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H1011 Quizzes

Quiz 3: Exponential and logarithmic functions
Question 1 Questions
Write 3sinx + 4cosx in the form Rsin(x + α).
a)
7sin(x + 0.64)
 b)
7sin(x + 0.93)
c)
5sin(x + 0.64)
 d)
5sin(x + 0.93)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Rcosα = 3, Rsinα = 4, R = 32 + 42 = 5, α = tan143 = 0.93
Write 1 2cosx 3 2 sinx in the form Rsin(x + α).
a)
sin(x + π 6 )
 b)
sin(x π 6 )
c)
sin(x + 5π 6 )
 d)
sin(x 5π 6 )

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Rcosα = 3 2 , Rsinα = 1 2, R = 1 3 + 3 4 = 1,tanα = 1 3 and α is in the 2nd quadrant.
Choice (d) is incorrect
Which of the following graphs indicates that y is proportional to x2.
a)
PIC
 b)
PIC
c)
PIC
 d)
PIC

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
The experimental data t 1 2 3 4 5 ̲ ̲ ̲ ̲ ̲ ̲ y0.93.68.114.422.5 appears to show that y is proportional to t2. Find the relationship between y and t.
a)
y = 0.9t2
 b)
y = 2.7t2
c)
y = 0.9t
 d)
y = 2.7t

Choice (a) is correct!
Scaling the data gives t2 = T 1 4 9 16 25 ̲ ̲ ̲ ̲ ̲ ̲ y 0.93.68.114.422.5 A scatter plot of y against T is a straight line through (0,0) with slope 3.6 0.9 4 1 = 0.9.
therefore,y = 0.9T = 0.9t2.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Given the experimental data t1.11.62.12.63.1 ̲ ̲ ̲ ̲ ̲ ̲ y2.94.45.97.48.9 which of the following statements is correct ?
a)
y is proportional to t.
b)
y is proportional to t2.
c)
y is a linear function of t.
d)
There is no recognisable relationship between y and t.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The equation of the line satisfying the data is y = 3t 0.4.
Choice (d) is incorrect
Solve 3ex = 5.
a)
x = ln5 3
 b)
x = ln5 ln3
c)
x = ln5 + ln(3)
 d)
x = ln5 ln3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
3ex = 5 ex = 5 3 lnex = ln 5 3 x = ln5 ln3.
Solve 1 3ln8x3 = 2.5 .
a)
x = e2.5 2
 b)
x = 7.5 ln83
c)
x = 3e7.5ln 8
 d)
x = 1 3e7.58

Choice (a) is correct!
1 3ln8x3 = 2.5 ln[(2x)3]13 = 2.5 ln2x = 2.5 2x = e2.5 x = e2.5 2 .
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Simplify e2 ln x3 .
a)
2x3
 b)
e2 + x32
c)
x3
 d)
2x3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
e2 ln x3 = e2 ln x32 = eln(x32)2 = eln x3 = x3.
Choice (d) is incorrect
It is expected that the set of experimental data x0.81.4 2.0 2.6 3.2 3.8 ̲ ̲ ̲ ̲ ̲ ̲ ̲ y 1.05.516.035.165.5109.7 follows a power law. Find the best approximation for the given data.
a)
y = 102e7.5x
 b)
y = 2 × 103e7.5x
c)
y = 1.1x13
 d)
y = 2x3

Choice (a) is incorrect
This equation does not follow a power law.
Choice (b) is incorrect
This equation does not follow a power law.
Choice (c) is incorrect
Choice (d) is correct!
We do a log-log transformation and to 2 decimal places we find lnx = X 0.220.340.690.961.161.34 ̲ ̲ ̲ ̲ ̲ ̲ ̲ lny = Y 0 1.702.773.564.184.70 A scatterplot shows that the relationship between X and Y is linear.You should draw the scatterplot to verify this. So putting Y = mX + b we find m = 3 thus Y = 3X + b. Now 0 = 0.66 + b therefore y = e0.66x3 2x3.
It is expected that the set of experimental data x 1 2 3 4 5 ̲ ̲ ̲ ̲ ̲ ̲ y 3.08.515.624.033.5 follows a power law. Find the best approximation for the given data.
a)
y = 1.1ex
 b)
y = 1.2e0.9x
c)
y = 3x32
 d)
y = 3x23

Choice (a) is incorrect
This equation does not follow a power law.
Choice (b) is incorrect
This equation does not follow a power law.
Choice (c) is correct!
We do a log-log transformation and to 2 decimal places we find lnx = X 0 0.691.101.391.61 ̲ ̲ ̲ ̲ ̲ ̲ lny = Y 1.102.142.753.183.51 A scatterplot shows that the relationship between X and Y is linear. You should draw the scatterplot to verify this. So letting Y = mX + b we find m = 1.5 and b = 1.1. Thus y = e1.1x1.5 3x1.5 = 3x32.
Choice (d) is incorrect