IB Mathematics Analysis and Approaches Topic SL…

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IB Mathematics Analysis and Approaches Topic SL… | RevisionDojo

Paper

Level

Difficulty

Question 1

SLPaper 2

View answer

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

(a)

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

[3]

(b)

Show that the line intersects $C$ at two distinct points for every real value of $m$ .

[2]

(c)

Show that $(x_P-x_Q)^2=m^2+76$ .

[3]

(d)

Given that $PQ=11$ , find the possible values of $m$

[4]

Question 2

SLPaper 2

View answer

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

(a)

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

[6]

(b)

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

[3]

(c)

Solve $q(x)=g(x)$ for $x\ge0$ , giving solutions correct to $3$ d.p., and the number of solutions.

[6]

Question 3

SL & HLPaper 1

View answer

For $k \in \mathbb{R}$ , consider the function $f$ given by

$f(x) = (k-1)x^{2} + 2kx + (k+2).$

(a)

Show that $b^{2} - 4ac = 4(2-k)$ .

[2]

(b)

all values of $k$ for which $f(x) \ge 0$ for every real number $x$ .

[4]

Question 4

SL & HLPaper 1

View answer

Consider the equation

$\left(x^{2}-5x+7\right)^{x^{2}-4x+3} = 1,$

where $x$ is real.

(a)

Find the values of $x$ for which the exponent is zero, and verify that the equation holds for each of them.

[4]

(b)

Find the values of $x$ for which the base is equal to $1$ .

[3]

(c)

Show that $x^{2}-5x+7 > 0$ for every real $x$ , and hence write down the complete solution set of the equation, showing that it has exactly three elements.

[6]

Question 5

SLPaper 2

View answer

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

(a)

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

[6]

(b)

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

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

[3]

(c)

Solve $q(x)=g(x)$ for $x\ge0$ to three decimal places. the number of solutions on $x\ge0$ .

[6]

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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.

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Paper

Level

Difficulty

Question 1

SLPaper 2

View answer

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

(a)

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

[3]

(b)

Show that the line intersects $C$ at two distinct points for every real value of $m$ .

[2]

(c)

Show that $(x_P-x_Q)^2=m^2+76$ .

[3]

(d)

Given that $PQ=11$ , find the possible values of $m$

[4]

Question 2

SLPaper 2

View answer

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

(a)

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

[6]

(b)

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

[3]

(c)

Solve $q(x)=g(x)$ for $x\ge0$ , giving solutions correct to $3$ d.p., and the number of solutions.

[6]

Question 3

SL & HLPaper 1

View answer

For $k \in \mathbb{R}$ , consider the function $f$ given by

$f(x) = (k-1)x^{2} + 2kx + (k+2).$

(a)

Show that $b^{2} - 4ac = 4(2-k)$ .

[2]

(b)

all values of $k$ for which $f(x) \ge 0$ for every real number $x$ .

[4]

Question 4

SL & HLPaper 1

View answer

Consider the equation

$\left(x^{2}-5x+7\right)^{x^{2}-4x+3} = 1,$

where $x$ is real.

(a)

Find the values of $x$ for which the exponent is zero, and verify that the equation holds for each of them.

[4]

(b)

Find the values of $x$ for which the base is equal to $1$ .

[3]

(c)

Show that $x^{2}-5x+7 > 0$ for every real $x$ , and hence write down the complete solution set of the equation, showing that it has exactly three elements.

[6]

Question 5

SLPaper 2

View answer

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

(a)

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

[6]

(b)

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

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

[3]

(c)

Solve $q(x)=g(x)$ for $x\ge0$ to three decimal places. the number of solutions on $x\ge0$ .

[6]

Unlock all 33 questions with a free account

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

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 SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank

Unlock all 33 questions with a free account

Unlock all 33 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 SL 2.7—solutions of Quadratic Equations and Inequalities, Discriminant and Nature of Roots Questionbank

Unlock all 33 questions with a free account

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

Unlock all 33 questions with a free account

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