"Full Coverage": Vectors This worksheet is designed to cover one question of each type seen in past papers, for each GCSE Higher Tier topic. This worksheet was automatically generated by the DrFrostMaths Homework Platform: students can practice this set of questions interactively by going to www.drfrostmaths.com/homework, logging on, Practise → Past Papers/Worksheets (or Library → Past/Past Papers for teachers), and using the ‘Revision’ tab. Question 1 Categorisation: Add to vectors. [Edexcel GCSE(9-1) Mock Set 2 Spring 2017 3F Q20ai, 3H Q4ai] 1 3 π = ( ) and π = ( ) 4 2 (a) Write down as a column vector (i) π + π .......................... Question 2 Categorisation: [IGCSE only?] Find the magnitude (i.e. length) of a vector: The π magnitude of ( ) is √ππ + ππ . π [Edexcel IGCSE Jan2015-4H Q19c] 5 π=( ) −2 1 π=( ) 7 −7 π=( ) 0 Work out the magnitude of π . Give your answer as a surd. .......................... Question 3 Categorisation: Appreciate scalar multiplication of vectors. [Edexcel IGCSE Jan2015-4H Q19b] 5 π=( ) −2 1 π=( ) 7 −7 π=( ) 0 Write, as a column vector, 3π − π .......................... www.drfrostmaths.com Question 4 Categorisation: Determine the coordinates of a point using an initial coordinate and subsequent translation by one or more vectors. [Edexcel IGCSE Jan2015(R)-4H Q15b] Here is the parallelogram ABCD. → 1 → 5 π΄π· = ( ) , π΄π΅ = ( ) 2 3 The point A has coordinates (4, 2). Work out the coordinates of the point C. .......................... Question 5 Categorisation: Appreciate that two vectors with the same magnitude and same direction are the same vector. [Edexcel GCSE June2003-5H Q23aii] The diagram shows a regular hexagon π΄π΅πΆπ·πΈπΉ with centre π . → → ππ΄ = 6π ππ΅ = 6π → Express in terms of π and/or π the vector πΈπΉ . → πΈπΉ = .......................... www.drfrostmaths.com Question 6 βββββ between two points using an appropriate path. Categorisation: Find a vector πΏπ [Edexcel GCSE June2006-6H Q13aii] π΄π΅πΆπ· is a parallelogram. π΄π΅ is parallel to π·πΆ . π΄π· is parallel to π΅πΆ . → π΄π΅ = π© → π΄π· = πͺ → Express π΅π· in terms of π© and πͺ . → π΅π· = .......................... Question 7 Categorisation: Appreciate that multiplying a vector by a scalar multiplies its magnitude by this scalar. [Edexcel GCSE Nov2006-5H Q22c Edited] OAB is a triangle. B is the midpoint of OR. Q is the midpoint of AB. → ππ = 2π → ππ΄ = π → ππ΅ = π → → ππ = 4 ππ The length of PQ is 3 cm. (c) Find the length of PR. .......................... cm www.drfrostmaths.com Question 8 Categorisation: Determine a vector where the vector between points is a more complex combination of multiple vectors. [Edexcel GCSE Nov2008-4H Q26a] → ππ = 2π + π → ππ = 4π + 3π → (a) Express the vector ππ in terms of π and π Give your answer in its simplest form. → ππ = .......................... Question 9 Categorisation: Determine a vector involving a fraction of the distance between two points. [Edexcel GCSE Nov2013-1H Q24a] OACB is a parallelogram. → → → → ππ΄ = π and ππ΅ = π π· is the point such that π΄πΆ = πΆπ· . The point π divides π΄π΅ in the ratio 2 : 1. → Write an expression for ππ in terms of a and b. .......................... www.drfrostmaths.com Question 10 Categorisation: Determine a vector involving multiple fractions. [Edexcel IGCSE May2014(R)-3H Q21] → → OABC is a parallelogram. ππ΄ = π and ππΆ = π 1 P is the point on AB such that π΄π = 4 π΄π΅ 2 Q is the point on OC such that ππ = 3 ππΆ → Find, in terms of π and π , ππ . Give your answer in its simplest