OCR FP1 AS 2017 December — Question 7 7 marks

Exam BoardOCR
ModuleFP1 AS (Further Pure 1 AS)
Year2017
SessionDecember
Marks7
TopicVector Product and Surfaces
TypeVector product properties and identities
DifficultyStandard +0.8 This question requires understanding of matrix composition, shear transformations, and solving a matrix equation by multiplying two unknown shear matrices and equating to the given form. While the concepts are standard FP1 material, students must correctly set up the product of two shear matrices with unknown parameters and solve the resulting system of equations, which involves more algebraic manipulation and conceptual understanding than routine matrix multiplication exercises.
Spec4.03d Linear transformations 2D: reflection, rotation, enlargement, shear
A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of \(s\) the matrices which represent each of the shears. [7]
AnswerMarks Guidance
Answer: \(\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}\) or \(\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}\)B1 Correct form of either shear matrix
Answer: \(\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix} \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}\)M1 Must be correct order. Must be different parameters.
Answer: \(\begin{pmatrix} 1 & a \\ b & 1 + ab \end{pmatrix}\)A1ft
Answer: \(ab + 1 = 0\)A1 Comparing
Answer: \(s = a\) and \(t = b\) and so \(t = -\frac{1}{s}\)A1 Comparing and deducing
Answer: Shear parallel to x-axis: \(\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}\)A1
Answer: Shear parallel to y-axis: \(\begin{pmatrix} 1 & 0 \\ -\frac{1}{s} & 1 \end{pmatrix}\)A1 If both matrices correct but not clearly identified then A1A0. 
Answer: $\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}$ or $\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}$ | B1 | Correct form of either shear matrix

Answer: $\begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix} \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}$ | M1 | Must be correct order. Must be different parameters.

Answer: $\begin{pmatrix} 1 & a \\ b & 1 + ab \end{pmatrix}$ | A1ft |

Answer: $ab + 1 = 0$ | A1 | Comparing

Answer: $s = a$ and $t = b$ and so $t = -\frac{1}{s}$ | A1 | Comparing and deducing

Answer: Shear parallel to x-axis: $\begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$ | A1 |

Answer: Shear parallel to y-axis: $\begin{pmatrix} 1 & 0 \\ -\frac{1}{s} & 1 \end{pmatrix}$ | A1 | If both matrices correct but not clearly identified then A1A0.

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A transformation is equivalent to a shear parallel to the $x$-axis followed by a shear parallel to the $y$-axis and is represented by the matrix $\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}$.

Find in terms of $s$ the matrices which represent each of the shears. [7]

\hfill \mbox{\textit{OCR FP1 AS 2017 Q7 [7]}}
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