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CE.M40 Maths: Scholarship Challenge Paper 2

ISEB COMMON ENTRANCE CE.M40 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Time allowed: 60 minutes. Answer ALL 18 questions. Write your answers in the spaces provided. You ARE allowed to use a calculator. You must show all your working; marks are available for a correct method as well as for the final answer, and method marks are awarded generously throughout. The number of marks for each question, or part-question, is shown in brackets. This is the second Scholarship Challenge Paper, harder again than the first: expect demanding multi-step problem solving, proof-style algebraic and geometric reasoning with justification, and harder number and algebra work throughout. Topics include: successive percentage change, algebraic fractions, isosceles triangle angle algebra, ratio and proportion, simultaneous equations formed from a word problem, composite area (rectangle and semicircle), algebraic proof (multiples of 6), inverse proportion, quadratic sequences, right-angled trigonometry combined with Pythagoras, forming and solving a quadratic equation from a word problem, indices and standard form, conditional probability without replacement, composite volume (cylinder minus cone), combined inequalities with integer solutions, angle reasoning with parallel lines and justification, a multi-stage time and speed problem, and a formal divisibility proof.

A cafe increases the price of a flat white from £2.80 by 15%. A month later, the new price is reduced by 10%. Work out the final price of the flat white, giving your answer to the nearest penny. (3 marks)

(Total for Question 1 is 3 marks)

Solve the equation (2x - 1)/3 - (x + 2)/4 = 1, giving your answer as a decimal. (4 marks)

(Total for Question 2 is 4 marks)

Triangle ABC is isosceles, with AB = AC. Angle BAC = (2x + 10) degrees, and angle ABC = angle ACB = (3x - 5) degrees each. Find the value of x, and hence find the size of angle BAC. (4 marks)

Figure (to be drawn): Triangle ABC (not drawn to scale) with AB = AC marked as equal sides, angle BAC labelled at the top vertex A, and the two base angles at B and C marked as equal.

(Total for Question 3 is 4 marks)

Amara, Ben and Chidi share a sum of money in the ratio 2:3:5. Ben receives £18 more than Amara. Work out the total sum of money shared. (4 marks)

(Total for Question 4 is 4 marks)

On Monday, a cinema sells 3 adult tickets and 2 child tickets for a total of £32.50. On Tuesday, it sells 2 adult tickets and 5 child tickets for a total of £40. By forming and solving a pair of simultaneous equations, find the cost of one adult ticket and the cost of one child ticket. (5 marks)

(Total for Question 5 is 5 marks)

A garden is in the shape of a rectangle 14 m long and 9 m wide, with a semicircular flower bed of diameter 9 m attached along one of the short (9 m) sides. Calculate the total area of the garden including the flower bed, giving your answer to 3 significant figures. (5 marks)

Figure (to be drawn): A garden shape: a rectangle 14 m by 9 m with a semicircle of diameter 9 m attached along one of the short (9 m) sides. Not drawn to scale.

(Total for Question 6 is 5 marks)

Prove that the sum of any three consecutive even numbers is always a multiple of 6. (4 marks)

(Total for Question 7 is 4 marks)

The time, T hours, taken to fill a swimming pool varies inversely with the number of pumps, p, used. Using 3 pumps, it takes 8 hours to fill the pool.

(a)Find a formula for T in terms of p. (2 marks)(2)

(b)Find the time taken to fill the pool using 5 pumps, giving your answer in hours and minutes. (3 marks)(3)

(Total for Question 8 is 5 marks)

The nth term of a sequence is given by the expression n² + 2n - 5.

(a)Find the 6th term of the sequence. (2 marks)(2)

(b)Show that the difference between the (n+1)th term and the nth term is 2n + 3. (3 marks)(3)

(Total for Question 9 is 5 marks)

A vertical flagpole AB stands on horizontal ground. Point C is on the ground, 15 m from the base B of the flagpole, with the angle of elevation of the top A of the flagpole from C equal to 38 degrees.

