OCR MEI Further Pure Core 2020 November — Question 2 6 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2020
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMatrices
TypeSolving matrix equations for unknown matrix
DifficultyStandard +0.3 This is a straightforward matrix multiplication problem requiring students to multiply two matrices, equate to the identity matrix, and solve simultaneous equations for three unknowns. The conceptual twist about whether N is the inverse (requiring recognition that M is non-square) adds minimal difficulty. Slightly easier than average as it's mostly mechanical computation with standard techniques.
Spec4.03b Matrix operations: addition, multiplication, scalar4.03o Inverse 3x3 matrix
2
  1. The matrices \(\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a \\ 1 & b & 0 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { c c } b & - 5 \\ - 1 & c \\ - 1 & 1 \end{array} \right)\) are such that \(\mathbf { M } \mathbf { N } = \mathbf { I }\).
    Find \(a , b\) and \(c\).
  2. State with a reason whether or not \(\mathbf { N }\) is the inverse of \(\mathbf { M }\).
Question 2:
AnswerMarks Guidance
2(a) −1 – a = 1 ⇒ a = −2
​ ​ ​ ​
c + a = 0 ⇒ c = 2
​ ​ ​ ​​
−5 + bc = 1 ⇒ b = 3
AnswerMarks
​ ​ ​ ​B1
M1
A1
A1ft
A1
AnswerMarks
[5]1.1a
1.1
1.1
1.1
AnswerMarks
1.1matrix multiplication
3 correct equations
their value of a from their
equations
AnswerMarks Guidance
all 3 values correctSoi
2(b) M is not a square matrix and so has no inverse
​B1
[1]2.4 MN=/ NM
different orders
PPMMTT
Y420/01 Mark SchemeNovember 2020
3 DR
M1 3.1a must be in the form k can award M1A1 if integral is
√4−x2
9 fully correct before limits
k=/ 1 substituted
A1 1.1 k arcsin(3x/2)
​ ​​
= 1(arcsin1[−arcsin0]) M1 1.1
3 2
A1 1.1
OR
M1 for suitable substitution
A1 an equivalent expression that
can be integrated
M1 substitution of correct limits
A1 of their variable
[4]
4 (a) 2 x 3 ​– 5 x + 7 = 0
​ ​ ​ ​
[α+β+γ = 0],βγ +αγ +αβ =− 5, αβγ =− 7 B1 1.1a
2 2
B1 3.1a
M1 1.1
A1 1.1
[4]
7
Question 2:
2 | (a) | −1 – a = 1 ⇒ a = −2
​ ​ ​ ​
c + a = 0 ⇒ c = 2
​ ​ ​ ​​
−5 + bc = 1 ⇒ b = 3
​ ​ ​ ​ | B1
M1
A1
A1ft
A1
[5] | 1.1a
1.1
1.1
1.1
1.1 | matrix multiplication
3 correct equations
their value of a from their
equations
all 3 values correct | Soi
2 | (b) | M is not a square matrix and so has no inverse
​ | B1
[1] | 2.4 | MN=/ NM
different orders
PPMMTT
Y420/01 Mark SchemeNovember 2020
3 DR
M1 3.1a must be in the form k can award M1A1 if integral is
√4−x2
9 fully correct before limits
k=/ 1 substituted
A1 1.1 k arcsin(3x/2)
​ ​​
= 1(arcsin1[−arcsin0]) M1 1.1
3 2
A1 1.1
OR
M1 for suitable substitution
A1 an equivalent expression that
can be integrated
M1 substitution of correct limits
A1 of their variable
[4]
4 (a) 2 x 3 ​– 5 x + 7 = 0
​ ​ ​ ​
[α+β+γ = 0],βγ +αγ +αβ =− 5, αβγ =− 7 B1 1.1a
2 2
B1 3.1a
M1 1.1
A1 1.1
[4]
7
2
\begin{enumerate}[label=(\alph*)]
\item The matrices $\mathbf { M } = \left( \begin{array} { c c c } 0 & 1 & a \\ 1 & b & 0 \end{array} \right)$ and $\mathbf { N } = \left( \begin{array} { c c } b & - 5 \\ - 1 & c \\ - 1 & 1 \end{array} \right)$ are such that $\mathbf { M } \mathbf { N } = \mathbf { I }$.\\
Find $a , b$ and $c$.
\item State with a reason whether or not $\mathbf { N }$ is the inverse of $\mathbf { M }$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2020 Q2 [6]}}
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