Quadratics - IB Math AA Question Bank

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Quadratics

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

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Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

A quadratic function has $x$ -intercepts $(-2,0)$ and $(4,0)$ . Its graph passes through the point $(1,-9)$ .

Find the function in factorized form.

[3]

Write down the equation of the axis of symmetry.

[1]

Find the $y$ -intercept of the graph of $f$ .

[1]

Question 2

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

The quadratic function $f$ is given by

f(x)=2x^2-12x+11

Write $f(x)$ in the form $a(x-h)^2+k$ .

[2]

State the coordinates of the vertex of the graph of $y=f(x)$ .

[1]

Find the two zeros of $f$ .

[2]

Question 3

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the quadratic function $f(x)=2x^2-12x+13$ , for $x\in\mathbb{R}$ .

Express $f(x)$ in the form $a(x-h)^2+k$ .

[2]

Write down the coordinates of the vertex of the graph of $f$ .

[1]

Solve the inequality $f(x)\leq 3$ .

[3]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The height, $h$ metres, of a ball above the ground $t$ seconds after it is thrown is modelled by
$h(t)=-5t^2+20t+25$ , where $t\geq 0$ .

Express $h(t)$ in vertex form.

[2]

Find the maximum height reached by the ball and the time at which it occurs.

[2]

Find the time at which the ball first reaches the ground.

[1]

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Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the quadratic equation $x^2-2kx+3k-2=0$ , where $k\in\mathbb{R}$ .

Find the discriminant in terms of $k$ , giving your answer in factorized form.

[2]

Find the values of $k$ for which the equation has two equal real roots.

[1]

Find the values of $k$ for which the equation has no real roots.

[2]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Consider the inequality $(x-1)(x-5)\leq 3$ , where $x\in\mathbb{R}$ .

Find the roots of the equation $(x-1)(x-5)=3$ .

[2]

Hence solve the inequality.

[2]

Question 7

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A ball is thrown vertically upwards from the top of a platform. Its height above the ground, $h$ metres, after $t$ seconds is modelled by

h(t)=-4.9t^2+18t+12, \quad t\ge0

Find the greatest height reached by the ball.

[2]

Find the time at which the ball reaches the ground.

[2]

Find the time interval during which the ball is at least $20$ metres above the ground.

[2]

Question 8

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Consider the quadratic equation

(k+2)x^2-4kx+(k-1)=0

where $k\in\mathbb{R}$ and $k\ne -2$ .

Find an expression for the discriminant in terms of $k$ .

[2]

Hence determine the nature of the roots of the equation for all possible values of $k$ .

[3]

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Question 9

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The graph of $y=x^2-5x-6$ is shown.

Find the $x$ -intercepts of the graph.

[2]

Hence solve the inequality $x^2-5x-6<0$ .

[2]

Find the minimum value of $x^2-5x-6$ .

[1]

Question 10

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The quadratic equation $x^2-mx+m+3=0$ has roots $\alpha$ and $\beta$ . One root is twice the other.

Write down $\alpha+\beta$ and $\alpha\beta$ in terms of $m$ .

[2]

Find the possible values of $m$ .

[4]

Question 11

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $k\in\mathbb{R}$ , consider the function $f_k(x)=x^2+2kx+k+3$ , where $x\in\mathbb{R}$ .

Find the minimum value of $f_k(x)$ in terms of $k$ .

[2]

Determine the values of $k$ for which $f_k(x)>0$ for all real $x$ .

[3]

Question 12

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $a\in\mathbb{R}$ , let $f_a(x)=x^2-2(a+1)x+a^2+4a$ .

Express $f_a(x)$ in vertex form.

[2]

The vertex of the graph of $f_a$ is $(H,K)$ . Find a linear relation between $H$ and $K$ .

[2]

Determine the values of $a$ for which the graph of $f_a$ has two distinct $x$ -intercepts.

[2]

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Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A quadratic function $f$ has a vertex at $(2,-3)$ . Its two $x$ -intercepts are $6$ units apart, and the coefficient of $x^2$ is positive.

Find the two $x$ -intercepts of the graph of $f$ .

[2]

Find $f(x)$ in the form $a(x-p)(x-q)$ .

[3]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The line $y=mx+1$ intersects the parabola $y=x^2-2x+5$ , where $m\in\mathbb{R}$ .

Find the quadratic equation in $x$ whose roots give the $x$ -coordinates of the intersection points.

