Mathematics Extension 2

Consider the following statement: “If an animal is a herbivore, then it does not eat meat.” Which of the following is the converse of this statement?

A. If an animal is a herbivore, then it eats meat.
B. If an animal is not a herbivore, then it eats meat.
C. If an animal eats meat, then it is not a herbivore.
D. If an animal does not eat meat, then it is a herbivore.

Consider the following statement about real numbers: “Whichever positive number $r$ you pick, it is possible to find a number $x$ greater than $1$ such that $\frac{\ln x}{x^3}<r$.” When this statement is written in the formal language of proof, which of the following is obtained?

A. $\forall x>1\;\exists r>0\;\frac{\ln x}{x^3}<r$ B. C. D.

The polynomial $P(z)$ has real coefficients and $P(0)=-1$. The imaginary number $\alpha$ and the real number $\beta$ satisfy $P(\alpha)=0$, $P(\beta)=0$ and $P'(\beta)=0$.

The degree of $P(z)$ is at least:

(A) $2$

(B) $3$

(C) $4$

(D) $5$

Given $a\in\mathbb{Z}$, prove that if $3a^2-4a+5$ is even, then $a$ is odd.

Prove that $\sqrt{23}$ is irrational.

Prove that for all real numbers $x$ and $y$, where $x^2+y^2\ne0$, $\frac{(x+y)^2}{x^2+y^2}\le2.$

Given $a$, $b$, $c$ are positive real numbers, hence or otherwise, prove that

Given $a$, $b$, $c$ are positive real numbers, hence or otherwise, prove that

Prove by contradiction that there are no rational solutions to the equation

Given $a$, $b$, $c$ are positive real numbers, prove that

The converse of “if it miaows it’s a cat” is

A. If it’s a cat it miaows.

B. If it’s not a cat it doesn’t miaow.

C. If it doesn’t miaow it isn’t a cat.

D. If it’s a cat it doesn’t miaow.

The following proof aims to establish that $-4=0$.

Let $a=-4$.

$\Rightarrow a^2=16$ and $4a+4=-12$ (Line 1)

$\Rightarrow a^2+4a+4=4$ (Line 2)

$\Rightarrow (a+2)^2=2^2$ (Line 3)

$\Rightarrow a+2=2$ (Line 4)

$\Rightarrow a=0$

At which line is the implication incorrect?

A. Line 1
B. Line 2
C. Line 3
D. Line 4

The contrapositive of “if it miaows it’s a cat” is

A. If it’s a cat it miaows.

B. If it’s not a cat it doesn’t miaow.

C. If it doesn’t miaow it isn’t a cat.

D. If it’s a cat it doesn’t miaow.

Let $A$, $B$, $P$ be three points in three-dimensional space with $A\ne B$.

If $P$ is on the line $AB$, then there exists a real number $\lambda$ such that $\overrightarrow{AP}=\lambda\overrightarrow{AB}$.

Which of the following is the contrapositive of this statement?

A. If for all real numbers $\lambda$, $\overrightarrow{AP}=\lambda\overrightarrow{AB}$, then is on the line . B. If for all real numbers , , then is not on the line . C. If there exists a real number such that , then is on the line . D. If there exists a real number such that , then is not on the line .

Consider the statement:

“If $z$ is even, then $x$ and $y$ are either both even or both odd.”

Which of the following is equivalent?

A. If $x$ and $y$ are either both even or both odd, then $z$ is even.

B. If $z$ is odd, then exactly one of $x$ and $y$ is even.

C. If $x$ and $y$ are either both even or both odd, then $z$ is odd.

D. If exactly one of $x$ and $y$ is even, then $z$ is odd.

Consider the statement $P$.

$P$: For all integers $n\ge1$, if $n$ is a prime number then $\dfrac{n(n+1)}2$ is a prime number.

Which of the following is true about this statement and its converse?

A. The statement $P$ and its converse are both true.
B. The statement $P$ and its converse are both false.
C. The statement $P$ is true and its converse is false.
D. The statement $P$ is false and its converse is true.

Which of the following is the negation of the statement:

“$\forall p\in P$, $p$ is of the form $4m+1\Rightarrow p$ can be written as a sum of two squares”?

A. $\forall p\in P$, $p$ is of the form $4m+1$ and $p$ cannot be written as a sum of two squares.
B. $\exists p\in P$, is not of the form and can be written as a sum of two squares. C. , is not of the form or cannot be written as a sum of two squares. D. , is of the form and cannot be written as a sum of two squares.

Use proof by contrapositive to prove that: If $a^2-2a+7$ is even, then $a$ is odd.

For real numbers $a,b\ge0$, prove that
$\frac{a+b}{2}\ge\sqrt{ab}.$

Given the complex number $z=e^{i\theta}$, show that
$w=\frac{\bar z^2-1}{\bar z^2+1}$
is purely imaginary.

Prove by contraposition that if $n^2$ is not divisible by $18$, then $n$ is not divisible by $6$.

Prove that for all integers $n$ with $n\ge3$, if $2^n-1$ is prime, then $n$ cannot be even.

Prove that if $a$ is divisible by $5$, then $a^2-5a$ is divisible by $50$.

(You may assume that the product of two consecutive numbers is even.)

Using (i), prove that $\sqrt{18}$ is irrational.