form. → ππ = .......................... Question 11 Categorisation: Prove that three points form a straight line. [Edexcel IGCSE May2015(R)-3H Q18b Edited] www.drfrostmaths.com PQR is a triangle. The midpoint of PQ is W. X is the point on QR such that QX : XR = 2 : 1 PRY is a straight line. → → ππ = π and ππ = π R is the midpoint of the straight line PRY. Use a vector method to show that WXY is a straight line. Question 12 Categorisation: Prove that two vectors are parallel. [Edexcel IGCSE May2016-4H Q22b Edited] 1 πππ is a triangle. π is the point on ππ such that ππ = 4 ππ π is the midpoint of ππ, π is the midpoint of ππ → ππ = π© → ππ = πͺ Use a vector method to prove that ππ is parallel to ππ www.drfrostmaths.com Question 13 Categorisation: As above. [Edexcel IGCSE Jan2014-3H Q21b Edited] ABCD is a trapezium. AB is parallel to DC. → π΄π΅ = 12π → π΄π· = 3π → π·πΆ = 18π 1 3 πΈ is the point on the diagonal π·π΅ such that π·πΈ = π·π΅ . Show by a vector method that π΅πΆ is parallel to π΄πΈ . Question 14 Categorisation: Determine the scalar multiplier that one vector is of another. [Edexcel GCSE(9-1) Mock Set 2 Spring 2017 1H Q20] ππ΄πΆπ· is a trapezium and ππ΄πΆπ΅ is a parallelogram.π΅ is the midpoint of ππ· .π is the midpoint of πΆπ·. → → ππ΄ = π and ππ΅ = π → → Given that π΅π = π × ππΆ where π is a scalar, use a vector method to find the value of π . .......................... www.drfrostmaths.com Question 15 Categorisation: Show that a vector is a particular expression. [Edexcel GCSE June2009-4H Q23b Edited] → → ππ΄π΅ is a triangle. ππ΄ = π ππ΅ = π π is the point on π΄π΅ such that π΄π: ππ΅ = 3: 2 → Show that ππ = π₯(2π + 3π) where π₯ is a fraction to be found. π₯ = .......................... Question 16 Categorisation: As above, but where no diagram is provided. [Edexcel IGCSE Jan2016(R)-3H Q19d] → ππ = 3π + 4π → ππ = −4π + 2π 2 The point π lies on ππ such that ππ = 3 ππ 1 The point π lies on ππ such that ππ = 3 ππ → Show that ππ = ππ where π is a constant. State the value of π . π = .......................... www.drfrostmaths.com Question 17 Categorisation: Determine a vector involving an extended line. [Edexcel GCSE March2012-3H Q23b] → → ABCDEF is a regular hexagon, with centre O. ππ΄ = π ππ΅ = π The line AB is extended to the point K so that AB : BK = 1 : 2 → Write the vector πΆπΎ in terms of π and π . Give your answer in its simplest form. → πΆπΎ = .......................... Question 18 Categorisation: Determine a vector when a fraction involved is greater than 1. [Edexcel GCSE Jun2015-2H Q27] π΄π΅πΆ is a straight line. π΄π΅: π΅πΆ = 2: 5 → ππ΄ = 2a + b → ππ΅ = 3a + 2b → Express ππΆ in terms of a and b. Give your answer in its simplest form. .......................... www.drfrostmaths.com www.drfrostmaths.com Answers Question 1 4 ( ) 6 Question 2 √29 Question 3 10 ( ) 21 Question 4 (10,7) Question 5 → πΈπΉ = 6π Question 6 → π΅π· = π − π Question 7 12 cm Question 8 → ππ = 2π + 2π Question 9 1 π 3 2 +3π Question 10 → 5 ππ = 12 π − π Question 11 1 2 1 βββββββ = − π + π = (−π + 2π) ππ 3 3 3 www.drfrostmaths.com βββββββ = −π + 2π ππ βββββββ ππ is a multiple of βββββββ ππ ∴ parallel. π is a common point. ∴ πππ is a straight line. Question 12 βββββ = 1 π ππ 2 βββββ = π ππ βββββ is a multiple of ππ βββββ therefore parallel. ππ Question 13 → π΅πΆ = 3(2π + π) βββββ π΄πΈ = 2(2π + π) βββββ is a multiple of βββββ π΅πΆ π΄πΈ ∴ parallel. Question 14 π = 0.5 Question 15 1 π₯=5 Question 16 π= 11 3 Question 17 → πΆπΎ = 2π − π Question 18 11 π 2 9 + 2π www.drfrostmaths.com
