on a WMA12 paper state that permitted calculators must not have 'the facility for symbolic algebra manipulation, differentiation and integration.' What does this mean in practice for a definite-integral question that asks for an exact answer?

A claim is written as "$g(n)$ holds for all positive integers $n$." Why can this claim be disproved by one failing value of $n$, while proving it true needs every value checked?

The circle $C$ has equation $(x-2)^2 + (y+3)^2 = 20$. Give the radius of $C$ exactly, in simplified surd form.

For the geometric series $45 + 15 + 5 + \cdots$, find the sum of all the terms after the first two.

Solve $2^x = 20$, giving your answer to 3 significant figures.

At a stationary point, the second derivative is found to be negative. What does this tell you about the point?

A curve is tabulated at seven equally spaced x-values, from $x = 1$ to $x = 4$, ready for use with the trapezium rule. What is the strip width $h$?

Which of these claims can be proved by exhaustion, as the technique is defined on this paper?

A chord of a circle has midpoint $M(3, 4)$. The circle's centre is at $(7, 1)$. What is the gradient of the chord?

A geometric series has first term $4$ and common ratio $3$. Find the sum of the first $7$ terms.

$\log_4(x+9) - \log_4 x$ can be written as a single logarithm. Which is correct?

A function has $\dfrac{dy}{dx} > 0$ for every $x$ in the open interval $(2, 5)$. What can be said about $f$ on this interval?

The trapezium rule is used with 5 strips, giving six ordinates $y_0$ to $y_5$. Inside the bracket of the compressed formula, which ordinates are doubled?

$f(x) = x^3 - 4x^2 + 7x - 3$. What is the remainder when $f(x)$ is divided by $(x - 2)$?

$A$ and $B$ are the endpoints of a diameter of a circle. $P$ is another point on the circle. The gradient of $AP$ is $\frac{2}{5}$. What must the gradient of $BP$ be?

In $S_n = \frac{a(1-r^n)}{1-r}$, which single feature of the expression decides whether a sum to infinity exists?

Which of these is a correct simplification of $\log_5(x+2)$?

Find and classify the stationary point of $y = x^2 - 6x + 5$.

A student's trapezium-rule estimate for an area comes out lower than the true value, found separately. Which of these is a complete reason the estimate is an underestimate?

$(2x + 1)$ is a factor of a polynomial $f(x)$. Which statement must therefore be true?

To find where the line $x - 2y - 8 = 0$ meets a circle, one route substitutes $y = \frac{x-8}{2}$ into the circle's equation, and another substitutes $x = 2y+8$. Which statement about the two routes is correct?

In the standard proof that $S_n = \frac{a(1-r^n)}{1-r}$, why is the series multiplied by $r$ before the two lines are subtracted?

Given that $\log_a y = 3$, what is $(\log_a y)^2$?

A curve has $\dfrac{dy}{dx} = (x-1)(x-4)$. For which values of $x$ is the curve decreasing?

A curve is tabulated at nine equally spaced x-values, from $x = 2$ to $x = 6$, for use with the trapezium rule. What is $h$?

A cubic $f(x)$ is divided by $(x - 4)$. Which of these could the remainder be?

A sequence is defined by $x_1 = 4$ and $x_{n+1} = x_n + 7$. What is the value of $x_4$?

The first 8 terms of a sequence form an arithmetic series with first term $30$ and common difference $-2$. From the 8th term onwards, the terms form a geometric series with common ratio $\frac{1}{2}$. What is the sum of all the terms of the sequence?

A candidate solving $6^{3x+2} = 50$ correctly reaches $3x + 2 = \log_6 50$. Which is the correctly finished answer for $x$?

A stationary point is found, and the second-derivative test gives $\dfrac{d^2y}{dx^2}=0$ there — inconclusive. Which method correctly determines the point's nature and is confirmed, on a real WMA12 mark scheme, to earn full credit?