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Specialist Maths — Question 12

QCAA 2025, Paper 1 · 6 marks

Q12 · 2025 · Technology-freeSimple familiar6 marks

QUESTION 12 (6 marks)

A line, ll, is given by the equation x+1−2=y−32=z−2.\frac{x+1}{-2}=\frac{y-3}{2}=z-2.
a)
Given the point (−1,3,a)(-1,3,a) lies on the line, determine the value of aa.
[1 mark]
b)
Determine a vector, d\mathbf d, in the direction of the line.
[1 mark]
c)
A plane, φ\varphi, is given by the equation x−y+4z=8x-y+4z=8. Given the point (b,b,−2b)(b,b,-2b) lies on the plane, determine the value of bb.
[1 mark]
d)
Determine a vector, n\mathbf n, that is normal to the plane.
[1 mark]
e)
Show that d⋅n=0\mathbf d\cdot\mathbf n=0.
[1 mark]
f)
Consider the statement: “The line, ll, is perpendicular to the plane, φ\varphi.” Use your result from Question 12e) to comment on the reasonableness of the statement.
[1 mark]
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