Edexcel F3 2018 Specimen — Question 6 9 marks

Exam BoardEdexcel
ModuleF3 (Further Pure Mathematics 3)
Year2018
SessionSpecimen
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVectors: Lines & Planes
TypeCartesian equation of a plane
DifficultyChallenging +1.2 This is a Further Maths question requiring finding a plane equation via cross product (standard technique) and using the scalar triple product formula for tetrahedron volume (less routine but a known formula). The multi-step nature and Further Maths context place it above average, but both parts follow established procedures without requiring novel geometric insight.
Spec4.04b Plane equations: cartesian and vector forms4.04j Shortest distance: between a point and a plane
The coordinates of the points \(A\), \(B\) and \(C\) relative to a fixed origin \(O\) are \((1, 2, 3)\), \((-1, 3, 4)\) and \((2, 1, 6)\) respectively. The plane \(\Pi\) contains the points \(A\), \(B\) and \(C\).
  1. Find a cartesian equation of the plane \(\Pi\). [5]
The point \(D\) has coordinates \((k, 4, 14)\) where \(k\) is a positive constant. Given that the volume of the tetrahedron \(ABCD\) is 6 cubic units,
  1. find the value of \(k\). [4]
Question 6:
AnswerMarks
6(a)(cid:167)(cid:16)2(cid:183) (cid:167) 1(cid:183) (cid:167) 3 (cid:183)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
AB (cid:32) 1 , AC (cid:32) (cid:16)1 , BC (cid:32) (cid:16)2
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
1 3 2
AnswerMarks
(cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185)Two correct vectors in Π
Can be negatives of those shownB1
i j k (cid:167)4(cid:183)
(cid:168) (cid:184)
(cid:16)2 1 1 (cid:32) 7
(cid:168) (cid:184)
(cid:168) (cid:184)
1 (cid:16)1 3 1
AnswerMarks
(cid:169) (cid:185)M1: Attempt cross product of two
vectors lying in Π (At least one
AnswerMarks
no. to be correct.)M1 A1
A1: Correct normal vector
(cid:167)4(cid:183) (cid:167)1(cid:183)
(cid:168) (cid:184) (cid:168) (cid:184)
7 (cid:120) 2 (cid:32)4(cid:14)14(cid:14)3
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184)
1 3
AnswerMarks Guidance
(cid:169) (cid:185) (cid:169) (cid:185)Attempt scalar product with their
normal and a point in the planedM1
4x(cid:14)7y(cid:14)z(cid:32)21Cao (oe) A1
(5)
Alternative 1
a(cid:14)2b(cid:14)3c(cid:32)d
(cid:16)a(cid:14)3b(cid:14)4c(cid:32)d
AnswerMarks Guidance
2a(cid:14)b(cid:14)6c(cid:32)dCorrect equations B1
4 1 1
a(cid:32) d, b(cid:32) d, c(cid:32) d
AnswerMarks
21 3 21M1: Solve for a, b and c in terms
of dM1 A1
A1: Correct equations
AnswerMarks Guidance
d (cid:32)21(cid:159)a(cid:32)..., b(cid:32)..., c(cid:32)...Obtains values for a, b, c and d M1
4x(cid:14)7y(cid:14)z(cid:32)21Cao (oe) A1
(5)
Alternative 2: Using r(cid:32)a(cid:14)sb(cid:14)tc where b and c are vectors in (cid:51)
AnswerMarks Guidance
Two correct vectors in the planeSee main scheme B1
(cid:167)x(cid:183) (cid:167)1(cid:183) (cid:167)(cid:16)2(cid:183) (cid:167) 1 (cid:183)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:32) y (cid:32) 2 (cid:14)s 1 (cid:14)t (cid:16)1
Eg r (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
z 3 1 3
AnswerMarks
(cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185)M1
x(cid:32)1(cid:16)2s(cid:14)t
y (cid:32)2(cid:14)s(cid:16)t
AnswerMarks Guidance
z (cid:32)3(cid:14)s(cid:14)3tDeduce 3 correct equations A1
4x(cid:14)7y(cid:14)z(cid:32)21M1: Eliminate s, t
A1: CaoM1 A1
(5)
AnswerMarks Guidance
QuestionScheme Marks
AnswerMarks Guidance
6(b)AD AB(cid:117)AC Attempt suitable triple product
(cid:167)4(cid:183) (cid:167)k(cid:16)1(cid:183)
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:32) 7 (cid:120) 2 (cid:32)4k(cid:16)4(cid:14)14(cid:14)11
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184)
1 11
(cid:169) (cid:185) (cid:169) (cid:185)
1
(cid:63) (cid:11)4k(cid:14)21(cid:12)(cid:32)6
AnswerMarks
61
M1: Set (their triple product) = 6
AnswerMarks
6dM1
A1
A1: Correct equation
15
k (cid:32)
AnswerMarks Guidance
4Cao (oe) A1
(4)
Alternative
Area ABC
1 1
= AB AC (cid:32) 6 11
AnswerMarks
2 2Attempt area ABC and distance
