DP IB Maths: AA HL

Topic Questions

IBMaths: AA HLDPTopic Questions2. Functions2.8 Inequalities

2.8 Inequalities

DownloadView
Medium
Hard
Very Hard
Medium
Hard
Very Hard
1a
Sme Calculator2 marks
(a)
Sketch the graph of the function f open parentheses x close parentheses equals x open parentheses x minus 2 close parentheses open parentheses x minus 4 close parentheses squared.
Mark your sketch clearly with the x-coordinates of the x-axis intercepts.

How did you do?

1b
Sme Calculator1 mark
(b)
Write down the solution to the inequality f open parentheses x close parentheses less or equal than 0.

 

How did you do?

1c
Sme Calculator2 marks
(c)
Briefly explain how the graph shows that there are no real solutions to the inequality f open parentheses x close parentheses plus 10 less or equal than 0.

How did you do?

Did this page help you?

2
Sme Calculator4 marks

Consider the function defined by g left parenthesis x right parenthesis equals e to the power of negative open vertical bar x close vertical bar end exponent.

(i)
Sketch the graph of y equals g open parentheses x close parentheses.
(ii)

Solve the inequality g open parentheses x close parentheses greater or equal than 0.5 using exact values.

How did you do?

Did this page help you?

3a
Sme Calculator2 marks

Consider the function f left parenthesis x right parenthesis equals 6 x cubed minus 19 x squared plus 16 x minus 4.

(a)
Fully factorise f open parentheses x close parentheses.

How did you do?

3b
Sme Calculator4 marks
(b)
Solve
(i)
f open parentheses x close parentheses less than 0
(ii)
f open parentheses 2 x close parentheses less than 0
(iii)
f open parentheses x minus 3 close parentheses less than 0
(iv)
open vertical bar f open parentheses x minus 3 close parentheses close vertical bar less or equal than 0

How did you do?

Did this page help you?

4
Sme Calculator4 marks

Find the values of k such that the equation k x equals k x squared plus k minus 2 has real solutions and the equation open parentheses 4 k minus 3 close parentheses x squared plus 2 k x plus 1 equals 0 has no real solutions.

 

How did you do?

Did this page help you?

5a
Sme Calculator3 marks
(a)
Giving answers to three significant figures, find the set of values of x that satisfy
open vertical bar sin open parentheses 2 x to the power of degree close parentheses close vertical bar greater or equal than 1 minus x over 360 
for 0 less or equal than x less or equal than 360.

How did you do?

5b
Sme Calculator1 mark
(b)
Explain how your answer to part (a) would differ if the domain of x was changed from 0 less or equal than x less or equal than 360 to x element of straight real numbers.

 

How did you do?

Did this page help you?

6a
Sme Calculator1 mark
(a)
Write down the set of values of x for which e to the power of 2 x end exponent less or equal than e to the power of x.

How did you do?

6b
Sme Calculator2 marks
(b)
Find the set of values of x for which 

          (i)    e to the power of 2 open parentheses x minus 4 close parentheses end exponent less or equal than e to the power of x minus 4 end exponent 

          (ii)    e to the power of 5 open parentheses x minus 4 close parentheses end exponent greater or equal than e to the power of negative open parentheses x minus 4 close parentheses end exponent

 

How did you do?

6c
Sme Calculator2 marks
(c)
Find the set of values of x for which
(i)
2 ln left parenthesis x minus 4 right parenthesis less or equal than ln left parenthesis x minus 4 right parenthesis
(ii)
negative ln left parenthesis x minus 4 right parenthesis greater or equal than 5 ln left parenthesis x minus 4 right parenthesis.

How did you do?

Did this page help you?

7a
Sme Calculator4 marks

Consider the functions

f open parentheses x close parentheses equals fraction numerator a over denominator a minus x end fraction and g open parentheses x close parentheses equals fraction numerator 2 a over denominator x minus 2 a end fraction 

where a is real constant such that a not equal to 0. 

(a)  In terms of the constant a, find 

          (i)     the values of x for which f open parentheses x close parentheses and g open parentheses x close parentheses are undefined, 

          (ii)    the x-coordinate of any intersections between the graphs of y equals f open parentheses x close parentheses and y equals g open parentheses x close parentheses.

