TMUA 2020 Paper 2

Find the complete set of values of $k$ for which the line $y=x-2$ crosses or touches the curve $y=x^2+kx+2$.

Given that $\tan \theta =2$ and $180^{\circ }<\theta <360^{\circ }$, find the value of $\cos \theta$.

A student makes the following claim: for all integers $n$, the expression $4\left(\frac{9n+1}{2}-\frac{3n-1}{2}\right)$ is divisible by 3. Here is the student's argument: $$4\left(\frac{9n+1}{2}-\frac{3n-1}{2}\right)=2\left(2\left(\frac{9n+1}{2}-\frac{3n-1}{2}\right)\right)\text{\hspace{1em}(I)}$$ $$=2(9n+1-3n-1)\text{\hspace{1em}(II)}$$ $$=2(6n)\text{\hspace{1em}(III)}$$ $$=12n\text{\hspace{1em}(IV)}$$ $$=3(4n)\text{\hspace{1em}(V)}$$ which is always a multiple of 3. So the expression $4\left(\frac{9n+1}{2}-\frac{3n-1}{2}\right)$ is always divisible by 3. Which one of the following is true?

Consider the following statement: every positive integer $N$ that is greater than 6 can be written as the sum of two non-prime integers that are greater than 1. Which of the following is/are counterexample(s) to this statement? $$\text{I}N=5\text{II}N=7\text{III}N=9$$

Which one of the following shows the graph of $y=\frac{2^x}{1+2^x}$? (Dotted lines indicate asymptotes.)

The function $\mathrm{f}(x)$ is defined for all real values of $x$. Which of the following conditions on $\mathrm{f}(x)$ is/are necessary to ensure that $${\int }_{-5}^0\mathrm{f}(x)\mathrm{d}x={\int }_0^5\mathrm{f}(x)\mathrm{d}x$$Condition I: $\mathrm{f}(x)=\mathrm{f}(-x)$ for $-5\le x\le 5$

Condition II: $\mathrm{f}(x)=c$ for $-5\le x\le 5$, where $c$ is a constant

Condition III: $\mathrm{f}(x)=-\mathrm{f}(-x)$ for $-5\le x\le 5$

Consider the following conditions on a parallelogram $PQRS$, labelled anticlockwise:

I: length of $PQ$ = length of $QR$

II: The diagonal $PR$ intersects the diagonal $QS$ at right angles

III: $\angle PQR=\angle QRS$

Which of these conditions is/are individually sufficient for the parallelogram $PQRS$ to be a square?

A student is asked to prove whether the following statement $(\ast )$ is true or false: $(\ast )$ For all real numbers $a$ and $b$, $|a+b|<|a|+|b|$. The student's proof is as follows: Statement $(\ast )$ is false. A counterexample is $a=3$, $b=4$, as $|3+4|=7$ and $|3|+|4|=7$, but $7<7$ is false. Which of the following best describes the student's proof?

A student wishes to evaluate the function $\mathrm{f}(x)=x\sin x$, where $x$ is in radians, but has a calculator that only works in degrees. What could the student type into their calculator to correctly evaluate $\mathrm{f}(4)$?

The real numbers $a$, $b$, $c$ and $d$ satisfy both $$0<a+b<c+d$$ and $$0<a+c<b+d.$$ Which of the following inequalities must be true?

I: $a<d$

II: $b<c$

III: $a+b+c+d>0$

A spiral line is drawn as shown. The spiral starts near the origin and expands outward in a rectangular pattern. The spiral pattern continues indefinitely. Which one of the following points is not on the spiral line?$$\text{A }(99,100)\text{B }(99,-100)\text{C }(-99,100)\text{D }(-99,-100)$$ $$\text{E }(100,99)\text{F }(100,-99)\text{G }(-100,99)\text{H }(-100,-99)$$

Which one of A-F correctly completes the following statement? Given that $a<b$, and $\mathrm{f}(x)>0$ for all $x$ with $a<x<b$, the trapezium rule produces an overestimate for ${\int }_a^b\mathrm{f}(x)\mathrm{d}x$ ...

