Simple Proof & Reasoning (DP IB Analysis & Approaches (AA): HL): Exam Questions

Show that $\left(4x-1\right)\left(2x+3\right)-{\left(2x+1\right)}^2=2\left(2x-1\right)\left(x+2\right)$.

Show that ${\left(a-b\right)}^2-{\left(a+b\right)}^2=-4ab$.

Prove that the sum of any three consecutive integers is divisible by 3.

Prove that the square of any even integer is divisible by 4.

Factorise $n^2+3n+2$.

Hence show that $n^3+3n^2+2n=n(n+1)(n+2).$

Given that $n$ is even, write down whether $\left(n+1\right)$ and $\left(n+2\right)$ are odd or even.

Hence deduce whether $n^3+3n^2+2n$ is odd or even. Justify your answer.

Show that ${\left(3n+2\right)}^2-{\left(n+2\right)}^2=8n^2+8n$, where $n\in \mathrm{\mathbb{Z}}$.

Hence, or otherwise, prove that ${\left(3n+2\right)}^2-{\left(n+2\right)}^2$ is divisible by 8.

Show that $x^2-3x+3>0$ for all values of $x$.

Let $z=x+y\mathrm{i}$, where $x,y\in \mathrm{ℝ}$.

(i) Show that $z{z}^{*}=\left|z\right|\left|z^{*}\right|$.

(ii) Show that, for $x>0$, $\text{arg}\left(z\right)+\text{arg}\left(z^{*}\right)=0$.

Prove that the sum of any three consecutive even integers is divisible by 6.

Prove that $\left(2q-1\right)\left(q-3\right)-3{\left(q-4\right)}^2=-q^2+17q-45$.

Prove that the square of any odd integer is odd.

Consider two consecutive positive integers, $n$ and $n+1$.

Show that the difference of their squares is equal to the sum of the two integers.

Show that $\frac{1}{n+1}+\frac{1}{n^2+n}=\frac{1}{n}.$

Consider the function $f\left(x\right)=x^2-10x+17$. Show that $f\left(x\right)\ge -8$ for all values of $x$.

Prove that the exterior angle in any triangle is equal to the sum of the two opposite interior angles. You may use the diagram below to help.

Consider the function $f\left(x\right)=5x^2+4x+1$. Show that $f\left(x\right)>0$ for all values of $x$.

The product of three consecutive integers is added to the middle integer. 

Prove that the result is a perfect cube.

The three statements below are false.

In each case, verify that the statement is false by use of a counterexample and state an alternative domain that would make the statement true.

(i) $n^2>2n,n\in {\mathrm{\mathbb{Z}}}^{+}$.

(ii) $2^n-1$ is a prime number for $n\in \mathrm{\mathbb{N}},1<n\le 4$.

(iii) $5^n>3^n+4^n,n\in {\mathrm{\mathbb{Z}}}^{+}$.

Three of the four statements below are false.

Eliminate the false statements by providing a counterexample and thus deduce the true statement.

(i) ${\left(x-1\right)}^2\ne {\left(x+1\right)}^2,x\in \mathrm{ℝ}$.

(ii) Every $\left(4n\right)$th triangular number is even, $n\in \mathrm{\mathbb{N}}$.

(iii) $2\text{ln}x>\text{ln}2x,x\in \mathrm{ℝ},x>0$.

(iv) The product of any two distinct positive integers is greater than their sum.

Prove that the sum of the squares of any two consecutive odd integers is even.

Prove that there are no non-zero real values of $a$ and $b$ such that ${\left(a+b\mathrm{i}\right)}^2=a+b\mathrm{i}$.

The sum of squares of two consecutive integers is 313.  Find the possible values of the integers.

Prove that the sum of the cubes of any two consecutive odd integers is divisible by 4.

Prove that $\frac{a^2-a-6}{a+4}\times \frac{a^2-16}{a^2+2a}=a-7+\frac{12}{a}.$

State any values of $a$ for which this mathematical statement does not hold true.

Prove that the product of any two odd integers is odd.

Determine, with appropriate reasoning, whether the following statements are true or false:

(i) Given $n\in \mathrm{\mathbb{Z}}$ and $n^2$ is divisible by 4, then $n$ is divisible by 4.

(ii) Given $n\in \mathrm{\mathbb{Z}}$ then $n^2-1$ is a prime number.

(iii) Given $n\in \mathrm{\mathbb{Z}}$ and $n^2$ is divisible by 3, then $n$ is divisible by 3.

(iv) Given an integer is a multiple of 8 and of 6, then it is a multiple of 48.

Write down $a,b,c$ and $d$ from smallest to largest, given $a,b,c,d\in \mathrm{ℝ}$ and $c>d$, $a<d$ and $a>b$.

Write down $p,q,r$ and $s$ from smallest to largest, given $p,q,r,s\in \mathrm{ℝ}$ and

$p>q$

$r-s<q-p$

$p+q=r+s.$

Prove $\frac{x}{1+x}<\frac{y}{1+y},x,y\in \mathrm{ℝ}$,  given $0\le x<y$.

The function $f\left(n\right)$ is given as $f\left(n\right)=n^3+n^2+17$, where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Find $f\left(1\right)$, $f\left(2\right)$ and $f\left(3\right)$.

Show that not all values of $f\left(n\right)$ are prime.

Prove that there are no integers $p$ and $q$ that satisfy the equation

$4p^2-q^2=49$

Show that

${}^n\mathrm{C}_k={}^{n-1}\mathrm{C}_k+{}^{n-1}\mathrm{C}_{k-1}$

for all integers $n$ and $k$ with $1\le k\le n-1$.

Given that the graph of  $y=x^4-10x^3+37x^2-60x+36$ touches the $x$-axis at the point with coordinates $\left(2,0\right)$ , prove that $y\ge 0$ for all real values of $x$ .