Eton College — King's 13+ Maths Scholarship A 2022 (Interactive)

Q1(a): Fully simplify the following expression: 5y + 16 - 8y - 4

Q1(b): Fully simplify the following expression: (34)y - y/2

Q1(c): Fully simplify the following expression: (y * y * y * y * y) / y^3

Q2(a): Find the value of the following, giving your answer as a reduced, mixed fraction: 115 13 + 62 25 - 71 49

Q2(b): Find the value of the following, giving your answer as a reduced, mixed fraction: (8198 / 3 67) * 1549

Q2(c): Find the value of the following, giving your answer as a reduced, mixed fraction: (3 23 - 2)^2

Q3(a): Find the value of the following, giving your answer as a decimal: 0.018 * 0.0045

Q3(b): Find the value of the following, giving your answer as a decimal: 0.2 - (0.2)^3

Q3(c): Find the value of the following, giving your answer as a decimal: 0.403 / 0.062

Q4(a): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is odd.

Q4(b): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is a multiple of 5.

Q4(c): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is a prime number.

Q4(d): A grid contains the numbers: 2, 15, 60, 12, 47, 9, 51, 19, 34. I choose a number at random from this grid. I am equally likely to choose any of these numbers. Giving your answer as a fully simplified fraction, find the probability that the number I choose is not a multiple of 3.

Q5(a): If p = 6, q = -2 and r = 12, find the value of the following expression, fully simplifying your answer: p^2 - 2q/r

Q5(b): If p = 6, q = -2 and r = 12, find the value of the following expression, fully simplifying your answer: (p^3 + q^3) / (p + q)

Q6: Solve the following simultaneous equations: 5x + 7y = 11 and 4x + 3y = 14. Give the values of x and y.

Q7: Solve the following equation for x. Give your answer as a reduced, mixed fraction: (2x + 1)/4 + (x - 1)/10 = 2

Q8: There are 8 more boys than girls in a football squad of 32. What is the ratio of girls to boys in the squad in simplified form?

Q9: Annie is 44 years old and her father is 36 years older than her. How many years ago was Annie's father three times her age?

Q10: Megan, Flossy and Emma are sisters. Megan is 80% taller than Emma. Flossy is 25% taller than Emma. Find the percentage by which Megan is taller than Flossy.

Q11(a): I visit three stores with some pocket money. I spend one fifth of my pocket money in the first shop and one quarter of what remains in the second shop. What fraction of the original amount is left over to spend in the third shop?

Q11(b): I am choosing a new washing machine. The standard model costs £200 and uses 17 12 pence of electricity per hour in operation. The energy-saving model costs £550 but only uses 3 12 pence of electricity per hour in operation. I operate my washing machine for twenty hours a week, on average. If I buy the energy-saving model, after how many weeks would my reduced electricity costs balance out the additional purchase price?

Q11(c): I run a cake stall on a Friday afternoon and spend 10 hours making 60 cakes. The ingredients for each cake cost me £1.60 and I charge £6 an hour for my labour. The sale price for a cake is 75% more than the total cost of making it. I have a special offer this week which gives 20% discount on my normal sale price. How much will a customer pay for one of my cakes this week?

Q12(a): A triangle ABC is made up of four smaller triangles. EBF and DFC are isosceles triangles. Angle EFD is x degrees and is acute. Calculate the size of angle EAD in terms of x. (refer to original PDF for figure)

Q12(b)(i): A shape is formed by joining an isosceles triangle ABE and a parallelogram BCDE. ABC is a straight line and angle BCD is ((90 - 5y)/4) degrees. Find angle BAE in terms of y, fully simplifying your answer. (refer to original PDF for figure)

Q12(b)(ii): Using the same shape from Q12(b), the line AF intersects BE at X as shown. It is given that DC = 17 cm and the ratio of the lengths AX and XF is 3:2. Given that the area of triangle ABE is 78 cm^2, calculate the area of the parallelogram BCDE. (refer to original PDF for figure)

Q13(a)(i): A triangle OAB is cut out of a circle with centre O. OA and OB are radii of length 35 cm and the chord AB is of length 56 cm. Find the area of the triangle OAB. (refer to original PDF for figure)

Q13(a)(ii): Using 227 as an approximation for pi, calculate an estimate for the shaded area (the circle minus the triangle OAB), to the nearest 10 square centimetres. The circle has centre O and radius 35 cm. (refer to original PDF for figure)

Q13(b)(i): Pentagon ABCDE is irregular but all five sides are each 34 cm long. Diagonal AC is 66 cm and is parallel to side ED. What is the value of x, where the area of triangle ABC is 33*sqrt(x) cm^2 and x is a positive whole number less than 100?

Q13(b)(ii): Pentagon ABCDE has all five sides of length 34 cm. Diagonal AC = 66 cm and AC is parallel to ED. What is the area of quadrilateral ACDE in square centimetres? Show that it is a whole number.

Q13(b)(iii): By making a suitable estimate for sqrt(x) from part (i), find the area of pentagon ABCDE correct to the nearest 10 square centimetres.

Q14(a): If m = (7 + n)/2 and n = (1 + m)/2, find the value of (m + n)/2.

Q14(b)(i): x, y and z are positive numbers such that x * y = 6, y * z = 27, and z * x = 2. Find the value of (xyz)^2.

Q14(b)(ii): x, y and z are positive numbers such that x * y = 6, y * z = 27, and z * x = 2. Find the values of x, y and z.

Q14(c): Find the values of u, v and w given: u + v = -3, v + w = -2, w + u = 6.

Q15(a): X is a three-digit whole number. The sum of its digits is 12. If the second and third digits (the tens digit and the units digit) are switched, the resulting number is 45 more than X. Find all possible values for X. [Hint: let the letters a, b and c represent the three digits of X and start by finding two equations involving a, b and c.]

Q15(b)(i): The numbers 1327231 and 394493 are both palindromic: they read the same when the order of their digits is reversed. Find the largest five-digit palindromic whole number which is divisible by 3 but not divisible by 9.

Q15(b)(ii): Of all possible five-digit palindromic whole numbers divisible by 15, Y is the largest and Z is the smallest. Find the difference between Y and Z.