Test of Mathematics for University Admission

TMUA Logic & proof questions

12 original TMUA logic & proof questions in the real exam style, each with a step-by-step worked solution and an explanation of why the tempting wrong answers are wrong. Try each one before opening the solution.

Methods on this page: Find a counterexampleNecessary vs sufficient conditionsNegate “for every” and “there is”Test statements about derivativesA counterexample makes the 'if' part true and the 'then' part false'Only if', converse and contrapositive'For every' and 'there exists': the order mattersFind the first line where the solution set changesMust it be true? Look for a counterexample, then justifyGuarantees: build the worst case, then add one

Question 1TMUA Paper 2Warm-up

Consider the statement: “For every integer

n

, if

n^2

is divisible by

4

, then

n

is divisible by

4

.” Which value of

n

is a counterexample?

A $n = 0$
B $n = 2$
C $n = 3$
D $n = 4$
E $n = 8$

Show the answer and worked solution

Answer: B · Method: Find a counterexample

A counterexample must make the “if” part true and the “then” part false.
For $n = 2$ : $n^2 = 4$ is divisible by $4$ , but $2$ is not. The others either make the conclusion true or the condition false.

Why the tempting wrong answers are wrong

C: For $n = 3$ , $n^2 = 9$ isn't divisible by $4$ , so the condition is false and the statement isn't tested at all.

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Question 2TMUA Paper 2Warm-up

For a positive integer

n

, the condition “

n

is a multiple of 4” is ______ for “

n

is a multiple of 12”.

Anecessary but not sufficient
Bsufficient but not necessary
Cnecessary and sufficient
Dneither necessary nor sufficient

Show the answer and worked solution

Answer: A · Method: Necessary vs sufficient conditions

Sufficient means: if $n$ is a multiple of 4, then $n$ is a multiple of 12. This is false: $n = 4$ satisfies the first but not the second.
Necessary means: if $n$ is a multiple of 12, then $n$ is a multiple of 4. This is true: every multiple of 12 is a multiple of 4.
So it's necessary but not sufficient.

Why the tempting wrong answers are wrong

B: You've swapped them. P is sufficient for Q if P guarantees Q; P is necessary for Q if Q can't happen without P.

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Question 3TMUA Paper 2Warm-up

The real numbers

a

and

b

satisfy

0 < a < b

. Which of the following must be true?

ab < b^2

a + b > 0

III

\dfrac{a + b}{2} < b

Anone of them
BI only
CII only
DIII only
EI and II only
FI and III only
GII and III only
HI, II and III

Show the answer and worked solution

Answer: H · Method: Must it be true? Look for a counterexample, then justify

I is true: multiply $a < b$ by $b > 0$ .
II is true: both are positive.
III is true: it's the midpoint of $a$ and $b$ .

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Question 4TMUA Paper 2Warm-up

Which one of the following functions

f

provides a counterexample to this statement?

f'(x) > 0

for all real

x

, then

f(x) > 0

for all real

x

A $f(x) = 2^x$
B $f(x) = x^3$
C $f(x) = x^2 + 1$
D $f(x) = 2^x - 1$
E $f(x) = 2^x + 1$

Show the answer and worked solution

Answer: D · Method: A counterexample makes the 'if' part true and the 'then' part false

A counterexample must make the 'if' part true and the 'then' part false.
$f(x) = 2^x$ : it is increasing and always positive, so the claim holds.
$f(x) = x^3$ : $f'(0) = 0$ , so the 'if' part fails.
$f(x) = x^2 + 1$ : $f'(x) = 2x$ is not always positive.
$f(x) = 2^x - 1$ : $2^x - 1$ is increasing, but $f(-1) = -\tfrac{1}{2}$ .
$f(x) = 2^x + 1$ : it is increasing and always positive.

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Question 5TMUA Paper 2Standard

Which of the following is the negation of: “Every student in the class passed at least one exam”?

AEvery student failed at least one exam.
BNo student passed any exam.
CAt least one student failed every exam.
DAt least one student passed every exam.
EEvery student failed every exam.

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Answer: C · Method: Negate “for every” and “there is”

Negating “for every student, there is an exam they passed” gives “there is a student for whom there is no exam they passed”.
In other words, at least one student failed every exam.

Why the tempting wrong answers are wrong

A: This is a different statement: a student can fail one exam and still pass another.
B: This is far stronger than the negation: one student failing everything is enough.

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Question 6TMUA Paper 2Standard

Consider the statement: “For every positive integer

n

n^2 + n + 41

is prime.” Which value of

n

is a counterexample?

A $n = 1$
B $n = 10$
C $n = 20$
D $n = 39$
E $n = 40$

Show the answer and worked solution

Answer: E · Method: Find a counterexample

A counterexample needs $n^2 + n + 41$ to be not prime.
For $n = 40$ : $40^2 + 40 + 41 = 40 \times 41 + 41 = 41 \times 41$ , which is not prime.
The other values give $43, 151, 461, 1601$ , which are all prime.

