Modulus & Inequalities - IB Math AA Question Bank

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Modulus & Inequalities

Practice exam-style IB Math AA questions for Modulus & Inequalities, aligned with the syllabus and grouped by topic.

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Question 1

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

Consider the inequality $|3x-2|<x+4$ .

State the necessary condition on $x+4$ .

[1]

Hence solve the inequality.

[4]

Question 2

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

For a function $f$ , it is known that

f(X)\geq0 \quad \text{if and only if} \quad -4\leq X\leq0 \text{ or } X\geq3

Let $h(x)=f(2x-5)$ .

Find the values of $x$ corresponding to $X=-4$ , $X=0$ and $X=3$ under the transformation $X=2x-5$ .

[2]

Hence solve $h(x)\geq0$ .

[2]

Question 3

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

Let $f(x)=x^2-3x-4$ , where $x\in\mathbb{R}$ .

Write down an expression for $f(|x|)$ .

[1]

Solve $f(|x|)<0$ .

[3]

Question 4

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2+1$ and $g(x)=x^3+x^2-3x+3$ , where $x\in\mathbb{R}$ .

Show that $g(x)-f(x)=(x+2)(x-1)^2$ .

[2]

Hence solve $g(x)\geq f(x)$ .

[3]

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Question 5

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the equation $|x+2|+|2x-1|=7$ .

Write down the critical values of $x$ at which the modulus expressions change form.

[1]

Solve the equation.

[4]

Question 6

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2-4x+3$ , where $x\in\mathbb{R}$ .

Sketch the graph of $y=|f(x)|$ , clearly showing the intercepts and the coordinates of any turning points.

[3]

Solve $|f(x)|=5$ .

[3]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^3-3x$ and let $h(x)=f(2x-1)$ .

Find the zeros of $h$ .

[2]

Solve $h(x)\geq0$ .

[3]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2-2x-3$ , where $x\in\mathbb{R}$ .

Rewrite $[f(x)]^2\leq9$ as a double inequality involving $f(x)$ .

[1]

Hence solve $[f(x)]^2\leq9$ .

[4]

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Question 9

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the inequality $|x-2|\geq x^2-4$ .

Solve the inequality for $x<2$ .

[2]

Solve the inequality for $x\geq2$ .

[2]

Hence state the solution set.

[1]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=x^2+1$ and $g(x)=x^3-x^2-5x+7$ , for $x\in\mathbb{R}$ .

Show that $g(x)-f(x)=(x+2)(x-1)(x-3)$ .

[2]

Hence solve $g(x)\geq f(x)$ .

[3]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=(x+3)(x-1)(x-4)$ , where $x\in\mathbb{R}$ .

Solve $|f(x)|=f(x)$ .

[3]

State the domain of the function $h(x)=\dfrac{1}{f(x)}$ and the equations of its vertical asymptotes.

[2]

Question 12

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Solve the equation $|2x-5|+|x+1|=8$ .

[5]

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Question 13

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Solve the equation $|x^2-4|=x+2$ .

[5]

Question 14

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The graph of $y=|x^2-4|$ is intersected by the horizontal line $y=k$ , where $k\in\mathbb{R}$ .

State the coordinates of the local maximum and the local minima of $y=|x^2-4|$ .

[2]

Determine the range of values of $k$ for which the equation $|x^2-4|=k$ has exactly four distinct real solutions.

[2]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=1+\dfrac{2}{x-1}$ , for $x\neq1$ . Let $h(x)=f(|x|)$ .

Write $h(x)$ as a piecewise function and state its domain.

[3]

Find the equations of the asymptotes of $y=h(x)$ and solve $h(x)=3$ .

[3]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2-4$ , where $x\in\mathbb{R}$ .

Sketch the graph of $y=\dfrac{1}{f(x)}$ , showing all asymptotes and the $y$ -intercept.

[4]

Solve $\dfrac{1}{f(x)}\geq\dfrac15$ .

[3]

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Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the inequality

\left|\frac{x+1}{x-2}\right|<2

where $x\neq2$ .

Find the values of $x$ for which $\left|\dfrac{x+1}{x-2}\right|=2$ .

[2]

Hence solve the inequality.

[3]

Question 18

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2-2x-3$ , where $x\in\mathbb{R}$ .

Sketch the graph of $y=f(|x|)$ , showing the intercepts and the coordinates of any local maxima or minima.

