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2017 2 Ked Ktek

The document contains a mathematics examination with various problems related to functions, curves, differential equations, and series. It includes tasks such as finding limits, determining continuity, sketching curves, calculating volumes, and solving differential equations. The exam is divided into two sections, with Section A requiring answers to all questions and Section B allowing the choice of one question.

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10 views3 pages

2017 2 Ked Ktek

The document contains a mathematics examination with various problems related to functions, curves, differential equations, and series. It includes tasks such as finding limits, determining continuity, sketching curves, calculating volumes, and solving differential equations. The exam is divided into two sections, with Section A requiring answers to all questions and Section B allowing the choice of one question.

Uploaded by

laikulim
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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2017-2-KED-KTED

Section A [ 45 marks]
Answer all questions in this section

1. The function f is defined by

{
4
+ k , x ≤−2
f ( x )= x
¿ ln (x+ 3), x>−2

(a) Find ¿ and ¿ [3 marks]

(b) Find k if lim ¿ x →−2 f (x) exists.


Hence, determine whether f is continuous at x=−2 [3
marks]

2． The parametric equations of a curve are x= +1 and y= −1 . The curve


2
t 4
t
crosses the
2

x -axis at A. Find the equation of the tangent to the curve at a point A.


[7 marks]

3. The equation of two curves are given by y 2=2 x and y = .


2 8
x
(a) Sketch the two curves on the same coordinate axes.
[5 marks]
(b) Find the coordinate of the points of intersection of the two curves.
[3 marks]
(c) Calculate the volume of the solid formed when the region bounded by the
two curves and the line x=4 is revolved through π radians about the x -axis.
[5 marks]

4. Solve the differential equation +(tan x ) y =tan x sec x, with condition y=2 when
dy 3
dx
x= .
π

Express your answer in the form of y=f (x ). [7


3

marks]

5. Show that the first three terms in the Maclaurin Series for sin(sin x ) is

1
[4
3 5
x x
x− + −…
3 10
marks] Hence or otherwise find x−sin ( sin x ) .
lim 3
x →0 x
[3 marks]

6. Use the trapezium rule with five ordinates to find an approximate value for

.
9 x

∫4 9
dx
0

Give your answer correct to one decimal place.


[5 marks]

Section B [15 marks]


Answer one question only in this section
7. The diagram shows the diagram y= .
lnx
x
y
A

0 x

The curve has a stationary points at A.

(a) State the ¿ and lim ¿ x →+∞ . [2 marks]


lnx
x

(b) Find the coordinates of the stationary point, A, of the curve, giving your
answer in exact form.
[3 marks]
Hence , show that x e ≤ e x for all positive values of x . [3
marks]
State the intervals for which
(i)
dy
>0,
dx

2
(ii) [2
dy
<0

marks]
dx

(c) Find the point of inflexion of the curve.


[5 marks]

8. Sketch the graph of y=3 ln x and y=2 x +8−x2 on the same coordinate axes.
Hence, show that the equation 3 ln x + x 2−2 x−8=0 has only one real root.
[5
marks]
Show that the equation
x=1+ √ 9−3 ln x
can be rearranged in the form
3 ln x + x −2 x−8=0 . [2 marks]
2

With x 1=3, without performing the iteration processes, show that the iterative
formula

x n+1=1+ √ 9−3 ln x n

converges to the root. [3


marks]
Hence, estimate the root correct to 3 decimal places.
[5 marks]

“If criticism makes you angry and compliments make you happy, bad company will come
your way and good friends will shy away.” -- Confucianism

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