[1]

Determine the values of $m$ for which the line is tangent to the parabola.

[2]

Determine the values of $m$ for which the line does not intersect the parabola.

[2]

Question 15

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A rectangular garden is to be built against a straight wall. Fencing is needed for the other three sides only. The total length of fencing available is $30$ metres. The side parallel to the wall has length $x$ metres and each perpendicular side has length $y$ metres.

Write down an equation connecting $x$ and $y$ .

[1]

Show that the area $A\ \text{m}^2$ of the garden is given by $A=15x-\frac{1}{2}x^2$ .

[2]

Find the maximum possible area of the garden.

[3]

Question 16

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A quadratic model $y=ax^2+bx+c$ is fitted to three measured points, shown in the table.

x	y
1	5.5
3	2.5
6	11.5

Using the data in the table, determine the quadratic model in the form $y=ax^2+bx+c$ .

[3]

Find the minimum value predicted by this model.

[2]

State the value of $x$ at which this minimum occurs.

[1]

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Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The line $y=mx+1$ is tangent to the parabola $y=x^2-4x+7$ .

Show that $m$ satisfies $m^2+8m-8=0$ .

[3]

Hence find the possible values of $m$ .

[2]

For the positive value of $m$ , find the point of tangency.

[1]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The quadratic equation

x^2-6x+k=0

has two distinct real roots $\alpha$ and $\beta$ .

Write down $\alpha+\beta$ and $\alpha\beta$ in terms of $k$ .

[1]

Given that $\alpha^2+\beta^2=20$ , determine the value of $k$ .

[2]

For this value of $k$ , form the quadratic equation with roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ .

[2]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The graph of a quadratic function $f$ has vertex $(2,-5)$ and passes through the point $(7,20)$ .

Determine $f(x)$ in the form $a(x-h)^2+k$ .

[3]

Find the exact $x$ -intercepts of the graph of $f$ .

[2]

Hence find the distance between the two $x$ -intercepts.

[1]

Question 20

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The quadratic function $f$ has vertex $(2,-9)$ and passes through the point $(0,3)$ .

I.

Find $f(x)$ in the form $a(x-h)^2+k$ .

[2]

II.

Hence express $f(x)$ in the form $ax^2+bx+c$ .

[2]

I.

Find the $x$ -intercepts of the graph of $y=f(x)$ . If you did not obtain $f(x)=3(x-2)^2-9$ , use this expression.

[2]

II.

Solve the inequality $f(x)<3$ . If you did not obtain $f(x)=3(x-2)^2-9$ , use this expression.

[2]

Determine the range of $f$ for the restricted domain $0\le x\le 5$ .

[2]

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Question 21

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The height, $h$ metres, of a small rocket above the ground $t$ seconds after launch is modelled by

h(t)=-4t^2+16t+5,\quad t\ge 0

I.

Express $h(t)$ in the form $a(t-p)^2+q$ .

[2]

II.

State the maximum height of the rocket and the time at which it occurs.

[2]

Find the time at which the rocket reaches the ground. If you did not obtain $h(t)=-4(t-2)^2+21$ , use this expression.

[3]

I.

Find the interval of time for which the rocket is at least $17$ metres above the ground. If you did not obtain $h(t)=-4(t-2)^2+21$ , use this expression.

[2]

II.

Find the second time at which the rocket is at its launch height.

[1]

Question 22

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

An arch is modelled by a parabola. The height of the arch, $H$ metres, above a horizontal floor is given by a quadratic function of the horizontal distance $x$ metres from the left end. The arch meets the floor at $x=0$ and $x=12$ , and its greatest height is $18$ metres.

I.

Write down the equation of the axis of symmetry of the arch.

[1]

II.

Show that the height of the arch may be written as $H=\frac12 x(12-x)$ .

[3]

A horizontal beam is placed at a height of $10$ metres above the floor. Find the horizontal width of the arch at this height. If you did not obtain $H=\frac12 x(12-x)$ , use this expression.

[4]

A rectangular vehicle is $6$ metres wide and passes centrally under the arch. Determine the greatest possible height of the vehicle.

[2]

Question 23

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A quadratic function $f$ has $x$ -intercepts $(-1,0)$ and $(5,0)$ , and passes through the point $(1,-12)$ .

I.

Find $f(x)$ in factorized form.

[2]

II.

Find the coordinates of the vertex of the graph of $y=f(x)$ .