[4]

Justify why $f(|x|)$ is an even function.

[1]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the inequality

|3x\arccos x|>1

Use a GDC to solve the inequality for $-1\leq x\leq1$ .

[5]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Solve the inequality $|x-4|\leq2|x+1|-3$ .

[5]

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Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=(x-2)e^{-x}$ , $x\in\mathbb{R}$ .

Function	Domain
$f(x)=(x-2)e^{-x}$	$x\\in\\mathbb{R}$

State the equation of the vertical asymptote of the graph of $y=\dfrac{1}{f(x)}$ .

[1]

Use a GDC to solve $|f(x)|<0.1$ .

[4]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=\ln x-\dfrac{x}{3}$ , where $x>0$ .

Use a GDC to solve $|f(x)|\leq0.5$ .

[5]

Question 23

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Solve the inequality $\dfrac{x+2}{x-1}\geq x$ .

[6]

Question 24

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(X)=(X-3)(X-2)(X+2)$ , where $X\in\mathbb{R}$ . A second function is defined by $h(x)=f(1-2x)$ , where $x\in\mathbb{R}$ .

I.

Find the intervals for which $f(X)\leq0$ .

[3]

II.

Hence solve $h(x)\leq0$ .

[1]

I.

Show that $h(x)=-2(x+1)(2x+1)(2x-3)$ .

[3]

II.

Hence solve $h(x)>0$ . If you did not obtain the result in part (b)(i), use $h(x)=-2(x+1)(2x+1)(2x-3)$ .

[2]

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Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the function $F(x)=|x+4|+2|x-1|-(x+10)$ , where $x\in\mathbb{R}$ .

I.

Write down the critical values of $x$ at which the modulus expressions change form.

[1]

II.

Express $F(x)$ without modulus signs on each of the intervals $x<-4$ , $-4\leq x<1$ and $x\geq1$ .

[3]

I.

Solve the equation $|x+4|+2|x-1|=x+10$ .

[3]

II.

Hence solve the inequality $|x+4|+2|x-1|\leq x+10$ . If you did not obtain the equation solutions in part (b)(i), use $x=-2$ and $x=4$ as boundary points.

[2]

III.

Find the length of the solution interval found in part (b)(ii).

[1]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $F(x)=|x-2|+|x+2|$ and $L(x)=x+5$ , where $x\in\mathbb{R}$ .

I.

Express $F(x)$ as a piecewise function.

[3]

II.

Find the points of intersection of $y=F(x)$ and $y=L(x)$ .

[1]

I.

Solve $F(x)\leq L(x)$ .

[2]

II.

Find the exact area of the finite region enclosed by the graphs of $y=F(x)$ and $y=L(x)$ . If you did not obtain the intersection points in part (a)(ii), use $x=-1$ and $x=5$ .

[4]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $p\in\mathbb{R}$ . Consider the equation $|x-1|=px+3$ .

For $x<1$ , find the solution in terms of $p$ and state when it is valid.

[2]

For $x\geq1$ , find the solution in terms of $p$ and state when it is valid.

[2]

Determine the values of $p$ for which the equation has exactly one solution.

[2]

Question 28

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=x^2-5x+4$ , for $x\in\mathbb{R}$ .

Consider the graph of $y=|f(x)|$ .

I.

Find the coordinates of the local maximum and local minima of $y=|f(x)|$ .

[3]

II.

Solve $|f(x)|\leq5$ .

[3]

Define $h$ by $h(x)=|f(2x-3)|$ .

I.

Write down the values of $x$ for which $h(x)=0$ .

[2]

II.

Hence solve $h(x)\leq5$ .

[2]

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Question 29

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=\dfrac{x^2-4}{x-5}$ , where $x\neq5$ . Define $h(x)=\dfrac{1}{f(|x|)}$ .

Investigate the domain and asymptotes of $h$ .

I.

State the domain of $h$ .

[2]

II.

Show that $h(x)=\dfrac{|x|-5}{x^2-4}$ , and state the equations of all vertical asymptotes.

[3]

Solve $h(x)\geq0$ .

[4]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=x^2-2x+1$ and $g(x)=x^3+2x^2-7x+4$ , for $x\in\mathbb{R}$ .

Compare the graphs of $y=f(x)$ and $y=g(x)$ .

I.

