A gambler bets on one of the integers from 1 to 6. Three fair dice are then rolled. If the gambler’s number appears times (), he wins $ . If his number fails to appear, he loses $1. Calculate the gambler’s expected winnings
Random Sec 4 Differentiations
B6
let
Sub to
Thus, it is a min point.
C7
let
(NA).
There are no stationary points for this curve.
C8
Let
Evaluate with a calculator…
Vectors Question #4
Another interesting vectors question.
The fixed point has position vector a relative to a fixed point . A variable point has position vector r relative to . Find the locus of if r (r – a) = 0.
Vectors Question #3
This is a question a student sent me a few days back, and I shared with my class.
Find the Cartesian equation of the locus of all points (plane) that is equidistant of the plane and plane.
The following should aid students to visualise.
Sidenote: I think Vectors is a very important topic for 9758 as its applications are wide. Students should do their best to understand the topic. I will share a few more applied questions next week when I have time.
A little reminder to students doing Calculus now
When , it implies we have a stationary point.
To determine the nature of the stationary point, we can do either the first derivative test or the second derivative.
The first derivative test:
Students should write the actual values of and in the table.
We use this under these two situations:
1. is difficult to solve for, that is, is tough to be differentiated
2.
The second derivative test:
Other things students should take note is concavity and drawing of the derivative graph.
Release of A’levels Results 2016
For students who took A’levels in 2016, please note that information for the release of A’levels Results 2016 can be found in the following!
Grade Profile (i.e. Number of As you need to get into courses for)
P.S. Results does not define you. When one door closes, another opens.
Checklist for Vectors
Many schools have been doing vectors recently. Thought I’ll share a little summary/ checklist I have done for my students.
Basic Concepts
 Operations on Vectors
 Addition & Subtraction
 Scalar multiplication
 Dot Product (Scalar)
 a • a = a^{2}
 If a ⊥ b, then a • b = 0
 a • b = b • a
 Cross Product (Vector)
 a × a = 0
 a × b = − b × a
 Unit Vectors
 Parallel Vectors ( a = kb )
 Collinear Vectors ( Parallel with a common point )
 Ratio Theorem ( Found in MF26)
 Midpoint Theorem
 Directional Cosines
 Angles between two Vectors
 Length of Projection
 Perpendicular Distance
Lines
 Equations
 Vector Form ( l : r = a + λb, λ∈ ℜ )
 Parametric Form
 Cartesian Form
 Line & Line
 Parallel ( Directions are parallel to each other. )
 Same ( Same Equations )
 Intersecting ( There is a unique solution for λ and μ. )
 Skewed ( Not parallel AND not Intersecting. )
 Angle between two lines ( Angle between their directions )
 Point & Line
 Foot of Perpendicular
 Perpendicular (Shortest) distance
 Point on Line
Planes
 Equations
 Parametric Form ( π : r = a + λb + μc, λ, μ ∈ ℜ )
 Scalar Product Form ( r • n = a • n = d )
 Cartesian Form
 Point & Plane
 Foot of Perpendicular
 Perpendicular (Shortest) distance
 Distance from O to Plane
 Point on Plane
 Reflection of Point
 Line & Plane
 Relationships
 Parallel
 Line intersects Plane entirely ( Infinite Solutions )
 Do not intersect ( No Solution )
 Not Parallel
 Intersects at a point ( One Solution )
 Parallel
 Intersection Point
 Angle between Line & Plane
 Reflection of Line
 Relationships
 Plane & Plane
 Relationships
 Parallel
 Same ( Infinite Solutions )
 Do not intersect ( No Solution )
 Not Parallel
 Intersects at a line ( Infinite Solutions )
 Parallel
 Intersection Line ( Use of GC )
 Angle between two Planes ( Angle between their normals )
 Relationships
Thoughts on A’levels H2 Mathematics 2016 Paper 2
I’ll keep this short since we are all busy. One thing about paper 1 we saw, there were many unknowns.
So topics which I think will come out…
Differentiation – I think a min/max problem will come out, possibly with r and h both not given and asked to express r in terms of h. But students should revise a on the properties of curves with differentiation; given a curve equation with an unknown, for instance , find the range of k such that there is stationary points.
Complex Number – Loci will definitely come out. I’m saying they will combine with trigonometry.
Integration – Modulus integration hasn’t really been tested. Else a question on Area/ Volume could be tested, and I’ll say they need students to do some
Conics too.
For statistics, my students should have gotten the h1 stats this year. And if it’s an indicator, then it should not be a struggle.
I expect PnC and probability to be combined. Conditional Probability in a poisson question should be tested too, so do revise it well. For hypothesis testing, students should be careful of their formula and read really carefully about the alternative hypothesis. Also, :9 know that the formulas for poison PDF and binompdf are both given in mf15. Lastly, know when to use CLT.
All the best!
Thoughts about 2016 A’levels H2 Mathematics Paper
I’ve covered some things in classes, with sufficient revisions and final lap papers set. So I thought we have a little breakdown. And of course, we should review what was weeded away in the 9740 H2 Mathematics Syllabus. After all, this is the very LAST time the can test them.
 Recurrence Relations. I’ve harped on this last year too. Conjectures! Conjectures! You can read more about it here. An a question involving conjecture should start with a recurrence relation, then a conjecture, ending with a recurrence MI. Students should know how to do both and recurrence MI. Yes, they are different.

Loci. You guys are the lucky last batch to do Loci. So please buy a protractor and compass. Draw them, as one of my student put it, surgically. If need be, use a graph paper (why not?). Harder loci for example can require students to draw for example, . You should not have trouble measuring this angle, because you could not even be able to do it. Students should be able to draw such angles with ease. One little note about Loci, will definitely their geometrical descriptions. Many students can draw this, but stumble to describe them.

Vectors. Truth be told, I’m still waiting for a question involving vectors in 3D, to land in A’levels. An example can be the HCI Prelims Paper 1 Question 6, which can be found here.

Poisson Distribution. I don’t know what this topic has been axed. So students should ready for one big Poisson Distribution questions, I say give it 1214 marks. And it should be tested with conditional probability. I’ll practice either Demand & Supply or Inflow & Outflow questions. An example can be the one found in NYJC Prelims Paper 2 Question 11, which can be found here.

Correlation & Regression. Ever wondered what the means in the GC? Well, is being removed from the syllabus as well. It hasn’t surfaced before, so maybe it shall finally make its one and only LAST presence felt this year. Students should familiarise themselves with the use of equation, which can be found in the MF15. I know many of you probably have not seen it before.

Hypothesis Testing. Students should review definitions of level of significance and pvalue. Also understand what you may conclude from a ZTest, using the results of a Ttest. A little small part that students can think about, is why use a small sample size? After all, we know that have a sufficiently large allows us to perform CLT and then use a ZTest.

Trigonometry. After it appeared in 2011 for a trigonometry MI, the product to sum formulas is still a problem for most students. I highly doubt its coming out again with MI, but its can easily come out again with complex numbers. An example can be this.
More examples and discussion will be made in class.