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OCR B (MEI) AS Further Maths Exam Q8 (2020) - MATRIX DETERMINANT AND REFLECTION

Hello, I was doing the 2020 OCR B (MEI) AS FURTHER MATHS CORE PURE PAPER and I was puzzled by the mark scheme.

Question 8 asks you to use determinants to “investigate whether N can represent a reflection”, where N is [k+10kk+2]\begin{bmatrix} k+1 & 0 \\k & k+2 \end{bmatrix}

The reason I am puzzled is that in the MS they set the determinant = -1, whereas I set it as <0. My reasoning was that if the determinant is negative, then the transformation includes a reflection. So, if we set (k+1)(k+2) < 0, and find a range of values of k for which the statement is true, then we have proven that N can indeed represent a reflection under those conditions. Since (k+1)(k+2) < 0 when -2 < k < -1, then N can indeed represent a reflection. Not for the MS though…

Could someone tell me what is wrong with my reasoning?

Thank you

PS: I have attached the question and the relevant mark scheme

Screen Shot 2024-04-26 at 19.37.03.png Screen Shot 2024-04-26 at 19.37.19.png
(edited 2 months ago)
Reply 1
Original post by skypestro
Hello, I was doing the 2020 OCR B (MEI) AS FURTHER MATHS CORE PURE PAPER and I was puzzled by the mark scheme.
Question 8 asks you to use determinants to “investigate whether N can represent a reflection”, where N is [k+10kk+2]\begin{bmatrix} k+1 & 0 \\k & k+2 \end{bmatrix}
The reason I am puzzled is that in the MS they set the determinant = -1, whereas I set it as <0. My reasoning was that if the determinant is negative, then the transformation includes a reflection. So, if we set (k+1)(k+2) < 0, and find a range of values of k for which the statement is true, then we have proven that N can indeed represent a reflection under those conditions. Since (k+1)(k+2) < 0 when -2 < k < -1, then N can indeed represent a reflection. Not for the MS though…
Could someone tell me what is wrong with my reasoning?
Thank you
PS: I have attached the question and the relevant mark scheme
Screen Shot 2024-04-26 at 19.37.03.png Screen Shot 2024-04-26 at 19.37.19.png

id imagine because it is purely a reflection, not a reflection and enlargement so that is why you set det(M)=-1 and then work from there rather than give a range because a range would include a reflection AND enlargement
Reply 2
Original post by aryanh
id imagine because it is purely a reflection, not a reflection and enlargement so that is why you set det(M)=-1 and then work from there rather than give a range because a range would include a reflection AND enlargement

Thank you :smile:

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