Proof & Reasoning (DP IB Analysis & Approaches (AA): SL): 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 $x^2-3x+3>0$ for all values of $x$.

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

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 $x$.

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

Prove that the square of any odd integer is odd.

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

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

Prove that the product of any two odd integers 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 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 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 $1\le k\le n-1$.