Quod Erat Demonstrandum
Solutions to F.4 Mathematics Easter Holiday Assignment (2010-03-31)
1. [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
2. [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
3. [pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
4.
(a) [pic]
[pic]
[pic]
[pic]
[pic] ([pic]it is given that [pic])
[pic]
[pic]
= 0 (given)
(b) [pic]
[pic]
[pic]
[pic] (by (a), [pic])
[pic]
[pic] (by (a), [pic])
[pic]
[pic]
= 0 (by (a))
5. Wrong question
6. Wrong question, change the following: replace “[pic]” by “[pic]”, and replace
[pic] by [pic]
Solution
Consider the sum and product of roots, we have
[pic]
The second equation will be [pic] ( [pic]
Hence [pic] and [pic] (since x lies on the second quadrant)
By the first equation in the system, [pic] ( [pic] ( [pic]
7. Consider the sum and product of roots, we have
[pic]
Squaring the first equation, yield
[pic] ( [pic]
By the second equation, yield
[pic]
[pic] ( [pic] or [pic]
But, for the original equation has real roots, [pic] ( [pic] ( [pic]
Hence k = 2 should be rejected.
ANS: k = −8
8. [pic] ( [pic]
( [pic]
( [pic] or [pic] (rejected since A lies on the second quadrant)
( [pic]
Hence, [pic]
9. [pic]
[pic]
[pic]
[pic]
[pic]
10.
(a) sinx + 2cosx = 0
[pic]
[pic]
[pic]
[pic] or [pic]
[pic] or [pic]
(b) sin2x = cos3x
[pic]
[pic]
[pic] ( [pic]
[pic] ( [pic]
[pic] ( [pic]
[pic] ( [pic]
[pic] ( [pic]
(c) cos2x + 3sinx = 3
[pic]
[pic]
[pic]
[pic] or [pic] (rejected)
[pic] or [pic]
[pic] or [pic]
(d) 2sinx + cosx = 1
Squaring,
[pic]
[pic]
[pic]
[pic]
[pic] or [pic]
[pic] or [pic]
[pic] or [pic] or [pic]
11.
(a) cos2x + sin3x
[pic]
[pic]
[pic]
[pic]
= 0
(b) sin3A
[pic]
[pic]
[pic]
(c) Not-good-set question, the equation should not be 4x3 + 2x2 ( 3x ( 1 = 0 (confusing with the
fact that x = 54o), instead, we may use any symbol except x, like, 4y3 + 2y2 ( 3y ( 1 = 0.
Solution
[pic] (by (a))
[pic] (by (b), put A = 54o)
[pic]
[pic]
Hence, [pic] is a root of
4y3 + 2y2 ( 3y ( 1 = 0.
(d) [pic]
( [pic] (by synthetic division, factor theorem, … whatever)
( [pic] or [pic] (rejected because y = sin54o ( −1)
Hence sin54o = [pic] or [pic] (rejected)
Thus, sin54o = [pic].
12. Consider the sum and product of roots, we have
[pic]
[pic]
Let [pic], then by above,
[pic]
( [pic]
( [pic]
( [pic]
For [pic], [pic]
For [pic], [pic]
13. Maximum value of y is 4 ( 4 = A(1) + B ( A + B = 4
Minimum value of y is −2 ( −2 = A(−1) + B ( −A + B = −2
Hence A = 3, B = 1.
Least positive period = ( ( k( = 2( ( k = (
The curve passes ([pic], 4) ( [pic] ( [pic] (say)
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