How did you do?

7b
Sme Calculator3 marks
(b)
In the case a greater than 0, find the set of values of x in terms of a for which 

          (i)     f open parentheses x close parentheses greater than g open parentheses x close parentheses, 

          (ii)    f open parentheses x close parentheses less than g open parentheses x close parentheses.

How did you do?

7c
Sme Calculator2 marks
(c)
Repeat questions (b) (i) and (ii) in the case a less than 0.

How did you do?

Did this page help you?

8a
Sme Calculator3 marks

f left parenthesis x right parenthesis equals x cubed minus 11 x squared plus 40 x minus 48

(a)
(i)
Show that x equals 3 is a root of the function .
(ii)

Hence fully factorise f open parentheses x close parentheses.

How did you do?

8b
Sme Calculator3 marks
(b)
Solve the inequality x cubed minus 10 x squared plus 32 x minus 32 less or equal than left parenthesis x minus 4 right parenthesis squared.

How did you do?

8c
Sme Calculator2 marks
(c)
Solve the inequality open parentheses x plus a close parentheses open parentheses x plus b close parentheses squared greater than 0, where a and b are constants such that a greater than b greater than 0.  Give your answers in terms of a and b.

 

How did you do?

Did this page help you?

9a
Sme Calculator3 marks
(a)
Consider the functions defined by  f open parentheses x close parentheses equals x squared minus a and g open parentheses x close parentheses equals a minus x squared,  where a is a positive constant. Solve the inequality f open parentheses x close parentheses less than g open parentheses x close parentheses,  giving your answer in terms of a.

How did you do?

9b
Sme Calculator2 marks
(b)
Consider the functions defined by  p open parentheses x close parentheses equals x cubed minus b and  q open parentheses x close parentheses equals b minus x cubed,  where b is a real constant. Solve the inequality p open parentheses x close parentheses less than q open parentheses x close parentheses,  giving your answer in terms of b.

 

How did you do?

9c
Sme Calculator3 marks
(c)
Consider the functions defined by h open parentheses x close parentheses equals x to the power of n minus m and j open parentheses x close parentheses equals m minus x to the power of n,  where m greater than 0
and n element of straight integer numbers to the power of plus. Write down, in terms of m and n, the solution to the inequality h open parentheses x close parentheses less than j open parentheses x close parentheseswhen
(i)
n is even,
(ii)
n is odd.

How did you do?

Did this page help you?

10
Sme Calculator3 marks

Consider the functions defined by f open parentheses x close parentheses equals a plus ln space b x and g open parentheses x close parentheses equals a minus ln space b x where a and b are positive constants. Show that

f left parenthesis x right parenthesis less than g left parenthesis x right parenthesis for  0 less than x less than 1 over b.   .

How did you do?

Did this page help you?

1a
Sme Calculator3 marks

Consider the functionsspace f left parenthesis x right parenthesis equals 3 x squared plus x minus 2 and g left parenthesis x right parenthesis equals negative 2 x squared plus 3 x plus 5.

a)
Sketch the graph of the functionspace f left parenthesis x right parenthesis on the axes provided, labelling the vertex as well as the x- and y-intercepts.
q1a_2-8_medium_ib-aa-hl-maths     

How did you do?

1b
Sme Calculator4 marks
b)
Solve the inequality f left parenthesis x right parenthesis less than g left parenthesis x right parenthesis.

How did you do?

Did this page help you?

2
Sme Calculator4 marks

Solve the inequality 5 x squared minus 8 x minus 48 greater or equal than 2 x squared plus 4 x minus 12.

How did you do?

Did this page help you?

3a
Sme Calculator2 marks

Consider the inequality  fraction numerator x squared minus 3 x minus 10 over denominator x minus 1 end fraction less than 0.

a)
Explain why you need to consider the cases x less than 1 comma x equals 1 and x greater than 1 separately when rearranging the inequality to find a solution. 

  

How did you do?

3b
Sme Calculator5 marks
b)
Solve the inequality.

How did you do?

Did this page help you?

4a
Sme Calculator2 marks

The functions and are defined such that begin mathsize 16px style f left parenthesis x right parenthesis equals fraction numerator x plus 4 over denominator 2 x minus 1 end fraction end style and g left parenthesis x right parenthesis equals 2 x minus 4. 

