1 Write ( ) p r q pqr r 2 1 3 2 3 1 - - ` j in the form p q r a b c, where a, b and c are constants. [3]

1 2 0 5 3 − − p q r 3 B1 for 0 = a B1 for 5 = − b B1 for 2 3 = − c To earn marks, you must correctly identify the constant 'a' as 2, 'b' as -5, and 'c' as -3/2. Each correct constant earns one mark.

3 O P B A C c b a The diagram shows the quadrilateral OABC such that a OA = , b OB = and c OC = . The lines OB and AC intersect at the point P, such that : : AP PC 3 2 = . (a) Find OP in terms of a and c. [3] (b) Given also that : : OP PB 2 3 = , show that b c a 2 3 2 = + . [2]

3(a) = −  AC c a B1 May be implied 3 5 = +   OP AC a or 2 5 −  AC c M1 Maybe implied, for correct use of ratio ( ) 3 5 = +   OP theirAC a or ( ) 2 5 −  theirAC c 2 3 5 5 = +  OP a c A1 Allow unsimplified 3(b) 2 5 =  OP b oe B1 2 2 3 5 5 5 = + b a c 2 2 3 = + b a c B1 Dep on previous B mark for equating vectors and rearrangement to obtain AG Alternative 2 3 3 5 5 5 = + + b a c b (B1) Need a clear indication of the method used, in the form of a correct unsimplified statement. 2 2 3 = + b a c (B1) Dep for simplification to obtain AG

7 (a) O A B C a b c The diagram shows triangle OAC, where OA a = , OB b = and OC c = . The point B lies on the line AC such that : : AB BC m n = , where m and n are constants. (i) Write down AB in terms of a and b. [1] (ii) Write down BC in terms of b and c. [1] (iii) Hence show that ( ) n m m n a c b + = + . [2] (b) Given that ( ) ( ) 2 1 1 4 7 1 4 2 m n m + - - = + - e e e o o o, find the value of each of the constants m and n. [4]

7(a)(i) − b a B1 7(a)(ii) − c b B1 7(a)(iii) =   nAB mBC M1 For substitution of their (i) and (ii) into =   nAB mBC ( ) + = + n m m n a c b A1 For correct manipulation to obtain the given answer 7(b) 2 4 4 4 4 λ μ λ − + = + or 7 7 2 2 λ μ λ + − = − − M1 For equating like components at least once, allow unsimplified M1 Dep for solving their equations to obtain both λ and μ 5 μ = A1 10 λ = − A1 (a)(i) To earn this mark, you must correctly express vector AB as b - a. (a)(ii) To earn this mark, you must correctly express vector BC as c - b. (a)(iii) To earn these marks, you must substitute your expressions for AB and BC into the relationship nAB = mBC and then correctly manipulate the equation to arrive at the given expression for b. (b) To earn these marks, you must equate the corresponding components of the given vector equation to form two simultaneous equations. Then, you need to correctly solve these equations to find the values of m and n.

6 The points A(5, 4 - ) and C(11, 6) are such that AC is the diagonal of a square, ABCD. (a) Find the length of the line AC. [2] (b) (i) The coordinates of the centre, E, of the square are (8, y). Find the value of y. [1] (ii) Find the equation of the diagonal BD. [3] (iii) Given that the x‑coordinate of B is less than the x‑coordinate of D, write EB and ED as column vectors. [2]

6(a) 2 2 (11 5) (6 4) − + −− oe M1 11.7 or 11.66[19...] rot to 4 or more figs A1 6(b)(i) [y = ] 1 B1 6(b)(ii) 6 4 11 5 −− = − AC m or 10 6 nfww oe B1 1 10 6 − = BD m their oe M1 y – their 1 = 3( 8) 5 − − x oe isw A1 FT their 1 from (b)(i) and their perpendicular gradient 6(b)(iii) 5 3 −       and 5 3     −   B2 B1 for either If 0 scored, SC1 for −5i + 3j and 5i −3j

7 The vector p has magnitude 39 and is in the direction i j 5 2 1 - + . The vector q has magnitude 34 and is in the direction i j 15 8 - . (a) Write both p and q in terms of i and j. [4] (b) Find the magnitude of p q + and the angle this vector makes with the positive x-axis. [4]

7(a) [p =] 15 36 − + i j isw B2 B1 for multiplier 2 2 39 5 12 + soi or unit vector 2 2 5 12 5 12 − + + i j [q =] 30 16 − i j isw B2 B1 for multiplier 2 2 34 15 8 + soi or unit vector 2 2 15 8 15 8 − + i j soi 7(b) [p + q =] 15 20 + i j or 15 20       soi B1 2 2 15 20 25   + = + =     p q B1 FT their( p + q) of the form      x y or xi + yj where x ≠ 0, y ≠ 0 53.1[°] or 53.13[01…] rot to 2 or more dp OR 0.927 [rads] or 0.9272[95…] rot to 4 or more sf B2 M1 FT their(p + q) of the form       x y or xi + yj where x ≠ 0, y ≠ 0 and x ≠ y for 20 15 tan(...) = their their oe or 15 25 cos(...) = their their oe or 20 25 sin(...) = their their oe (a) To earn marks, you must first calculate the magnitude of the direction vectors for both p and q, then use these magnitudes and the given overall magnitudes to find the scalar multipliers, and finally multiply the direction vectors by these scalars to express p and q in terms of i and j. (b) You earn marks by first adding the vectors p and q, then calculating the magnitude of the resultant vector, and finally using trigonometry (e.g., tan, sin, or cos) to find the angle the resultant vector makes with the positive x-axis.

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Vectors in two dimensions past paper questions

Revise O-Level Additional Mathematics (4037) — Vectors in two dimensions 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.

32 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.

Vectors in two dimensions in O-Level Additional Mathematics 4037 exams, 2020–2025

Across 2020–2025, Vectors in two dimensions accounted for 31 questions worth 241 marks in Cambridge O Level Additional Mathematics (4037) papers.