Figure (to be drawn): A vertical flagpole AB stands on horizontal ground. Point C is on the ground 15 m from B, with the angle of elevation of A from C marked as 38 degrees. Point D is on the ground on the opposite side of B from C, with BD = 20 m and angle ABD = 90 degrees. Not drawn to scale.

(a)Calculate the height of the flagpole, giving your answer to 1 decimal place. (2 marks)(2)

(b)Point D is on the ground on the opposite side of B from C, with BD = 20 m and angle ABD = 90 degrees. Calculate the straight-line distance AD, giving your answer to 1 decimal place. (3 marks)(3)

(Total for Question 10 is 5 marks)

A rectangular lawn is 3 m longer than it is wide. The area of the lawn is 88 m^2. Form an equation in terms of the width w, and solve it to find the width of the lawn (w > 0). (5 marks)

(Total for Question 11 is 5 marks)

Simplify and evaluate the following.

(a)Simplify (3 x 10⁴) x (5 x 10^-2), giving your answer in standard form. (2 marks)(2)

(b)Solve 2^x = 128 to find the value of x. (2 marks)(2)

(Total for Question 12 is 4 marks)

A bag contains 5 red counters and 7 blue counters. Two counters are drawn at random from the bag, one after the other, without replacement.

(a)Find the probability that both counters are red. (2 marks)(2)

(b)Find the probability that the two counters are different colours. (3 marks)(3)

(Total for Question 13 is 5 marks)

A solid is made from a cylinder of radius 4 cm and height 10 cm, with a cone of the same radius and a height of 6 cm removed from one flat end; the cone's circular base coincides with one circular face of the cylinder, with its apex pointing inward along the axis.

Figure (to be drawn): A solid cylinder of radius 4 cm and height 10 cm with a cone of the same radius and a height of 6 cm removed from one flat end, apex pointing into the cylinder along its axis. Not drawn to scale.

(a)Calculate the volume of the cylinder. (2 marks)(2)

(b)Calculate the volume of the cone removed. (2 marks)(2)

(c)Hence find the volume of the remaining solid, giving your answer to 3 significant figures. (2 marks)(2)

(Total for Question 14 is 6 marks)

A number n is such that 3(n - 2) < 2n + 7, and also n > 9.

(a)Solve the inequality 3(n - 2) < 2n + 7 to find the range of possible values of n. (3 marks)(3)

(b)Given also that n > 9, and that n is an integer, list all possible values of n. (2 marks)(2)

(Total for Question 15 is 5 marks)

In the diagram, lines PQ and RS are parallel. A straight transversal line crosses PQ at point A and RS at point B. The angle QAB is (3x) degrees, and the angle SBA, on the same side of the transversal, is (x + 40) degrees.

Figure (to be drawn): Two parallel lines PQ and RS are crossed by a straight transversal through points A (on PQ) and B (on RS). The angle QAB is marked 3x degrees at A, and the angle SBA is marked (x + 40) degrees at B, on the same side of the transversal. Not drawn to scale.

(a)State the relationship between angle QAB and angle SBA, giving a reason. (1 mark)(1)

(b)Form and solve an equation to find the value of x. (3 marks)(3)

(c)Hence find the size of angle QAB, and state whether it is acute or obtuse. (1 mark)(1)

(Total for Question 16 is 5 marks)

Priya cycles from her home to the library, a distance of 9 km, at an average speed of 15 km/h. She stays at the library for 40 minutes, then cycles back home along the same route at an average speed of 12 km/h.

