Originally posted by tangjing:
Wah you long term memory, the numbers remember until so exact.
My friend tell me plot graph for that question. How did you play around with the calculator? That's the question I didn't know how to do in the paper. Horrorrr. Question 9 (iii) and (v). They are linked, so too bad for me =/ I know how to do (iv) though.
Then I also spotted a careless mistake after the paper.
Question 11, the sketching of the graph. I went to draw both the graph below and above the x-axis when you're only required to draw the one above. Sigh =/
Ok now let me recall all the various questions...
Qn 11)
I got the graph wrong by including the bottom part too. 1 mark gone. Anyway for the 2nd part, it is possible to show the expression although the step is tedious (I did and cancelled my original answer twice. Did in pencil on the qn paper before copying the solution onto my paper later on). Integration one was rather easy. Just need to eliminate the cos t term by multiplying by dt/du which is 1/cos t.
Qn 10)
I believe the majority were asking about the values for n which S > 4a. Ok at first, u have to find ur d in terms of a. Use ur ar - a = 3d to get the result that d = -1/9 a. Then sub this expression back into the inequality. You will get a quadratic inequality of something like an^2 - 19an + 72a < 0. You can divide by a as a > 0 as defined by the question. Therefore u get n^2 - 19n + 72<0. Solve for n which should be in the range of (5+,13+), therefore n should be 6 to 13 as n is an integer.
Qn 9)
2nd part: When X1=0, X will increase towards alpha. X1 = 1, X will decrease towards alpha. X1 = 2, X will increase to infinity.
3rd part: Proving that X(n+1)>Xn for Xn between values of alpha and beta. To prove it, first express X(n+1)-Xn = 1/3 * e^(Xn) - Xn. Let Xn be x, therefore expression will give you 1/3 of the equation of the graph (e^x - 3x). When Xn is between alpha and beta, e^x - 3x is always less than zero. Since 1/3 * anything negative is less than zero, therefore X(n+1) - Xn < 0, X(n+1) < Xn. You can prove the other case using this method too.
That's all for qn 9, 10 and 11.
*EDIT* Range of values for n should be 6 to 13, not 5 to 13. Sorry if I made anyone panic.