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Question 7TMUA Paper 2Standard

For positive integers

n

, let A be “

n

is a multiple of 3” and B be “

n

is a multiple of 9”. Which of the following statements is/are true?

IIf

n

is not a multiple of 3, then

n

is not a multiple of 9.

IIIf

n

is a multiple of 9, then

n

is a multiple of 3.

III

n

is a multiple of 3 if and only if

n

is a multiple of 9.

Anone of them
BI only
CII only
DIII only
EI and II only
FI and III only
GII and III only
HI, II and III

Show the answer and worked solution

Answer: E · Method: 'Only if', converse and contrapositive

A ⇒ B is false: $n = 3$ satisfies the first but not the second. B ⇒ A is true: every multiple of 9 is a multiple of 3.
I means the contrapositive of B ⇒ A, so it's true exactly when B ⇒ A is: true.
II means B ⇒ A: true.
III means both A ⇒ B and B ⇒ A: false.

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Question 8TMUA Paper 2Standard

Let

S = \{1, 2, 3, 4, 5\}

. In each statement,

a

and

b

may be equal. Which of the following statements is/are true?

IFor every

a

S

, there is a

b

S

such that

ab > 5

IIFor every

b

S

, there is an

a

S

such that

a^2 \ge b

IIIThere is an

a

S

such that

a \le b

for every

b

S

Anone of them
BI only
CII only
DIII only
EI and II only
FI and III only
GII and III only
HI, II and III

Show the answer and worked solution

Answer: G · Method: 'For every' and 'there exists': the order matters

I is false: $a = 1$ has no $b$ that works.
II is true: $b = 1$ has $a = 1$ , $b = 2$ has $a = 2$ , $b = 3$ has $a = 2$ , $b = 4$ has $a = 2$ , $b = 5$ has $a = 3$ .
III is true: $a = 1$ works for every $b$ .

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Question 9TMUA Paper 2Standard

A student attempts to solve the equation

x^3 = 25x

as follows:

(I)

x^2 = 25

(II)

x^2 - 25 = 0

(III)

(x - 5)(x + 5) = 0

(IV)

x = 5 \text{ or } x = -5

Which of the following best describes the student's attempt?

AThe argument is correct.
BThe argument is incorrect, and the first error occurs on line (I).
CThe argument is incorrect, and the first error occurs on line (II).
DThe argument is incorrect, and the first error occurs on line (III).
EThe argument is incorrect, and the first error occurs on line (IV).

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Answer: B · Method: Find the first line where the solution set changes

Check each line has exactly the same solutions as the line before it.
Line (I) is the first error. Dividing both sides by $x$ loses the solution $x = 0$ .

Why the tempting wrong answers are wrong

A: Test a value: the final line doesn't give the same solutions as the original equation.
C: Line (II) follows correctly from the line before it. The mistake happens earlier.

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Question 10TMUA Paper 2Standard

A drawer contains 7 red and 3 blue socks. Socks are taken out at random, without looking. What is the smallest number that must be taken out to be certain of getting at least one sock of every colour?

A $2$
B $4$
C $7$
D $8$
E $9$

Show the answer and worked solution

Answer: D · Method: Guarantees: build the worst case, then add one

Worst case: every sock except the colour with the fewest (3) comes out first, which is 7 socks.
The next one must be the missing colour, so $8$ .

Why the tempting wrong answers are wrong

A: You might keep drawing the same colour. Think about the worst case.
B: The worst case leaves out the smallest colour until last.

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Question 11TMUA Paper 2Challenging

Let

f(x) = x^3

. Consider the statements:
I:

f

is strictly increasing on all real numbers.
II:

f'(x) > 0

for all real

x

.
III: The graph of

f

has no stationary points.
Which of these statements are true?

Anone of them
BI only
CII only
DI and II only
EI and III only
FI, II and III

Show the answer and worked solution

Answer: B · Method: Test statements about derivatives

I is true: if $a < b$ then $a^3 < b^3$ .
II is false: $f'(0) = 0$ . III is false: $x = 0$ is a stationary point (a point of inflection).
So a function can be strictly increasing even where its derivative is zero.

Why the tempting wrong answers are wrong

D: $f'(0) = 0$ , so II fails, even though $f$ is strictly increasing.

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Question 12TMUA Paper 2Challenging

b

and

c

are real constants. Consider the statement:

(∗) The equation

x^2 + bx + c = 0

has two distinct real roots.

The condition

c < 0

is ______ for (∗).

Anecessary but not sufficient
Bsufficient but not necessary
Cnecessary and sufficient
Dneither necessary nor sufficient

Show the answer and worked solution

Answer: B · Method: Necessary vs sufficient conditions

(∗) holds exactly when $b^2 > 4c$ .
Sufficient: yes. Whenever the condition holds, so does (∗).
Necessary: no. $x^2 - x = 0$ has two distinct real roots, but $b = -1$ , $c = 0$ doesn't satisfy the condition.

Why the tempting wrong answers are wrong

A: You've swapped them. Sufficient: the condition guarantees the roots. Necessary: the roots can't happen without the condition.

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