Question bank

TMUA-P2-PRF2-001

To prove a claim about $|x|$ for all real $x$ by cases, which splitting of the cases is complete and non-overlapping?

TMUA-P2-PRF2-002

A proof that $n^2 + n$ is even for every integer $n$ splits into cases. Which pair of cases completes the proof?

TMUA-P2-PRF2-003

Prove: for every integer $n$, the remainder of $n^2$ on division by $4$ is $0$ or $1$. Which case argument is valid?

TMUA-P2-PRF2-004

To prove an identity for $|x - 1| + |x + 1|$ valid for all real $x$, a student proposes cases. Which case split is exhau…

TMUA-P2-PRF2-005

A student “proves” $n(n+1)$ is even for all integers $n$ by writing: “If $n$ is even then $n(n+1)$ is even.” What is wro…

TMUA-P2-PRF2-006

A lecturer proves that $n^3 - n$ is divisible by $3$ for every integer $n$ by splitting into remainders modulo $3$, writ…

TMUA-P2-PRF2-007

To solve $|x-3|=5$ by cases, which split is complete and correct?

TMUA-P2-PRF2-008

To prove $n^2+n$ is even for every integer $n$, which case split works?

TMUA-P2-PRF2-009

Solve $|2x-1|=3$ by cases.

TMUA-P2-PRF2-010

Solve $|x-2|<3$ by cases.

TMUA-P2-PRF2-011

Which describes all real $x$ satisfying $x^2>4$?

TMUA-P2-PRF2-012

To prove every integer square is $0$ or $1$ modulo $4$, which argument is correct?

TMUA-P2-PRF2-013

Solve $x^2-5|x|+6=0$.

TMUA-P2-PRF2-014

Solve $|x-1|+|x-3|=4$.

TMUA-P2-PRF2-015

Solve $(x-1)(x-3)<0$.

TMUA-P2-PRF2-016

Let $n$ be odd. Which completes a proof that $8$ divides $n^2-1$?

TMUA-P2-PRF2-017

To prove $|a+b| \le |a|+|b|$ for all real $a,b$, which case analysis is sufficient?

TMUA-P2Z-PRF2-001

Which argument correctly proves by cases that for every integer $n$, the square $n^2$ has the same parity as $n$?

TMUA-P2Z-PRF2-002

Prove: if the integer $n$ is not divisible by $3$ then $n^2$ leaves remainder $1$ on division by $3$. Which case argumen…

TMUA-P2Z-PRF2-003

To prove $|xy| = |x||y|$ for all real $x$ and $y$ by cases, which splitting of the cases is exhaustive, with each case c…

TMUA-P2Z-PRF2-004

A seminar proof claims that the cube of every integer leaves remainder $0$, $1$ or $8$ on division by $9$, splitting int…