Thinking [email protected] #7

JC Mathematics, Mathematics

<blockquote>[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 <a href=”https://theculture.sg/our-team/tutor-ks/”>here</a>.</blockquote>


This is an application question for hypothesis testing from the 9758 H2 Mathematics Specimen Paper 2 Question 10.

The average time required for the manufacture of a certain type of electronic control panel is 17 hours. An alternative manufacturing process is trialled, and the time taken, t hours, for the manufacture of each of 50 randomly chosen panels using the alternative process, in hours, is recorded. The results are summarized as follows

n = 50
\sum t = 835.7
\sum t^2 = 14067.17

The Production Manager wishes to test whether the average time taken for the manufacture of a control panel is different using the alternative process, by carrying out a hypothesis test.
(i) Explain whether the Production Manager should use a 1-tail or a 2-tail test.
(ii) Explain why the Production Manager is able to carry out a hypothesis test without knowing anything about the distribution of the times taken to manufacture the control panels.
(iii) Find unbiased estimates of the population mean and variance, and carry out the test at the 10% level of significance for the Production Manager.
(iv) Suggest a reason why the Production Manager might be prepared to use an alternative process that takes a longer average time than the original process.
The Finance Manager wishes to test whether the average time taken for the manufacture of a control panel is shorter using the alternative process. The Finance Manger finds that the average time taken for the manufacture of each of the 40 randomly chosen control panels, using the alternative process, is 16.7 hours. He carries out a hypothesis test at 10% level of significance.
(v) Explain, with justification, how the population variance of the times will affect the conclusion made by the Finance Manager.

Thinking Math@TheCulture #6

Thinking [email protected] #6

JC Mathematics, Mathematics

<blockquote>[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 <a href=”https://theculture.sg/our-team/tutor-ks/”>here</a>.</blockquote>


This is a very interesting vectors question from a recent JC BT

Shown in the diagram is a methane molecule consisting of a carbon atom, G, with four hydrogen atoms, A, B, C, and D, symmetrically placed around it in three dimensions, such that the four hydrogen atoms form the vertices of a regular tetrahedron.

Methane

Treat A, B, C, D, and G as points. The coordinates of A, B, C, and D are given by (5, -2, 5), (5, 4, -1), (-1, -2, -1) and (-1, 4, 5) respectively, By considering the line DG and the symmetrical properties of methane, find the bond angle of methane, that is, \angle DGA.

Thinking Physics@TheCulture #5

Thinking [email protected] #5

JC Physics

[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 Aaron. More of him can be found here.


Given a chance to counter Sir Isaac Newton’s famous quote of “What goes up must come down”, do you think it is true in all scenarios?

June Crash Course

June Crash Course

Chemistry, JC Chemistry, JC Mathematics, Mathematics

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.

For 2 hr lessons, they are priced at $70.

Lessons will be held at:
Newton Apple Learning Hub
Blk 131, Jurong Gateway Road #03-263/265/267 Singapore 600131
Tel: +65 6567 3606

For math enquiries, you may contact Mr. Teng at +65 9815 6827.

For chem enquiries, you may contact Ms. Chan at +65 93494384.

For GP enquiries, you may contact Ms. Chen at +65 91899133.

Thinking Physics@TheCulture #4

Thinking [email protected] #4

JC Physics

[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 Aaron. More of him can be found here.


Can you explain how a privacy screen protector works on your smartphone?

Thinking Math@TheCulture #4

Thinking [email protected] #4

JC Mathematics, Mathematics

[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 1993 Paper 1.

The positive integers, starting at 1, are grouped into sets containing 1, 2, 4, 8, \ldots integers, as indicated below, so that the number of integers in each set after the first is twice the number of integers in the previous set.

\{ 1 \}, \{ 2, 3 \}, \{ 4, 5, 6, 7 \}, \{ 8, 9, 10, 11, 12, 13, 14, 15 \}, \ldots

(i) Write down the expressions, in terms of r for

(a) the number of integers in the r^{th} set,

(b) the first integer in the r^{th} set,

(c) the last integer in the r^{th} set.

(ii) Given that the integer 1,000,000 occurs in the r^{th} set, find the integer value of r.

(iii) The sum of all the integers in the 20^{th} set is denoted by S, and the sum of all the integers in all of the first 20 sets is denoted by T. Show that S may be expressed as 2^{18}(3 \times 2^{19} - 1).

Hence, evaluate \frac{T}{S}, correct to 4 decimal places.

 

Population problems eventually solve themselves-government meddling only makes things worse. Discuss

Population problems eventually solve themselves-government meddling only makes things worse. Discuss

JC General Paper

Government intervention solves population problems such as population decline, which will be left unresolved if left to the masses. With a preference for smaller families and a general unwillingness to start a family in today’s modern society, negative or zero population growth often ensues. These have detrimental impact on affected countries, such as a fall in tax revenues, a smaller workforce and a high dependence of an ageing population on the working population. As these socioeconomic perspectives are entrenched in the minds of young urban professionals, these population problems are incapable of eventually solving themselves. In this case, government intervention is beneficial. In developed countries like Italy and Spain, where fertility rates stand at a meagre 1.25, new generations are unable to replace past generations thus leading to population decline. The implementation of pro-natal policies could possibly help to increase the incentive for couples to procreate and boost total population numbers. Implemented measures include longer maternity and paternity leave in Switzerland, as well as cash incentives in Singapore. Another method of boosting population growth is through the relaxation of immigration policies, which allows for an influx of permanent residents.

Population problems such as the rampant spread of diseases are also combated more efficiently and effectively through government intervention. If left to solve by itself, this results in a higher death toll and increased spread of illnesses. The successful results of government intervention is exemplified through the World Health Organization and governments’ collaboration to wipe out smallpox, which was deadly enough to kill one in every four infected persons. With public health measures to increase hygiene standards and mandatory vaccinations, smallpox was eradicated worldwide in the 1800s.

Despite the effectiveness of government intervention in solving population problems, some policies and measures undoubtedly create new problems for countries. Firstly, policies to reduce overpopulation are often successful to the extent that they eventually lead to population decline. This is evident in Singapore, which, due to the overwhelming success of the “stop at two” policy, currently faces a replacement rate of 1.25. This has led to national concerns of unsustainable population growth and the possibility of a population decline in the near future. Furthermore,  the policy of migration to solve population problems has led to social segregation in some countries.

Thinking Physics@TheCulture #3

Thinking [email protected] #3

JC Physics

[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 Aaron. More of him can be found here.


In designing a circuit, an engineer needs to use five 5KΩ resistors to design a resistors network of approximately 4.3KΩ. How should he place the resistors to achieve that resistance?

Thinking Math@TheCulture #3

Thinking [email protected] #3

JC Mathematics, Mathematics

[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.

Thinking Physics@TheCulture #2

Thinking [email protected] #2

JC Physics

[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 Aaron. More of him can be found here.


Assuming an object is moving in a circular motion in a polar coordinate system given by x = r \text{cos} \theta and y = r \text{sin} \theta. Can you derive the formula of the centripetal acceleration? Hint: look at one of the axes and think of the direction and what is centripetal acceleration