[3]

Solve the inequality $f(x)\le 0$ . If you did not obtain $f(x)=\frac32(x+1)(x-5)$ , use this expression.

[2]

The line through the vertex and the $y$ -intercept of the graph of $f$ is denoted by $L$ . Find the equation of $L$ .

[3]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

A quadratic model $y=ax^2+bx+c$ passes through the three points $(0,5)$ , $(2,-3)$ and $(5,0)$ .

	1	2	3
x	0	2	5
y	5	-3	0

I.

Write down the value of $c$ .

[1]

II.

Find the values of $a$ and $b$ .

[3]

Express the model in vertex form and state its minimum value. If you did not obtain $y=x^2-6x+5$ , use this expression.

[3]

Determine the values of $x$ for which the model predicts $y\le 0$ .

[2]

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Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The quadratic equation $(k-1)x^2-2kx+k+3=0$ , where $k\neq 1$ , has roots $\alpha$ and $\beta$ . The roots differ by $2$ .

Write down $\alpha+\beta$ and $\alpha\beta$ in terms of $k$ .

[2]

Find the possible values of $k$ .

[4]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the quadratic equation $x^2-2(k+1)x+k^2-1=0$ , where $k\in\mathbb{R}$ .

Find the discriminant in terms of $k$ .

[2]

Determine the values of $k$ for which the equation has two distinct positive real roots.

[4]

Question 27

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The cross-section of a pedestrian arch is modelled by a parabola. The arch meets the ground at points $A(0,0)$ and $B(12,0)$ , where distances are measured in metres. The greatest height of the arch is $5$ metres.

I.

Write down the equation of the axis of symmetry of the parabola.

[1]

II.

Find an equation for the height $h$ of the arch in terms of the horizontal distance $x$ from $A$ .

[4]

I.

Find the two values of $x$ for which the height of the arch is $3.2$ metres.

[2]

II.

Hence find the horizontal width of the part of the arch that is at least $3.2$ metres high.

[2]

A rectangular vehicle of height $H$ metres is driven centrally through the arch. Show that the maximum possible width $w$ metres of the vehicle is given by $w=12\sqrt{1-\dfrac{H}{5}}$ , for $0\leq H\leq5$ .

[2]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A company sells headphones. If $x$ headphones are produced and sold in one week, the selling price is modelled by $p=50-0.025x$ dollars per headphone. The weekly cost, in dollars, is modelled by $C=900+18x$ . The factory can produce at most $1000$ headphones per week.

I.

Write down an expression for the weekly revenue $R$ in terms of $x$ .

[1]

II.

Show that the weekly profit $P$ is given by $P=-0.025x^2+32x-900$ .

[2]

I.

Find the number of headphones that should be produced and sold to maximize the weekly profit.

[2]

II.

Find the maximum weekly profit.

[2]

I.

Find the break-even values of $x$ , according to the model.

[2]

II.

Taking account of the factory capacity and the fact that $x$ must be a whole number, determine the values of $x$ for which the company makes a profit.

[2]

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Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A sensor records a signal strength $S$ at time $t$ seconds. The signal is modelled by a quadratic function $S(t)=at^2+bt+c$ , for $0\leq t\leq9$ . Three recorded values are shown in the table.

time t [s]	signal strength S
0	2.1
4	5.7
9	1.2

I.

Use the data to determine the values of $a$ , $b$ and $c$ .

[3]

II.

Write the model in vertex form.

[1]

I.

Find the maximum signal strength predicted by the model.

[1]

II.

Find the times at which the signal strength is $4$ .

[2]

A technician wants the signal strength to be at least a threshold value $T$ for exactly $4$ seconds. Determine the greatest possible value of $T$ .

[4]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A farmer uses $80$ metres of fencing to make three identical rectangular pens side by side. Each pen has length $x$ metres and width $y$ metres. The pens share two internal fences, each of length $y$ metres.

I.

Show that $6x+4y=80$ .

[1]

II.

Show that the total area $A$ of the three pens is given by $A=60x-\dfrac{9}{2}x^2$ .

[3]

I.

Find the value of $x$ that gives the maximum total area.

[1]

II.

Find the corresponding value of $y$ and the maximum total area.

[2]

The farmer changes the design to make $n$ identical pens side by side, using the same $80$ metres of fencing. Deduce the maximum possible total area in terms of $n$ .

