All solutions here are **suggested**. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions. Sorry, due to copyrights issues, I cannot share the paper.

Numerical Answers (workings/explanations are after the numerical answers.)

Question 1:

Question 2:

Question 3:

Question 4:

Question 5: ; refer to solutions below for explanation.

Question 6:

Question 7:

Question 8:

Question 9:

Question 10:

Question 11:

(i) [Maximum marks: 2]

(ii) [Maximum marks: 2]

Required angle

Note/ Comment: For (i), if students simplified, it is ok since the questions simply asked for a vector normal to the plane.

Differentiating with respect to ,

Let

, where

Note/ Comment: Slightly tedious but fully manageable. A more elegant approach will be to consider

and differentiate from here.

(i) [Maximum marks: 3]

(ii) [Maximum marks: 4]

Note/ Comment: Quite a standard question. Students can verify their answers using the standard series too.

(i) [Maximum marks: 4]

We can first convert all the given complex numbers to exponential form.

*Another approach is to straightaway solve for and .

(ii) [Maximum marks: 3]

, and

* Another approach is to observe that

Note/ Comment: This can be tedious if the student did not use exponential form to get around things. For (ii), the question has stated “possible values” which is a hint that there is more than one answer to be expected. Aside from it, students should use GC to check their answers.

(i) [Maximum marks: 2]

We have that , so is parallel to and where is a scalar.

(ii) [Maximum marks: 3]

R is a set of points lying on a line that passes through position vector and is parallel to direction vector , where is a scalar parameter.

(iii) [Maximum marks: 4]

R is a set of point lying on a plane that passes through position vector and is perpendicular to normal vector , where units is the shortest displacement R is from the origin.

Note/ Comment: Quite a standard question, from the specimen paper and also from 2017 A’levels paper. Students should be able to answer this comfortably.

By comparing the coefficients,

Other root and and

Note/ Comment: Actually this question did not prohibit the GC, so the last part could be done easily.

(i) [Maximum marks: 1]

(ii) [Maximum marks: 4]

(iii) [Maximum marks: 4]

Note/ Comment: Students should use GC to check their answers.

(a)(i) [Maximum marks: 2]

(a)(ii) [Maximum marks: 3]

(b)(i) [Maximum marks: 2]

(b)(ii) [Maximum marks: 5]

Smallest possible .

Note/ Comment: Standard question.

(i) [Maximum marks: 1]

Consider the lines and , since the gradient , where is the angle the line makes with the horizontal and the two lines and are perpendicular, i.e., radians spaced from each other. We have that .

(ii) [Maximum marks: 3]

Since , required set

(iii) [Maximum marks: 3]

Note: Right side of the graph. The asymptote is y=0, I do not know why wordpress cut it off. 🙁

(iv) [Maximum marks: 5]

Using GC,

(i) [Maximum marks: 2]

(ii) [Maximum marks: 4]

, where is an arbitrary constant.

In the long run, , The number of sheep will decrease and approach 0.

(iii) [Maximum marks: 2]

(iv) [Maximum marks: 4]

where is an arbitrary constant.

(v) [Maximum marks: 2]

In the long run,

Note/ Comment: This question is quite difficult to interpret, but once you understand, it should be manageable.

(i) [Maximum marks: 3]

Let and , then

(ii) [Maximum marks: 3]

Differentiate with respect to ,

Let

(iii) [Maximum marks: 1]

There are other external factors such as the strength of the kick, or wind, etc.

Note: I personally cannot think of a mathematical reason. Please share with me if you do. So I guess anything logical is fine.

(iv) [Maximum marks: 3]

.

When

(v) [Maximum marks: 2]

I wanna read the comment

you will have better luck at reddit

For the DE question, if i got the equation wrong but the steps in ii correct and deduced that overtime population approaches 0, will i get any mark.

you should. But the model will probably be wrong since math in general does not carry ecf.

does a wrong DE but correct way of solving DE get marks, thanks!

Given the marks distribution for the question, quite tough.. :/ Since the method to solve the DE was variable separable.

regarding question 2, if i did all steps correctly just during differentiation i carelessly left out the dy dx on the product side of the equation, will there be ecf? or will the 6 marks just fly away like that 🙁 thank you for your reply and the answers!

it won’t just fly away. if your method to find tangents, etc are correct. there will be marks.