CIE AS Maths: Pure 2

Topic Questions

ASMaths: Pure 2CIETopic Questions2. Logs & Exponentials2.1 Logarithmic & Exponential Functions

2.1 Logarithmic & Exponential Functions

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1
Sme Calculator3 marks

Write down the value of:

(i)

33 

(ii)

4-2 

(iii)
90.5

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2
Sme Calculator3 marks

Sketch the graph with equation y equals a to the power of x space comma a greater than 1, stating the coordinates of the point where the graph intersects the y-axis and the equation of any asymptotes.

Also state whether this equation would represent exponential growth or decay.

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3
Sme Calculator3 marks

The following equations can be used for exponential models.

State whether each one would represent exponential growth or exponential decay.

(i)

y equals 3 to the power of negative 2 x end exponent

(ii)

y equals 20 open parentheses 2 close parentheses to the power of x

(iii)
y equals 30 a to the power of negative x end exponent where a greater than 1

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4
Sme Calculator3 marks

Write down the value of  in the following statements:

(i)

3 to the power of a equals 27

(ii)

a to the power of begin inline style 1 third end style end exponent equals 5

(iii)
4 a squared equals 64

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5
Sme Calculator3 marks

Write down the value of  in the following statements:

(i)

log subscript 3 a equals 4

(ii)

log subscript a space end subscript 216 equals 3

(iii)
log subscript 2 space end subscript 128 equals a

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6
Sme Calculator1 mark

Solve the equation

2 to the power of x equals 16

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7a
Sme Calculator2 marks

Solve the equation x squared minus 12 x plus 27 equals 0.

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7b
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Hence, or otherwise, solve the equation open parentheses 3 to the power of x close parentheses squared minus 12 open parentheses 3 to the power of x close parentheses plus 27 equals 0.

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8
Sme Calculator2 marks

Solve the equation

2 log subscript 3 space 9 equals 5 x minus 6

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9
Sme Calculator3 marks

Sketch the graph of y equals e to the power of x, clearly showing the coordinates of the point where the graph intercepts the y-axis and stating the equations of any asymptotes.

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10
Sme Calculator2 marks

Given y equals e to the power of 2 x end exponent:

(i)

Write down an expression for fraction numerator d y over denominator d x end fraction.

(ii)
Find the gradient of y equals e to the power of 2 x end exponent at the point where x equals 0.

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11
Sme Calculator2 marks

Use a calculator to find the value of

(i)

5 space log subscript 3 space 7

(ii)
2 space log subscript 2 space 3plus 3 space log subscript 3 space 2

giving your answers to four significant figures.

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12
Sme Calculator3 marks

Solve the equation e to the power of 2 x end exponent minus 16 equals 0, giving your answer in the form a ln a where a is an integer.

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1a
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals 4 to the power of x and y equals 5 to the power of x.

Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

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1b
Sme Calculator1 mark

Write down an equation for the graph that is a reflection of y equals 4 to the power of x in the y-axis.

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2a
Sme Calculator3 marks
(i)

Sketch the graph of y equals 0.4 to the power of x.

(ii)
State whether this graph indicates exponential growth or exponential decay.

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2b
Sme Calculator1 mark

Find the value of x when y equals 0.064.

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3a
Sme Calculator1 mark

Find the value of log space 1000 plus log space 10000

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3b
Sme Calculator1 mark

Write down the value of a spacein the statement 6 to the power of log subscript 6 space a end exponent equals 36.

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3c
Sme Calculator2 marks

Evaluate fraction numerator 2 log subscript 4 space 64 plus 3 to the power of log subscript 2 space 8 end exponent minus log subscript 5 space 5 over denominator log space 100 end fraction.

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4a
Sme Calculator2 marks

Solve 2 space log space 1000 equals x space log subscript 16 space 4.

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4b
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Solve 3 space log subscript 4 space x equals log subscript 4 space x plus 3 space log subscript 5 space 25.

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5
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Solve 2 open parentheses 2 to the power of 2 x end exponent close parentheses plus 4 equals 9 open parentheses 2 to the power of x close parentheses.

