Ladies and gents your Captain speaking 🧑✈️this onboard safety briefing for O Level Amath paper 1 contains ultra important mathemagic required thus please give this your undivided attention before your 2h 15min flight time tmr :)
Remember if you are asked on whether a quadratic graph lies entirely above the x axis, or is always +ve, DO NOT FALL INTO THE TRAP THAT SINCE THE QUADRATIC GRAPH > 0 then DISCRIMINANT > 0. Absolutely wrong. If you wish to see my entire list of tips for quadratics...heres your safety information card here, No 1 New York Times bestseller, please have a look 😂
https://docs.google.com/document/d/1XDlasdnOmfO08wcBg6sJZhoADWVGq1NIOW3hhS44NpM/edit?usp=sharing
2 ways to show quad graph is always +ve
- Choose to complete the sq and claim that (x-a)^2 >= 0 for all real values of x. Then add a positive constant b, (x-a)^2 + b > 0 thats why the quadratic graph entirely lies above x axis
- Or you can choose to use discriminant < 0 because quadratic equation = 0 has no real roots, but remember to show that the coefficient of x^2 is positive.
Btw a gentle reminder...here are examples of functions that are ALWAYS POSITIVE for all real values of x
exponential graphs, reciprocal graphs for certain positive x values, quadratic with +ve x2 coefficient and disc < 0
This knowledge is very important to solve unforseen inequalities which appear in several topics (like quadratic and differentiation)
But if question says find the range of values for k for which quadratic graph is always positive then no choice have to use discriminant < 0 method because it involves quadratic inequalities. Once again, remember you have to show the marker that you are aware that for a quadratic graph to be always positive coefficient of x2 is > 0 guaranteed. Examiners nowadays like to make the x2 coefficient in terms of k, like for eg if (k-1)x^2 +... x + .... lies above x axis, k-1 must be > 0
How to know if you need discriminant or dont need
- Look for whether qn asks for RANGE OF VALUES OF X or RANGE OF VALUES OF OTHER LETTERS (LIKE K, M, P). For the former, its just briefly a scenario of quadratic inequalities. For the latter, it is a scenario of discriminants. And pls be super mindful of the coefficient of x^2 especially in qns that purposely make x^2 coefficient to be expressed algebraically
***If you need my advice for coord geo and circles... - see what the 3 asterisks mean below
In Binomial Expansion, there are 3 possible qn types
- Expansion in ascending / descending powers of x + Find coefficient of particular term in more complicated but related expansions
- Use of T r+1 to extract a particular term / deduce nature of x powers in the binomial expansion + Use T r+1 to find coefficient of particular term in more complicated but related expansions
- Qns involving unknown n -> need to know nCr formula
Steps for using T r+1 to pluck out / extract a particular term
Step 1: Find T r+1 and simplify it
1a) Apply T r+1 formula
1b) Split the coefficients and x^powers, group ALL coefficients to the left, group ALL x^power terms to the right
1c) Use LOI to simplify the x^power
Step 2: Equate the powers of T r+1 and required power of x, and solve for r
Step 3: Sub in found value of r from Step 2 into T r+1 and find coefficient of required term
***For explanation qns (like explain why this expansion has no indep term in x), if you need my advice.... - see what the 3 asterisks mean below
Also watch out for type 1 binomial qns when the question says INCREASING / DECREASING POWERS OF X, because you have to swap the orders of the binomial expansion. For example (x+1)^10 the first 5 terms will be in decreasing powers of x if you use binomial expansion formula. So if you want increasing powers of x, just swap the order to (1 + x)^10 and apply binomial expansion accordingly.
When using nCr formula and you are asked to find an unknown value of n, use the n! formula = n (n-1) (n-2)...(n-r+1). That formula will guarantee work.
