A-level H2 Mathematics (9758) Suggested Solutions (2017)

Here is the suggested solutions for H2 Mathematics (9758). They are all typed in LaTeX, so if it does not render, please leave a comment and let me know. Thank you.

The suggested solutions for H2 Mathematics (9740) is here.

Students of mine should obtain the modified A’levels Paper, and the solutions to the additional questions can be found here.

Year 2017

MF26

2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1:
Question 2:
Question 3:
Question 4:
Question 5:
Question 6: $\mu = 1.69, \sigma^2 = 0.0121$
Question 7: $0.254; 0.194; 0.908$
Question 8: $40320; 0.0142; \frac{1}{4}$
Question 9: $\text{r}=0.978; a=0.182, b=2.56$; $293 Question 10: $0.0336; \bar{y}=0.64, s^2 = 0.0400$; Sufficient evidence. Question 11: $\frac{48+x}{80+x}, \frac{32+x}{80+x}; x= 16; \frac{25}{32}; \frac{7}{16}; \frac{341}{8930}$ Question 12: $0.773; 0.0514; 0.866; 0.362$ Relevant materials MF26 KS Comments 2017 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. This is answers for H2 Mathematics (9740). H2 Mathematics (9758), click here. Numerical Answers (click the questions for workings/explanation) Question 1: $2 \sqrt{15}; xy=6$ Question 2: $d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42$ Question 3: $(\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1 , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0$ Question 4: $15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}$ Question 5: $0.647;~ 0.349;~k=2.56$ Question 6: $955514880;~ 1567641600;~ \frac{1001}{3876}$ Question 7: $31.8075, 0.245;~ p = 0.0139$; Do not reject $h_0$, Not necessary. Question 8: Model (D); $a \approx 4.18, b \approx 74.0;~ r \approx 0.981$ Question 9: $0.632;~ 1.04 \times 10^{-4};~ 0.472;~ 0.421;~ 0.9408$ Question 10: $0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7$ Relevant materials MF26 KS Comments Solutions to the modified A’levels Questions Students of mine who have been diligently doing the modified TYS I sent them, and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, these are modified to cater to the 9758 syllabus… The rest of the solutions (that are questions from the original TYS) can be found here. 2012/P1/Q10 2012/P2/Q2 20112/P2/Q7 2012/P2/Q7 2017 A-level H2 Mathematics (9758) Suggested Solutions All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Paper 1 Paper 2 Relevant materials MF26 Comments on 2017 Paper 2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. This is answers for H2 Mathematics (9740). H2 Mathematics (9740), click here. Numerical Answers (click the questions for workings/explanation) Question 1: $2 \sqrt{15}; xy=6$ Question 2: $d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42$ Question 3: $(\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1 , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0$ Question 4: $15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}$ Question 5: $\frac{5}{12}, \frac{5}{14}, \frac{5}{28}, \frac{1}{21};~ \mathbb{E}(T) = \frac{20}{7}, \text{Var}(T) = \frac{75}{98};~ 0.238$ Question 6: $955514880;~ 1567641600;~ \frac{1001}{3876}$ Question 7: $31.8075, 0.245;~ p = 0.0139$; Do not reject $h_0$, Not necessary. Question 8: Model (D); $a \approx 4.18, b \approx 74.0;~ r \approx 0.981$ Question 9: $0.632;~ 1.04 \times 10^{-4};~ 0.458;~ 0.421;~ 0.9408$ Question 10: $0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7$ Relevant materials MF26 KS Comments H2 Mathematics (9740) 2016 Prelim Papers So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. 🙂 Here are the Prelim Paper 2016. Have fun! Here is the MF26. Quick Summary (Probability) University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful. Random Variables Suppose $X$ is a random variable which can takes values $x \in \chi$. $X$ is a discrete r.v. is $\chi$ is countable. $\Rightarrow p(x)$ is the probability of a value of $x$ and is called the probability mass function. $X$ is a continuous r.v. is $\chi$ is uncountable. $\Rightarrow f(x)$ is the probability density function and can be thought of as the probability of a value $x$. Probability Mass Function For a discrete r.v. the probability mass function (PMF) is $p(a) = P(X=a)$, where $a \in \mathbb{R}$. Probability Density Function If $B = (a, b)$ $P(X \in B) = P(a \le X \le b) = \int_a^b f(x) ~dx$. And strictly speaking, $P(X = a) = \int_a^a f(x) ~dx = 0$. Intuitively, $f(a) = P(X = a)$. Properties of Distributions For discrete r.v. $p(x) \ge 0 ~ \forall x \in \chi$. $\sum_{x \in \chi} p(x) = 1$. For continuous r.v. $f(x) \ge 0 ~ \forall x \in \chi$. $\int_{x \in \chi} f(x) ~dx = 1$. Cumulative Distribution