Inequalities (DP IB Analysis & Approaches (AA): HL): Exam Questions

(a) Sketch the graph of the function .
Mark your sketch clearly with the -coordinates of the -axis intercepts.

(b) Write down the solution to the inequality .

(c) Briefly explain how the graph shows that there are no real solutions to the inequality .

Consider the function defined by .

(i) Sketch the graph of .

(ii) Solve the inequality  using exact values.

Consider the function

(a) Fully factorise .

(b) Solve

(i)

(ii)

(iii)

(iv)

Find the values of  such that the equation  has real solutions and the equation  has no real solutions.

(a) Giving answers to three significant figures, find the set of values of  that satisfy

for

(b) Explain how your answer to part (a) would differ if the domain of was changed from  to .

(a) Write down the set of values of  for which .

(b) Find the set of values of for which 

          (i)     

          (ii)    

(c) Find the set of values of  for which

(i)

(ii)

Consider the functions

and  

where  is real constant such that  

(a)  In terms of the constant , find 

          (i)     the values of  for which  and  are undefined, 

          (ii)    the -coordinate of any intersections between the graphs of and .

(b) In the case , find the set of values of  in terms of  for which 

          (i)     , 

          (ii)    .

(c) Repeat questions (b) (i) and (ii) in the case .

(a)

(i) Show that  is a root of the function . (ii) Hence fully factorise .

(b)

Solve the inequality 

(c) Solve the inequality , where  and are constants such that .  Give your answers in terms of  and .

(a) Consider the functions defined by   and ,  where  is a positive constant.
Solve the inequality ,  giving your answer in terms of .

(b) Consider the functions defined by   and  ,  where  is a real constant.
Solve the inequality ,  giving your answer in terms of .

(c) Consider the functions defined by  and ,  where  and .
Write down, in terms of  and , the solution to the inequality when

(i) is even,

(ii) is odd.

Consider the functions defined by  and  where  and  are positive constants.
Show that

 for  .   .

Consider the functions defined by and where .
Given that  only for find the values of a and b.

The function defined by  can be factorised into the form  where  and  are positive integers such that .

(a) Find the values of a and b.

(b) Determine the set of values of that satisfy

(i) ,

(ii)

(iii)

(c) Determine the smallest positive value  such that the solution to the inequality   is a single interval.

The function  is such that 

Find a polynomial, of the lowest degree possible, that satisfies the condition .

(a) Sketch the graph of where

Label any intersections with the coordinate axes and state the equations of any vertical asymptotes.

(b) Find the values of  that satisfy

(i)

(ii)

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

The function  is defined by .
Find the largest domain of  such that the graph of lies within the region R.
Give answers as exact values where appropriate.

(a) Consider the graphs with equations

. 

Explain why the two graphs do not intersect.

(b) Consider the graphs with equations

.

(i) Find the coordinates of any points of intersections between the two graphs.

(ii) Hence, or otherwise, solve the inequality

Consider the functions defined by  and All three functions have the domain  

(a) On the same diagram, sketch the graphs of and .

(b) Find the set of values of which satisfy the inequality .

Find the exact values for  such that