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

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

For$f\left(x\right)=x^2-10x+17$, prove 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$f\left(x\right)$ is positive for all values of $x.$

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.

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

Prove that the square of an odd number is always odd.

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

Prove that the sum of any three consecutive even numbers is a multiple of 6.

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

Prove that the result is a perfect cube.

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 three statements below are false.

In each case verify 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) $5^n>3^n+4^n,n\in {\mathrm{\mathbb{Z}}}^{+}$

(i) Prove that

$\frac{a}{\left({\displaystyle \frac{b}{c}}\right)}=\frac{ac}{b}$

(ii) Specify any cases for which the relation in part (a)(i) is not valid.

Prove that $(p-q{)}^2=(q-p{)}^2$ for all numbers$p$ and $q.$

Prove that the product of two odd numbers is odd.

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 four.

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

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

Prove the binomial coefficient identity 

$\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}n-1\\ k\end{array}\right)+\left(\begin{array}{c}n-1\\ k-1\end{array}\right).$

Prove that the sum of all integers between 600 and 1400 (inclusive) that are not divisible by 7 is equal to $685885.$

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 $0\le x<y$.

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 $x$ .

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

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

The function $f\left(n\right)$ is given as $f\left(n\right)=n^3+n^2+17$  where $n$ is an integer.

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

Prove that $f\left(n\right)$ is not prime for all values of $n$.