TMUA Paper 2 practice: Mathematical Reasoning

Let $x$ be a real number.

Which one of the following conditions is necessary but not sufficient for $x^2>4$?

Consider the statement:

S: Every student who passed the test attended the revision session.

Which of the following statements is/are logically equivalent to S?

I: Any student who did not attend the revision session did not pass the test.

II: Every student who attended the revision session passed the test.

III: A student passed the test only if they attended the revision session.

Consider the claim:

For every real number $x$, if $x^2>x$ then $x>1$.

Which of the following values of $x$ is a counterexample to the claim?

A student wants to prove the following statement by contradiction.

For every integer $n$, if $n^2+2n$ is odd, then $n$ is odd.

Which of the following should the student assume at the start of the proof?

Consider the claim:

For every non-zero real number $x$, $x+\frac{1}{x}\ge 2$.

Here is an attempted proof.

(I) Let $x$ be any non-zero real number. Since a square is never negative, $(x-1{)}^2\ge 0$.

(II) Expanding the bracket gives $x^2-2x+1\ge 0$.

(III) Adding $2x$ to both sides gives $x^2+1\ge 2x$.

(IV) Dividing both sides by $x$ gives $x+\frac{1}{x}\ge 2$.

(V) Since $x$ was any non-zero real number, the claim is proved.

Which of the following best describes this proof?

Consider the claim:

For every integer $n$, if $n^2$ is a multiple of 3, then $n$ is a multiple of 3.

A student offers this argument.

'Suppose $n$ is a multiple of 3, so $n=3k$ for some integer $k$. Then $n^2=9k^2=3(3k^2)$, which is a multiple of 3. This proves the claim.'

Which of the following best describes the argument?

Each of five cards has a whole number on one side and a colour, either red or blue, on the other side. The cards lie on a table, and the faces you can see are:

4, 7, red, blue, 9

You are told this rule:

'A card is red on one side only if it has an even number on the other side.'

Which cards must be turned over to check whether the rule holds for all five cards?

Consider the statement:

For every integer $n\ge 2$, there is a prime number $p$ with $n<p<2n$.

Which of the following is the negation of this statement?

In this question $b$ and $c$ are real numbers.

Which of the following statements is/are true?

I: If the equation $x^2+bx+c=0$ has two distinct real roots, then $c<0$.

II: If $c<0$, then the equation $x^2+bx+c=0$ has two distinct real roots.

III: If the equation $x^2+bx+c=0$ has two distinct positive roots, then $b<0$ and $c>0$.

Consider the claim:

The equation $\sin 2x=\sin x$ has exactly two solutions with $0^{\circ }\le x<{360}^{\circ }$.

Here is an attempted proof.

(I) Using the double-angle formula, the equation becomes $2\sin x\cos x=\sin x$.

(II) Dividing both sides by $\sin x$ gives $2\cos x=1$.

(III) So $\cos x=\frac{1}{2}$.

(IV) For $0^{\circ }\le x<{360}^{\circ }$, the solutions of $\cos x=\frac{1}{2}$ are $x={60}^{\circ }$ and $x={300}^{\circ }$.

(V) Hence the equation has exactly two solutions in this range.

Which of the following best describes the attempted proof?

In this question $a$, $b$ and $c$ are positive real numbers, none of which is equal to 1.

Which of the following statements is/are true?

I: If ${\log }_{a}b>1$, then $b>a$.

II: For all such $a$ and $b$, ${\log }_a(b^2)=({\log }_{a}b{)}^2$.

III: There exist such $a$, $b$ and $c$ for which ${\log }_a(b+c)={\log }_{a}b+{\log }_{a}c$.

The line $y=mx$ and the circle $(x-3{)}^2+y^2=5$ are drawn on the same axes.

Which one of the following conditions on $m$ is sufficient, but not necessary, for the line to meet the circle at two distinct points?

A sequence is defined by $u_1=k$ and $u_{n+1}=u_n^2-2u_n+2$ for $n\ge 1$, where $k$ is a real number.

Which of the following statements is/are true?

I: The sequence is constant if $k=2$.

II: For every value of $k$, $u_n\ge 1$ for all $n\ge 2$.

III: If $u_2=u_3$, then $k=1$ or $k=2$.

In this question $f$ and $g$ are polynomials.

Which of the following statements is/are true?

I: If ${\int }_0^{1}f(x)\mathrm{d}x=0$, then $f(x)=0$ for at least one value of $x$ with $0\le x\le 1$.

II: If ${\int }_0^{1}f(x)\mathrm{d}x>{\int }_0^{1}g(x)\mathrm{d}x$, then $f(x)>g(x)$ for every $x$ with $0\le x\le 1$.

III: If ${\int }_0^{2}f(x)\mathrm{d}x=2{\int }_0^{1}f(x)\mathrm{d}x$, then $f$ is a constant function.

Let $f(x)=x^3-3x$.

Which of the following statements is/are true?

I: There is a real number $a$ such that $f(x)\ge a$ for every real number $x$.

II: For every real number $a$, there is a real number $x$ such that $f(x)>a$.

III: There is a real number $x$ such that $f(x)>a$ for every real number $a$.

The real number $x$ makes exactly one of the following three statements true.

P: $x^2>4$

Q: $x^3>8$

R: $x>1$

Which of the following describes all the possible values of $x$?

Let $f$ be a polynomial. Consider the following three conditions.

P: $f^{\mathrm{\prime }}(x)>0$ for every real number $x$.

Q: $f(b)>f(a)$ for all real numbers $a$ and $b$ with $b>a$.

R: $f^{\mathrm{\prime }}(x)\ge 0$ for every real number $x$.

Which one of the following is true?

Consider the claim:

For every odd integer $n$, $n^2-1$ is divisible by 16.

Here is an attempted proof.

(I) Let $n$ be odd. Then $n=2k+1$ for some integer $k$.

(II) So $n^2-1=4k^2+4k=4k(k+1)$.

(III) Of the two consecutive integers $k$ and $k+1$, one is even, so $k(k+1)$ is even. Write $k(k+1)=2m$, where $m$ is an integer.

(IV) Then $n^2-1=4\times 2m=8m$.

(V) Now $m=\frac{k(k+1)}{2}$, and one of $k$ and $k+1$ is even, so $m$ is even.

(VI) Writing $m=2j$ gives $n^2-1=16j$, which is divisible by 16.

Which of the following best describes this proof?

In this question $p$ and $q$ are real numbers. Consider the equation $x^2+px+q=0$ and the condition

C: $q(1+p+q)<0$.

Which of the following is true?

The positive integers $a$, $b$ and $c$ satisfy $a^2+b^2=c^2$.

Which of the following statements must be true?

I: At least one of $a$ and $b$ is even.

II: At least one of $a$, $b$ and $c$ is a multiple of 3.

III: If $a$ is prime, then $c=b+1$.