[3]

Question 31

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Two functions are defined by $f(x)=x^2-6x+c$ and $g(x)=-\dfrac{1}{2}x^2+4x+1$ , where $c$ is a real constant.

I.

Show that the $x$ -coordinates of any points of intersection satisfy $1.5x^2-10x+c-1=0$ .

[2]

II.

Write down the discriminant of this quadratic equation in terms of $c$ .

[1]

I.

Find the value of $c$ for which the graphs are tangent.

[2]

II.

State the number of points of intersection when $c=20$ .

[1]

For $c=5$ , solve the inequality $f(x)\leq g(x)$ , giving your answer in interval notation.

[4]

Question 32

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A decorative panel is made in the shape of a parabola with vertical axis of symmetry. In a coordinate system with origin at the lowest point of the panel, the curve passes through the points $(-2.4,1.8)$ and $(2.4,1.8)$ , where distances are in metres.

I.

Explain why the equation of the curve may be written in the form $y=ax^2$ .

[1]

II.

Find the value of $a$ .

[2]

III.

Write down the equation of the curve.

[1]

I.

Find the width of the panel at a height of $1.0$ metre above the lowest point.

[2]

II.

Find the height of the panel at a horizontal distance of $1.5$ metres from the axis of symmetry.

[1]

A rectangular sign of height $0.80$ metres is placed centrally with its lower edge at the lowest point of the panel. Determine the greatest possible width of the sign.

[3]

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Question 33

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the equation

(k-1)x^2+2kx+k+3=0

where $k\in\mathbb{R}$ .

Find the values of $k$ for which the equation is quadratic and has two equal real roots.

[4]

For $k=1$ , solve the resulting equation.

[2]

Question 34

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The graphs of

y=x^2-3x+1

and

y=2x+k

intersect at two points whose $x$ -coordinates differ by $4$ .

Show that the $x$ -coordinates of the points of intersection satisfy $x^2-5x+(1-k)=0$ .

[2]

Find the value of $k$ .

[3]

For this value of $k$ , find the coordinates of the two points of intersection.

[2]

Question 35

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A family of quadratic functions is defined by

f_k(x)=x^2-2kx+3k

where $k\in\mathbb{R}$ .

Find the coordinates of the vertex of the graph of $y=f_k(x)$ in terms of $k$ .

[2]

Find the values of $k$ for which the graph of $y=f_k(x)$ does not meet the $x$ -axis.

[3]

For the values of $k$ found in part (b), determine the greatest possible value of the $y$ -coordinate of the vertex.

[2]

Question 36

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

Consider the quadratic equation

x^2-2(k-1)x+k^2-5=0

where $k\in\mathbb{R}$ .

I.

Find the discriminant in terms of $k$ , giving your answer in factorized form.

[3]

II.

Hence determine the values of $k$ for which the equation has two distinct real roots, two equal real roots, and no real roots.

[2]

Solve the equation when $k=1$ .

[2]

Determine the values of $k$ for which both roots of the equation are real and positive.

[4]

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Question 37

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $t\in\mathbb{R}$ , consider the family of quadratic functions

f_t(x)=x^2-2tx+4t-3

I.

Find the coordinates of the vertex of the graph of $y=f_t(x)$ in terms of $t$ .

[3]

II.

The vertex is denoted by $(H,K)$ . Show that $K=-(H-2)^2+1$ .

[2]

Determine the values of $t$ for which the graph of $y=f_t(x)$ has two distinct $x$ -intercepts.

[3]

Determine the values of $t$ for which $f_t(x)>0$ for all $x\in\mathbb{R}$ .

[2]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The line $L_k$ has equation $y=kx+1$ , where $k\in\mathbb{R}$ . The parabola $P$ has equation $y=x^2-4x+7$ .

I.

Show that the $x$ -coordinates of the points of intersection of $L_k$ and $P$ satisfy $x^2-(k+4)x+6=0$ .

[2]

II.

Determine the values of $k$ for which $L_k$ is tangent to $P$ .

[3]

For $k=-4+2\sqrt{6}$ , find the point of tangency. If you did not obtain $x^2-(k+4)x+6=0$ , use this equation.

[3]

Determine the values of $k$ for which $L_k$ intersects $P$ at two points whose $x$ -coordinates are both positive.

[3]

Question 39

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The quadratic equation

x^2-kx+(k+1)=0

has roots $\alpha$ and $\beta$ , where $k\in\mathbb{R}$ .

I.

