Date | May 2018 | Marks available | 6 | Reference code | 18M.1.SL.TZ1.S_9 |
Level | Standard Level | Paper | Paper 1 | Time zone | Time zone 1 |
Command term | Find | Question number | S_9 | Adapted from | N/A |
Question
Point A has coordinates (−4, −12, 1) and point B has coordinates (2, −4, −4).
The line L passes through A and B.
Show that
Find a vector equation for L.
Point C (k , 12 , −k) is on L. Show that k = 14.
Find .
Write down the value of angle OBA.
Point D is also on L and has coordinates (8, 4, −9).
Find the area of triangle OCD.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
correct approach A1
eg
AG N0
[1 mark]
any correct equation in the form r = a + tb (any parameter for t) A2 N2
where a is or and b is a scalar multiple of
eg r r
Note: Award A1 for the form a + tb, A1 for the form L = a + tb, A0 for the form r = b + ta.
[2 marks]
METHOD 1 (solving for t)
valid approach (M1)
eg
one correct equation A1
eg −4 + 8t = 12, −12 + 8t = 12
correct value for t (A1)
eg t = 2 or 3
correct substitution A1
eg 2 + 6(2), −4 + 6(3), −[1 + 3(−5)]
k = 14 AG N0
METHOD 2 (solving simultaneously)
valid approach (M1)
eg
two correct equations in A1
eg k = −4 + 6t, −k = 1 −5t
EITHER (eliminating k)
correct value for t (A1)
eg t = 2 or 3
correct substitution A1
eg 2 + 6(2), −4 + 6(3)
OR (eliminating t)
correct equation(s) (A1)
eg 5k + 20 = 30t and −6k − 6 = 30t, −k = 1 − 5
correct working clearly leading to k = 14 A1
eg −k + 14 = 0, −6k = 6 −5k − 20, 5k = −20 + 6(1 + k)
THEN
k = 14 AG N0
[4 marks]
correct substitution into scalar product A1
eg (2)(6) − (4)(8) − (4)(−5), 12 − 32 + 20
= 0 A1 N0
[2 marks]
A1 N1
[1 marks]
METHOD 1 ( × height × CD)
recognizing that OB is altitude of triangle with base CD (seen anywhere) M1
eg sketch showing right angle at B
or (seen anywhere) (A1)
correct magnitudes (seen anywhere) (A1)(A1)
correct substitution into A1
eg
area A1 N3
METHOD 2 (subtracting triangles)
recognizing that OB is altitude of either ΔOBD or ΔOBC(seen anywhere) M1
eg sketch of triangle showing right angle at B
one correct vector or or or (seen anywhere) (A1)
eg ,
(seen anywhere) (A1)
one correct magnitude of a base (seen anywhere) (A1)
correct working A1
eg
area A1 N3
METHOD 3 (using ab sin C with ΔOCD)
two correct side lengths (seen anywhere) (A1)(A1)
attempt to find cosine ratio (seen anywhere) M1
eg
correct working for sine ratio A1
eg
correct substitution into A1
eg
area A1 N3
[6 marks]