$\mathrm{f}(x)$ is a function for which $${\int }_0^3(\mathrm{f}(x){)}^2\mathrm{d}x+{\int }_0^3\mathrm{f}(x)\mathrm{d}x={\int }_0^1\mathrm{f}(x)\mathrm{d}x.$$ Which of the following claims about $\mathrm{f}(x)$ is/are necessarily true?

I: $\mathrm{f}(x)\le 0$ for some $x$ with $1\le x\le 3$

II: ${\int }_0^3\mathrm{f}(x)\mathrm{d}x\le {\int }_0^1\mathrm{f}(x)\mathrm{d}x$

An arithmetic sequence $T$ has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero integers. Property P is: for some positive integer $m$, the sum of the first $m$ terms of the sequence is equal to the sum of the first $2m$ terms of the sequence. For example, when $a=11$ and $d=-2$, the sequence $T$ has property P, because $11+9+7+5=11+9+7+5+3+1+(-1)+(-3)$, i.e. the sum of the first 4 terms equals the sum of the first 8 terms. Which of the following statements is/are true?

I: For $T$ to have property P, it is sufficient that $ad<0$.

II: For $T$ to have property P, it is necessary that $d$ is even.

Which one of the following is a necessary and sufficient condition for $$\sum _{k=1}^n\sin \left(\frac{k\pi }{3}\right)=\frac{\sqrt{3}}{2}$$ to be true?

The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial $\mathrm{f}$: $$\frac{\mathrm{d}}{\mathrm{d}x}\left({\int }_0^x\mathrm{f}(t)\mathrm{d}t\right)=\mathrm{f}(x).$$ A student calculates $\frac{\mathrm{d}}{\mathrm{d}x}\left({\int }_x^{2x}{t}^2\mathrm{d}t\right)$ as follows:

(I) ${\int }_x^{2x}{t}^2\mathrm{d}t={\int }_0^{2x}{t}^2\mathrm{d}t-{\int }_0^x{t}^2\mathrm{d}t$

(II) By FTC, $\frac{\mathrm{d}}{\mathrm{d}x}\left({\int }_0^x{t}^2\mathrm{d}t\right)=x^2$

(III) By FTC, $\frac{\mathrm{d}}{\mathrm{d}x}\left({\int }_0^{2x}{t}^2\mathrm{d}t\right)=(2x{)}^2=4x^2$

(IV) So $\frac{\mathrm{d}}{\mathrm{d}x}\left({\int }_x^{2x}{t}^2\mathrm{d}t\right)=4x^2-x^2$

(V) giving $\frac{\mathrm{d}}{\mathrm{d}x}\left({\int }_x^{2x}{t}^2\mathrm{d}t\right)=3x^2$

Which of the following best describes the student's calculation?

A set of six distinct integers is split into two sets of three. The first set of three integers has a mean of 10 and a median of 8. The second set of three integers has a mean of 12 and a median of 9. What is the smallest possible range of the set of all six integers?

In this question, $\mathrm{f}(x)=a{x}^3+b{x}^2+cx+d$ and $\mathrm{g}(x)=p{x}^3+q{x}^2+rx+s$ are cubic polynomials. If $\mathrm{f}(x)-\mathrm{g}(x)>0$ for every real $x$, which of the following is/are necessarily true?

I: $a>p$

II: if $b=q$ then $c=r$

III: $d>s$

Nine people are sitting in the squares of a 3 by 3 grid, one in each square. Two people are called neighbours if they are sitting in squares that share a side. (People in diagonally adjacent squares, which only have a point in common, are not called neighbours.) Each of the nine people in the grid is either a truth-teller who always tells the truth, or a liar who always lies. Every person in the grid says: 'My neighbours are all liars'. Given only this information, what are the smallest number and the largest number of people who could be telling the truth?

$x$ is a real number and $\mathrm{f}$ is a function. Given that exactly one of the following statements is true, which one is it?

A: $x\ge 0$ only if $\mathrm{f}(x)<0$

B: $x<0$ if $\mathrm{f}(x)\ge 0$

C: $x\ge 0$ only if $\mathrm{f}(x)\ge 0$

D: $\mathrm{f}(x)<0$ if $x<0$

E: $\mathrm{f}(x)\ge 0$ only if $x\ge 0$

F: $\mathrm{f}(x)\ge 0$ if and only if $x<0$