Vector equations of lines Test

{"questions":[{"content":"A line passes through (2, −1, 3) with direction (4, 0, −2). Which equation is correct?<br>[[choice-vec-q1]]","widgets":{"choice-vec-q1":{"type":"choice","options":["r = (2, −1, 3) + λ(4, 0, −2)","r = (4, 0, −2) + λ(2, −1, 3)","r = λ(6, −1, 1)","r = (2, 1, 3) + λ(4, 0, 2)"],"answer":[0],"explanations":["[M1] Identify a and d. [A1] r = (2, −1, 3) + λ(4, 0, −2).","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Use fixed point plus parameter times direction."],"explanation":"[M1] Identify a and d. [A1] r = (2, −1, 3) + λ(4, 0, −2).","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"For r = (3, 1, 2) + λ(−1, 2, 4), the z-coordinate at λ = 2 is [[input-vec-q2]]","widgets":{"input-vec-q2":{"type":"input","answer":"10"}},"hints":["Compute 2 + 4λ."],"explanation":"[A1] z = 2 + 4(2) = 10.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["A1"],"follow_through":false}},{"content":"Which condition proves two direction vectors d and e are parallel?<br>[[choice-vec-q3]]","widgets":{"choice-vec-q3":{"type":"choice","options":["e = kd for some non-zero scalar k","d · e = 0","|d| = |e| only","d + e = 0 only"],"answer":[0],"explanations":["[R1] State the scalar-multiple condition.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Parallel vectors are scalar multiples."],"explanation":"[R1] State the scalar-multiple condition.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["R1"],"follow_through":false}},{"content":"Solving two line equations gives λ = 2 and μ = −1 from x and y, but z gives 7 = 9. What follows?<br>[[choice-vec-q4]]","widgets":{"choice-vec-q4":{"type":"choice","options":["The lines do not intersect","The lines meet at z = 8","The lines are coincident","The parameter values should be averaged"],"answer":[0],"explanations":["[M1] Compare all coordinates. [R1] The contradiction means no common point; if directions are non-parallel, the lines are skew. [FT] A consistent remaining-coordinate check can follow through earlier parameter values.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["A common point must satisfy every coordinate."],"explanation":"[M1] Compare all coordinates. [R1] The contradiction means no common point; if directions are non-parallel, the lines are skew. [FT] A consistent remaining-coordinate check can follow through earlier parameter values.","assessment":{"paper_mode":"paper_3","calculator_mode":"optional","gdc_skill":"","marks":["M1","R1"],"follow_through":true}},{"content":"A GDC solves two coordinate equations for the parameters. What written evidence is still needed?<br>[[choice-vec-q5]]","widgets":{"choice-vec-q5":{"type":"choice","options":["The vector equations and a check in the remaining coordinate","Only the decimal parameters","A histogram","No written evidence"],"answer":[0],"explanations":["[M1] State the simultaneous equations. [R1] Substitute in the remaining coordinate before concluding.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Show what the calculator solved and verify consistency."],"explanation":"[M1] State the simultaneous equations. [R1] Substitute in the remaining coordinate before concluding.","assessment":{"paper_mode":"paper_2","calculator_mode":"optional","gdc_skill":"matrix","marks":["M1","R1"],"follow_through":false}}],"mix":1}