# Notes for past papers

### Notes for past MAT papers

These notes have been written for candidates attempting MAT past papers from 2007-2020. Most of the notes refer to syllabus changes which were made in January 2018. These notes were last updated in 2022 for candidates preparing for MAT 2022 and may be updated again in the future.

#### MAT 2021

[No notes]

#### MAT 2020

Q4 There was unfortunately a typo in the printed question booklet; an odd function was defined as one for which $f(-x)=-f(-x)$, which is too many minus signs. This has been fixed for the version on the MAT website. The Solutions document outlines how marks were awarded to candidates who used the printed definition and also to candidates who used the normal definition of an odd function.

#### MAT 2019

[No notes]

#### MAT 2018

[No notes]

#### MAT 2017

Q1G This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1H This question assumes knowledge of the remainder theorem. MAT questions from 2018 onwards do not assume knowledge of the remainder theorem.

Q1J This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. We could set a roughly equivalent question in degrees by changing the options to

(a) $\int_0^{2} (x^2-4) \sin^8x\,\mathrm{d}x$

(b) $\int_0^{360^\circ} (2+\cos x )^3\,\mathrm{d}x$

(c) $\int_0^{180^\circ} \sin^{100} (x)\,\mathrm{d}x$

(d) $\int_0^{180^\circ}(3-\sin x)^6\,\mathrm{d}x$

(e) $\int_0^{360^\circ}108(\sin x-1)\,\mathrm{d}x$

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. If the function $\cos^{-1}$ returns angles measured in degrees, then the “area that neither can reach” in (iii) would be $\frac{1}{45^\circ}\cos^{-1}\left(\frac{\sqrt{\pi}}{2}\right)-\sqrt{\frac{4-\pi}{\pi}}$.

#### MAT 2016

Q1D This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

#### MAT 2015

Q1C This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The statements would now be written as

I. $\sin(90^\circ+x)=\cos(90^\circ-x)$

II. $2+2\sin(x)-\cos^2(x)\geq 0$

III. $\sin(x+270^\circ)=\cos(180^\circ-x)$

IV. $\sin(x)\cos(x)\leq \frac{1}{4}$

Q1E This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1F This question assumes knowledge of the trapezium rule. MAT questions from 2018 onwards do not assume knowledge of the trapezium rule.

Q1G This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The labels on the axes would now be $90^\circ$, $180^\circ$, $270^\circ$, instead of $\pi/2$, $\pi$, and $3\pi/2$.

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. If the function $\tan^{-1}$ returns angles measured in degrees, then “the value of $p$ minimising the lop-sidedness of circle $B$” in part (iv) satisfies the equation $\frac{\pi}{180^\circ}(p^2+1)\tan^{-1}\left(\frac{1}{p}\right)-p=\frac{\pi}{6}$. The note below the equation would now be written as “Note that $\tan^{-1}(x)$ is sometimes written as $\arctan(x)$ and is the value of $\theta$ in the range $-90^\circ<\theta<90^\circ$ such that $\tan(\theta)=x$.”

#### MAT 2014

Q3 This question assumes knowledge of the trapezium rule. MAT questions from 2018 onwards do not assume knowledge of the trapezium rule.

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \alpha<\beta\leq\pi/2$ would now be written as $0\leq \alpha\leq\beta\leq90^\circ$. Similarly in part (v), the range $0\leq \beta<\alpha\leq\pi/2$ would now be written as $0\leq \beta\leq\alpha\leq90^\circ$.

#### MAT 2013

Q1B This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The line $x=\pi$ would now be written as $x=180^\circ$.

Q1G This question assumes knowledge of the remainder theorem. MAT questions from 2018 onwards do not assume knowledge of the remainder theorem.

Q1H This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. Option (a) would now be written as $\frac{\sin(180^\circ\times\sqrt{2}/\pi)}{\sqrt{2}}$.

