UCL CENTRE FOR LANGUAGES
& INTERNATIONAL EDUCATION (CLIE)

UPC Practice Entry 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? The 20th term is divisible by 2 The 30th term is divisible by 3 The 40th term is divisible by 4 The 50th term is divisible by 5

	0
	1
	2
	3
	4
Try again. Look for a pattern in when the n^th term of the sequence is odd/even.

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
Try again. Multiply the equation by ½.

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
Try again. By using the equation b³=b²-1 you can reduce the powers of the terms in the equation.

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
	2^1/2
	2^1/2 − 1
	2^1/2 + 1
	(2^1/2 + 1)/2
Try again. Write down an equation relating the area of the small circle to the area of the large circle, and then solve it for y/x.

5.	If x = 2log_t(t²) + log_(t³)(5) then t^x =

	t⁴(5)^1/3
	t⁴ + (5)^1/3
	t⁴+(log_t5)/3
	(t⁴log_t5)/3
	None of A to D is correct
Try again. t^x can be written as (t^2log_t(t))(t^log_t(5)/3). Now use properties of logarithms.

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

	Cannot be calculated
	34cm
	36cm
	42cm
	46cm
Try again. Extend one of the lines to form an equilateral triangle with sides of length 12cm adjacent to a parallelogram.

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
Try again. Let the mass of a bag of sugar be x and the mass of the box be y. Now write down 2 equations both using x and y.

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
Try again. Substitute x with a number.

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
Try again. Write down all the integers from 1 to 25, and group together any numbers that multiply to 10, 20, 30, etc or 100. Each such group will add another zero.

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
Try again. Let x be the price that the shopkeeper paid for the product and let y be the amount that he sold it for. Set up an equation relating x and y and then solve for y/x.

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
Try again. If D is the distance to school then write an equation in terms of D and the running/walking speed, then solve the equation.

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)
Try again. Work through the possible answers to find the correct one.

13.	How many pairs of integers (x,y) satisfy either of the following two equations: 2^x+2^y=2^(x+y) 3^x3^y=3^xy

	0
	1
	2
	3
	infinite
Try again. You need x and y that satisfy x+y=xy or 3^x+3^y=3^x3^y. Solve for x and 2^xand consider what values are possible if both x and y must be integers.

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

	2x/3
	x(2 − 2^1/2)
	x
	(2^1/2/2)x
	2x(2^1/2 − 1)
Try again. If the middle piece of each side of the square is y, we get a right-angled triangle all of whose sides can be expressed in terms of x and y. Then apply Pythagoras's Theorem.

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 ?

	(3^1/2+ 1)/2
	1/(3)^1/2
	1/9
	3^1/2
	3^1/2 − 1
Try again. Express the area of the small square in terms of the area of the large square.

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)
Try again. Write cot(2θ) and in terms of tan(2θ) and use the double angle formula for tangent.

17.	If a is a constant and 4^x −5(2^x) = a is true for at least one real number x then what can be concluded about the number a?

	a ≥ -25/2
	a ≥ -25/4
	a ≤ -25/2
	a ≤ -25/4
	-25/2 ≤ a ≤ 25/4
Try again. If 4^x is written 2^2x then the equation becomes a quadratic equation in 2^x . Consider the discriminant.

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]
Try again. Consider if this is a monotonically increasing or monotonically decreasing function.

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

	1/(1 −k)
	k/(1 −k)
	1/(1 −k²)
	k/(1 −k²)
	k²/(1 −k²)
Try again. The region is a parallelogram

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
Try again. If you find the equation of the line through the vertex and centre, and then the equation of the normal line through the centre, the points can be checked.

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)
	(x²+ y²)/(2x + 2y)
	(x³ + y³⁾/4xy
	(x + y)/xy
	Not enough information is provided to solve the problem
Try again. ABC and EFC are similar triangles, as are DCB and EFB.

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

	0
	1
	2
	4
	infinite
Try again. Factorise the expression. If the value is prime then one of the factors must be 1 or -1.

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 + 2^1/2)
	[4 + 2(2)^1/2)/16
	[2(2)^1/2− 1]/7
	2^1/2/4
	(1 + 2^1/2)/8
Try again. The circles must lie along the diagonal of the paper. The distance from the centre of one circle to the nearby corner can be calculated in terms of r, and hence the length of the diagonal can be calculated in terms of r.

24.	If a>1 and a^x− a^x ⁻² = 19, then what is the value of x?

	log_a(1+365^0.5)-log_a(2)
	log_a(a²+19)-log_a(365^0.5)
	log_a(a²-1)-log_a(19)-2
	log_a(a-1)-log_a(19)-2
	log_a(19)+2-log_a(a²-1)
Try again. Use rules of exponents and logs.

25.	Let d(x)=cos⁴x −sin⁴x How many of the functions below are equal to d(x) for all x∈(0,π/2)? 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
Try again. Trig identities and factorization show that these functions are all the same.

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 13^1/2 cm?

	BD
	BE
	EM
	FM
	None of these
Try again. The suggested lengths can all be calculated. The cosine rule is helpful.

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

	n + n^1/2
	n + 2(n)^1/2 + 1
	n²+ 1
	n²+ n
	n² + 2n + 1
Try again. If n is k², then the next perfect square will be (k + 1)²

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
Try again. One corner of the triangle must be in a corner of the square. Each angle outside the triangle at this corner is 15 degrees.

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 ?

	3^1/2 : 4^1/2 : 5^1/2
	(1 + 2^1/2) : 6^1/2: (1 + 3^1/2)
	2 : 6^1/2 : (1 + 3^1/2)
	3 : 4 : 5
	6^1/2 : (1 + 3^1/2) : (2^1/2 + 3^1/2)
Try again. The sine rule can be useful here

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
Try again. Calculate the number of miles travelled, divide by 100 and multiply by x.