Given that space f has the largest possible valid domain, 

a)
State the domain and range of space f.

How did you do?

4b
Sme Calculator4 marks
b)
Solve the inequalityspace f left parenthesis x right parenthesis less or equal than g left parenthesis x right parenthesis.

How did you do?

Did this page help you?

5a
Sme Calculator3 marks

Consider the function f left parenthesis x right parenthesis equals negative 2 space sin space x in the interval negative 2 pi less or equal than x less or equal than 2 pi.

a)
Sketch a graph of the function over the given interval on the axes provided, labelling all x-intercepts as well as local minima and maxima. q5a_2-8_medium_ib-aa-hl-maths

How did you do?

5b
Sme Calculator4 marks
b)
Solve the inequalityspace f left parenthesis x right parenthesis greater than 1.

How did you do?

Did this page help you?

6
Sme Calculator4 marks

Solve the inequality begin mathsize 16px style fraction numerator 3 x minus 2 over denominator 5 end fraction plus 3 greater than fraction numerator 4 x minus 4 over denominator 5 end fraction end style

How did you do?

Did this page help you?

7a
Sme Calculator4 marks

Consider the functionsf left parenthesis x right parenthesis equals x squared minus 9 plus 4 over x and g left parenthesis x right parenthesis equals negative x plus 5. space

a)
Sketch the graphs of space f left parenthesis x right parenthesis and g left parenthesis x right parenthesis, clearly labelling any points of intersection or asymptotes. 

How did you do?

7b
Sme Calculator3 marks
b)
Determine the values of x such that space f left parenthesis x right parenthesis greater or equal than g left parenthesis x right parenthesis.

How did you do?

Did this page help you?

8a
Sme Calculator4 marks

Consider two functions, f left parenthesis x right parenthesis equals ln left parenthesis x plus 3 right parenthesis plus 4 and g left parenthesis x right parenthesis equals e to the power of x minus 3 end exponent. 

a)
Sketch both functions on the axes below, clearly labelling the asymptotes and points of intersection.

q8a_2-8_medium_ib-aa-hl-maths

How did you do?

8b
Sme Calculator2 marks
b)
Hence or otherwise, solve the inequalityspace f left parenthesis x right parenthesis greater or equal than g left parenthesis x right parenthesis.

How did you do?

Did this page help you?

9a
Sme Calculator4 marks

Consider the polynomial q left parenthesis x right parenthesis equals x cubed minus 8 x squared plus 19 x minus 12. 

a)
Given that left parenthesis x minus 4 right parenthesis is a factor of q left parenthesis x right parenthesis, determine the x-intercepts of q left parenthesis x right parenthesis.

How did you do?

9b
Sme Calculator3 marks
b)
Hence or otherwise, solve the inequality x cubed plus 19 x less or equal than 8 x squared plus 12.

How did you do?

Did this page help you?

10
Sme Calculator3 marks

Consider the two functionsspace f left parenthesis x right parenthesis equals 2 space sin invisible function application 2 x and g left parenthesis x right parenthesis equals cos space x comma both having the domain 0 less or equal than x less or equal than 2 pi. 

Solve the inequality f left parenthesis x right parenthesis greater or equal than g left parenthesis x right parenthesis.

How did you do?

Did this page help you?

1
Sme Calculator4 marks

Consider the functions defined by f left parenthesis x right parenthesis equals x squared minus 6 a x plus b plus 10 and g left parenthesis x right parenthesis equals a x plus 2 b plus 3 commawhere a comma b element of Z to the power of plus. Given that f left parenthesis x right parenthesis less or equal than g left parenthesis x right parenthesis only for space 2 less or equal than x less or equal than 5 comma find the values of a and b.

 

How did you do?

Did this page help you?

2a
Sme Calculator3 marks

The function defined by f left parenthesis x right parenthesis equals x to the power of 4 minus 12 x cubed plus 46 x squared minus 60 x plus 25 can be factorised into the form f left parenthesis x right parenthesis equals left parenthesis x minus a right parenthesis squared left parenthesis x minus b right parenthesis squared comma where a and b are positive integers such that a less than b.

