Question bank

TMUA-P2-ERR1-001

A “proof” that $1 = 2$ reads: (1) Let $a = b$. (2) Then $a^2 = ab$. (3) So $a^2 - b^2 = ab - b^2$. (4) Factorising gives…

TMUA-P2-ERR1-002

A proof claims $\sqrt{(-4)^2} = -4$: (1) $\sqrt{(-4)^2} = \sqrt{16}$. (2) $\sqrt{16} = 4$. (3) Hence $\sqrt{(-4)^2} = -4…

TMUA-P2-ERR1-003

A student claims: (1) $\frac{ab}{a} = b$ for all numbers $a, b$. (2) Put $a = 0$, $b = 5$. (3) Hence $\frac{0}{0} = 5$.…

TMUA-P2-ERR1-004

A “proof” for acute $A, B$ reads: (1) Suppose $\sin A = \sin B$. (2) “Cancel” $\sin$ to get $A = B$. (3) Hence equal sin…

TMUA-P2-ERR1-005

A student solves an inequality: (1) Suppose $x < 3$. (2) Divide both sides by $x$ to get $1 < \frac{3}{x}$. (3) Hence ev…

TMUA-P2-ERR1-006

A viral video “proves” that $-3 = 3$ with the following line-numbered argument: line one notes that $(-3)^2 = 9$ and $3^…

TMUA-P2-ERR1-007

A student “proves” $n^2 + n$ is odd for every integer $n$: “If $n$ is even, $n = 2k$, so $n^2 + n = 4k^2 + 2k = 2(2k^2 +…

TMUA-P2-ERR1-008

A “proof” that the sum of two odds is odd reads: (1) Assume odd $m, n$ with $m + n$ odd. (2) Write $m = 2j+1$, $n = 2k+1…

TMUA-P2-ERR1-009

A six-line algebra proof submitted for a scholarship problem claims to show that $x = 5$ follows from the equation $(x -…

TMUA-P2-ERR1-010

A flawed lemma in a student journal claims that from the true statement “For every real $x$ there exists a real $y$ with…

TMUA-P2-ERR1-011

A purported proof claims $1 = 2$: (1) Let $a = b = 1$. (2) Then $a^2 = ab$. (3) So $a^2 - b^2 = ab - b^2$, i.e. $(a-b)(a…

TMUA-P2-ERR1-012

A purported proof of “If $n^2$ is odd then $n$ is odd” runs: (1) Assume $n$ is odd, so $n = 2k+1$. (2) Then $n^2 = 4k^2+…

TMUA-P2-ERR1-013

A purported proof claims “If $x < y$ then $x^2 < y^2$” for all reals: (1) Assume $x < y$. (2) Multiply both sides by $x$…

TMUA-P2-ERR1-014

A purported proof claims “All prime numbers are odd”: (1) Recall a prime has no divisors other than $1$ and itself. (2)…

TMUA-P2-ERR1-015

A purported proof claims “The product of two irrational numbers is irrational”: (1) Let $a$ and $b$ be irrational. (2) W…

TMUA-P2-ERR1-016

A student imitates the $\sqrt{2}$ proof: (1) Assume $\sqrt{2} = p/q$ for some integers $p$, $q$ with $q \ne 0$. (2) Then…