Giving Your Answer in an Exact Form. 3

1 Solve the inequality

giving your answer in an exact form. [3]

Hence solve . [2]

2 The sequence of numbers , where n = 0, 1, 2, 3, …, is such that = and .

Proof by induction that, for

. [5]

3 The functions f and g are defined as follows:

(a) Define in a similar form. [3]

(b) State the value of a such that the range of g is . [1]

4 A curve is defined by the parametric equations

, , where

(i) Show that the tangent to the curve at any point with parameter t has equation [3]

(ii) Find the gradient of the tangent to the curve at . Hence determine the acute angle between this tangent and the line. [3]

5 Robert took a study loan of $100 000 from a bank on 1st January 2010. The bank charges an annual interest rate of 10% on the outstanding loan at the end of each year. After his graduation, Robert pays the bank $x at the beginning of each month. The first payment is made on 1st January 2014. Let denote the amount owed by Robert at the end of nth year after 2013, where .

(i) Find . [1]

(ii) Show that = , where k is a constant to be determined. [4]

(iii) Given that Robert owes the bank less than $1000 at the end of 2020, find the minimum value of x, giving your answer to the nearest dollar. [3]

6 (a) Find . [3]

(b) Use the substitution to find . [5]

7 It is given that the function has the Maclaurin’s series and satisfies , where a and b are real constants.

(i) Show that b = 2 and find the value of a. [4]

(ii) Find the series expansion of in ascending powers of x, up to and including the term in . [3]

(iii) State the equation of the normal to the curve at x = 0. [1]

8 (i) Express in the form . [2]

(ii) Hence find . [3]

Give a reason why the series is convergent, and state its limit. [2]

(iii) Use your answer to part (ii) to find . [2]

9 On a single Argand diagram, sketch the loci given by

(i) ,

(ii) ,

(iii) . [7]

Hence, or otherwise, find the range of values of and arg [3]

10 A file is downloaded at r kilobytes per second from the internet via a broadband connection. The rate of change of r is proportional to the difference between r and a constant. The initial value of r is 348. If r is 43, it remains at this constant value.

(i) Show that . [2]

(ii) Hence obtain an expression for r in terms of k and t. [4]

The total amount of data downloaded, I kilobytes, in time t seconds, is given by

(iii) Given that there is no data downloaded initially, find I in terms of k and t. [2]

(iv) It is given that a file with a size of 5700 kilobytes takes 90 seconds to download. Find the value of . [2]

(v) Explain what happens to the value of r in the long run. [1]

The diagram above shows part of the structure of a modern art museum designed by Marcus, with a horizontal base OAB and vertical wall OADC. Perpendicular unit vectors i, j, k are such that i and k are parallel to OA and OC respectively.

The walls of the museum BCD and ABD can be described respectively by the equations

and , where .

(i) Write down the distance of A from O. [1]

(ii) Find the vector equation of the intersection line of the two walls BCD and ABD. [3]

(iii) Marcus wishes to repaint the inner wall ABD. Find the area of this wall. [3]

Suppose Marcus wishes to divide the structure into two by adding a partition such that it intersects with the walls BCD and ABD at a line. This partition can be described by the equation , where .

(iv) Find the values of . [2]

(v) Another designer, Jenny, wishes to construct another partition which is described by the equation , where . State the relationship between Jenny’s and Marcus’ partitions. [1]

Deduce the number of intersection point(s) between the walls BCD, ABD, and Jenny’s partition. [1]

12 The curves and have equations and , where respectively. Describe the geometrical shape of C1. [1]

(a) State a sequence of transformations which transforms the graph of to the graph of . [3]

(b) (i) Sketch and on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes. [6]

(ii) Show algebraically that the x-coordinates of the points of intersection of and satisfy the equation

. [2]

(iii) Deduce the number of real roots of the equation in part (ii). [1]

Qtn / Solutions
1. /

, is always positive for all real values of x.
For
Replace x by ,

2. / Let P(n) be the proposition.
When n = 0,
LHS of P(0) = (given)
RHS of P(0) =
P(0) is true.
Assume P(k) is true for some
i.e. .
Show that P(k+1) is true
i.e. .
When n = k + 1,
LHS of P(k+1) =
=
=
= = RHS of P(k+1)
Since P(0) is true & P(k) is true P(k + 1) is also true, hence by mathematical induction P(n) is true for all
3
(a)
(b)
(c) /

a = 2
, .
Since , gf exists.

Df Rf Rgf
4(i)
(ii) /

Equation of tangent:

When ,
Let be the acute angle between the two lines.

Alternative Solution

5(i) /
(ii) /

:
:

(iii) / n = 7 at end of 2020

Least x to the nearest dollar = $2271
6(a) /
(b) /

7i /

ii /
iii / Gradient of normal =
Equation of normal:

8(i) /

(ii) /
(iii) / , hence the series in (ii) converges.

(iv) /

9. /

Method 1: ;
Method 2: QR is the perpendicular bisector, so PQ = (radius)

10(i) / Let the constant be a.

Given when ,

(shown)
(ii) /

When , .
.

(iii) /
When , .

(iv) / Given and ,

Solving using GC,
or (NA)
(v) /
If t becomes larger, ,
Hence r would be reduced to a steady 43 kilobytes per second in the long run.
11i / OA = 14
ii / Plane ABD

OR

iii /

iv /
OR
The 3 planes intersect at the line

v / Their partitions are parallel to each other.
There is no intersection point.
12 / Ellipse
(a) /

Method 1:
Sequence of transformations:
1) Scale // to x-axis by factor a.
2) Translate in the positive x-direction by 2 units.
Method 2:
Sequence of transformations:
1) Translate in the positive x-direction by units.
2) Scale // to x-axis by factor a.
(bi) /

(bii) / Sub into :

--- (*)
(shown)
Hence the x-coordinate of the points of intersection of
and satisfy equation (*).
(b)
(iii) / From (ii), number of intersection points between and gives the number of real roots of the equation (*).
From the graphs, there are 2 points of intersection between and . Hence 2 real roots.

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