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0<\theta<\pi/2$ would now be written as $0<\theta<90^\circ$.

#### MAT 2012

Q1C This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. Option (c) would now be written as $\left( 3 \sin 60^\circ \right)^2$

Q1F This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The expression for $T$ might instead have been written as $$T=\left(\int_{-90^\circ}^{90^\circ} \cos x\,\mathrm{d}x\right)\times\left(\int_{180^\circ}^{360^\circ} \sin x\,\mathrm{d}x\right)\times\left(\int_{0}^{22.5^\circ} \frac{1}{\cos 3x}\,\mathrm{d}x\right)$$

#### MAT 2011

Q1D This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1F This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<\pi$ would now be written as $0\leq \theta<180^\circ$. The options would be written as (a) $0< \theta < 60^\circ$ (b) $45^\circ<\theta<135^\circ$ (c) $0<\theta<90^\circ$ (d) all values of $\theta$.

Q1I This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q3 This question should have been written to specify that $a<b$, as in the diagram.

#### MAT 2010

Q1C This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1F This question assumes knowledge of the trapezium rule. MAT questions from 2018 onwards do not assume knowledge of the trapezium rule.

Q3 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. It is possible to rephrase this question in terms of degrees (for example “you may assume that the derivative of $\sin x$ is $(180/\pi) \cos x$), but it’s a real mess if you do. This question would probably not be set with the new syllabus.

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. For part (iv), the area in the Solutions document is written in terms of radians; if $\theta$ is measured in degrees then replace $\theta$ in the Solutions document with $\pi \theta/180^\circ$ instead. In part (v), the area of the region inside both $C$ and $T$ equals $$\frac{27}{35}+\frac{\pi \theta}{90^\circ}$$ if $\theta$ is measured in degrees.

Q5 This question refers to “the last date before today”. The date of the test was 03/11/2010. Your answer may differ from that in the Solutions document if you are attempting this question sufficiently far in the future.

#### MAT 2009

Q1E This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1H This question assumes knowledge of the trapezium rule. MAT questions from 2018 onwards do not assume knowledge of the trapezium rule.

#### MAT 2008

Q1C This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1D This question assumes knowledge of the remainder theorem. MAT questions from 2018 onwards do not assume knowledge of the remainder theorem.

Q1F This question assumes knowledge of the trapezium rule. MAT questions from 2018 onwards do not assume knowledge of the trapezium rule.

Q1J This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

#### MAT 2007

Q1C This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q1G This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The equation should now be written as $y=2^{-x}\sin^2(180^\circ x^2/\pi)$.

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. In part (ii), the expression relating $A$ and $B$ would now be written as $A(\theta)=B(90^\circ-\theta)$, and you would be asked to calculate $A(45^\circ)$. In part (iii), the equation would now be written as $A(60^\circ)=\sqrt{3}-\frac{\pi}{3}$

#### MAT Specimen 1

Q1F This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq x<2\pi$ would now be written as $0\leq x<360^\circ$.

Q2 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. Part (ii) would now be written as “Give the roots when $\theta=60^\circ$. In parts (iii) and (iv), the range $0\leq \theta<2\pi$ would now be written as $0\leq \theta<360^\circ$.

Q4 This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq \alpha, \beta, \gamma\leq \pi/2$ would now be written as $0\leq \alpha, \beta, \gamma\leq 90^\circ$.

#### MAT Specimen 2

Q1C This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The options would now be written as (a) $\tan(225^\circ)$ (b) $\sin^2(225^\circ)$ (c) $\log_{10}\left(\frac{5\pi}{4}\right)$ (d) $\log_{2}\left(\frac{5\pi}{4}\right)$ Naturally, the note in brackets in the question would be “all angles are given in degrees” instead of “all angles are given in radians”.

Q1E This question was written in terms of radians. MAT questions from 2018 onwards are written in terms of degrees. The range $0\leq x<2\pi$ would now be written as $0\leq x<360^\circ$.