CAIE Further Paper 1 2024 November — Question 1 10 marks

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2024
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear transformations
TypeCombined transformation matrix product
DifficultyStandard +0.3 This is a straightforward Further Maths question on linear transformations requiring standard techniques: composing transformation matrices, finding invariant lines by solving (M-I)x=0, finding inverse matrices, and using determinants for area scaling. All parts follow routine procedures with no novel insight required, making it slightly easier than average even for Further Maths.
Spec4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03o Inverse 3x3 matrix
The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).
  1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
  2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]
The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).
  1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
  2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]
Question 1:
AnswerMarks
1(a)k 0
 
AnswerMarks Guidance
0 1B1 [Stretch parallel to the x-axis, scale factor k
(k0)]. (Allow without identification.)
1 k
 
AnswerMarks Guidance
0 1B1 [Shear, x-axis fixed, with (0,1) mapped to
(k,1).]
(Allow without identification.)
1 kk 0 k k
M=  = 
AnswerMarks Guidance
0 10 1 0 1M1 A1 Correct order for M1, must have identified
which matrix gives which transformation, AG.
4
AnswerMarks
1(b)k kx kx+ky
  = 
AnswerMarks Guidance
0 1y  y B1 x X
Transforms   to  
y Y 
AnswerMarks Guidance
kx+ky=xM1 X x
Sets  = 
Y  y
1−k
y= x oe
AnswerMarks
kA1
3
AnswerMarks
1(c)11 −k
M−1=
 
AnswerMarks Guidance
k0 k B1 (An alternative is possible.)
1
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
1(d) k
B1.
k 0k =1
AnswerMarks
3A1
2
AnswerMarks Guidance
QuestionAnswer Marks 
Question 1:
--- 1(a) ---
1(a) | k 0
 
0 1 | B1 | [Stretch parallel to the x-axis, scale factor k
(k0)]. (Allow without identification.)
1 k
 
0 1 | B1 | [Shear, x-axis fixed, with (0,1) mapped to
(k,1).]
(Allow without identification.)
1 kk 0 k k
M=  = 
0 10 1 0 1 | M1 A1 | Correct order for M1, must have identified
which matrix gives which transformation, AG.
4
--- 1(b) ---
1(b) | k kx kx+ky
  = 
0 1y  y  | B1 | x X
Transforms   to  
y Y 
kx+ky=x | M1 | X x
Sets  = 
Y  y
1−k
y= x oe
k | A1
3
--- 1(c) ---
1(c) | 11 −k
M−1=
 
k0 k  | B1 | (An alternative is possible.)
1
Question | Answer | Marks | Guidance
--- 1(d) ---
1(d) | |k|=3k2 | M1 | Uses that detM=k. Without modulus is SC
B1.
k 0k =1
3 | A1
2
Question | Answer | Marks | Guidance
The matrix $\mathbf{M}$ represents the sequence of two transformations in the $x$-$y$ plane given by a stretch parallel to the $x$-axis, scale factor $k$ ($k \neq 0$), followed by a shear, $x$-axis fixed, with $(0, 1)$ mapped to $(k, 1)$.

\begin{enumerate}[label=(\alph*)]
\item Show that $\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}$. [4]

\item The transformation represented by $\mathbf{M}$ has a line of invariant points.

Find, in terms of $k$, the equation of this line. [3]
\end{enumerate}

The unit square $S$ in the $x$-$y$ plane is transformed by $\mathbf{M}$ onto the parallelogram $P$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find, in terms of $k$, a matrix which transforms $P$ onto $S$. [1]

\item Given that the area of $P$ is $3k^2$ units$^2$, find the possible values of $k$. [2]
\end{enumerate}

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q1 [10]}}
This paper (7 questions)
View full paper
Q1 10 Q2 6 Q3 10 Q4 8 Q5 13 Q6 13 Q7 15