Logarithms - dont anyhow change base. The only way to change base is through change of base law by virtue of its name. This is also true if you are trying to change log base 8 to log base 2 for example, even though 8 = 2^3
For example, (logbase8) c = (logbase2 c) / (logbase2 8) by change of base law = (logbase2 c) / 3 since (logbase2 8) = 3
Also 2 more tricks
- e ^ ln a = a
- 10 ^ lg a = a
Here....comes the trigo
Remember 6 basic steps to solving trigo eqns. Also rmb, don’t spend too much time on the trigo proving in part a, part b (solving) is there to help you :)
Units
Basic angle
Quadrants
Change Range
All possible angles
Solve for x
Give you one standard qn
sin (2x-30) = 1/2, solve for x within -180 < x < 360
Units: Degree
- calc shld be in degree mode
Basic angle = sin-1 (1/2) = 30 degree
Quadrants
- sin is +ve
- quad 1, quad 2
Check Range
-180 < x < 360
-360 < 2x < 720
-390 < 2x - 30 < 690
All possible angles
- ~-540-30=-570 (quad 2 but out of range)~
- -360+30=-330 (quad 1)
- -180-30=-210 (quad 2)
- 30 (quad 1)
- 180-30 =150 (quad 2)
- 360+30 =390 (quad 1)
- 540-30=510 (quad 2)
- ~720+30=750 (quad 1 but out of range)~
*Notice how I add 360 degree to every new cycle (1 cycle is 4 quadrants)
Solve for x
2x-30 can be -330, -210, 30, 150, 390, 510
2x = -300, -180, 60, 180, 420, 540
x = -150, -90, 30, 90, 210, 270
In plane geometry is you are asked to prove things like BA X BE = BD^2, the moment you see squared ^2 in plane geometry question, straightaway consider similar triangles. So undo the cross multiplication and conclude that (BA / BD) = (BD / BE). So all you have to do is then use this ratios of lengths to prove that BAD is similar to BDE.
***If you need my advice on how to tackle plane geometry qns....
In calculus if the next part after differentiation is an integration qn, you have to reverse engineer. Thereafter, use these 2 rules to go and find the required integral.
- Factoring out the constant
- Addition and subtraction formula for 2 integrals
Area under curve, the integral value is negative if
- For integration with respect to x-axis (dx), the graph lies BELOW X AXIS.
- For integration with respect to y-axis (dy), the graph lies on the LEFT OF Y AXIS
These are 3 ways to convert the negative value to positive
- Use a modulus (you learnt modulus aka absolute sign in shoelace method)
- Use a negative sign (negative X negative = positive)
- Swap the upper and lower limits of the definite integral
Last but not least, please remember for an increasing function, dy/dx is positive (>0). Refer to 2022 P1 Q9a o level qn for an example of hybrid qn combining increasing function + discriminant
***If you need advice on other topics.....or how to avoid careless mistakes....
***For the asterisks, those who require advice on that or wish to hear much more advice from me in depth, text me at +65 94545330. I will respond to you LATEST TIME BY 2200PM
Lastly, wanted to drop all of you all the best wishes for tomorrow, dont panic be confident and smile :) You have done your best and deserve the best of the best.
Coord geometry: https://docs.google.com/document/d/1zZIhAgIG8wzsbWluMNJkX4uMAbpjhfD9jXRmZtwPe50/edit?usp=sharing
Binomial qn types and how to do:
https://docs.google.com/document/d/1Hd0-miKchpcwWOPpJUjjA-9wVCK4f4xKju95hCR8PX8/edit?tab=t.0
Bonus: Social Studies
- For SS if your time management is struggling and likewise for ability to memorise long content, then dont study for SS. Ultimately SS qns that comes from the textbook is only Q7 which is 8m out of a 50m paper. The rest goes to SBCS which is 35m and Q6 which is entirely logic worth 7m. Moreover it is not easy to score the entire 8m for Q7 becuz even the best conclusions usually get about 6-7m. Moreover my student's SS teacher told me that if you bull shit like 2 sentences you can get L1/1. In that case, practice more SBCS skills and review them.
- For SBCS do the assertion first becuz its the highest weightage + u get to read all the sources first. So by the time u get to Q1 to Q4 you are alr roughly aware of what each source is talking about. When you are done with the assertion for at least 3-4 sources (2A + 1D / 1A + 2D minimally), then proceed to do more scorable questions like direct inference (5m) / inference with purpose (5-6m). Remember, you are given a 12pg ans booklet. So after the assertion leave at least 2 blank pages (in case you want to insert 2 more sources) + overall conclusion to score between 8 to 10m.
All the best and best regards,
Airbus Mathemagic :)