Write down $\alpha+\beta$ and $\alpha\beta$ in terms of $k$ .

[2]

II.

Given that $\alpha^2+\beta^2=6$ , determine the possible values of $k$ .

[4]

Show that only one of the values of $k$ found in part (a)(ii) gives real roots for the original equation.

[2]

For the value of $k$ which gives real roots, form the quadratic equation with roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ .

[3]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $k\in\mathbb{R}$ , consider the family of quadratic functions $f_k(x)=x^2-2kx+k^2-4k+1$ . The graph of $y=f_k(x)$ has vertex $V_k$ .

I.

Express $f_k(x)$ in vertex form.

[2]

II.

Find the coordinates of $V_k$ and hence find the equation of the locus of $V_k$ .

[2]

I.

Determine the values of $k$ for which the graph has two distinct $x$ -intercepts.

[2]

II.

For $k>\dfrac{1}{4}$ , show that the distance between the two $x$ -intercepts is $2\sqrt{4k-1}$ .

[2]

The two $x$ -intercepts and $V_k$ form a triangle. Determine the value of $k$ for which the area of this triangle is $27$ .

[4]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The quadratic equation $x^2-(2k+1)x+k^2-3=0$ has roots $\alpha$ and $\beta$ , where $k\in\mathbb{R}$ .

I.

Write down $\alpha+\beta$ and $\alpha\beta$ in terms of $k$ .

[2]

II.

Find the discriminant in terms of $k$ .

[2]

I.

Determine the values of $k$ for which the equation has two distinct positive real roots.

[3]

Given that $\alpha^2+\beta^2=30$ , determine the value of $k$ for which the equation has real roots.

[4]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The graphs of $y=x^2+ax+4$ and $y=-x^2+2x+a$ intersect at points whose $x$ -coordinates satisfy a quadratic equation. Here $a\in\mathbb{R}$ .

I.

Show that the $x$ -coordinates of the points of intersection satisfy $2x^2+(a-2)x+(4-a)=0$ .

[2]

II.

Find the discriminant of this equation in terms of $a$ .

[1]

I.

Find the values of $a$ for which the graphs are tangent.

[2]

II.

Determine the values of $a$ for which the graphs intersect at two distinct points.

[2]

For $a=6$ , find the coordinates of the two points of intersection and hence find the distance between them.

[5]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A family of quadratic functions is defined by

f_t(x)=x^2-2tx+t^2-4t+3

where $t\in\mathbb{R}$ . The graph of $y=f_t(x)$ has vertex $V_t$ . The diagram shows several members of the family and the line on which their vertices appear to lie.

I.

Express $f_t(x)$ in vertex form and state the coordinates of $V_t$ .

[2]

II.

Hence show that the vertices of all graphs in the family lie on a straight line.

[1]

Determine the values of $t$ for which the graph of $y=f_t(x)$ has two distinct $x$ -intercepts.

[3]

I.

Show that the line found in part (a)(ii) intersects the graph of $y=f_t(x)$ at $x=t$ and $x=t-4$ .

[2]

II.

Hence determine the distance between the two intersection points, showing that it is independent of $t$ . If you did not obtain the two values in part (c)(i), use $x=t$ and $x=t-4$ .

[1]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A stream of water from a fountain is modelled by

h_k(x)=-0.2x^2+kx+1

where $x$ is the horizontal distance in metres from the nozzle, $h_k(x)$ is the height in metres above the ground, and $k>0$ is a parameter controlled by the nozzle angle. A low wall is located $4$ metres from the nozzle and has height $3$ metres.

Express $h_k(x)$ in vertex form and write down the maximum height in terms of $k$ .

[3]

I.

Determine the least value of $k$ for which the stream clears the wall.

[2]

II.

For $k=1.3$ , calculate the horizontal distance from the nozzle to the splash point.

[2]

The fountain is redesigned so that the stream must clear the wall and land between $8$ m and $10$ m from the nozzle, inclusive. Determine the possible values of $k$ .

[4]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A simplified model for the temperature $T$ degrees Celsius in a laboratory between 02:00 and 08:00 is

T_m(x)=m(x-2)(x-8)+20

where $x$ is the time in hours after midnight and $m>0$ . The graph is used to estimate the time interval during which the temperature is at or below a specified level. The graph shows the curves from $x=0$ to $x=10$ ; portions with $x<2$ or $x>8$ are extrapolations outside the model's stated validity interval.