(a)Calculate the time, in minutes, that Priya takes to cycle to the library. (2 marks)(2)

(b)Calculate the time, in minutes, that Priya takes to cycle back home. (2 marks)(2)

(c)Priya leaves home at 09:52. Work out the time she arrives back home. (2 marks)(2)

(Total for Question 17 is 6 marks)

Prove that n³ - n is divisible by 6 for every positive integer n. (7 marks)

(Total for Question 18 is 7 marks)

Mark scheme · CE.M40 Maths: Scholarship Challenge Paper 2

Question 1

M1 2.80 x 1.15 (= 3.22) oe
M1 'their 3.22' x 0.9 (= 2.898) oe
A1 £2.90 cao
Answer: £2.90

Question 2

M1 multiplies every term by 12, the LCM of 3 and 4
M1 correctly clears both fractions: 4(2x-1) - 3(x+2) = 12
M1 expands and simplifies to 5x - 10 = 12
A1 x = 4.4 oe (= 22/5) cao
Answer: x = 4.4

Question 3

M1 angle sum of a triangle: (2x+10) + (3x-5) + (3x-5) = 180 oe
M1 simplifies correctly to 8x = 180
A1 x = 22.5
A1 angle BAC = 2(22.5) + 10 = 55 degrees, ft their x
Answer: x = 22.5, angle BAC = 55 degrees

Question 4

M1 recognises Ben's share minus Amara's share = 1 part (3 - 2) oe
M1 1 part = £18
M1 total number of parts = 2 + 3 + 5 = 10
A1 £180 cao
Answer: £180

Question 5

M1 forms two correct equations: 3a + 2c = 32.5 and 2a + 5c = 40, where a and c are the ticket prices
M1 attempts to eliminate one variable using consistent scaling of both equations
M1 correct elimination working, e.g. 15a + 10c = 162.5 and 4a + 10c = 80, leading to 11a = 82.5
A1 a = 7.5 (adult ticket £7.50)
A1 c = 5 (child ticket £5.00), ft their a
Answer: Adult ticket = £7.50, child ticket = £5.00

Question 6

M1 area of rectangle = 14 x 9 (= 126) oe
M1 radius of semicircle = 4.5 (half of diameter 9)
M1 area of semicircle = 0.5 x π x 4.5² (= 31.8...) oe
M1 adds 126 + 'their semicircle area'
A1 158 m² (3 sf) awrt
Answer: 158 m² (3 s.f.)

Question 7

M1 lets the three consecutive even numbers be 2n, 2n+2 and 2n+4, for integer n
M1 sums to 2n + (2n+2) + (2n+4) = 6n + 6
A1 = 6(n+1)
B1 since n+1 is an integer, 6(n+1) is a multiple of 6 for every integer n, so the sum of any three consecutive even numbers is always a multiple of 6 (conclusion stated, dependent on correct algebra)
Answer: 6n + 6 = 6(n+1), a multiple of 6 for all integers n

Question 8

(a) M1 T = k/p oe, with k found from 8 = k/3
(a) A1 T = 24/p cao
(a) Answer: T = 24/p
(b) M1 substitutes p = 5 into their formula T = 24/p
(b) M1 T = 4.8 hours oe
(b) A1 4 hours 48 minutes cao
(b) Answer: 4 hours 48 minutes

Question 9

(a) M1 substitutes n = 6: 6² + 2(6) - 5 (= 36+12-5)
(a) A1 43 cao
(a) Answer: 43
(b) M1 expands (n+1)² + 2(n+1) - 5 = n² + 4n - 2 oe
(b) M1 subtracts n² + 2n - 5 correctly from their expanded expression
(b) A1 = 2n + 3 (shown), as required
(b) Answer: 2n + 3 (shown)

Question 10

(a) M1 AB = 15 x tan(38 degrees) oe
(a) A1 11.7 m (1 dp) awrt
(a) Answer: 11.7 m
(b) M1 recognises angle ABD = 90 degrees and sets up Pythagoras: AD² = AB² + BD²
(b) M1 AD² = 11.719² + 20² (using unrounded or ft height from part a) (= 537.3...)
(b) A1 23.2 m (1 dp) awrt, ft their height from part a
(b) Answer: 23.2 m

Question 11

M1 forms the equation w(w+3) = 88, i.e. w² + 3w - 88 = 0
M1 attempts to factorise or uses the quadratic formula correctly
A1 (w+11)(w-8) = 0 oe, or correct substitution into the quadratic formula
A1 w = 8 or w = -11
B1 w = 8 selected, rejecting the negative solution since w must be positive, with reason stated
Answer: w = 8 m

