TMUA 2019 Paper 2

Find the coefficient of the $x^4$ term in the expansion of $$x^2{\left(2x+\frac{1}{x}\right)}^6$$

$(2x+1)$ and $(x-2)$ are factors of $2x^3+p{x}^2+q$.

What is the value of $2p+q$?

$a$, $b$ and $c$ are real numbers.

Given that $ab=ac$, which of the following statements must be true?

- I: $a=0$
- II: $b=0$ or $c=0$
- III: $b=c$

Consider the following conjecture:

If $N$ is a positive integer that consists of the digit 1 followed by an odd number of 0 digits and then a final digit 1, then $N$ is a prime number.

Here are three numbers:
- I: $N=101$ (which is a prime number)
- II: $N=1001$ (which equals $7\times 11\times 13$)
- III: $N=10001$ (which equals $73\times 137$)

Which of these provide(s) a counterexample to the conjecture?

Consider the following statement about the positive integers $a$, $b$ and $n$:

$(\ast )$: $ab$ is divisible by $n$

The condition 'either $a$ or $b$ is divisible by $n$' is

A student attempts to solve the equation $$\cos x+\sin x\tan x=2\sin x-1$$ in the range $0\le x\le 2\pi$.

The student's attempt is as follows: $$\cos x+\sin x\tan x=2\sin x-1$$ $$\text{So}\cos x-\sin x+\sin x\tan x-\sin x=-1\text{\hspace{1em}(I)}$$ $$\text{So}(\sin x-\cos x)(\tan x-1)=-1\text{\hspace{1em}(II)}$$ $$\text{So}\sin x-\cos x=-1\text{or}\tan x-1=-1\text{\hspace{1em}(III)}$$ $$\text{So}(\sin x-\cos x{)}^2=1\text{or}\tan x=0\text{\hspace{1em}(IV)}$$ $$\text{So}2\sin x\cos x=0\text{or}\tan x=0\text{\hspace{1em}(V)}$$ $$\text{So}x=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi \text{\hspace{1em}(VI)}$$Which of the following best describes this attempt?

For which one of the following statements can the fact that $12^2+16^2=20^2$ be used to produce a counterexample?

$a$, $b$ and $c$ are real numbers with $a<b<c<0$.

Which of the following statements must be true?

- I: $ac<ab<a^2$
- II: $b(c+a)>0$
- III: $\frac{c}{b}>\frac{a}{b}$

A large circular table has 40 chairs round it.

What is the smallest number of people who can be sitting at the table already such that the next person to sit down must sit next to someone?

$PQRS$ is a quadrilateral, labelled anticlockwise.

Which one of the following is a necessary but not sufficient condition for $PQRS$ to be a parallelogram?

An arithmetic series has $n$ terms, all of which are integers.

The sum of the series is 20.

Which of the following statements must be true?

- I: The first term of the series is even.
- II: $n$ is even.
- III: The common difference is even.

Most students in a large college study Mathematics. A teacher chooses three different students at random, one after the other.

Consider these three probabilities:$$R=\mathrm{P}(\mathrm{At}\mathrm{least}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{students}\mathrm{chosen}\mathrm{studies}\mathrm{Mathematics})$$ $$S=\mathrm{P}(\mathrm{The}\mathrm{second}\mathrm{student}\mathrm{chosen}\mathrm{studies}\mathrm{Mathematics})$$ $$T=\mathrm{P}(\mathrm{All}\mathrm{three}\mathrm{of}\mathrm{the}\mathrm{students}\mathrm{chosen}\mathrm{study}\mathrm{Mathematics})$$Which of the following is true?

A student approximates the integral ${\int }_a^b{\sin }^{2}xdx$ using the trapezium rule with 4 strips. The resulting approximation is an overestimate.

Which of the following is/are necessarily true?

- I: If the student approximates ${\int }_{-b}^{-a}{\sin }^{2}xdx$ in the same way, the result will be an overestimate.
- II: If the student approximates ${\int }_a^b{\cos }^{2}xdx$ in the same way, the result will be an underestimate.

Consider the following statements about the polynomial $\mathrm{p}(x)$, where $a<b$:

- I: $\mathrm{p}(a)\le \mathrm{p}(b)$
- II: ${\mathrm{p}}^{\prime }(a)\le {\mathrm{p}}^{\prime }(b)$
- III: ${\mathrm{p}}^{\prime \prime }(a)\le {\mathrm{p}}^{\prime \prime }(b)$

Which of these statements is a necessary condition for $\mathrm{p}(x)$ to be increasing for $a\le x\le b$?

The numbers $a$, $b$ and $c$ are each greater than 1.

The following logarithms are all to the same base: $$\log (a{b}^{2}c)=7$$ $$\log (a^{2}b{c}^2)=11$$ $$\log (a^2{b}^2{c}^3)=15$$What is this base?

The graph of the quadratic $$y=p{x}^2+qx+p$$ where $p>0$, intersects the $x$-axis at two distinct points.

In which one of the following graphs does the shaded region show the complete set of possible values that $p$ and $q$ could take?

A multiple-choice test question offered the following four options relating to a certain statement:

- A: The statement is true if and only if $x>1$
- B: The statement is true if $x>1$
- C: The statement is true if and only if $x>2$
- D: The statement is true if $x>2$

Given that exactly one of these options was correct, which one was it?

Consider the following inequality: $$(\ast ):a|x|+1\le |x-2|$$ where $a$ is a real constant.

Which one of the following describes the complete set of values of $a$ such that $(\ast )$ is true for all real $x$?

Find the value of the expression $$\sqrt{8-4\sqrt{2}+1}+\sqrt{9-12\sqrt{2}+8}$$

When the graph of the function $y=\mathrm{f}(x)$, defined on the real numbers, is reflected in the $y$-axis and then translated by 2 units in the negative $x$-direction, the result is the graph of the function $y=\mathrm{g}(x)$.

When the graph of the same function $y=\mathrm{f}(x)$ is translated by 2 units in the negative $x$-direction and then reflected in the $y$-axis, the result is the graph of the function $y=\mathrm{h}(x)$.

Which one of the following conditions on $y=\mathrm{f}(x)$ is necessary and sufficient for the functions $\mathrm{g}(x)$ and $\mathrm{h}(x)$ to be identical?