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Confidential Section A (45 Marks) : 2020-2-NS-KGV

This document contains a multi-part calculus exam with the following questions: 1) Find the value of a if the limit of a function f(x) exists as x approaches 2, and determine if f is continuous. 2) Find the equation of the normal to a parametric curve at a given point. 3) Express a rational function in partial fractions and evaluate two integrals. 4) Transform a differential equation using a substitution and solve for a particular solution. 5) Derive a Maclaurin series and use it to estimate an integral. 6) Use the trapezium rule to estimate two integrals.

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145 views2 pages

Confidential Section A (45 Marks) : 2020-2-NS-KGV

This document contains a multi-part calculus exam with the following questions: 1) Find the value of a if the limit of a function f(x) exists as x approaches 2, and determine if f is continuous. 2) Find the equation of the normal to a parametric curve at a given point. 3) Express a rational function in partial fractions and evaluate two integrals. 4) Transform a differential equation using a substitution and solve for a particular solution. 5) Derive a Maclaurin series and use it to estimate an integral. 6) Use the trapezium rule to estimate two integrals.

Uploaded by

voon sj
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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NS KGV 2020

CONFIDENTIAL*
2020-2-NS-KGV

Section A [45 marks]


Answer all questions in this section
x −1

1. The function f is defined by f(x) =


{ x +2
ax 2 −1 ,
, 0≤ x < 2
x¿2 , where a  .

lim
Find the value of a, if x→ 2 f(x) exists. [3 marks]

With this value of a, determine whether f is continuous at x = 2. [2 marks]

2. The parametric equations of a curve are x = t2 – 2, y = t3 – 3. Find the equation of the normal to the
curve at the point where the parameter t = 2. [4
marks]

17  x
3. a) Express (4  3 x)(1  2 x) in partial fractions. Hence, or otherwise, show that

1
2
17  x 1
 (4  3x)(1  2 x) dx  6 (19 ln 2  9 ln 3)
1

3 . [5 marks]

1
2
x2
 1
dx
b) By using the substitution x  sin  , find
0
(1  x )2 2
. (A result in decimal form is not required)

[6 marks]

4. Using the substitution y=vx , where v is a function of x, show that the differential equation
dy 1 dv 1−4 v 2
x − y= x 2 − y 2 =
dx 4 can be transformed into dx 4 . [3
marks]
Hence, find the particular solution of the differential equation which satisfies the condition that

y = 0 when x=2 . [5 marks]

d3y d2y dy
1 (1  4 x ) 3  16 x 2  8  0
2

5. a) Given y  tan 2 x , show that dx dx dx [3 marks]


CONFIDENTIAL*

1 3
Hence, obtain the Maclaurin’s series for tan 2 x up to and including the term in x . [3 marks]
1
5

 tan
1
2 xdx
b) Use the series expansion in (a) to estimate the value of 0 and giving your answer as a
fraction. [4 marks]

π
3

∫ ln(2+cos x)dx,
6. Use the trapezium rule with 5 ordinates, estimate the value of 0 giving your
answer to 3 significant figures. [5
marks]
π
3

∫ ln(2+cos x)8 dx ,
Hence, find 0 giving 3 significant figures in your answer. [2 marks]

Section B [15 marks]


Answer any one question in this section

2 x−2
7. The equation of a curve is y=x e . Find the coordinates of the stationary points and
2 x−2 2
determine their nature. Sketch the curves y=x e and y=x on the same axes.
[9 marks]
Calculate the area of the region bounded between the two curves. [6 marks]

8. a) At time t = 0, there are 8000 fish in a lake. At time t days, the birth rate of fish is equal to
one-fiftieth of the number N of fish present. Fish are taken from the lake at the rate of 100 per day.
Modeling N as a continuous variable, show that
dN
50 =N −5000
dt
Solve the differential equation to find N in terms of t. Hence, find the time taken for the
population of fish in the lake to increase to 11000. [8 marks]

b) Show that the equation xe x  1 has only one real root between 0 and 1. By taking
x 0=0.6 as the first approximation, use Newton-Raphson method to find the root of the equation
correct to three significant figures. [7 marks]

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