UCL CENTRE FOR LANGUAGES
& INTERNATIONAL EDUCATION (CLIE)

UPC Placement Tests

Test: Maths 1



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1.

A sequence begins 1, 3, 4, 7, 11, ... 

Each term (after the first two terms) is equal to the sum of the two previous terms.

How many of the following statements are true?

  1. The 20th term is divisible by 2
  2. The 30th term is divisible by 3
  3. The 40th term is divisible by 4
  4. The 50th term is divisible by 5

0

1

2

3

4

2.

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

divided by 4

divided by 2

unchanged

multiplied by 2

multiplied by 4

3.

Let b be a real number such that b2=b3+1. Which of the following is false?

b3-b4=b

b2=b4+b+1

b4+b3+b2=1

b3+b+1=b3+b2+1+1/b

b3+b2+1=2b3+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 ?

1

21/2

21/2 − 1

21/2 + 1

(21/2 + 1)/2

5.

If x = 2logt(t2) + log(t3)(5) then tx =

t4(5)1/3

t4 + (5)1/3

  t4+(logt5)/3

(t4logt5)/3

None of A to D is correct

6.

Find the perimeter of the figure below, if possible.


Cannot be calculated

34cm

36cm

42cm

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?

1.2 kg

1.6 kg

2.0 kg

3.2 kg

Can't be sure

8.

If x9+512=(x+2)(a8x8+a7x7+...+a1x+a0) then what is the value of a8+a7+...+a1+a0?

1

19

57

17

171

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?

1

3

4

5

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?

16

24

25

30

84

11.

A boy walks at 4 km per hour and runs at 6 km per hour.

If he runs to school instead of walking, then he saves 3 minutes and 45 seconds.

How far does he live from school?

0.125 km

0.375 km

0.75 km

6.9 km

7.5 km

12.

We can write the number 384 as 4(2)4 where the brackets denotes a negative digit, so that 4(2)4 means 4×100−2×10+4.

How could we write 1988 in this way?

2 (1) 0 (2)

2 0 0 (2)

2 (1) 2 (2)

2 (1) (1) (2)

2 0 (1) (2)

13.

How many pairs of integers (x,y) satisfy either of the following two equations:

  • 2x+2y=2(x+y)
  • 33y=3xy

0

1

2

3

infinite

14.

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

2x/3

x(2 − 21/2)

x

(21/2/2)x

2x(21/2 − 1)

15.

In the diagram below there are two squares. The smaller square has sides of length x. The larger square has sides of length x+y.

The area of the small square is one third of the area of the large square.

What is the value of x/y ?

(31/2 + 1)/2

1/(3)1/2

1/9

31/2

31/2 − 1

16.

Let cot(θ)=a.

What is the value of cot(θ)+cot(2θ)?

(3a2-1)/(2a)

(1+3a2)/(2a)

(a2+2a -1)/(2a)

(a+1)2/(2a)

(a2+1)/(2a)

17.

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

1 or log23

1 or log32

1 or −3/2

1 or log42

None of A to D is correct

18.

Let y=(0.2x)/(x+1). If x takes all values in the range [1,∞) then what range of values does y take?

[0.1, 0.2)

(0, 0.1]

(0, 0.2)

[0, 0.1]

[0.1, 0.2]

19.

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

1/(1 −k)

k/(1 −k)

1/(1 −k2)

k/(1 −k2)

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?

P, Q and R

P only

P and R only

Q only

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:

xy/(x + y)

(x+ y2)/(2x + 2y)

(x3 + y3)/4xy

(x + y)/xy

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

0

1

2

4

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?

1/(2 + 21/2)

[4 + 2(2)1/2)/16

[2(2)1/2 − 1]/7

21/2/4

(1 + 21/2)/8

24.

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

5

6

7

8

9

25.

Let d(x)=cos4x −sin4x.


How many of the functions below are equal to d(x) for all values of x?


e(x) = cos(2x)


f(x) = 1 − 2sin2x


g(x) = (cosx + sinx)(cosx − sinx)


h(x) = (1 − sinx)(1 + tanx)(1 + sinx)(1 − tanx)

0

1

2

3

4

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?

BD

BE

EM

FM

None of these

27.

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

n + n1/2

n + 2(n)1/2 + 1

n+ 1

nn

n2 + 2n + 1

28.

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

(3a4)1/2/4

[2(3)1/2 −3]a2

(3a4)1/2/2

(3a4)1/2

[3(3)1/2 −3]a2

29.

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

31/2 : 41/2 : 51/2

(1 + 21/2) : 61/2 : (1 + 31/2)

2 : 61/2 : (1 + 31/2)

3 : 4 : 5

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 ?

x(kd + mb)/100

(k + m)x/100

100(k + m)/x

100(kd + mb)/x

bdkmx/100