(a)

(b)

Cultivating Champions, Moulding Success

(a)(i)

Let

(a)(ii)

*Students are expected to prove that gives the maximum area.

(b)(i)

latex x$-axis

Sub into

(b)(iii)

When the line is a tangent to C,

When

When

(i)

When

Equation of tangent:

Equation of normal:

(ii)

Hence, tangent cuts curve again at

(iii)

At Q,

At R,

(i)

For tangents to be parallel to the -axis,

Sub into

Thus, equations of tangents which are parallel to -axis are

(ii)

units per second.

Please do check through the solutions on your own, especially for questions that we did not have chance to properly discuss during class. You may whatsapp me too if you have a burning question.

Note: You should not spend more than 180mins on the entire exercise.

June Revision Exercise 3 Q1

June Revision Exercise 3 Q2

June Revision Exercise 3 Q3

June Revision Exercise 3 Q4

June Revision Exercise 3 Q5

June Revision Exercise 3 Q6

June Revision Exercise 3 Q7

June Revision Exercise 3 Q8

June Revision Exercise 3 Q9

June Revision Exercise 3 Q10

June Revision Exercise 3 Q11

This question was sent in by a student, which I think is rather interesting given 2015 A’levels P1 Q3 had something similar.

Use a definite integral to evaluate

,

given your answer correct to 5 decimal places.

Hint: Recall that the sum of area under the graph can be found using integration.

I came across the integral a few days ago. Pretty interesting, many ways to solve this integral, be it graphically or by substitution.

?

Answer:

Hint: Consider the graph of , it should give a semicircle which centre is

Many students seem to struggle when they see an integral with modulus as they do not know where to begin with. The first thing they should note is that, we cannot evaluate an integral with modulus directly, that means, we must remove (address) the modulus first.

So let’s see how we should approach such questions, considering , we must first know what range of values of for which is negative, in this case, let us assume that for , where .

Then

We break the integral up into two parts, adding a negative sign to the integral part for which . Students can relate this to reflecting about the x-axis to make it a positive area.

Students may want to reference this recent ACJC Prelim 2015 Question to see if they can do it.

This is question that was tested in ACJC H2 Math Prelim P1 2015. A few of my students know how to answer it but were uncertain how to express it.

Most students were concerned with (ii) of the question.

I think the easiest way to prove this, is to first avoid writing too much and attempt to show it mathematically.

while

Thus will be smaller in magnitude.

Students can also attempt to justify using the area under graph but they must express the answers in words carefully.