2023 May/June · Paper 22 · Q108 marksVector geometry
2022
2022 Oct/Nov · Paper 12 · Q5 (a, b, c)8 marksVector algebra and unit vectors · Vector geometry
2022 May/June · Paper 12 · Q4 (a, b)5 marksVector algebra and unit vectors
2022 May/June · Paper 22 · Q6 (a, b)7 marksVector algebra and unit vectors · Velocity and position vectors
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school. “Paper 42” means Paper 4, variant 2.
Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1Velocity and position vectors · Core · Paper 2 style · 3 marks
(a)A fishing boat leaves a jetty on Keenjhar Lake. At time t = 0 hours it is at the point with position vector (2i + 5j) km relative to the jetty, and it moves with constant velocity (4i − 3j) km/h. Write down the position vector of the boat t hours later.[1]
(b)Find the position vector of the boat when t = 2.5.[1]
(c)Find the speed of the boat.[1]
Show answer
(a) r = (2 + 4t)i + (5 − 3t)j
B1 r = (2i + 5j) + t(4i − 3j)
(b) 12i − 2.5j
B1 2 + 10 = 12, 5 − 7.5 = −2.5
(c) 5 km/h
B1 √(4² + 3²) = 5
Q2Velocity and position vectors · Core · Paper 1 style · 4 marks
(a)A particle moves with speed 26 m/s in the direction of the vector 5i − 12j. Find its velocity vector.[2]
(b)At t = 0 the particle is at the point with position vector (−3i + 7j) m. Find its position vector when t = 4 s.[2]
Q3Velocity and position vectors · Core · Paper 2 style · 5 marks
(a)A camera drone in a park in Islamabad moves in a straight line with constant velocity. At t = 0 it is at the point with position vector (−6i + 8j) m and at t = 5 s it is at (18i − 2j) m. Find the velocity vector of the drone.[2]
(b)Find the speed of the drone.[1]
(c)Find the angle between the direction of motion and the vector i.[2]
Show answer
(a) (4.8i − 2j) m/s
M1 displacement = 24i − 10j; A1 ÷ 5
(b) 5.2 m/s
B1 √(4.8² + 2²) = √27.04 = 5.2
(c) 22.6° (below i)
M1 tan⁻¹(2/4.8); A1 22.6°
Q4Velocity and position vectors · Core · Paper 2 style · 4 marks
(a)In this question i is a unit vector due east and j a unit vector due north; distances are in kilometres. At 12:00 a ferry near Karachi is at the point with position vector (3i + 4j) km relative to a lighthouse L. It moves with constant velocity (−5i + 12j) km/h. Find the position vector of the ferry t hours after 12:00.[1]
(b)Find the time at which the ferry is due north of L, and its distance from L at that time.[3]
Show answer
(a) (3 − 5t)i + (4 + 12t)j
B1
(b) 12:36; 11.2 km
M1 3 − 5t = 0; A1 t = 0.6 h = 36 min, 12:36; A1 4 + 12(0.6) = 11.2 km
Q5Velocity and position vectors · Core · Paper 1 style · 4 marks
(a)A delivery van moves with constant velocity across a flat plain. At 08:00 it is at the point with position vector (4i − 2j) km and at 08:15 it is at (7i + 2j) km. Find the velocity of the van in km/h and its speed.[3]
(b)Assuming the velocity does not change, find the position vector of the van at 09:00.[1]
Show answer
(a) (12i + 16j) km/h; 20 km/h
M1 displacement 3i + 4j in 0.25 h; A1 12i + 16j; B1 20 km/h
(b) 16i + 14j
B1 (4i − 2j) + 1 × (12i + 16j)
Q6Velocity and position vectors · Standard · Paper 2 style · 5 marks
