1 (a) On the Venn diagrams below, shade the region indicated. A B C A B C ( ) A B C + , ( ) ' A B C , + [2] (b) On the Venn diagram below, draw sets P, Q and R such that P R 1 , Q R 1 and P Q + Q = .  [2]

1(a) E B1 E B1 1(b) E R P Q B2 B1 for ⊂ P R and ⊂ Q R B1 for ∩ = ∅ P Q

1 (a) On the Venn diagrams below, shade the region indicated. A A C B B C A B C + + l l l ( ) A B C , + [2] (b)  { : } x x 0 360 ° ° G G = { : . } cos P x x 2 0 5 = = { : . } sin Q x x 0 5 = = Find P Q + . [3]

1(a) E B1 E B1 1(b) o o o o {30 ,150 , 210 , 330 } P = B1 May be seen or implied in a Venn diagram Allow without set notation o o {30 ,150 } Q = B1 May be seen or implied in a Venn diagram Allow without set notation o o {30 ,150 } P Q ∩ = B1 Dep on both previous B marks Must be in set notation (a) To earn marks, you must correctly shade the specified regions on the Venn diagrams. (b) You need to correctly identify the sets P and Q by finding the x-values that satisfy each trigonometric equation, then find their intersection, ensuring the final answer is in set notation.

1 In a group of 145 students, the numbers studying mathematics, physics and chemistry are given below. All students study at least one of the three subjects. x students study all 3 subjects 24 students study both mathematics and chemistry 23 students study both physics and chemistry 28 students study both mathematics and physics 50 students study chemistry 75 students study physics 80 students study mathematics (i) Using the Venn diagram, find the value of x. [4] Mathematics Physics Chemistry (ii) Find the number of students who study mathematics only. [1]

1(i) M1 for a Venn diagram showing at least 4 correct ‘parts’ in terms of x A1 for all 7 ‘parts’ correct in terms of x on a Venn diagram or in working. May be implied by a correct equation. 80 24 23 3 145 + + + − + + = x x x 50 28 28 24 145 + + + − + + = x x x 75 28 24 3 145 + + + − + + = x x x ( ) 50 80 75 23 28 24 145 + + − + + + = x or equivalents M1 for forming an equation in x using sum of ‘parts’ = 145 or ( ) 50 80 75 23 28 24 145 + + − + + + = x Equations must be seen 15 = x A1 from correct working only 1(ii) 43 B1ft for their x plus 28

4 It is given that . A 5 4 2 1 = - e o (i) Find A-1. [2] (ii) Hence find, in radians, the acute angles x and y such that , . tan tan tan tan x y x y 5 2 12 4 7 + = - = [5]

4(i) 1 2 1 4 5 13 − −   −   −   oe B2 B1 for 1 13 − B1 for 1 2 4 5 − −     −   4(ii) 1 2 12 1 4 5 7 13       −    26 1 13 13   =     M1 for pre-multiplication by their inverse from (i) M1 for correct method for matrix multiplication 2 1  =   A1 1.11 = x B1 π 4 = y or 0.785 B1 (i) To get marks, you must correctly find the determinant of matrix A and then use it to find the inverse matrix, ensuring all elements are correct. (ii) You need to pre-multiply the given column matrix by your inverse matrix from part (i) and correctly perform the matrix multiplication. Then, solve the resulting equations for tan x and tan y, and finally find the acute angles x and y in radians.

8 (a) Five teams took part in a competition in which each team played each of the other 4 teams. The following table represents the results after all the matches had been played. Team Won Drawn Lost A 2 1 1 B 1 3 0 C 1 1 2 D 0 1 3 E 3 0 1 Points in the competition were awarded to the teams as follows 4 for each match won, 2 for each match drawn, 0 for each match lost. (i) Write down two matrices whose product under matrix multiplication will give the total number of points awarded to each team. [2] (ii) Evaluate the matrix product from part (i) and hence state which team was awarded the most points. [2] (b) It is given that A 1 2 1 4 = - c m and B 5 1 0 2 = - c m. (i) Find A-1. [2] (ii) Hence find the matrix C such that AC = B. [3]

8(a)(i) Either 2 1 1 4 1 3 0 2 1 1 2 0 0 1 3 3 0 1                  or 2 1 1 3 4 1 1 2 0 1 3 0                 B2 For correct matrices in correct order or B1 if one correct matrix and a slip in one element of the other matrix Or ( ) 2 1 1 0 3 4 2 0 1 3 1 1 0 1 0 2 3 1           or ( ) 2 1 1 0 3 4 2 1 3 1 1 0       B2 For correct matrices in correct order or B1 if one correct matrix and a slip in one element of the other matrix 8(a)(ii) 10 10 6 2 12                 or ( ) 10 10 6 2 12 Team E M1 For matrix multiplication of their (i), with at least 2 elements correct, must be in correct form, may be unsimplified A1 All correct and identifying team E 8(b)(i) 4 1 1 2 1 6     −   B2 B1 for 1 6 and B1 for 4 1 2 1     −   8(b)(ii) C = A-1B M1 For pre-multiplication by their inverse from (i) C 4 1 5 0 1 2 1 1 2 6    =    − −    M1 Dep for matrix multiplication, using their inverse from (i), at least 2 elements correct 21 2 1 9 2 6 −   =   − −   oe A1

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Sets & Matrices (2016-2019 syllabus only) past paper questions

Revise O-Level Additional Mathematics (4037) — Sets & Matrices (2016-2019 syllabus only) with real Cambridge past-paper questions grouped by topic, each shown with its marking scheme. Working one topic at a time targets exactly the skill each question tests.

33 free questions from 2016–2019, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.

Sets & Matrices (2016-2019 syllabus only) in O-Level Additional Mathematics 4037 exams, 2020–2025

Across 2020–2025, Sets & Matrices (2016-2019 syllabus only) accounted for 1 questions worth 3 marks in Cambridge O Level Additional Mathematics (4037) papers.