# UCL University Preparatory Certificate for Science & Engineering

## Maths Sample Test

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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 ?

 A. \$0 B. \$25 C. \$50 D. \$75 E. \$225

2) In the diagram PQ = PR, ∠QMP = 80° and ∠QPM = 30°. What is ∠RPM ?

 A. 20o B. 35o C. 30o D. 10o E. 15o

3) What is the value of 21 x 101 – 21 – 101 ?

 A. 2001 B. 2000 C. 1999 D. 1998 E. none of these

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 ?

 A. 3 B. 6 C. 8 D. 12 E. 18

5) What is the sum of all the first 20 even numbers ?

 A. 420 B. 105 C. 110 D. 210 E. 410

6) equals

 A. 2 B. 1/2 C. 1/16 D. 1/ √2 E. √2

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?

 A. 45% B. 50.8% C. 38.2% D. 51% E. none of these.

8) (1550 + 1551 + 1552 + .... + 1648 + 1649) – (150 + 151 + 152 + .... + 248 + 249) =

 A. 14000 B. 1401400 C. 16000 D. 140000 E. 160000

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 ?

 A. 1/2 B. √2 −1 C. 7/16 D. √3− √2 E. √2/3

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?

 A. when n is odd B. when n is even C. when n is a multiple of 3 D. always E. never

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 trapezium?

 A. 1/9 B. 2/9 C. 2/(3√3 ) D. 1/(6√3 ) E. 1/3

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 side ?

 A. 10.0cm B. 10.5cm C. 11cm D. 11.5cm E. 12cm

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 ?

 A. 6cm2 B. 4cm2 C. 5cm2 D. 3cm2 E. 8cm2

14) In the diagram PQ = PR = RS. What is the size of angle x?

 A. 600 B. 1080 C. 900 D. 800 E. 200

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 ?

 A. divide by 40 B. divide by 8 C. multiply by 5 D. multiply by 320 E. divide by 5

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?

 A. 3 : 4 B. 4 : 2 C. 4 : 3 D. 2 : 3 E. 8 : 5

17) Which of these numbers is odd ?

 A. 14 + 1 B. 34 + 2 C. 54 + 3 D. 74 + 5 E. 114 + 7

18) The sum of six consecutive odd numbers is 60. What is the smallest of the six numbers ?

 A. 1 B. 2 C. 3 D. 4 E. 5

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 ?

 A. 2.5 B. 3.6 C. 4.6 D. 5.8 E. 6.2

20) If |x| < 0.5, (4 - 8x) - ½ =

 A. ½ (1 + x + 3x2/2 + ...) B. ½ (1 – x + 3x2/2 + ...) C. ½ (1 + x – 3x2/2 + ...) D. 2 (1 + x + 3x2/2 + ...) E. ½ (1 + x + 3x2/2)

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]

 A. all three equations are true for all real numbers p and q. B. equations (i) and (iii) are true for all real number p. C. the three equations are true only when p and q are integers. D. equations (i) and (ii) are true for all real number p and q. E. equations (ii) and (iii) are false for most values of p and q.

22) If y = xex then dy/dx =

 A. xex B. x + ex C. x + xex D. ex + xex E. None of these

23) If c is a constant, what is (2 − 3x2)2 dx ?

 A. - 24x2 + 36x3 + c B. 4x − 12x3 + 9x5 + c C. 4 − 4x2 + 9x4/4 + c D. 4x − 4x3 + 9x5/5 + c E. None of these

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 ?

 A. 15 B. 14 C. 13 D. 12 E. 11

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 ?

 A. 2/3 B. 3/4 C. 4/5 D. 5/6 E. 7/8

26) If y = sin(x2) then dy/dx =

 A. cos(x2) B. 2x.sin(x2) C. 2sin(x) D. 2x.cos(x) E. 2x.cos(x2)

27) (x − 1)(x4 + 1)(x2 + 1)(x + 1) =

 A. x8 + 1 B. x8 + x6 + x4 + x2 + 1 C. x8 - 1 D. x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1 E. x8 - x6 + x4 - 1

28) If x2 - 3x + 1 = 0, what is the value of x2 + (1/x)2 ?

 A. 7 B. (7 −3√5 )/2 C. 9 D. (7 +3√5 )/2 E. 10

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 ?

 A. 3 5 → 4 B. 3 6 → 4 C. 3 8 → 4 D. 4 6 → 4 E. 4 8 → 4

30) Consider the triangle of numbers below :

X and Y equal

 A. 10 and 32 B. 5 and 19 C. 6 and 13 D. 4 and 19 E. 6 and 19