Show that $g(x)-f(x)=(x+3)(x-1)^2$ .

[2]

II.

Hence solve $g(x)\geq f(x)$ .

[3]

Let $F(x)=f(2x+1)$ and $G(x)=g(2x+1)$ .

I.

Find a factorised expression for $G(x)-F(x)$ .

[2]

II.

Solve $G(x)<F(x)$ .

[2]

Question 31

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $A=x^2-2x-8$ and $B=x+2$ .

Consider the equation $|A|=|B|$ .

I.

Show that $x=-2$ , $x=3$ and $x=5$ satisfy $|x^2-2x-8|=|x+2|$ .

[3]

II.

Explain why $x=-2$ is a repeated boundary point when solving inequalities involving $|A|$ and $|B|$ .

[2]

Solve $|x^2-2x-8|\geq|x+2|$ .

[4]

Question 32

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(X)=(X+2)(X-3)$ and define $h(x)=|f(3-2x)|$ , for $x\in\mathbb{R}$ .

Investigate the transformation from $y=f(X)$ to $y=h(x)$ .

I.

Show that $h(x)=|4x^2-10x|$ .

[2]

II.

Find the coordinates of the local maximum of $y=h(x)$ between its two zeros.

[3]

Solve $h(x)\leq6$ .

[3]

The horizontal line $y=k$ , where $k>0$ , intersects the graph of $y=h(x)$ at exactly four distinct points. Determine the possible values of $k$ .

[2]

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Question 33

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The function $f$ is defined by $f(x)=\dfrac{x-1}{x+2}$ , for $x\neq-2$ .

I.

State the equations of the asymptotes of $y=f(x)$ and the intercepts with the axes.

[3]

II.

Sketch $y=|f(x)|$ , clearly showing any asymptotes and intercepts.

[2]

I.

Find an expression for $\dfrac{1}{f(x)}$ and state its domain.

[2]

II.

Solve $|f(x)|\leq\dfrac12$ .

[4]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=x^2-6x+5$ , where $x\in\mathbb{R}$ , and let $h(x)=f(|x|)$ .

I.

Write $h(x)$ in terms of $|x|$ and find its zeros.

[2]

II.

Find the coordinates of the local maximum and local minima of $y=h(x)$ .

[3]

I.

Solve $h(x)\leq0$ .

[2]

II.

State the domain and the vertical asymptotes of $y=\dfrac{1}{h(x)}$ .

[2]

III.

Hence solve $\dfrac{1}{h(x)}<0$ . If you did not obtain the zeros in part (a)(i), use $x=-5,-1,1,5$ .

[1]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=x^2-4x$ , where $x\in\mathbb{R}$ , and let $q(x)=[f(x)]^2$ .

I.

Write $f(x)$ in completed-square form and state its minimum value.

[2]

II.

Find the coordinates of the local maxima and local minima of $y=q(x)$ .

[2]

I.

Solve $q(x)\leq16$ .

[4]

II.

Determine the number of distinct real solutions of $q(x)=k$ , for each possible range of $k\in\mathbb{R}$ .

[3]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(X)=(X+4)(X-2)(X-5)$ , where $X\in\mathbb{R}$ . The function $h$ is defined by $h(x)=f(3-2x)$ .

I.

Solve $f(X)\geq0$ .

[3]

II.

Hence solve $h(x)\geq0$ .

[1]

I.

Show that $h(x)=-2(x+1)(2x-1)(2x-7)$ .

[3]

II.

Hence solve $|h(x)|=-h(x)$ . If you did not obtain the expression in part (b)(i), use $h(x)=-2(x+1)(2x-1)(2x-7)$ .

[2]

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Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $a\geq0$ , define $F_a(x)=|x-a|+|x+a|$ , where $x\in\mathbb{R}$ .

I.

Show that $F_a(x)=2a$ for $|x|\leq a$ and $F_a(x)=2|x|$ for $|x|\geq a$ .

[3]

II.

For $a=3$ , solve $F_a(x)=8$ .

[1]

I.

Determine the number of distinct real solutions of $F_a(x)=8$ for each possible value of $a\geq0$ .

[3]

II.

Solve $F_a(x)\leq8$ in terms of $a$ .

[3]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(t)=(t+1)(t-2)(t-4)$ , where $t\in\mathbb{R}$ . Define $p(x)=f(|x|)$ .

I.

