IB Mathematics Analysis and Approaches Topic SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots and mirrors Paper 1, 2, 3 style where relevant.

Get instant solutions, detailed explanations, and build confidence with questions aligned to IB examiner expectations.

Paper

Level

Question 1

SLPaper 1

A line through $A (- 4, 3)$ with gradient $m$ intersects $C:\ y=x^2+8x$ at points $P$ and $Q$ . The distance $P Q = 11$ .

Show the $x$ -coordinates satisfy $x^2+(8-m)x+(-4m-3)=0.$

[4]

Determine the condition on $m$ for two distinct intersections.

[2]

Show $(x_{P} - x_{Q})^{2} = m^{2} + 76$ .

[3]

Solve for $m$ using $P Q = 11$ .

[4]

Question 2

SLPaper 2

Let $p \in R$ and $q_{p} (x) = x^{2} - (p + 8) x + (2 p + 8)$ .

Find all real values of $p$ for which $q_{p}$ has two distinct real roots and both roots are greater than $1$ .

[6]

Let $p = 3$ (not necessarily satisfying the condition in the previous part) so that $q (x) = x^{2} - 11 x + 14$ , and $g (x) = ln (x + 2) + 1$ . Find the exact roots of $q (x) = 0$ .

[3]

Solve $q (x) = g (x)$ for $x \geq 0$ , giving solutions correct to $3$ d.p., and justify the number of solutions.

[6]

Question 3

SLPaper 2

For a real parameter $p$ , consider the quadratic $q_p(x)=x^2-(p+2)x+(2p-5).$

Find all real values of $p$ for which $q_{p}$ has two distinct real roots and both roots are greater than $0$ .

[6]

Let $p = 4$ so that $q (x) = x^{2} - 6 x + 3$ , and let $g (x) = ln (x + 2) + 1$ (domain $x > - 2$ ).

Find the exact roots of $q (x) = 0$ .

[3]

Solve $q (x) = g (x)$ for $x \geq 0$ to three decimal places. Justify the number of solutions on $x \geq 0$ .

[6]

Question 4

SLPaper 1

Let $f (x) = 3 x^{2} - 12 x + 9$ , for $x \in R$ .

Write down the value of $f (0)$ .

[1]

Solve the equation $f (x) = 0$ .

[3]

The function $f$ can be written in the form $f (x) = a (x - h)^{2} + k$ .

Find the values of $a$ , $h$ and $k$ .

[3]

For the graph of $f$ , write down:

the coordinates of the vertex;

the equation of the axis of symmetry.

[2]

The graph of a function $g$ is obtained from the graph of $f$ by a reflection in the $x$ -axis, followed by a translation by the vector $(- 2 1)$ .

Find $g (x)$ , giving your answer in the form $g (x) = p x^{2} + q x + r$ .

[4]

Question 5

SLPaper 1

Let $f (x) = 25 - x^{2}$ , for $x \in R$ .

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

[2]

The following diagram shows part of the graph of

f

. Rectangle

A B C D

is drawn with

A

and

B

on the

x

-axis and

C

and

D

on the graph of

f

. Let

O A = a

, where

O

is the origin.

Show that the area of

A B C D

50 a - 2 a^{3}

[3]

Hence find the value of $a > 0$ such that the area of $A B C D$ is a maximum.

[4]

Find the maximum area of $A B C D$ .

[2]

Let $g (x) = (x - 3)^{2} + k$ , for $x \in R$ , where $k$ is a constant.

Show that when the graphs of $f$ and $g$ intersect, $2 x^{2} - 6 x + (k - 16) = 0$ .

[2]

Given that the graphs of $f$ and $g$ intersect only once, find the value of $k$ .

[3]

IB Mathematics Analysis and Approaches SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank

Unlock all 19 questions with a free account

IB Mathematics Analysis and Approaches SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank

Unlock all 19 questions with a free account