Question 12

(a) M1 3 x 5 = 15 and 10⁴ x 10^-2 = 10² oe
(a) A1 1.5 x 10³ cao (must be given in standard form)
(a) Answer: 1.5 x 10³
(b) M1 recognises 128 = 2⁷ oe
(b) A1 x = 7 cao
(b) Answer: x = 7

Question 13

(a) M1 5/12 x 4/11 oe
(a) A1 5/33 cao oe
(a) Answer: 5/33
(b) M1 P(red then blue) = 5/12 x 7/11 and P(blue then red) = 7/12 x 5/11 oe
(b) M1 sums the two probabilities: 35/132 + 35/132 = 70/132
(b) A1 35/66 cao oe (allow the equivalent unsimplified 70/132)
(b) Answer: 35/66

Question 14

(a) M1 π x 4² x 10 oe
(a) A1 160pi cm³ (= 502.7 cm³ to 1 dp) oe
(a) Answer: 160 π cm³ (approx 502.7 cm³)
(b) M1 (1/3) x π x 4² x 6 oe
(b) A1 32pi cm³ (= 100.5 cm³ to 1 dp) oe
(b) Answer: 32 π cm³ (approx 100.5 cm³)
(c) M1 160pi - 32pi = 128pi oe, i.e. subtracts the cone volume from the cylinder volume (ft their a and b)
(c) A1 402 cm³ (3 sf) awrt
(c) Answer: 402 cm³ (3 s.f.)

Question 15

(a) M1 expands: 3n - 6 < 2n + 7
(a) M1 rearranges correctly to isolate n
(a) A1 n < 13 cao
(a) Answer: n < 13
(b) M1 combines n < 13 (ft from part a) with n > 9 to give 9 < n < 13
(b) A1 n = 10, 11, 12 (all three values, cao)
(b) Answer: n = 10, 11, 12

Question 16

(a) B1 angle QAB and angle SBA are co-interior (allied) angles and sum to 180 degrees, since PQ is parallel to RS (reason required for the mark)
(a) Answer: angle QAB + angle SBA = 180 degrees (co-interior angles, PQ parallel to RS)
(b) M1 3x + (x + 40) = 180 oe
(b) M1 simplifies correctly to 4x + 40 = 180
(b) A1 x = 35 cao
(b) Answer: x = 35
(c) B1 angle QAB = 3(35) = 105 degrees, which is obtuse since it is greater than 90 degrees, ft their x
(c) Answer: angle QAB = 105 degrees (obtuse)

Question 17

(a) M1 9/15 oe
(a) A1 36 minutes cao (0.6 hours)
(a) Answer: 36 minutes
(b) M1 9/12 oe
(b) A1 45 minutes cao (0.75 hours)
(b) Answer: 45 minutes
(c) M1 total time = 36 + 40 + 45 = 121 minutes oe, ft their (a) and (b)
(c) A1 11:53 cao, ft
(c) Answer: 11:53

Question 18

M1 factorises n³ - n = n(n² - 1) oe
M1 factorises further using difference of two squares: n² - 1 = (n-1)(n+1), so n³-n = (n-1)n(n+1)
B1 recognises that (n-1), n and (n+1) are three consecutive integers
B1 among any two consecutive integers, one is even, so the product (n-1)n(n+1) is divisible by 2
B1 among any three consecutive integers, exactly one is a multiple of 3, so the product is divisible by 3
M1 since the product is divisible by both 2 and 3, which are coprime, it is divisible by 2 x 3 = 6
A1 conclusion clearly stated: hence n³ - n is divisible by 6 for every positive integer n
Answer: n³ - n = (n-1)n(n+1), a product of three consecutive integers, which is always divisible by 6

Maths: Scholarship Challenge Paper 2 - Worksheets, Questions and Revision