Find the zeros of $p$ .

[2]

II.

Explain why $p$ is an even function.

[2]

I.

Solve $p(x)\geq0$ .

[3]

II.

The function $s$ is defined by $s(x)=\dfrac{1}{p(x)}$ . State the domain of $s$ and solve $s(x)<0$ . If you did not obtain the zeros in part (a)(i), use $x=-4,-2,2,4$ .

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=\ln(x+3)$ , for $x>-3$ , and let $g(x)=0.15x^2-0.7$ . Define $h(x)=g(x)-f(x)$ .

Use a GDC to investigate the intersections of the two curves.

I.

Find the $x$ -coordinates of the points of intersection of $y=f(x)$ and $y=g(x)$ .

[3]

II.

Hence solve $g(x)\geq f(x)$ .

[2]

Let $h(x)=g(x)-f(x)$ .

I.

Find the minimum value of $h(x)$ for $x>-3$ .

[3]

II.

The graph of $y=g(x)$ is translated vertically upwards by $k$ units. Determine the least value of $k$ for which the translated graph is on or above the graph of $y=f(x)$ for all $x>-3$ .

[2]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider the equation and inequality involving two modulus expressions
$|x+1|+|2x-5|$ and the quadratic expression $x^2-2$ .

Solve the equation $|x+1|+|2x-5|=x^2-2$ .

I.

Write down the critical values of $x$ and the three corresponding intervals that must be considered.

[2]

II.

Solve the equation $|x+1|+|2x-5|=x^2-2$ , giving exact answers.

[4]

Hence solve the inequality $|x+1|+|2x-5|\leq x^2-2$ .

[4]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=x^3-4x$ , for $x\in\mathbb{R}$ .

Consider the equation $[f(x)]^2=9$ .

I.

Show that the solutions of $f(x)=3$ are $x=-1$ and $x=\dfrac{1\pm\sqrt{13}}{2}$ .

[3]

II.

Find the solutions of $f(x)=-3$ .

[3]

Hence solve $[f(x)]^2\leq9$ .

[5]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=\ln x$ and $g(x)=2-\dfrac{x}{2}$ , for $x>0$ . Also define $d(x)=g(x)-f(x)$ .

Use a GDC to compare the two functions.

I.

Find the $x$ -coordinate of the point of intersection of $y=f(x)$ and $y=g(x)$ .

[2]

II.

Hence solve $g(x)\geq f(x)$ .

[2]

Let $d(x)=g(x)-f(x)$ .

I.

Explain why $d$ is strictly decreasing on $x>0$ .

[2]

II.

Solve $|g(x)-f(x)|\leq1$ .

[4]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A sensor has signed error $e(t)=0.4t^2-3t+2$ , where $t$ is measured in hours and $0\leq t\leq8$ . The absolute error is modelled by $E(t)=|e(t)|$ .

Investigate the model for the absolute error.

I.

Find the times at which the signed error is zero.

[2]

II.

Find the maximum value of $E(t)$ on $0\leq t\leq8$ .

[3]

The sensor is considered to be within tolerance when $E(t)\leq1.5$ .

I.

Find the time intervals during which the sensor is within tolerance.

[4]

II.

Find the total length of time for which the sensor is within tolerance.

[2]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a>0$ , define $D_a(x)=|x-a|+|x+a|$ . This question investigates the shape of $y=D_a(x)$ and inequalities involving a sloping line.

I.

Write $D_a(x)$ as a piecewise-defined function.

[3]

II.

Sketch $y=D_3(x)$ , showing the coordinates of the two corner points.

[2]

Solve $D_3(x)\leq x+6$ .

[4]

For $0<m<2$ , solve $D_a(x)\leq mx+2a$ in terms of $a$ and $m$ .

[5]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a real parameter $t$ , define $h_t(X)=(X+2)(X-t)(X-4)$ and $H_t(x)=h_t(2x-1)$ . This question compares the sign of a cubic before and after a horizontal transformation.

Object	Description	Role
$h_t(X)$	A cubic depending on $t$	Object studied
$h_1(X)$	A special case of $h_t(X)$	Used in part (a)
$H_t(x)$	A transformed cubic	Used in parts (b) and (c)
Transformation	A linear change of variable	Relates $X$ and $x$

Solve $h_1(X)\geq0$ .

[3]

I.

Write down the three roots of $H_t(x)=0$ .