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6a
Sme Calculator3 marks

Sketch the graph of y equals 12 e to the power of negative x end exponent for x greater or equal than 0.
Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

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6b
Sme Calculator1 mark

Write down the gradient of y equals 12 e to the power of negative x end exponent at the point where x equals 0.

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7a
Sme Calculator2 marks

The function straight f open parentheses x close parentheses is defined by straight f open parentheses x close parentheses equals 3 straight e to the power of 2 straight x end exponent for x element of straight real numbers.

Find straight f open parentheses italic 2 x close parentheses.

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7b
Sme Calculator2 marks

Find straight f apostrophe open parentheses italic 2 x close parentheses.

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8
Sme Calculator3 marks

Solve 2 e to the power of 2 x end exponent equals e to the power of x plus 10, giving your answer to 3 significant figures.

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9a
Sme Calculator1 mark

Find the gradient of the curve y equals a e to the power of b x end exponent where and a and b are constants.

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9b
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At the point open parentheses 0 comma a close parentheses the gradient is 12, find b spacein terms of a.

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9c
Sme Calculator1 mark

Hence write down y in terms of a(and x) only.

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10a
Sme Calculator3 marks

Show that the equation e to the power of x minus e to the power of negative x end exponent equals 0 has only one real solution.

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10b
Sme Calculator2 marks

Explain why the equation e to the power of x plus e to the power of negative x end exponent equals 0 has no real solutions.

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1
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals 2 to the power of x and y equals 3 to the power of x, labelling any points where the graphs cross the coordinate axes and writing down the equation of any asymptotes.

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2a
Sme Calculator3 marks

Sketch the graph of y equals 2 to the power of negative x end exponent, stating whether this graph indicates exponential growth or exponential decay.

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2b
Sme Calculator1 mark

Find the exact value of y spacewhen x equals 3.

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3
Sme Calculator4 marks

Write down the value of a in the following statements.

(i)

log subscript a space 8 equals 3

(ii)

log space a equals 2

(iii)

ln space e cubed equals a

(iv)
log subscript 5 space a equals 1

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4a
Sme Calculator3 marks

Using a calculator, find to 3 significant figures

(i)

log subscript 2 space 5 plus log subscript 5 space 2

(ii)

log space 25 minus ln space 2

(iii)
log space 200 plus log subscript 5 space 50 minus log space 20 plus ln space 10

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4b
Sme Calculator2 marks

Solve 3 space log subscript 2 space 4 plus 3 x equals 5 space log subscript 6 space 216.

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5
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Solve 2 to the power of 2 x end exponent minus 24 open parentheses 2 to the power of x close parentheses plus 128 equals 0.

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6a
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals e to the power of x and y equals e to the power of negative x end exponent.
Label any points where the graphs intersect the coordinate axes.
Write down the equations of any asymptotes.

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6b
Sme Calculator1 mark

Write down the gradient of y equals e to the power of x at the point open parentheses 0 comma 1 close parentheses.

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7a
Sme Calculator1 mark

Given y equals e to the power of 4 x end exponent write down an expression for fraction numerator d y over denominator d x end fraction.

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7b
Sme Calculator1 mark

Given y equals 2 e to the power of 2 x end exponent write down an expression for fraction numerator d y over denominator d x end fraction.

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7c
Sme Calculator3 marks

Find the gradient y equals 3 e to the power of negative 2 x end exponent of at the point where x equals 3.
Give your answer in the form p e to the power of q, where p and q are integers to be found.

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8a
Sme Calculator2 marks
(i)
Write down the gradient function of y equals straight e to the power of negative 3 x end exponent

(ii)
Find the gradient of y equals straight e to the power of negative 3 x end exponent at the point where x equals 0.

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8b
Sme Calculator2 marks
(i)
In terms of e, write down the gradient of y equals e to the power of negative 3 x end exponent at the point where x equals 2.

(ii)
Find the value for x for which the gradient of y equals e to the power of negative 3 x end exponent is negative 3 e to the power of negative 12 end exponent.