I.

Expand $T_m(x)$ into standard form.

[1]

II.

State the equation of the axis of symmetry and the minimum temperature in terms of $m$ .

[2]

For $m=1.5$ , find the times at which the temperature is $12^\circ\text{C}$ , giving your answers to the nearest minute.

[3]

Determine the value of $m$ for which the temperature is at or below $12^\circ\text{C}$ for exactly $2$ hours.

[3]

For a general threshold temperature $H$ , where $H<20$ , show that the duration $D$ hours for which $T_m(x)\le H$ is

D=2\sqrt{9-\frac{20-H}{m}}

and state the condition on $m$ for this duration to exist.

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The fixed parabola

y=x^2-4x+1

is intersected by the family of lines

y=mx-3

where $m\in\mathbb{R}$ . The diagram shows some possible intersections.

I.

Show that the $x$ -coordinates of the intersection points satisfy

x^2-(m+4)x+4=0

[2]

II.

Determine the values of $m$ for which the line intersects the parabola at two distinct points.

[2]

For $m=5$ , find the exact $x$ -coordinates of the intersection points.

[2]

I.

Show that the square of the horizontal distance between the two intersection points is $(m+4)^2-16$ .

[2]

II.

Find the values of $m$ for which the horizontal distance between the intersection points is $3$ . If you did not obtain part (c)(i), use $(m+4)^2-16$ for the square of the distance.

[2]

Find the two tangent lines from the family and their points of tangency.

[2]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A rectangular courtyard is $30$ m by $18$ m. A uniform paved border of width $x$ metres is built around the inside edge, leaving a central rectangular grass region of area $A(x)$ square metres. The diagram shows the courtyard and grass region.

I.

Show that $A(x)=4x^2-96x+540$ .

[2]

II.

State the possible values of $x$ .

[1]

III.

Write down the axis of symmetry of the graph of $A(x)$ .

[1]

Calculate the width of the border when the grass area is $320\ \text{m}^2$ .

[3]

Determine the values of $x$ for which the grass area is at least $320\ \text{m}^2$ .

[2]

I.

For a general rectangular courtyard of dimensions $L$ by $W$ , where $L>W>0$ , show that the grass area is a decreasing function of $x$ throughout its physical domain.

[3]

II.

Hence find, in terms of $L$ , $W$ and $B$ , the greatest possible border width if the grass area must be at least $B$ , where $0<B<LW$ .

[2]

Question 48

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The parabolas $C_1$ and $C_2$ have equations

C_1: y=x^2-6x+5,\qquad C_2: y=-x^2+2kx-1

where $k\in\mathbb{R}$ .

I.

Show that the $x$ -coordinates of the points of intersection satisfy $x^2-(k+3)x+3=0$ .

[2]

II.

Determine the values of $k$ for which the two parabolas are tangent to each other.

[3]

For $k=-3+2\sqrt{3}$ , find the point of tangency. If you did not obtain $x^2-(k+3)x+3=0$ , use this equation.

[3]

Determine the values of $k$ for which the two parabolas intersect at two points whose $x$ -coordinates are both greater than $1$ .

[4]

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Question 49

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $a\in\mathbb{R}$ , $a\ne 0$ , consider the quadratic function

f_a(x)=ax^2-2(a+1)x+a+4

I.

Find the $x$ -coordinate of the vertex in terms of $a$ .

[2]

II.

The vertex is denoted by $(H,K)$ . Show that $H+K=3$ .

[2]

Determine the values of $a$ for which the graph of $y=f_a(x)$ opens upwards and its vertex is below the $x$ -axis.

[3]

Determine the values of $a$ for which the graph of $y=f_a(x)$ has no $x$ -intercepts.

[2]

For $a=\frac14$ , solve $f_a(x)\le 0$ .

[3]

Question 50

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $p\in\mathbb{R}$ , consider the quadratic function

q_p(x)=x^2-2px+p^2-p

I.

Find the coordinates of the vertex of the graph of $y=q_p(x)$ .

[2]

II.

Find the roots of $q_p(x)=0$ in terms of $p$ , stating the condition for the roots to be real.

[3]

Determine the values of $p$ for which the equation $q_p(x)=0$ has two distinct positive roots. If you did not obtain the roots $p\pm\sqrt{p}$ , use these roots.

