Logic & Proof

12 methods · 77 questions

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These items give two or three labelled claims and ask which are forced. Treat each claim in isolation: either give a one-line reason it must hold, or produce one concrete case where it fails. A single failing instance settles "must" as false, so you are building counterexamples, not theories. Then read off the option matching your true/false pattern.

Trigger: numbered statements I, II, III with options like "I and III only".

Instances:
(i) for an algebraic identity claim, expand to a square or factor to show it is forced for all values;
(ii) for a claim that can fail, plug in small integers or a sign-flipped value to break it;
(iii) decide the statements independently, the answer is just the resulting pattern.

Linked questions (3)

TMUA 2017 Paper 2 · Q13

The positive real numbers

a \times 1 0^{- 3}

b \times 1 0^{- 2}

and

c \times 1 0^{- 1}

are each in standard form, and

(a \times 1 0^{- 3}) + (b \times 1 0^{- 2}) = (c \times 1 0^{- 1}) .

Which of the following statements (I, II, III, IV) must be true?

I: a > 9 II: b > 9 III: a < c IV: b < c

AI only
BII only
CI and II only
DI and III only
EI and IV only
FII and III only
GII and IV only
HI, II, III and IV

Worked solution

Answer:B

Since all three are in standard form,

1 ⩽ a, b, c < 10

. The equation is

0.001 a + 0.01 b = 0.1 c

, i.e.

a + 10 b = 100 c

. Since

1 ⩽ c < 10

, we have

100 ⩽ 100 c < 1000

. Now

a < 10

, so

100 c = a + 10 b < 10 + 10 b

, giving

100 c - 10 < 10 b

, hence

10 c - 1 < b

. Since

c ⩾ 1

b > 9

. Thus II must be true. For I: if

b = 9.9

and

c = 1

, then

a = 100 - 10 (9.9) = 1

, so

a = 1 ⩽ 9

, and I is not necessarily true. For III:

a = 1 < 1 = c

is not guaranteed (taking

c = 1

a = 1

gives

a = c

, so III fails). For IV:

b > 9 ⩾ c

is possible (e.g.

b = 9.9

c = 1

), so IV (

b < c

) is not necessarily true and in fact is never true since

b > 9 ⩾ c

. So II must be true and only II. The answer is B.

TMUA 2018 Paper 2 · Q5

The two diagonals of the quadrilateral

Q

are perpendicular.

Consider the following statements:

I. One of the diagonals of

Q

is a line of symmetry of

Q

.

II. The midpoints of the sides of

Q

are the vertices of a square.

Which of these statements is/are necessarily true for the quadrilateral

Q

Aneither of them
BI only
CII only
DI and II

Worked solution

Answer:A

We look for counterexamples to each statement.

For I: draw two perpendicular diagonals, but place the off-diagonal vertices at different distances from each diagonal. This gives a quadrilateral with perpendicular diagonals where neither diagonal is a line of symmetry. So I is not necessarily true.

For II: draw a very elongated (long and skinny) quadrilateral with perpendicular diagonals. The midpoints of the sides form a rectangle, not necessarily a square. So II is not necessarily true.

Neither statement is necessarily true. Hence the answer is A.

TMUA 2018 Paper 2 · Q8

The diagram shows an example of a mountain profile.

This consists of upstrokes which go upwards from left to right, and downstrokes which go downwards from left to right. The example shown has six upstrokes and six downstrokes. The horizontal line at the bottom is known as sea level.

A mountain profile of order

n

consists of

n

upstrokes and

n

downstrokes, with the condition that the profile begins and ends at sea level and never goes below sea level (although it might reach sea level at any point). The example shown is a mountain profile of order 6.

Mountain profiles can be coded by using U to indicate an upstroke and D to indicate a downstroke. The example shown has code UDUUDUDUUDDD.

Which of the following statements is/are true?

I. If a valid code is written in reverse order, the result is always a valid code.

II. If each U in a valid code is replaced by D and each D by U, the result is always a valid code.

III. If U is added at the beginning of a valid code and D is added at the end of the code, the result is always a valid code.

illustrative

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

Worked solution

Answer:D

Statement I: Every valid code begins with U and ends with D. Reversing gives a code beginning with D, which immediately dips below sea level. So the reverse is never valid. Statement I is false.

Statement II: Replacing each U with D and each D with U transforms a code starting with U into one starting with D, again immediately going below sea level. Statement II is false.

Statement III: Adding U at the start and D at the end lifts the entire middle profile one unit above its original level. Since the original never goes below sea level, the lifted version also never does, and the new code still starts at sea level, ends at sea level, and never dips below. Statement III is always true.

Hence only statement III is true. The answer is D.