between DandΠM1
4k(cid:14)28(cid:14)14(cid:16)21
D to Π is
16(cid:14)49(cid:14)1
1 4k(cid:14)28(cid:14)14(cid:16)21
6 11 (cid:32)6
AnswerMarks
6 16(cid:14)49(cid:14)11
M1: Set (their area x their
3
AnswerMarks
distance) = 6dM1
A1
A1: Correct equation
15
k (cid:32)
AnswerMarks Guidance
4Cao (oe) A1
(4)
(9 marks)
AnswerMarks Guidance
QuestionScheme Marks 
Question 6:
--- 6(a) ---
6(a) | (cid:167)(cid:16)2(cid:183) (cid:167) 1(cid:183) (cid:167) 3 (cid:183)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
AB (cid:32) 1 , AC (cid:32) (cid:16)1 , BC (cid:32) (cid:16)2
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
1 3 2
(cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185) | Two correct vectors in Π
Can be negatives of those shown | B1
i j k (cid:167)4(cid:183)
(cid:168) (cid:184)
(cid:16)2 1 1 (cid:32) 7
(cid:168) (cid:184)
(cid:168) (cid:184)
1 (cid:16)1 3 1
(cid:169) (cid:185) | M1: Attempt cross product of two
vectors lying in Π (At least one
no. to be correct.) | M1 A1
A1: Correct normal vector
(cid:167)4(cid:183) (cid:167)1(cid:183)
(cid:168) (cid:184) (cid:168) (cid:184)
7 (cid:120) 2 (cid:32)4(cid:14)14(cid:14)3
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184)
1 3
(cid:169) (cid:185) (cid:169) (cid:185) | Attempt scalar product with their
normal and a point in the plane | dM1
4x(cid:14)7y(cid:14)z(cid:32)21 | Cao (oe) | A1
(5)
Alternative 1
a(cid:14)2b(cid:14)3c(cid:32)d
(cid:16)a(cid:14)3b(cid:14)4c(cid:32)d
2a(cid:14)b(cid:14)6c(cid:32)d | Correct equations | B1
4 1 1
a(cid:32) d, b(cid:32) d, c(cid:32) d
21 3 21 | M1: Solve for a, b and c in terms
of d | M1 A1
A1: Correct equations
d (cid:32)21(cid:159)a(cid:32)..., b(cid:32)..., c(cid:32)... | Obtains values for a, b, c and d | M1
4x(cid:14)7y(cid:14)z(cid:32)21 | Cao (oe) | A1
(5)
Alternative 2: Using r(cid:32)a(cid:14)sb(cid:14)tc where b and c are vectors in (cid:51)
Two correct vectors in the plane | See main scheme | B1
(cid:167)x(cid:183) (cid:167)1(cid:183) (cid:167)(cid:16)2(cid:183) (cid:167) 1 (cid:183)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:32) y (cid:32) 2 (cid:14)s 1 (cid:14)t (cid:16)1
Eg r (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184) (cid:168) (cid:184)
z 3 1 3
(cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185) (cid:169) (cid:185) | M1
x(cid:32)1(cid:16)2s(cid:14)t
y (cid:32)2(cid:14)s(cid:16)t
z (cid:32)3(cid:14)s(cid:14)3t | Deduce 3 correct equations | A1
4x(cid:14)7y(cid:14)z(cid:32)21 | M1: Eliminate s, t
A1: Cao | M1 A1
(5)
Question | Scheme | Marks
--- 6(b) ---
6(b) | AD AB(cid:117)AC | Attempt suitable triple product | M1
(cid:167)4(cid:183) (cid:167)k(cid:16)1(cid:183)
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:32) 7 (cid:120) 2 (cid:32)4k(cid:16)4(cid:14)14(cid:14)11
(cid:168) (cid:184) (cid:168) (cid:184)
(cid:168) (cid:184) (cid:168) (cid:184)
1 11
(cid:169) (cid:185) (cid:169) (cid:185)
1
(cid:63) (cid:11)4k(cid:14)21(cid:12)(cid:32)6
6 | 1
M1: Set (their triple product) = 6
6 | dM1
A1
A1: Correct equation
15
k (cid:32)
4 | Cao (oe) | A1
(4)
Alternative
Area ABC
1 1
= AB AC (cid:32) 6 11
2 2 | Attempt area ABC and distance
between DandΠ | M1
4k(cid:14)28(cid:14)14(cid:16)21
D to Π is
16(cid:14)49(cid:14)1
1 4k(cid:14)28(cid:14)14(cid:16)21
6 11 (cid:32)6
6 16(cid:14)49(cid:14)1 | 1
M1: Set (their area x their
3
distance) = 6 | dM1
A1
A1: Correct equation
15
k (cid:32)
4 | Cao (oe) | A1
(4)
(9 marks)
Question | Scheme | Marks
The coordinates of the points $A$, $B$ and $C$ relative to a fixed origin $O$ are $(1, 2, 3)$, $(-1, 3, 4)$ and $(2, 1, 6)$ respectively. The plane $\Pi$ contains the points $A$, $B$ and $C$.

\begin{enumerate}[label=(\alph*)]
\item Find a cartesian equation of the plane $\Pi$.
[5]
\end{enumerate}

The point $D$ has coordinates $(k, 4, 14)$ where $k$ is a positive constant.

Given that the volume of the tetrahedron $ABCD$ is 6 cubic units,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $k$.
[4]
\end{enumerate}

\hfill \mbox{\textit{Edexcel F3 2018 Q6 [9]}}
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