(a)Two motor boats A and B move on a lake with constant velocities. At t = 0, A is at (2i + 10j) km and B is at (14i − 2j) km. The velocity of A is (6i + 2j) km/h and the velocity of B is (2i + 6j) km/h. Find, in terms of t, the position vectors of A and of B at time t hours.[2]
(b)Show that A and B would collide, and find the time and the position vector of the point of collision.[3]
M1 equate i-components: 2 + 6t = 14 + 2t → t = 3; A1 j-components both 16 at t = 3 (so they collide); A1 20i + 16j
Q7Velocity and position vectors · Standard · Paper 2 style · 5 marks
(a)Particle P starts at the point with position vector (i + 2j) m and moves with velocity (3i + 4j) m/s. At the same time particle Q starts at (10i − j) m and moves with velocity (−i + 5j) m/s. Show that P and Q do not collide.[3]
(b)Find the distance between P and Q when t = 2.[2]
Show answer
(a) i-components equal at t = 2.25, j-components equal at t = 3: different times, so no collision
M1 1 + 3t = 10 − t → t = 2.25; M1 2 + 4t = −1 + 5t → t = 3; A1 conclusion
(b) √2 m (1.41 m)
M1 P = 7i + 10j, Q = 8i + 9j; A1 √(1² + 1²) = √2
Q8Velocity and position vectors · Standard · Paper 1 style · 5 marks
(a)i is due east and j is due north. Boat A leaves port O at 10:00 and moves with velocity (8i + 6j) km/h. Boat B leaves a point with position vector (4i + 20j) km at 11:00 and moves with velocity (12i − 4j) km/h. Write down the position vectors of A and B at time t hours after 10:00, where t ≥ 1.[2]
(b)Find the time at which B is due east of A, and the distance between the boats at that time.[3]
M1 6t = 24 − 4t; A1 t = 2.4 h → 12:24; A1 A at 19.2i + 14.4j, B at 20.8i + 14.4j, distance 1.6 km
Q9Velocity and position vectors · Standard · Paper 2 style · 5 marks
(a)i is due east and j is due north. A light aircraft flies with constant velocity (−120i + 160j) km/h. Find its speed and the bearing on which it is flying.[3]
(b)At 09:00 the aircraft is at (300i − 100j) km relative to an airfield. Find its position vector after it has flown 500 km.[2]
Show answer
(a) 200 km/h on a bearing of 323.1°
B1 √(120² + 160²) = 200; M1 tan⁻¹(120/160) = 36.9° west of north; A1 323.1°
(b) 300j km (due north of the airfield, 300 km away)
Q10Velocity and position vectors · Challenge · Paper 2 style · 8 marks
(a)i is due east and j is due north. A ship sails at 15 km/h on a bearing of 060°. Show that its velocity is (7.5√3 i + 7.5j) km/h.[2]
(b)At t = 0 the ship is at (−20i + 5j) km relative to a buoy O. Find the time at which the ship is due north of O, giving your answer in minutes correct to the nearest minute, and its distance from O then.[3]
(c)By writing the square of the distance from O as a quadratic in t, or otherwise, find the shortest distance between the ship and O.[3]
Show answer
(a) 15 sin 60° i + 15 cos 60° j = 7.5√3 i + 7.5j
M1 resolve using the bearing; A1 exact values
(b) 92 minutes; 16.5 km
M1 −20 + 7.5√3 t = 0; A1 t = 1.540 h ≈ 92 min; A1 5 + 7.5(1.540) = 16.5 km
(c) 14.3 km (at t ≈ 0.988 h)
M1 d² = (−20 + 12.99t)² + (5 + 7.5t)²; M1 minimise: t = (20×12.99 − 37.5)/225 = 0.988; A1 d = 14.3 km