Function For discrete r.v., the Cumulative Distribution Function (CDF) is $F(a) = P(X \le a) = \sum_{x \le a} p(x)$. For continuous r.v., the CDF is $F(a) = P(X \le a ) = \int_{- \infty}^a f(x) ~dx$. Expected Value For a discrete r.v. X, the expected value is $\mathbb{E} (X) = \sum_{x \in \chi} x p(x)$. For a continuous r.v. X, the expected value is $\mathbb{E} (X) = \int_{x \in \chi} x f(x) ~dx$. If $Y = g(X)$, then For a discrete r.v. X, $\mathbb{E} (Y) = \mathbb{E} [g(X)] = \sum_{x \in \chi} g(x) p(x)$. For a continuous r.v. X, $\mathbb{E} (Y) = \mathbb{E} [g(X)] = \int_{x \in \chi} g(x) f(x) ~dx$. Properties of Expectation For random variables $X$ and $Y$ and constants $a, b, \in \mathbb{R}$, the expected value has the following properties (applicable to both discrete and continuous r.v.s) $\mathbb{E}(aX + b) = a \mathbb{E}(X) + b$ $\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)$ Realisations of $X$, denoted by $x$, may be larger or smaller than $\mathbb{E}(X)$, If you observed many realisations of $X$, $\mathbb{E}(X)$ is roughly an average of the values you would observe. $\mathbb{E} (aX + b)$ $= \int_{- \infty}^{\infty} (ax+b)f(x) ~dx$ $= \int_{- \infty}^{\infty} axf(x) ~dx + \int_{- \infty}^{\infty} bf(x) ~dx$ $= a \int_{- \infty}^{\infty} xf(x) ~dx + b \int_{- \infty}^{\infty} f(x) ~dx$ $= a \mathbb{E} (X) + b$ Variance Generally speaking, variance is defined as $Var(X) = \mathbb{E}[(X- \mathbb{E}(X)^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$ If $X$ is discrete: $Var(X) = \sum_{x \in \chi} ( x - \mathbb{E}[X])^2 p(x)$ If $X$ is continuous: $Var(X) = \int_{x \in \chi} ( x - \mathbb{E}[X])^2 f(x) ~dx$ Using the properties of expectations, we can show $Var(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$. $Var(X)$ $= \mathbb{E} [(X - \mathbb{E}[X])^2]$ $= \mathbb{E} [(X^2 - 2X \mathbb{E}[X]) + \mathbb{E}[X]^2]$ $= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2$ $= \mathbb{E}[X^2] - \mathbb{E}[X]^2$ Standard Deviation The standard deviation is defined as $std(X) = \sqrt{Var(X)}$ Covariance For two random variables $X$ and $Y$, the covariance is generally defined as $Cov(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$ Note that $Cov(X, X) = Var(X)$ $Cov(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]$ Properties of Variance Given random variables $X$ and $Y$, and constants $a, b, c \in \mathbb{R}$, $Var(aX \pm bY \pm b ) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)$ This proof for the above can be done using definitions of expectations and variance. Properties of Covariance Given random variables $W, X, Y$ and $Z$ and constants $a, b, \in \mathbb{R}$ $Cov(X, a) = 0$ $Cov(aX, bY) = ab Cov(X, Y)$ $Cov(W+X, Y+Z) = Cov(W, Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z)$ Correlation Correlation is defined as $Corr(X, Y) = \dfrac{Cov(X, Y)}{Std(X) Std(Y)}$ It is clear the $-1 \le Corr(X, Y) \le 1$. The properties of correlations of sums of random variables follow from those of covariance and standard deviations above. June Crash Course The team at The Culture SG has been really busy and we have a lot of things prepared to help you guys work for that A. First up! Crash course for June… And we know it is a bit late to be announcing this on the site now, but we have really been caught up with preparing our students lately that we don’t have the time to properly update here. So here are the details for the Math Crash Course and the Chemistry Crash Course. P.S. For SCIENCE students who wish to chiong in October, please take note that the H2 Chem/ Phy/ Bio Paper 4 (practical) is in October. So better start soon! Here are the details! Click to view For 3 hr lessons, they are priced at$105.

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Thinking [email protected] #3

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here.

This is a question from 1976 A’levels Paper 2. I thought it is pretty interesting to discuss the question with a little extension.

(a) In how many ways can 5 copies of a book be distributed among 10 people, if no-one gets more than one copy?

(b) In how many ways can 5 different books be distributed among 10 people if each person can get any number of books?

So now, let us modify it a bit.

(c) In how many ways can 5 copies of a book be distributed among 10 people if each person can get any number of books?

Notice that the difference between (b) and (c) is that the book distributed is not identical. So for (c), we are pretty much distributing $r$ identical balls to $n$ distinct boxes. Whereas for (b) , we are pretty much distributing $r$ distinct balls to $n$ distinct boxes.