[2]

II.

Hence solve $H_3(x)\geq0$ .

[3]

Determine the values of $t$ for which $H_t(x)\geq0$ for every $x$ in the interval $0\leq x\leq2$ .

[4]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $f(x)=x^2-2x-8$ , where $x\in\mathbb{R}$ . This question compares the graphs of $y=f(x)$ , $y=|f(x)|$ and $y=[f(x)]^2$ .

Find the zeros of $f$ and the coordinates of its vertex.

[3]

Using the supplied graph, state all intercepts and local extrema of $y=|f(x)|$ , describe how it is obtained from $y=f(x)$ , and state its end behaviour.

[4]

Solve $[f(x)]^2\leq16$ .

[5]

Hence solve $|f(x)|\geq [f(x)]^2$ .

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Define $N(x)=\big||x-2|-3\big|$ . This question investigates a nested modulus function.

Write $N(x)$ as a piecewise-defined function.

[4]

Using the graph, state the range of $N(x)$ and the intervals on which $N(x)$ is increasing and decreasing.

[3]

Solve $N(x)\leq2$ .

[3]

Determine the number of real solutions of $N(x)=c$ for all real values of $c$ .

[4]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $S(x)=|x+1|+2|x-4|$ . This question investigates inequalities involving a weighted sum of distances on a number line.

Write $S(x)$ as a piecewise-defined function.

[3]

Solve $S(x)\leq9$ .

[3]

I.

For $p>0$ , define $T_p(x)=|x|+p|x-6|$ . Write $T_p(x)$ as a piecewise-defined function.

[3]

II.

Determine the value or values of $x$ for which $T_p(x)$ is a minimum, considering separately $0<p<1$ , $p=1$ and $p>1$ .

[3]

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Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $F(x)=(x+4)(x-2)^2$ . This question investigates inequalities involving a cubic with a repeated root and transformations of its input.

I.

Solve $F(x)\geq0$ .

[3]

II.

Solve $|F(x)|=F(x)$ .

[2]

Solve $F(3x-1)<0$ .

[3]

Solve $\frac{1}{F(|x|)}>0$ .

[4]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $f(x)=(x-2)^2-1$ , where $x\in\mathbb{R}$ . The function $r$ is defined by $r(x)=\dfrac{1}{f(x)}$ .

AI.

State the zeros of $f$ and hence the vertical asymptotes of $y=r(x)$ .

[2]

AII.

Find the coordinates of the turning point of $y=f(x)$ and state the horizontal asymptote of $y=r(x)$ .

[2]

BI.

Find the values of $x$ for which $r(x)=1$ .

[2]

BII.

Hence solve $r(x)\geq1$ . If you did not obtain the values in part (b)(i), use $x=2-\sqrt2$ and $x=2+\sqrt2$ .

[4]

Question 51

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $k>2$ , the graphs of $y=|x-1|$ and $y=k-|x+1|$ enclose a finite region.

I.

Show that the $x$ -coordinates of the points of intersection are $x=-\dfrac{k}{2}$ and $x=\dfrac{k}{2}$ .

[3]

II.

For $k=4$ , find the area of the enclosed region.

[1]

I.

Show that the area $A$ of the enclosed region is $A=\dfrac{(k-2)^2}{2}+2(k-2)$ .

[5]

II.

Given that the area of the enclosed region is $24$ , find $k$ . If you did not obtain the formula in part (b)(i), use $A=\dfrac{(k-2)^2}{2}+2(k-2)$ .

[3]

Question 52

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $R(x)=\left|\dfrac{x-3}{x+1}\right|$ , where $x\in\mathbb{R}$ and $x\neq-1$ .

I.

Solve $R(x)=2$ .

[3]

II.

Hence solve $R(x)>2$ . If you did not obtain the values in part (a)(i), use $x=-5$ and $x=\dfrac13$ as boundary points.

[2]

I.

For $m\in\mathbb{R}$ , solve $R(x)=m$ in terms of $m$ , where possible.

[3]

II.

Determine the number of distinct real solutions of $R(x)=m$ for all $m\in\mathbb{R}$ .

[2]

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Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=\dfrac{1}{x-1}$ and $g(x)=\dfrac{x+2}{x+4}$ , where $x\neq1,-4$ .

Compare $f$ and $g$ algebraically.

I.

