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MATH1011 Quizzes

Quiz 3: Exponential and logarithmic functions
Question 1 Questions
Write 3sinx + 4cosx in the form Rsin(x + α). Exactly one option must be correct)
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). Exactly one option must be correct)
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. Exactly one option must be correct)
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. Exactly one option must be correct)
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 ? Exactly one option must be 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. Exactly one option must be correct)
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 . Exactly one option must be correct)
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 . Exactly one option must be correct)
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. Exactly one option must be correct)
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. Exactly one option must be correct)
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