Question bank

TMUA-P2-ERR2-001

A student argues: “If $ab = ac$ then $b = c$, just divide both sides by $a$.” For which values does this implication fai…

TMUA-P2-ERR2-002

A detective reasons with $P$ = “it rained” and $Q$ = “the ground is wet”: “If $P$ then $Q$. $Q$ holds. Therefore $P$.” W…

TMUA-P2-ERR2-003

A student writes $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$. What is wrong?

TMUA-P2-ERR2-004

A student solves $-2x > 6$ by dividing both sides by $-2$ to get $x > -3$. What is the error?

TMUA-P2-ERR2-005

A student expands $(x + y)^2$ as $x^2 + y^2$, then “cancels” squares against roots elsewhere in the problem. Which pair…

TMUA-P2-ERR2-006

A ferry firm says a ticket ($T$) is necessary for boarding ($B$), but a customer with $T$ and no cabin reservation is tu…

TMUA-P2-ERR2-007

A student simplifies $\frac{x + 3}{3}$ as $x + 1$. Which evaluation exposes the illegal cancellation?

TMUA-P2-ERR2-008

A student writes $\log(a + b) = \log a + \log b$ for positive $a, b$, then “verifies” it with $a = b = 1$. What is the c…

TMUA-P2-ERR2-009

A sixth-former optimises a profit function $P(x)$ and argues as follows: Fermat's theorem says that if $P$ has a local m…

TMUA-P2-ERR2-010

A philosophy essay argues: “Everyone has a biological mother. Therefore there is someone who is everyone's biological mo…

TMUA-P2-ERR2-011

Sam solves $x(x-2)=3(x-2)$ by dividing both sides by $(x-2)$ and reports $x=3$ as the only solution. What is wrong?

TMUA-P2-ERR2-012

Mia solves $x^2=9$ and concludes $x=3$. What is the error?

TMUA-P2-ERR2-013

A student solves $2x(x+1)=x+1$ by cancelling $(x+1)$ to get $x=\tfrac12$ only. What is the full solution set?

TMUA-P2-ERR2-014

A student claims that if $\sin\theta=\sin(\pi/6)$ with $0 \le \theta < 2\pi$ then $\theta=\pi/6$ is the only solution. W…

TMUA-P2-ERR2-015

To solve $\sqrt{x+5}=x-1$, a student squares to get $x=4$ or $x=-1$ and reports both. What is correct?

TMUA-P2-ERR2-016

Tom claims $\sqrt{a^2}=a$ for all real $a$, and hence $\sqrt{(-5)^2}=-5$. What is the error?

TMUA-P2-ERR2-017

A student solves $\sin 2\theta=\sin\theta$ on $[0,2\pi)$ by writing $2\theta=\theta$, hence $\theta=0$ only. What is wro…

TMUA-P2-ERR2-018

A student solves $(2x-1)^2=(x+2)^2$ by taking square roots to get $2x-1=x+2$, hence $x=3$ only. What is the full solutio…

TMUA-P2-ERR2-019

A student solves $x^3-2x^2=x-2$ by dividing by $(x-2)$ to get $x^2=1$, hence $x=\pm 1$ only. What is the full solution s…

TMUA-P2-ERR2-020

Consider $(k-2)x=k^2-4$ with real parameter $k$. A student divides by $(k-2)$ and claims $x=k+2$ for every $k$. What is…

TMUA-P2-ERR2-021

A student solves $\sin 5x=\sin 3x$ by writing $5x=3x+2k\pi$, hence $x=k\pi$ only. What is the complete general solution?

TMUA-P2-ERR2-022

A student solves $\sin 2x=\tfrac12$ on $[0,2\pi]$ by writing $2x=\pi/6$, hence $x=\pi/12$ only. How many solutions are t…