Q11Velocity and position vectors · Challenge · Paper 1 style · 4 marks
(a)Particle A starts at the point with position vector (−4i + 9j) m and moves with constant velocity (3i − j) m/s. Particle B starts at the same time from (20i − 7j) m and moves with constant velocity (pi + qj) m/s. The particles collide when t = 4. Find p and q.[3]
(b)Find the exact speed of B.[1]
Show answer
(a) p = −3, q = 3
M1 A at t = 4: 8i + 5j; M1 20 + 4p = 8, −7 + 4q = 5; A1 p = −3, q = 3
(b) 3√2 m/s
B1 √(9 + 9) = 3√2
Q12Velocity and position vectors · Challenge · Paper 2 style · 6 marks
(a)Two remote-controlled boats move on a pond. Boat P starts at (2i + 3j) m with velocity (ai + 2j) m/s, where a is a constant. Boat Q starts at the same time at (11i − 5j) m with velocity (−i + 6j) m/s. Given that the boats collide, find the time of the collision and the value of a.[4]
(b)Find the position vector of the point of collision and the speed of P, correct to 3 significant figures.[2]
Q13Vector geometry · Core · Paper 1 style · 4 marks
(a)Relative to an origin O, the position vectors of A and B are a and b. M is the midpoint of AB and P lies on OB with OP : PB = 2 : 1. Find, in terms of a and b, the vector AB.[1]
(b)Find OM.[1]
(c)Find AP.[2]
Show answer
(a) b − a
B1
(b) ½(a + b)
B1 a + ½(b − a)
(c) ⅔b − a
M1 OP = ⅔b; A1 AP = ⅔b − a
Q14Vector geometry · Core · Paper 1 style · 4 marks
(a)Points A and B have position vectors 2i + 5j and 8i − j. Find AB and |AB| in exact form.[2]
(b)Point C lies on AB with AC : CB = 1 : 2. Find the position vector of C.[2]
Show answer
(a) AB = 6i − 6j; |AB| = 6√2
B1 B1
(b) 4i + 3j
M1 OA + ⅓AB; A1
Q15Vector geometry · Core · Paper 2 style · 4 marks
(a)OABC is a parallelogram with OA = 3i + j and OC = i + 4j. Find the vectors OB and AC.[2]
(b)Find the lengths of the two diagonals, correct to 3 significant figures.[2]
Show answer
(a) OB = 4i + 5j; AC = −2i + 3j
B1 OA + OC; B1 OC − OA
(b) OB = 6.40, AC = 3.61
B1 √41; B1 √13
Q16Vector geometry · Core · Paper 1 style · 4 marks
(a)The points P, Q and R have position vectors i + 2j, 4i + 8j and 6i + 12j. Find PQ and PR.[2]
(b)Hence show that P, Q and R lie on a straight line, and find the ratio PQ : QR.[2]
Show answer
(a) PQ = 3i + 6j; PR = 5i + 10j
B1 B1
(b) PR = (5/3)PQ, common point P so collinear; PQ : QR = 3 : 2
M1 PR is a multiple of PQ, sharing P; A1 QR = 2i + 4j so 3 : 2
Q17Vector geometry · Core · Paper 2 style · 4 marks
(a)Relative to O, A and B have position vectors 5i + j and i + 5j. Find AB.[1]
(b)Find the lengths OA, OB and AB in exact form and state what type of triangle OAB is.[3]
Show answer
(a) −4i + 4j
B1
(b) OA = OB = √26, AB = 4√2; isosceles
B1 √26 both; B1 √32 = 4√2; B1 isosceles
Q18Vector geometry · Standard · Paper 1 style · 4 marks
(a)Relative to O, the points A and B have position vectors a and b. C lies on AB such that AC : CB = 3 : 1. Show that OC = ¼a + ¾b.[2]
(b)D is the midpoint of OC. Find OD and BD in terms of a and b.[2]
Show answer
(a) OC = a + ¾(b − a) = ¼a + ¾b
M1 OA + ¾AB; A1 simplified
(b) OD = ⅛a + ⅜b; BD = ⅛a − ⅝b
B1 OD; B1 BD = OD − OB
Q19Vector geometry · Standard · Paper 1 style · 6 marks
