Q241 · Practice questionTechnology-freeComplex unfamiliar8 marks
QUESTION 241 (8 marks)
Three flat sensor screens are modelled by
where .
a)[2 marks]
Determine the determinant of the coefficient matrix and the values of a for which it is singular.
b)[4 marks]
Classify all parameter pairs (a,b) according to whether there is a unique common point, no common point, or infinitely many. Give the common point or intersection line where applicable.
c)[2 marks]
For a=3 and b=10, determine whether the common point lies on the sphere .
WORKED SOLUTION
8 marksPractice marking scheme
ANSWER
(a) ; singular when a=1. (b) If : . If a=1,b=6: line . If : no common point. (c) Yes: lies on the sphere.
Worked solution
(a) The coefficient rows are (1,1,1),(1,-1,1),(1,1,a). Subtract the first row from the third: (0,0,a-1). Expansion along this row gives . Hence the determinant vanishes exactly when a=1.
(b) Subtracting the first two equations gives y=2 and then x+z=4. Subtracting the first from the third gives . If a is not 1, z is uniquely determined, followed by x and y. If a=1,b=6, the third equation repeats the first, leaving the line . If a=1,b is not 6, the last reduced equation is contradictory.
(c) The unique case gives z=(10-6)/(3-1)=2, y=2 and x=2. Substitution into the sphere gives 4+4+4=12, so the point lies on its surface.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Calculate the determinant.
Part a: Find exactly the singular parameter value.
Part b: Reduce to y=2, x+z=4 and the parameter equation.
Part b: Give the unique case with all coordinates.
Part b: Give the infinite case with its full line.
Part b: Identify the inconsistent case.
Part c: Find the common point.
Part c: Check sphere incidence and conclude.
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