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9a
Sme Calculator3 marks

The function straight f open parentheses x close parentheses is defined by straight f open parentheses x close parenthesesequals 2 e to the power of 3 x end exponent for x element of straight real numbers

(i)

Find straight f open parentheses negative x close parentheses.

(ii)
On the same axes, sketch the graphs of y equals straight f open parentheses x close parentheses and y equals straight f open parentheses negative x close parentheses.
Label any points where the graphs intersect the coordinate axes.

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9b
Sme Calculator2 marks

Describe the transformation from y equals straight f open parentheses x close parentheses to y equals negative straight f open parentheses x close parentheses.

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10
Sme Calculator3 marks

Solve e to the power of 2 x end exponent minus 8 e to the power of x plus 15 equals 0, giving your answers to 3 significant figures.

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1a
Sme Calculator4 marks

On the same axes, sketch the graphs of y equals 0.3 to the power of x and y equals 0.5 to the power of x.

Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

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1b
Sme Calculator1 mark

Write down an equation for the graph that is a reflection of y equals 0.5 to the power of x in the y-axis.

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2a
Sme Calculator3 marks

Sketch the graph of y equals 0.2 to the power of negative x end exponent, stating whether this graph indicates exponential growth or exponential decay.

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2b
Sme Calculator1 mark

Find the value of x when y equals 625.

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3a
Sme Calculator2 marks

Without using a calculator, evaluate log subscript 4 space 128.
Show each stage of your solution carefully.

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3b
Sme Calculator2 marks

Evaluate fraction numerator 3 log subscript 6 space 216 minus ln space e to the power of 5 plus 4 to the power of log subscript 5 625 end exponent over denominator log space 10000 end fraction.

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4
Sme Calculator4 marks

Solve 3 to the power of 2 open parentheses x plus 1 close parentheses end exponent plus 3 equals 28 open parentheses 3 to the power of x close parentheses.

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5
Sme Calculator4 marks

Find two values of x for which log open parentheses x squared close parentheses equals open parentheses log space x close parentheses squared is true.

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6a
Sme Calculator3 marks

Sketch the graph of y equals 4 e to the power of x for x greater or equal than 0.
Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

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6b
Sme Calculator1 mark

Find the gradient of y equals 4 e to the power of x at the point where x equals 3, giving your answer correct to 3 significant figures.

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6c
Sme Calculator1 mark

The population growth of population, P, at time, t years, is modelled by the equation P equals 4 e to the power of t
Write down the initial population.

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7a
Sme Calculator2 marks

The function straight f open parentheses x close parentheses is defined by straight f open parentheses x close parenthesesequals 5 e to the power of 3 x end exponent for x element of straight real numbers.

Find straight f open parentheses 4 x close parentheses.

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7b
Sme Calculator2 marks

Find straight f apostrophe open parentheses 5 x close parentheses.

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8a
Sme Calculator1 mark

Find the gradient of the curve y equals begin inline style 1 over a end style e to the power of negative b x end exponent where a and b are constants.

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8b
Sme Calculator1 mark

State a condition on b to ensure y spacerepresents exponential decay.

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8c
Sme Calculator2 marks

At the point open parentheses 0 comma a close parentheses the gradient is 10. Find y in terms of a (and x) only.

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9a
Sme Calculator1 mark

A particle is travelling with velocity, v ms-1, at time t seconds.
The velocity of the particle is modelled v equals 0.3 e to the power of k t end exponent, where k is a constant.

Write down the initial velocity of the particle.

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9b
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Find an expression (in terms of k and t) for the acceleration of the particle.

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9c
Sme Calculator3 marks

After 12 seconds the velocity of the particle is 0.9 ms-1.
Find the value of k, giving your answer to 3 significant figures.

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9d
Sme Calculator1 mark

State a problem with the model for large values of t

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10
Sme Calculator4 marks

Solve open parentheses e to the power of x minus e to the power of negative x end exponent close parentheses squared equals 0.

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11
Sme Calculator5 marks

Solve the equation 2 e to the power of 3 x end exponent minus 11 e to the power of 2 x end exponent plus 12 e to the power of x equals 0, giving answers to 3 significant figures where appropriate.

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