Comprehensive assessment covering the language of proof, proof by contradiction, the triangle and absolute-value inequalities, the AM-GM and Cauchy-Schwarz inequalities, and mathematical induction (series, divisibility, inequality and strong induction).
The contrapositive of the statement "if $n^2$ is even, then $n$ is even" is:
The negation of "for all real $x$, $x^2 \ge 0$" is:
To disprove the statement "for all integers $n$, $n^2 + n + 41$ is prime", the correct approach is to:
A proof by contradiction that $\sqrt{2}$ is irrational begins by assuming that:
The converse of "if a quadrilateral is a square, then it is a rectangle" is:
The triangle inequality states that for all real numbers $a$ and $b$:
For two positive real numbers $a$ and $b$, the AM-GM inequality states that:
Equality in the inequality $\dfrac{a + b}{2} \ge \sqrt{ab}$ (for $a, b > 0$) holds if and only if:
In a proof by mathematical induction, after establishing the base case, the inductive step assumes the result holds for $n = k$ and then proves it holds for:
Which technique is used in the classical proof that there are infinitely many prime numbers?
The biconditional statement "$P$ if and only if $Q$" is logically equivalent to:
Using the AM-GM inequality, the minimum value of $x + \dfrac{1}{x}$ for $x > 0$ is:
Which of the following is a correct statement of the Cauchy-Schwarz inequality for real numbers $a_1, a_2, b_1, b_2$?
When proving that $n^3 - n$ is divisible by $6$ by induction, the expression $\big[(k+1)^3 - (k+1)\big] - \big[k^3 - k\big]$ simplifies to:
The negation of "every student passed the exam" is:
1. Write the contrapositive of "if $n$ is divisible by $4$, then $n$ is even". (1 mark)
2. Write the negation of "there exists a real number $x$ such that $x^2 = -1$". (1 mark)
3. Prove directly that the sum of two even integers is even. (2 marks)
4. Prove by contradiction that $\sqrt{2}$ is irrational. (3 marks)
5. Use the AM-GM inequality to prove that $a + \dfrac{1}{a} \ge 2$ for all $a > 0$, and state when equality holds. (2 marks)
6. Prove that $a^2 + b^2 \ge 2ab$ for all real numbers $a$ and $b$. (2 marks)
7. Prove by mathematical induction that $1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2}$ for all positive integers $n$. (3 marks)
8. Prove by induction that $n^3 - n$ is divisible by $6$ for all positive integers $n$. (3 marks)
9. Give a counterexample to disprove the claim "for all positive integers $n$, $2^n - 1$ is prime". (1 mark)
10. Prove that if $a > 0$ and $b > 0$, then $\dfrac{a}{b} + \dfrac{b}{a} \ge 2$. (2 marks)