HILLVIEW COLLEGE
TERM III 2018/2019
MATHEMATICS
I.A. – Unit 1 Module 3
𝟏
LOWER SIX
TIME: 1𝟐 HOURS
Attempt ALL questions. Give non-exact numerical answers correct to 3 significant figures or 1 decimal place
in the case of angles in degrees unless a different level of accuracy is specified.
1.
(a)
(b)
√𝑥+3 −2
𝑥−1
=
(i)
Show that
(ii)
Hence evaluate Lim
1
√𝑥+3 −2
𝑥→1 𝑥−1
[1]
4 − 9𝑥 𝑥 < −3
The functions f is defined by 𝑓(𝑥) = {
. Find
2𝑥 + 3 𝑥 ≥ −3
(i)
Lim+ 𝑓(𝑥)
𝑥→−3
(ii)
[2]
√𝑥+3+2
Lim 𝑓(𝑥)
[2]
[2]
𝑥→−3−
Deduce that 𝑓(𝑥) is discontinuous at 𝑥 = −3
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[2]
(c)
Differentiate from first principles y=cos 2x
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[5]
𝑘
Find the value of k if ∫2 6(1 − 𝑥)2 𝑑𝑥 = 52
2.
(a)
(b)
The rate of change of the area, Acm2, of a circle is 6t2 – 2t + 1
[5]
(i)
Write down a differential equation.
[2]
(ii)
Given that the area of the circle is 11cm2 when t = 2, find A in terms of t.
[5]
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(c)
6𝑥
Using the substitution 𝑢 = 3𝑥 + 1, find ∫ (3𝑥+1) 𝑑𝑥
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[5]
3.
8
(a)
Find the equation of the normal to the curve 𝑦 = 𝑥 − 6√𝑥, 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑤ℎ𝑒𝑟𝑒 𝑥 = 4 [5]
(b)
The function y = ax3 + bx2 -12x + 13 passes through (1, 0) and has a stationary point where
x = -1. Find
(i)
the value of a and of b
[5]
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(ii)
the type and the position of the stationary points.
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[5]
𝜋⁄
(c)
Evaluate∫𝜋⁄ 2 3cosx + 2sin2x dx.
4
(a)
[5]
6
(i)
𝑑𝑦
Show that if 𝑦 = 𝑠𝑒𝑐𝜃, then 𝑑𝜃 = 𝑠𝑒𝑐𝜃 𝑡𝑎𝑛𝜃
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[3]
(ii)
1
The parametric equations of a curve are 𝑥 = 1 + 𝑡𝑎𝑛𝜃, 𝑦 = 𝑠𝑒𝑐𝜃 𝑓𝑜𝑟 − 2 𝜋 < 𝜃 <
𝑑𝑦
𝜋
2
Find 𝑑𝑥 , simplifying your answer.
(iii)
[3]
Find the coordinates of the point in the curve at which the gradient is the curve is ½. [3]
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(b)
(i)
𝑑𝑦
−𝑘
The diagram shows a curve for which 𝑑𝑥 = 𝑥 3 , 𝑘 is a
constant. The curve passes through the points (1, 18)
and (4, 3).
16
Show that the equation of the curve is 𝑦 = 𝑥 2 + 2.
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[5]
16
2
(ii)
Find ∫ (𝑥 2 + 2) 𝑑𝑥
(iii)
Hence or otherwise find the volume, in terms of π, when the shaded region is rotated through
360° about the x – axis.
[5]
[5]
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BLANK PAGE
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