Question 8:
--- 8(a) ---
8(a) |  ( −6i+4j−k−(−10i+5j−4k )) | M1
e.g. 1 0 5 4 ( 4 3 )  r = − i + j − k  i − j + k
e.g. r 6 i 4 j k ( 4 i j 3 k )  = − + −  − + | A1
(2)
(b) | 3 3 6 3   + = −  = − | M1
p + 1 2 = 4  p = . . . or q − 3 = − 1  q = . . . | dM1
p = − 8 , q = 2 | A1
(3)
(c) | ( 4 i − j + 3 k ) ( 3 i − 4 j + k ) = 1 2 + 4 + 3 | M1
( 4 3 ) ( 3 4 ) 2 6 2 6 c o s 1 9 c o s . . .   i − j + k i − j + k = =  = | M1
19
(cos=)
26 | A1
(3)
(d) | 2
1 9  
A C A B s i n 2 6 1  = = −
2 6 | M1
3 9 1 0
o.e.
2 6 | A1
(2)
Total 10
Do not be concerned by the mechanics of the rearrangement in solving.
A1: Correct values
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Alt (b)
M1: Forms three correct equations 3+3= −6, p−4= 4, q+= −1
dM1: Solves simultaneously to find a value for p or q. Do not be concerned by the mechanics of the
rearrangement.
A1: Correct values
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(c)
M1: Attempts the scalar product using their direction vectors or may find another direction vector
which should be a multiple of the one for l1 or l
2
(condoning slips e.g. if they restart using
another point on l
2
to find a direction vector). Look for at least two correct products using
their l1 . May be implied by “±19”. May find the obtuse angle which can still score this mark.
M1: Completes the scalar product method using their direction vectors to find cos θ (acute or
obtuse). Attempts the magnitude of both of their direction vectors and uses these values in the
correct positions in the equation for cos θ
A1:
1
2
9
6
or exact equivalent. Do not accept fractions where the numerator or denominator is not
an integer (square roots must be evaluated). Isw if they proceed to find θ
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Alt (c) There will be many alternatives such as the one below. Send to review if unsure.
M1: The first mark should be for finding the required components to use in the cosine rule e.g.
attempts to find another point P on l
2
PMT
and attempts to find the lengths of AB, BP and PA
M1: The second mark should be using all the components required in the cosine rule. E.g.uses
their lengths for AB, BP and PA in the correct positions in the cosine rule.
A1: As above in the main scheme notes.
(d)
M1: Fully correct method to find the exact value for the length of AC.
May alternatively attempt e.g.
47
 
26
 3   13+3   
2 2 2
    97 25 47 25 59
 −4  •  −13−4  =0=−  AC =    AC =   +   +  
   
26 13 26 13 26
 1   6+   
59
 
 
26
 213 90 45
Note the coordinates for C are  − , , −  and they may use their coordinates for C
 26 13 26
with point A to find the distance AC. Condone one sign slip in the distance formula. i.e. there
must be sufficient evidence that they were attempting the differences between appropriate
coordinates.
Look out for other fully correct methods to find the exact value for the length of AC.
e.g. attempts to find BC and uses either trigonometry or Pythagoras’ Theorem.
315 35
A1: Correct value. Allow or 3
26 26