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Open book, calculators allowed. SHOW ALL OF YOUR WORKING.
1. [7 points ] Find all solutions of the equation jx , 10j = x2 , 10x. [Hint: I suggest you
begin by writing down the denition jaj = : : :]
Solution: Split into cases when (1 point)
(i) x , 10 0; and
(ii) x , 10 < 0.
Hence
Case (i) For x 10 we have jx , 10j = x , 10 so the equation is (1 point)
x , 10 = x2 , 10x:
Solve this equation by any method to get x = 1; 10 (1 point) but then
must observe that x = 1 is extraneous (1 point) because it doesn't lie in
the region x 10.
Case (ii) For x < 10 we have jx , 10j = ,(x , 10) = ,x +10 so the equation
becomes (1 point)
,x + 10 = x2 , 10x:
Solve this equation by any method to get x = ,1; 10 (1 point) but then
must observe that x = 10 is extraneous (1 point) because it doesn't lie
in the region x < 10.
Finally, the answer is therefore x = 10; ,1. (1 point)
2. [3 points ] Find two additional points on the line with slop m = 2 passing through the
point (,5; 4).
Solution: The equation of the line is (1 point)
y , 4 = 2 (x + 5); or equivalently y = 2x + 16:
Points on the line could be, e.g., (0; 16); (1; 18). (1 point for each).
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