H2 Maths Exam 2009

wee_ws SG

Hi,

I have not gotten the exam paper to analyse (only can do so this evening when I get the paper from my student), but someone more diligent than me has done so already :)

For those interested to find out, please read:

http://blog.tanwj.net/mathematics/h2-mathematics/topics-for-2009-a-level-h2-mathematics/

Thanks!

Cheers,
Wen Shih

wee_ws SG

Hi,

There are several good learning points one can pick up from Q4, which focuses on concepts of the topic functions:

1. Firstly, f is a piecewise function, which is defined by more than one formula. In the question, it has two formulae defined over different intervals.

Although piecewise functions are unfamiliar to students, students could fall back on an example they have encountered before, i.e. the modulus function f(x) = |x|, where x is real.

2. Secondly, f is periodic with period 4, since f(x) = f(x + 4) for all real x.

This could be difficult to understand at first glance (and in the chaos of exam), but one may recall that sin x = sin (x + 2 pi) meaning that the graph of sine function is periodic with period 2 pi. To put simply, the graph of sine function repeats indefinitely over an interval of 2 pi. For example, the section over 0 <= x <= 2 pi is repeated over 2 pi <= x <= 4pi.

Alternatively, one may first substitute values x = 0, 1, 2, 3, 4 into f(x) = f(x + 4):

f(0) = f(4),
f(1) = f(5),
f(2) = f(6),
f(3) = f(7),
f(4) = f(8).

Now seeing that the y-values are the same between the interval 0 <= x <= 4 and 4 <= x <= 8, one can then conclude periodicity by observation.

Thanks!

Cheers,
Wen Shih

wee_ws SG

Hi,

In Q11 (iii), the student is asked to find the area of a region between the curve and the positive x-axis, using the fact that he/she has found the answer to the integral of f(x) over 0 and n with respect to x.

Here, the student has encountered an instance of the improper definite integral (an extension of the ordinary and familiar definite integrals), because the right end-point is not bounded with a real number. In this case, the student is expected to consider the limit as n approaches infinity on the answer that was found earlier.

Let me provide another (simpler) example, for the purpose of learning.

Find the area of the region between the curve y = 1/(x + 1)^2 and the positive x-axis.

First, we find the definite integral of y with respect to x for the interval 0 <= x <= n. This gives 1 - 1/(n + 1).

Next, as n approaches infinity, we see that 1/(n + 1) approaches zero and so the limit is 1.

Thus, the require area is 1 sq unit.

Now it seems to me that this set of exam questions assesses the student's ability to extend existing knowledge to unfamiliar problems using concepts that the student already possesses.

Thanks!

Cheers,
Wen Shih

wee_ws SG

Hi,

In Q9 (Complex Numbers), we see that loci can be represented in cartesian forms.

A quick summary is given below for the sake of learning:

Locus of the form |z - (a + bi)| = r
has cartesian equation given by (x - a)^2 + (y - b)^2 = r^2.

Locus of the form |z - (a + bi) = |z - (c + di)|
has cartesian equation given by y - y1 = m(x - x1),
where (x1, y1) is the midpoint of (a, b) and (c, d) and
m is the gradient of the normal to the line segment joining (a, b) and (c, d).
Note that gradient of line segment = (d - b)/(c - a).

Locus of the form arg(z - (a + bi)) = theta
has cartesian equation given by y - b = (tan theta)(x - a).

It is also extremely practical for the student to know that roots of z^n = c can be represented as points along the circumference of a circle (centred at the origin) and separated by (2 pi)/n radians between them.

Thanks!

Cheers,
Wen Shih

wee_ws SG

Hi,

In paper 2, there is a stats question in which differentiation is being applied.

The line between pure maths and stats has been blurred.

Thanks!

Cheers,
Wen Shih

tut4nkh4m3n SG

paper 1 was different. that's all i have to say about it.

paper 2 was more 'standard' except the probabilty-differentiation question.

does anyone here know about how bell curve trends are usually like? will the required mark for A go up to as high as >75?

wee_ws SG

Hi,

I'd like to take an opporunity to share a question involving an arbitrary probability (paper 2 has this type of question too) and invite discussions.

Two events A and B are independent and defined in the same finite sample space. Given that P(A) = 1/4 and P(A union B) = p (p is non-negative and p < 1), find

(i) P(B),

(ii) P(A intersect B | A union B),

in terms of p.

Find a stronger condition for p such that both (i) and (ii) may always be valid.

Thanks!

Cheers,
Wen Shih

wee_ws SG

Hi,

Paper 1 Q10 is a question about Vectors.

What is particularly interesting is in the last part, where the student is asked to find the cartesian equation of the plane.

If one is not careful and gives the answer as x - y = -1, it is not complete...because nothing is said about the z-component.

The missing piece of information is to include the fact that z is any value in the real set. If not, x - y = -1 is just a line.

Recall that any point on the plane has three coordinates. So in our case, the point will look like (x, y, k) such that x - y = -1 and k is any real number. Points that lie on this plane are (12, 13, 0), (-4, -3, -90978376498305938) and many others, as long as the first two coordinates satisfy the condition that x - y = -1. We need not worry what z takes.

To add to learning, I found this read an insightful one for any student:

http://www.netcomuk.co.uk/~jenolive/vect16.html

Thanks!

Cheers,
Wen Shih

absol SG

i made that mistake for vectors -_-

Sputnik159 SG

Wah A got so high one meh?? i hope A is 70 and above looking at the 8 marks so many ppl lost in paper 1 lol. The stupid complex questions also headache.