(a)
Find the values of a and b.

How did you do?

2b
Sme Calculator3 marks
(b)
Determine the set of values of that satisfy
(i)
f open parentheses x close parentheses greater or equal than 0,
(ii)
f open parentheses negative x close parentheses greater or equal than 0 comma
(iii)
negative f open parentheses x close parentheses less than 0.

How did you do?

2c
Sme Calculator2 marks
(c)
Determine the smallest positive value k such that the solution to the inequality f open parentheses x close parentheses less or equal than k  is a single interval.

 

How did you do?

Did this page help you?

3
Sme Calculator5 marks

The function f is such that 

 f left parenthesis x right parenthesis greater or equal than 0 space space for space space x less or equal than 3 space space and space for space space 4 less or equal than x less or equal than 5 comma

f left parenthesis x right parenthesis less or equal than 0 space space for space space 3 less or equal than x less or equal than 4 space space and space for space space x greater or equal than 5.

Find a polynomial, of the lowest degree possible, that satisfies the condition f open parentheses 0 close parentheses equals 5.

How did you do?

Did this page help you?

4a
Sme Calculator3 marks
(a)
Sketch the graph of y equals f open parentheses x close parentheses where
f left parenthesis x right parenthesis equals fraction numerator left parenthesis x plus 2 right parenthesis left parenthesis x minus 4 right parenthesis left parenthesis x minus 6 right parenthesis over denominator left parenthesis x minus 1 right parenthesis left parenthesis x minus 5 right parenthesis end fraction    
       
Label any intersections with the coordinate axes and state the equations of any vertical asymptotes.

How did you do?

4b
Sme Calculator5 marks
(b)
Find the values of  x that satisfy
(i)
f left parenthesis x right parenthesis greater or equal than 0. space space space
(ii)
f left parenthesis vertical line x vertical line right parenthesis greater or equal than 0.

 

How did you do?

Did this page help you?

5
Sme Calculator6 marks

The region R is defined by the three straight lines given by the inequalities

y greater or equal than 1 comma space

y less or equal than 2 x plus 8 comma space

x plus y less or equal than 10.

The function f is defined by f open parentheses x close parentheses equals 2 plus fraction numerator 1 over denominator x minus 1 end fraction. Find the largest domain of f such that the graph of f lies within the region R. Give answers as exact values where appropriate.

How did you do?

Did this page help you?

6a
Sme Calculator1 mark
(a)
Consider the graphs with equations
y equals fraction numerator open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses over denominator x minus 1 end fraction and space y equals 6 minus x. 
Explain why the two graphs do not intersect.

How did you do?

6b
Sme Calculator3 marks
(b)
Consider the graphs with equations
y equals fraction numerator open parentheses x minus 6 close parentheses open parentheses x minus 1 close parentheses squared over denominator x minus 1 end fraction space and space y equals open parentheses 8 minus x close parentheses open parentheses x minus 1 close parentheses.
(i)

Find the coordinates of any points of intersections between the two graphs.

(ii)

Hence, or otherwise, solve the inequality

fraction numerator open parentheses x minus 6 close parentheses open parentheses x minus 1 close parentheses squared over denominator x minus 1 end fraction less or equal than open parentheses 8 minus x close parentheses open parentheses x minus 1 close parentheses.

How did you do?

Did this page help you?

7a
Sme Calculator3 marks

Consider the functions defined by f left parenthesis x right parenthesis equals square root of left parenthesis 9 minus x squared right parenthesis end root comma space g left parenthesis x right parenthesis equals 3 minus square root of left parenthesis 9 minus x squared right parenthesis end root and h open parentheses x close parentheses equals fraction numerator x plus 3 over denominator 2 end fraction.All three functions have the domain negative 3 less or equal than x less or equal than 3. 

(a)
On the same diagram, sketch the graphs of f comma space g spaceand h.

How did you do?

7b
Sme Calculator3 marks
(b)
Find the set of values of x which satisfy the inequality f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis greater than h left parenthesis x right parenthesis.

How did you do?

Did this page help you?

8
Sme Calculator6 marks

Find the exact values for x such that

fraction numerator x over denominator open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses end fraction greater or equal than x

 

How did you do?

Did this page help you?