(a)OAB is a triangle with OA = a and OB = b. M is the midpoint of OB and N lies on AB with AN : NB = 2 : 1. Find ON and AM in terms of a and b.[2]
(b)The lines ON and AM meet at X, where OX = λON and AX = μAM. Find two expressions for OX and hence find λ and μ.[4]
Q20Vector geometry · Standard · Paper 2 style · 5 marks
(a)Triangle ABC has vertices with position vectors OA = i + j, OB = 7i + 3j and OC = 3i + 9j. M is the midpoint of BC. Find AM and its length, correct to 3 significant figures.[3]
(b)G lies on AM with AG : GM = 2 : 1. Find the position vector of G.[2]
Show answer
(a) AM = 4i + 5j; |AM| = 6.40
B1 OM = 5i + 6j; B1 AM; B1 √41 = 6.40
(b) (11/3)i + (13/3)j
M1 OA + ⅔AM; A1
Q21Vector geometry · Standard · Paper 1 style · 6 marks
(a)OABC is a parallelogram with OA = a and OC = c. M is the midpoint of AB. Find OM and AC in terms of a and c.[2]
(b)The line OM meets the diagonal AC at X. By writing OX = λOM and AX = μAC, find λ and μ, and hence the ratio OX : XM.[4]
Q22Vector geometry · Challenge · Paper 1 style · 6 marks
(a)Relative to O, OA = a and OB = b. P lies on OA with OP : PA = 1 : 2, and Q lies on OB with OQ : QB = 3 : 1. The lines AQ and BP meet at R. Given that AR = λAQ and BR = μBP, express OR in terms of λ, a and b, and also in terms of μ, a and b.[2]
(b)Find λ and μ, and hence express OR in terms of a and b.[4]
Q23Vector geometry · Challenge · Paper 2 style · 6 marks
(a)Relative to O, A has position vector 6i + 2j and B has position vector 2i + 8j. P is the midpoint of OA and Q is the midpoint of AB. The lines OQ and BP meet at X. Write OX = λOQ and BX = μBP and find λ and μ.[4]
(b)Find the position vector of X and the distance OX correct to 3 significant figures.[2]
Show answer
(a) λ = ⅔, μ = ⅔
M1 OQ = 4i + 5j, BP = i − 7j; M1 4λ = 2 + μ and 5λ = 8 − 7μ; A1 A1
(b) (8/3)i + (10/3)j; 4.27
B1; B1 ⅔√41 = 4.27
Q24Vector geometry · Challenge · Paper 1 style · 6 marks
(a)Relative to O, OA = a and OB = b. The point C lies on AB and OC = ⅖a + kb. Find k and the ratio AC : CB.[3]
(b)The point D is such that OD = h × OC and BD is parallel to OA. Find h and express OD in terms of a and b.[3]
Show answer
(a) k = 3/5; AC : CB = 3 : 2
M1 C on AB ⇒ OC = (1 − t)a + tb, coefficients add to 1; A1 k = 3/5; A1 AC = ⅗AB so 3 : 2
(b) h = 5/3; OD = ⅔a + b
M1 BD = (2h/5)a + (3h/5 − 1)b; M1 b-coefficient = 0; A1 h = 5/3, OD = ⅔a + b
Q25Vector algebra and unit vectors · Core · Paper 1 style · 3 marks
(a)a = 3i − 4j and b = −2i + 5j. Find 2a + 3b.[1]
(b)Find |a| and the unit vector in the direction of a.[2]
Show answer
(a) 7j
B1 (6 − 6)i + (−8 + 15)j
(b) |a| = 5; ⅕(3i − 4j)
B1 B1
Q26Vector algebra and unit vectors · Core · Paper 2 style · 4 marks
(a)p = 7i + 24j. Find the vector of magnitude 75 in the same direction as p.[2]
(b)Find the vector q such that p + 2q = i − 6j.[2]
Show answer
(a) 21i + 72j
M1 |p| = 25; A1 3p
(b) q = −3i − 15j
M1 2q = −6i − 30j; A1
Q27Vector algebra and unit vectors · Standard · Paper 1 style · 5 marks
(a)a = 2i + 3j and b = i − 2j. Find the value of μ for which a + μb is parallel to i + j.[3]
(b)For this value of μ, find |a + μb| in exact form.[2]
