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UCL Undergraduate Preparatory Certificate for Science & Engineering

Maths - Sample Test



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1) A man sold two cameras for £240 each. One was sold for a profit of 20%, and one for a loss of 20%. What was his overall profit or loss?

A. Loss of £20
B. Loss of £10
C. Made no profit or loss
D. Profit of £10
E. Profit of £20

2) If 1/a = 1/b + 1/c and a and c are both doubled, the value of b will be

A. divided by 4
B. divided by 2
C. unchanged
D. multiplied by 2
E. multiplied by 4

3) Which of the following expressions has the largest value when T = - 0.25?

A. 41/T
B. − 4/T
C. 4T
D. 4/T2
E. 4/(−T)1/2

4) The area of a small circle with diameter x is half the area of a larger circle whose diameter is y. What is the value of y/x ?

A. 1
B. 21/2
C. 21/2 − 1
D. 21/2 + 1
E. (21/2 + 1)/2

5) If x = 2log39 + log275 then 3x =

A. 81(5)1/3
B. 81 + (5)1/3
C. 405/3
D. 135
E. None of A to D is correct

6) Find the perimeter of the figure below, if possible.

A. Cannot be calculated
B. 34cm
C. 36cm
D. 42cm
E. 46cm

7) A box containing three bags of sugar weighs 6.0 kg. When the same box contains five bags of sugar it weighs 9.2 kg. How heavy is the empty box?

A. 1.2 kg
B. 1.6 kg
C. 2.0 kg
D. 3.2 kg
E. Can't be sure

8) If y = cos2x sinx the derivative function, dy/dx is

A. cosx(2sin2x − cos2x)
B. cosx(sin2x − cos2x)
C. − 2cos2xsinx
D. 2cos2xsinx
E. cosx(1 − 3sin2x)

9) The symbol 25! denotes the product of all the whole numbers from 1 up to 25. If the actual value of 25! is calculated, how many zeros will be at the end?

A. 1
B. 3
C. 4
D. 5
E. 6

10) A product in a shop is reduced in price by 20%. At this reduced price the shopkeeper makes only 4% profit. What percentage profit (to the nearest whole percent) does the shopkeeper make at its normal selling price?

A. 16
B. 24
C. 25
D. 30
E. 84

11) A solid sphere with radius r fits exactly inside a cylinder, touching the sides, the top and the bottom. What fraction of the cylinder is empty?

A. 2/3
B. 1/3
C. 1/2
D. 3/4
E. It depends on the value of r.

12) We can write the number 384 as 4 2 4 where the bar denotes a negative digit, so that 424 means 4×100−2×10+4. How could we write 1988 in this way?

A. 2 1 0 2
B. 2 0 0 2
C. 2 1 2 2
D. 2 1 1 2
E. 2 0 1 2

13) 1 + 22 + 333 + 4 444 + 55 555 + 666 666 + 7 777 777 + 88 888 888 + 999 999 999 equals:

A. 1 097 393 685
B. 1 097 393 645
C. 1 097 389 685
D. 1 097 093 685
E. 1 077 393 685

14) Two squares of side 2x overlap to form a regular octagon. How long is each side of the octagon?

A. 2x/3
B. x(2 − 21/2)
C. x
D. (21/2/2)x
E. 2x(21/2 − 1)

15) The area of the small square is one third of the area of the large square. What is the value of x/y ?

A. (31/2 + 1)/2
B. 1/(3)1/2
C. 1/9
D. 31/2
E. 31/2 − 1

16) What is the value of cot (π/4) + cot (π/6) + cot (π/8) ?

A. 2 + 21/2 + 31/2
B. 21/2 −1
C. 1 + 31/2 + 21/2
D. 31/2 + 21/2
E. 1 + 31/2 + 2(2)1/2

17) If 4x −5(2x) = −6, then the value of x is:

A. 1 or log23
B. 1 or log32
C. 1 or −3/2
D. 1 or log42
E. None of A to D is correct

18) If x⊗y = (xy)1/2 then what is the value of (3⊗48)⊗9 ?

A. 6(3)1/2
B. 6(3)1/4
C. 6
D. 36
E. 108

19) The real number k lies between 0 and 1. The area of the quadrilateral formed by the lines y = kx, y = kx + 1, x = ky and x = ky + 1 is

A. 1/(1 −k)
B. k/(1 −k)
C. 1/(1 −k2)
D. k/(1 −k2)
E. k2/(1 −k2)

20) A square has centre (2,1). One vertex is at (5,6). Points P, Q, R have coordinates P := (−3, 4), Q := (−1, −4) and R := (5, −4). Which of the following are vertices of the square?

A. P, Q and R
B. P only
C. P and R only
D. Q only
E. P and Q only

21) In the diagram below, AB, EF and DC are perpendicular to BC. AEC and BED are straight lines. AB = x, EF = h and DC = y. Then h is:

A. xy/(x + y)
B. (x2 + y2)/(2x + 2y)
C. (x3 + y3)/4xy
D. (x + y)/xy
E. Not enough information is provided to solve the problem

22) The number of prime values of the polynomial n3 − 10n2 − 84n + 840 where n is an integer is:

A. 0
B. 1
C. 2
D. 4
E. infinite

23) A man has a square piece of paper where each side has length 1m. Two equal circles are to be cut from this paper. What is the radius, in metres, of the largest possible circles?

A. 1/(2 + 21/2)
B. [4 + 2(2)1/2)/16
C. [2(2)1/2 − 1]/7
D. 21/2/4
E. (1 + 21/2)/8

24) If 2x − 2x −2 = 192, then what is the value of x?

A. 5
B. 6
C. 7
D. 8
E. 9

25) Let

e(x) = cos4x −sin4x
f(x) = 1 − 2sin2x
g(x) = (cosx + sinx)(cosx − sinx)
h(x) = (1 − sinx)(1 + tanx)(1 + sinx)(1 − tanx)

If the values e(π/7), f(π/7), g(π/7) and h(π/7) are calculated, how many different answers will there be?

A. 1 (i.e. all the values are the same)
B. 2
C. 3
D. 4
E. Can't be done without a calculator.

26) A regular hexagon ABCDEF has sides of length 2 cm. M is the midpoint of AB. Which of the following line segments have length 131/2 cm?

A. BD
B. BE
C. EM
D. FM
E. None of these

27) If the number n is a perfect square, what is the next perfect square above it?

A. n + n1/2
B. n + 2(n)1/2 + 1
C. n2 + 1
D. n2 + n
E. n2 + 2n + 1

28) What is the area of the largest equilateral triangle which fits inside a square of side a?

A. (3a4)1/2/4
B. [2(3)1/2 −3]a2
C. (3a4)1/2/2
D. (3a4)1/2
E. [3(3)1/2 −3]a2

29) Triangle ABC is such that its three angles ∠A, ∠B, ∠C are in the ration 3:4:5. What is the ratio of its three sides BC:CA:AB ?

A.
31/2 : 41/2 : 51/2
B.
(1 + 21/2) : 61/2 : (1 + 31/2)
C.
2 : 61/2 : (1 + 31/2)
D.
3 : 4 : 5
E.
61/2 : (1 + 31/2) : (21/2 + 31/2)

30) A driver travels an average of k miles a day for the first d days of a journey and then m miles a day for the next b days. The car uses x litres of petrol per 100 miles. How many litres of petrol does the car use for the whole journey ?

A. x(kd + mb)/100
B. (k + m)x/100
C. 100(k + m)/x
D. 100(kd + mb)/x
E. bdkmx/100