Edexcel CP AS 2020 June — Question 1 6 marks

Exam BoardEdexcel
ModuleCP AS (Core Pure AS)
Year2020
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMatrices
TypeConditions for unique solution
DifficultyStandard +0.8 This question requires understanding of determinants, matrix singularity conditions, and geometric interpretation of linear systems. Part (a) involves setting det(A)=0 and solving a cubic/quadratic equation. Parts (b) and (c) require checking consistency via row reduction and interpreting three planes geometrically—going beyond routine matrix manipulation to require conceptual understanding of when systems have no solution vs infinitely many solutions.
Spec4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations
  1. A system of three equations is defined by
$$\begin{aligned} k x + 3 y - z & = 3 \\ 3 x - y + z & = - k \\ - 16 x - k y - k z & = k \end{aligned}$$ where \(k\) is a positive constant.
Given that there is no unique solution to all three equations,
  1. show that \(k = 2\) Using \(k = 2\)
  2. determine whether the three equations are consistent, justifying your answer.
  3. Interpret the answer to part (b) geometrically.
Question 1:
Part (a)
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\begin{vmatrix} k & 3 & -1 \\ 3 & -1 & 1 \\ -16 & -k & -k \end{vmatrix} = k(k+k)-3(-3k+16)-1(-3k-16)\)M1 2.1
Solves \(\det = 0 \Rightarrow 2k^2+12k-32=0\) or \(k^2+6k-16=0\); achieves \(k=2\) (\(k=-8\) must be rejected)A1 1.1b
Special Case:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Substitutes \(k=2\) into determinant: \(2(2+2)-3(-3\times2+16)-1(-3\times2-16)\)M1 2.1
Shows \(\det=0\), therefore when \(k=2\) there is no unique solutionA0 1.1b
Part (b)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Eliminates one variable to achieve two equations in two unknowns, e.g. \(5x+2y=1\), \(-10x-4y=-2\), \(20x+8y=4\)M1 3.1a
Correct pair of equations shownA1 1.1b
Must give a reason: e.g. two equations are a linear multiple of each other / shows they are the same equation, therefore the equations are consistentA1 2.4
Alternative:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Eliminates two different variables to form two equations; rearranges and substitutes into one of the original equations in three variables, e.g. \(2x+3\left(\dfrac{1-5x}{2}\right)-\left(\dfrac{-3-11x}{2}\right)=3\)M1 3.1a
Correct equations e.g. \(5x+2y=1\) and \(11y-5z=13\)A1 1.1b
Shows equations are a solution e.g. \(3=3\), therefore consistentA1 2.4
Part (c)
AnswerMarks Guidance
Answer/WorkingMark Guidance
The three planes form a sheafB1 2.2a
(6 marks total)
Question 1 (Planes/Simultaneous Equations):
Part (a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Finds the determinant of the matrix corresponding to the system of equationsM1 Complete method required
Sets determinant \(= 0\) and solves to achieve \(k = 2\) (\(k = -8\) must be rejected)A1
Special case: Uses \(k=2\) and finds determinant of matrix; shows determinant \(= 0\) and concludes no unique solutionM1A0
Part (b):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Eliminates one variable using two different pairs of equationsM1 Condone if different value of \(k\) used
Achieves two equations in the same two variablesA1
Shows equations are a linear multiple of each other, therefore consistentA1 Must give a reason
Alternative:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Eliminates one variable using two different pairs; substitutes into original equationM1
Achieves two correct equations in two different variablesA1
Shows equation works, therefore consistentA1
Part (c):
AnswerMarks Guidance
Answer/WorkingMark Guidance
The three planes form a sheafB1 Must have full marks in (b) to award 
## Question 1:

### Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{vmatrix} k & 3 & -1 \\ 3 & -1 & 1 \\ -16 & -k & -k \end{vmatrix} = k(k+k)-3(-3k+16)-1(-3k-16)$ | M1 | 2.1 |
| Solves $\det = 0 \Rightarrow 2k^2+12k-32=0$ or $k^2+6k-16=0$; achieves $k=2$ ($k=-8$ must be rejected) | A1 | 1.1b |

**Special Case:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $k=2$ into determinant: $2(2+2)-3(-3\times2+16)-1(-3\times2-16)$ | M1 | 2.1 |
| Shows $\det=0$, therefore when $k=2$ there is no unique solution | A0 | 1.1b |

---

### Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Eliminates one variable to achieve two equations in two unknowns, e.g. $5x+2y=1$, $-10x-4y=-2$, $20x+8y=4$ | M1 | 3.1a |
| Correct pair of equations shown | A1 | 1.1b |
| Must give a reason: e.g. two equations are a linear multiple of each other / shows they are the same equation, therefore the equations are **consistent** | A1 | 2.4 |

**Alternative:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Eliminates two different variables to form two equations; rearranges and substitutes into one of the original equations in three variables, e.g. $2x+3\left(\dfrac{1-5x}{2}\right)-\left(\dfrac{-3-11x}{2}\right)=3$ | M1 | 3.1a |
| Correct equations e.g. $5x+2y=1$ and $11y-5z=13$ | A1 | 1.1b |
| Shows equations are a solution e.g. $3=3$, therefore **consistent** | A1 | 2.4 |

---

### Part (c)

| Answer/Working | Mark | Guidance |
|---|---|---|
| The three **planes** form a **sheaf** | B1 | 2.2a |

**(6 marks total)**

# Question 1 (Planes/Simultaneous Equations):

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Finds the determinant of the matrix corresponding to the system of equations | M1 | Complete method required |
| Sets determinant $= 0$ and solves to achieve $k = 2$ ($k = -8$ must be rejected) | A1 | |

**Special case:** Uses $k=2$ and finds determinant of matrix; shows determinant $= 0$ and concludes no unique solution | M1A0 | |

## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Eliminates one variable using two different pairs of equations | M1 | Condone if different value of $k$ used |
| Achieves two equations in the same two variables | A1 | |
| Shows equations are a linear multiple of each other, therefore **consistent** | A1 | Must give a reason |

**Alternative:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Eliminates one variable using two different pairs; substitutes into original equation | M1 | |
| Achieves two correct equations in two different variables | A1 | |
| Shows equation works, therefore **consistent** | A1 | |

## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The three **planes** form a **sheaf** | B1 | Must have full marks in (b) to award |

---
\begin{enumerate}
  \item A system of three equations is defined by
\end{enumerate}

$$\begin{aligned}
k x + 3 y - z & = 3 \\
3 x - y + z & = - k \\
- 16 x - k y - k z & = k
\end{aligned}$$

where $k$ is a positive constant.\\
Given that there is no unique solution to all three equations,\\
(a) show that $k = 2$

Using $k = 2$\\
(b) determine whether the three equations are consistent, justifying your answer.\\
(c) Interpret the answer to part (b) geometrically.

\hfill \mbox{\textit{Edexcel CP AS 2020 Q1 [6]}}
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