[3]

Let the two roots be $\alpha$ and $\beta$ . Determine the value of $p$ for which $\frac{1}{\alpha}+\frac{1}{\beta}=4$ , given that the roots are positive.

[3]

Determine the values of $p$ for which the number $1$ lies strictly between the two roots of $q_p(x)=0$ .

[2]

Question 51

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $m\in\mathbb{R}$ , define $q_m(x)=x^2-2mx+m+2$ . A designer wants $q_m(x)$ to be non-negative for every real value of $x$ in a model.

AI.

Express $q_m(x)$ in vertex form.

[2]

AII.

Write down the minimum value of $q_m(x)$ in terms of $m$ .

[2]

BI.

Determine the values of $m$ for which $q_m(x)\geq0$ for all $x\in\mathbb{R}$ .

[3]

BII.

For which values of $m$ does $q_m(x)=0$ have exactly one real solution?

[1]

The designer instead only requires $q_m(x)\geq0$ for all $x$ in the interval $0\leq x\leq4$ . Determine the set of possible values of $m$ .

[4]

Question 52

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A family of parabolas is given by $f_t(x)=t(x-1)(x-5)+8$ , where $t$ is a non-zero real parameter. The vertex of the graph of $y=f_t(x)$ is denoted by $V_t$ .

I.

Write down the equation of the axis of symmetry of the graph of $y=f_t(x)$ .

[1]

II.

Find the coordinates of $V_t$ in terms of $t$ .

[3]

I.

Determine the values of $t$ for which the graph has two distinct $x$ -intercepts.

[3]

II.

Find the value of $t$ for which the graph touches the $x$ -axis.

[1]

For $t<0$ or $t>2$ , let the two $x$ -intercepts be $P$ and $Q$ . Show that the length $PQ$ is $2\sqrt{4-\dfrac{8}{t}}$ .

[4]

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Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $r\in\mathbb{R}$ , consider the quadratic equation $(r+1)x^2-2rx+r-3=0$ . When $r=-1$ , the equation is not quadratic.

I.

For $r\neq-1$ , find the discriminant in terms of $r$ .

[2]

II.

Determine the values of $r$ for which the equation is quadratic and has two distinct real roots.

[2]

I.

For $r\neq-1$ , write down the sum and product of the roots in terms of $r$ .

[2]

II.

Determine the values of $r$ for which the equation has two distinct positive real roots.

[3]

For the case $r=-1$ , solve the resulting equation. Hence explain why it must be considered separately from the discriminant analysis.

[4]

Question 54

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A quadratic curve $y=f_a(x)$ passes through the two fixed points $P(-1,6)$ and $Q(3.5,1.5)$ . It is written in the form

f_a(x)=ax^2+bx+c

where $a\ne0$ . The diagram shows several possible curves through $P$ and $Q$ .

I.

Show that $b=-1-\frac52a$ and $c=5-\frac72a$ .

[3]

II.

Write $f_a(x)$ in terms of $a$ only.

[1]

Determine the values of $a$ for which the graph is tangent to the $x$ -axis.

[4]

I.

For each value of $a$ found in part (b), find the point of tangency with the $x$ -axis. If you did not obtain values in part (b), use $a=\frac{2}{27}$ and $a=\frac23$ .

[2]

II.

Determine the values of $a$ for which the graph has no real $x$ -intercepts.

[2]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A vertical motion model is given by

s_v(t)=-t^2+vt+4-v

where $s_v(t)$ is the height in metres after $t$ seconds, $t\ge0$ , and $v$ is a real parameter. A horizontal beam is located at height $9$ metres.

I.

Write $s_v(t)$ in vertex form.

[2]

II.

Find the maximum height in terms of $v$ , assuming the vertex occurs for $t\ge0$ .

[2]

Determine the values of $v$ for which the object reaches the beam at least once.

[4]

I.

For $v=8$ , find the two times at which the object is at height $9$ metres.

[2]

II.

Find the value of $v$ for which the object just touches the beam, and state the time at which this occurs. If you did not obtain part (b), use $v^2-4v-20=0$ .

[3]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $0<d<5$ , a quadratic function has $x$ -intercepts at $5-d$ and $5+d$ , and has $y$ -intercept $20$ . It is denoted by $f_d$ . The diagram shows several such parabolas for different values of $d$ .

I.

Show that

f_d(x)=\frac{20}{25-d^2}(x-(5-d))(x-(5+d))

[2]

II.

State whether the parabola opens upwards or downwards.

[1]

III.

