WAEC 2022 · Paper 2 · Q7

(a)
The hypotenuse of a right-angled triangle is $39\text{ cm}$ long and the perimeter is $90\text{ cm}$ . Find the lengths of the other two sides.
Separate values with commas, e.g. 3, −2
(b)
Consider the following statements:

$P$ : There is no Mathematics student who is not clever. $Q$ : Every Physics student studies Mathematics.

(i) Draw a Venn diagram to represent the statements. (ii) Deduce whether the following statements are valid or not valid. (I) Every clever student is a Physics student. (II) All Physics students are clever. (III) Every Mathematics student studies Physics.
Model answer
$C$ = clever students, $M$ = Mathematics students, $P$ = Physics students. $P$ : every Mathematics student is clever, so $M$ lies inside $C$ ; $Q$ : every Physics student studies Mathematics, so $P$ lies inside $M$ . So all Physics students are clever (II valid), but a clever student need not study Physics (I not valid), and a Mathematics student need not study Physics (III not valid).

Worked solution (try it first)

Call the other two sides $a$ and $b$ .
The perimeter is 90 cm and the hypotenuse is 39 cm, so $a + b = 90 - 39 = 51$ , and $b = 51 - a$ .
By Pythagoras' theorem, $a^2 + b^2 = 39^2 = 1521$ .
Put in $b = 51 - a$ : $a^2 + (51 - a)^2 = 1521$ , so $a^2 + 2601 - 102a + a^2 = 1521$ .
Simplify: $2a^2 - 102a + 1080 = 0$ .
Divide by 2: $a^2 - 51a + 540 = 0$ .
Factorise: $(a - 15)(a - 36) = 0$ , so $a = 15$ or $a = 36$ .
Either way the two sides are 15 cm and 36 cm.
(Check: $15 + 36 + 39 = 90$ and $15^2 + 36^2 = 225 + 1296 = 1521 = 39^2$ ✓.)

$P$ : no Mathematics student is outside the clever students, so every Mathematics student is clever: draw the Mathematics circle inside the Clever circle.
$Q$ : every Physics student studies Mathematics: draw the Physics circle inside the Mathematics circle.
So the universal set of students holds three circles, Physics inside Mathematics inside Clever.

(I)** A clever student can be outside the Physics circle (for example, outside the Mathematics circle altogether).
So "every clever student is a Physics student" is not valid.
(II) The Physics circle is inside the Mathematics circle, which is inside the Clever circle, so every Physics student is clever: valid.
(III) A Mathematics student can be in the ring between the Physics and Mathematics circles, studying Mathematics but not Physics.
So "every Mathematics student studies Physics" is not valid.