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This is answers for H2 Mathematics (9758). H2 Mathematics (9740), click here.

Numerical Answers (click the questions for workings/explanation)

Question 1: ;

Question 2: or

Question 3: ; Maximum point

Question 4: ; translate the graph 4 units in negative y-direction and translate the graph 2 units in positive x-direction.

Question 5: ; ; or

Question 6:

Question 7: ;

Question 8: or ; ;

Question 9: ; ; ; ;

Question 10: ; ;

Question 11: ; ; ;

Let

or

Since is non-zero, .

(i)

Differentiating wrt x,

let

(ii)

Since

Differentiating wrt x,

Let ,

Thus, it is a maximum point.

(i)

for all

(ii)

Asymptotes are and

(iii)

First, translate the graph 4 units in negative y-direction.

Then, translate the graph 2 units in positive x-direction.

(i)

Let

—(1)

—(2)

—(3)

(ii)

Since is a strictly increasing function with no stationary point. Thus, it can only have one real root.

(iii)

or

(i)

Set of points lying on the line which passes through **a** and is parallel to **b**.

(ii)

Set of points on the plane which has normal vector, **n**.

is the displacement of the plane from Origin.

(iii)

Since , line is not parallel to plane, the solution represents the point of intersection between the line and plane.

(i)

(ii)

Since for , we have,

(a)

or

(b)

(i)

Comparing real and imaginary parts,

—(1)

—(2)

(1) + (2):

(ii)

Since all coefficients of are real, is a root is also a root.

(a)

(i)

(ii)

—(1)

—(2)

(2) – (1):

(b)

LHS

Using method of difference,

(c)

Let

Thus, it converges.

From MF26,

For students that do not understand what they are saying, you can find this rendition easier to comprehend.

For them to intersect,

for some and

—(1)

—(2)

—(3)

Solving,

(ii)

Let for some

Since discriminant

There are no solutions for

(iii)

Let

Let

gives minimum .

m.

(i)

(a)

(b)

, where is an arbitrary constant.

When

When ,

(ii)

, where is an arbitrary constant.

Let

When .

(iii)

As ,

When seconds.

### Relevant materials

MF26

### KS Comments

Firstly, to do well in this paper, student has to be quite intuitive, to be comfortable with the levels of unfamiliarity.

Q1. Simple expansion using MF26. If you used it carefully, it should provide some guidance to Q9(c) actually.

Q2. Simple graphings, using secondary school modulus function knowledge.

Q3. Students have to know how to use to find back the y-coordinate.

Q4. (a) is even easier if you simply did long division.

Q5. Remainder Theorem from Secondary School for (i). (ii), students need to be alert that when the gradient is ALWAYS positive, the function is strictly increasing, not just increasing.

Q6. Interesting question, that is similar to the Specimen Paper.

Q7. Use of Factor Theorem form MF26 will make this integration much comfortable. By parts work too.

Q8. Standard complex number practice question.

Q9. Very interesting questions. Especially (c), but like mentioned a keen student who did Q1 well, will realise the sum to infinity is simply from MF26.

Q10. Standard vectors questions. Just read carefully and it will be manageable.

Q11. Simple DE too. For the terminal velocity, just need to read that its “after a long time”.

Overall, a manageable paper.

Now things that have yet to come out…

Reciprocal Graph, Area/ Volume, Parametric Equations, Min/Max Problem, APGP, Function, Integration Techniques, Complex Number (Polar Form, Modulus, Argument), Vectors (Planes, Ratio Theorem), Small angle approximation