Write down the equation of the axis of symmetry.

[1]

Find the minimum value of $f_d(x)$ in terms of $d$ .

[2]

I.

Determine $d$ if the minimum value is $-45$ .

[2]

II.

For this value of $d$ , write $f_d(x)$ in vertex form.

[2]

Determine the possible values of $d$ for which the minimum value is less than $-80$ .

[3]

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Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For each real number $p$ , consider the quadratic equation

x^2-2px+p+2=0

When the roots are real, they are denoted by $\alpha$ and $\beta$ . The graph shows how the number of real roots changes as $p$ varies.

I.

Find the discriminant in terms of $p$ .

[1]

II.

Determine the values of $p$ for which the equation has two distinct real roots.

[2]

III.

State the values of $p$ for which the equation has equal real roots.

[1]

Show that, when the roots are real,

(\alpha-\beta)^2=4(p^2-p-2)

[3]

I.

Find the values of $p$ for which the roots differ by $6$ . If you did not obtain part (b), use $(\alpha-\beta)^2=4(p^2-p-2)$ .

[3]

II.

Prove that there is no value of $p$ for which the roots are real and both roots lie strictly between $0$ and $1$ .

[2]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A family of parabolas is given by

y=a(x-1)^2+2

where $a>0$ . A point $M$ lies on the graph, and its coordinates are $(x,y)$ . The area of the rectangle with opposite vertices $(1,2)$ and $M$ is studied for points on the right-hand branch of the parabola.

I.

Let $u=x-1$ , where $u>0$ . Express the vertical side length of the rectangle in terms of $u$ and $a$ .

[1]

II.

Hence write the area $R$ of the rectangle in terms of $u$ and $a$ .

[1]

III.

For $a=\frac12$ , find $x$ when $R=32$ .

[2]

A horizontal line $y=18$ intersects the parabola in two points. Determine the value of $a$ for which the distance between these two points is $8$ .

[3]

I.

Show that the line through $(1,2)$ and $M$ has gradient $au$ .

[2]

II.

For $a=2$ , find the point $M$ for which this gradient is $10$ . If you did not obtain part (c)(i), use gradient $au$ .

[2]

Deduce a relationship between the area $R$ of the rectangle and the gradient $g$ of the line through $(1,2)$ and $M$ , in terms of $a$ .

[3]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The concentration $C$ of a medicine in a patient, in arbitrary units, is modelled for $0\le t\le 8$ by

C_k(t)=-kt^2+6kt+2

where $k>0$ and $t$ is measured in hours. The medicine is considered effective when $C_k(t)\ge 10$ .

I.

Express $C_k(t)$ in vertex form.

[2]

II.

State the maximum concentration in terms of $k$ .

[1]

III.

Find the least value of $k$ for which the medicine is ever effective.

[1]

For $k=2$ , determine the time interval during which the medicine is effective.

[3]

I.

Show that, for $k>\frac89$ , the duration $D$ for which the medicine is effective is

D=2\sqrt{9-\frac8k}

[2]

II.

Find the value of $k$ for which the medicine is effective for exactly $4$ hours. If you did not obtain part (c)(i), use $D=2\sqrt{9-\frac8k}$ .

[2]

Evaluate the model by determining whether it is possible for the medicine to be effective for the entire interval $0\le t\le8$ .

[2]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two families of parabolas are defined by

f_a(x)=x^2+ax+1,\qquad g_a(x)=-x^2+2x+a

where $a\in\mathbb{R}$ . The intersections of their graphs depend on $a$ .

I.

Show that the $x$ -coordinates of the intersection points satisfy

2x^2+(a-2)x+(1-a)=0

[2]

II.

Find the discriminant of this quadratic in terms of $a$ .

[2]

III.

Determine the values of $a$ for which the two graphs intersect at two distinct points.

[1]

For the value of $a$ for which the graphs are tangent and $a>-2$ , find the point of tangency.

[4]

I.

Let the two intersection $x$ -coordinates be $r$ and $s$ . Show that

r+s=\frac{2-a}{2},\qquad rs=\frac{1-a}{2}

[2]

II.

Find the values of $a$ for which the two intersection $x$ -coordinates have product $-3$ . If you did not obtain part (c)(i), use $rs=\frac{1-a}{2}$ .

[2]

Determine the values of $a$ for which both intersection points are distinct and have positive $x$ -coordinates.

[2]

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Properties of Functions

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