Show that $g(x)-f(x)=\dfrac{x^2-6}{(x+4)(x-1)}$ .

[2]

II.

Solve $g(x)\geq f(x)$ .

[4]

Solve $|g(x)-f(x)|\leq1$ .

[6]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $p(x)=|2\sin x-1|$ and $q(x)=1+\cos x$ , for $0\leq x\leq2\pi$ .

Find the intersection points of the two graphs.

I.

For the case $2\sin x-1\geq0$ , solve $p(x)=q(x)$ on $0\leq x\leq2\pi$ .

[4]

II.

For the case $2\sin x-1<0$ , solve $p(x)=q(x)$ on $0\leq x\leq2\pi$ .

[3]

Hence solve $|2\sin x-1|\leq1+ \cos x$ for $0\leq x\leq2\pi$ .

[4]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $q(x)=|x\ln(x+2)|$ , where $x>-2$ . This question uses both graphical and numerical methods to investigate a modulus inequality.

I.

State the domain of $q$ .

[1]

II.

Use a GDC to solve $q(x)=1$ .

[4]

Hence solve $q(x)\leq1$ .

[3]

Determine the smallest value of $c$ such that the solution of $q(x)\leq c$ contains the whole interval $[-1,1]$ .

[4]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $f(x)=x^2-5x+6$ and define $R(x)=\frac{1}{f(x)}$ . This question investigates inequalities involving a reciprocal graph.

State the domain of $R$ and the equations of its vertical asymptotes.

[2]

Solve $R(x)\geq\frac12$ .

[4]

I.

For $k>0$ , show that the solutions of $R(x)=k$ are $x=\frac{5\pm\sqrt{1+\frac4k}}{2}$ .

[2]

II.

Hence solve $R(x)\geq k$ for $k>0$ .

[2]

Find the value of $k$ for which the total length of the two intervals in part (c)(ii) is $5$ .

[3]

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Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $f(x)=x^3-5x+4$ . This question compares $y=f(|x|)$ and $y=|f(x)|$ .

Find the zeros of $f$ and solve $f(x)\geq0$ .

[4]

Hence solve $f(|x|)\geq0$ .

[4]

Solve $|f(x)|=f(|x|)$ .

[5]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $p(x)=|e^{-x}-x|$ . This question uses technology to solve a modulus inequality and then studies the effect of a horizontal transformation.

I.

Use a GDC to solve $p(x)=0.5$ .

[3]

II.

Hence solve $p(x)\leq0.5$ .

[2]

Let $r(x)=p(2x-1)$ . Solve $r(x)\leq0.5$ . If you did not obtain the values in part (a), use $0.266249$ and $0.904674$ .

[3]

Suppose the solution of $p(X)\leq0.5$ is $\alpha\leq X\leq\beta$ . For $a>0$ , $b\in\mathbb{R}$ , deduce the solution of $p(ax+b)\leq0.5$ .

[4]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a\in\mathbb{R}$ , let $C_a$ be the graph $y=|x-a|$ and let $P$ be the parabola $y=x^2-1$ . This question investigates their intersections.

I.

For $a=0$ , solve $|x|=x^2-1$ .

[3]

II.

Hence solve $|x|\leq x^2-1$ .

[3]

Show that intersections of $C_a$ and $P$ must satisfy $x^2-x+a-1=0$ when $x\geq a$ , and $x^2+x-a-1=0$ when $x<a$ .

[2]

Prove that $C_a$ and $P$ have exactly two points of intersection for every real value of $a$ .

[4]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let $A(X)=|X-3|$ and $B(X)=\ln(X+4)$ , where $X>-4$ . This question investigates the inequality $A(X)\leq B(X)$ and then applies an inside transformation.

I.

Use a GDC to solve $|X-3|=\ln(X+4)$ .

[3]

II.

Hence solve $|X-3|\leq\ln(X+4)$ .

[3]

Solve $|2x-4|\leq\ln(2x+3)$ . Give your answer to 2 decimal places. If you did not obtain the values in part (a), use $1.327$ and $5.222$ (or use the unrounded GDC values).

[3]

Suppose $A(X)\leq B(X)$ has solution $\alpha\leq X\leq\beta$ . For $a<0$ , deduce the solution of $A(ax+b)\leq B(ax+b)$ in terms of $a$ , $b$ , $\alpha$ and $\beta$ .

[3]

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