UCL CENTRE FOR LANGUAGES & INTERNATIONAL EDUCATION (CLIE)
1) On a journey a certain weight of luggage is carried free, but there is a charge of $5 per
kilogram for any additional luggage above this weight. Laa-laa’s luggage, which weighs a
total of 70kg, is overweight and she is charged $200. If Po’s luggage weighs a total of 45kg,
what will she have to pay ?
2) In the diagram PQ = PR, ∠QMP = 80° and ∠QPM = 30°. What is ∠RPM ?
3) What is the value of 21 x 101 – 21 – 101 ?
4) In the diagram below how many different routes are there from S to T which do not go
through either of the points U and V more than once ?
5) What is the sum of all the first 20 even numbers ?
7) Mary’s height increased by 20% between her 5th birthday and her 10th birthday. It increased
by 15% between her 10th birthday and her 15th birthday, and it increased by 10% between
her 15th birthday and her 20th birthday. By how much did her height increase between her 5th
birthday and her 20th birthday?
8) (1550 + 1551 + 1552 + .... + 1648 + 1649) – (150 + 151 + 152 + .... + 248 + 249) =
9) A rectangular sheet of paper with sides 1 and 2 has been folded once as shown, so that one
corner just meets the opposite long edge. What is the value of the length d ?
10) When exactly is the value of the value of the product equal (1 + 1/2)(1 + 1/3)(1 + 1/4)....(1 + 1/n) equal
to an integer?
11) The diagram shows a trapezium ABCD, whose base length CD is twice the top length AB.
If CD is twice CE, what fraction does the area of the triangle take of the area of the
12) In a right angled triangle the longest side has length 13cm, and one of the shorter sides has
length 7cm. Which of the following approximations is closest to the length of the remaining
13) ABCD is a trapezium. The length of CE is ¼ the length of CD, and FD = CE. If the area of
each triangle is 1 cm2 what is the area of the trapezium ?
14) In the diagram PQ = PR = RS. What is the size of angle x?
15) Ima Divvy used her calculator and divided a number by 40 instead of 8. What should she do
now to obtain the correct answer ?
16) A pencil AB lying on a table is given a half turn about the end B (so that A moves to A')
and then a half-turn about A' (so that B moves to B'), and finally a half-turn about B' (so
that A' moves to C). The point D of the pencil is three-quarters of the way from A to B.
What is the ratio of the total distance moved by A and D?
17) Which of these numbers is odd ?
18) The sum of six consecutive odd numbers is 60. What is the smallest of the six numbers ?
19) The integer part of a positive number is the part before the decimal point; The decimal part
of a positive number is the part after the decimal point. For example the integer part of 3.72
is ‘3’ and the decimal part is ‘0.72’. Which of the following numbers has a decimal part
equal to exactly one-fifth of the integer part ?
20) If |x| < 0.5, (4 - 8x) - ½ =
21) For each real number x, let [x] be the biggest integer which is less than or equal to x. What
can you say about the following three equations ?
(i) [p + 3] = [p] + 3
(ii) [p + q] = [p] + [q]
(iii) [5p] = 5 [p]
22) If y = xex then dy/dx =
23) If c is a constant, what is (2 − 3x2)2 dx ?
24) Observe that 18 = 42 + 12 + 12 + 02. How many of the first fifteen positive integers can be
written as the sum of the squares of four integers ?
25) In the diagram below the smaller circle touches the larger circle and goes through the centre
of the larger circle. What fraction of the area of the larger circle is outside the smaller circle
26) If y = sin(x2) then dy/dx =
27) (x − 1)(x4 + 1)(x2 + 1)(x + 1) =
28) If x2 - 3x + 1 = 0, what is the value of x2 + (1/x)2 ?
29) The expression ”3 7 → 4 is a short way of writing the statement “it is possible to fit a
3-sided polygon and a 7-sided polygon together (without overlap) to make a 4-sided
polygon”. This statement is correct (as shown in the diagram below). Which of the
following is a statement which is NOT correct ?
30) Consider the triangle of numbers below :
X and Y equal