MC maths 92 36,34

92)
Q 34) Ans : C ( II, I, III )
Consider the slopes of the curves in the middle part.
For graph II, it has the steepest slope in the middle and it has the greatest number of data concentrated there. Thus its s.d. is the smallest.
For graph I, it has a less steep slope in the middle and the data are more dispersed. Thus its s.d. is greater than II.
For graph III, it has the least steep slope in the middle and the data are the most dispersed. Thus its s.d. is the greatest.
36) Ans : E
This Q is out of the syllabus now.
Let f ( x ) = 2x3 - x2 - x - 3
f ( -1 ) = 2 ( - 1 )3 - ( - 1 )2 - ( - 1 ) - 3 = - 5 < 0
f ( 1 ) = 2 ( 1 )3 - ( 1 )2 - 1 - 3 = - 3 < 0
So there are no roots in - 1 < x < 1.
f ( 0 ) = 2 ( 0 )3 - 02 - 0 - 3 = -3 < 0
f ( 2 ) = 2 ( 2 )3 - 22 - 2 - 3 = 7 > 0
So in 0 < x < 2, it must contain 1 root.
f ( 1 ) = - 3 < 0
f ( 3 ) = 2 ( 3 )3 - 32 - 3 - 3 = 39 > 0
So in 1 < x < 3, it must contain 1 root.
Hence the answer is II and III only.

2008-03-25 18:41:35 補充：
Yes, it's the remainder but it's not so related to how the answer is arrived.
We know that f ( 0 ) < 0 and f ( 2 ) > 0 and so think about the graph, you know that it must cut the x - axis in between and here we get that f ( x ) = 0 must contain a root in the range.

2008-03-25 18:42:14 補充：
It also works the same for 1 < x < 3.