IB Mathematics Analysis and Approaches Topic SL 2.4—key Features of Graphs, Intersections Using Technology Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 2.4—key Features of Graphs, Intersections Using Technology Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 2.4—key Features of Graphs, Intersections Using Technology with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 2.4—key Features of Graphs, Intersections Using Technology and mirrors Paper 1, 2, 3 style where relevant.

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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

A quadratic model $f (x) = a x^{2} + b x + c$ is fitted to three measured points $P (1, ln 5)$ , $Q (4, 1.5 + ln 10)$ , $R (7, 1 + ln 4)$ .

Determine $a, b, c$ (numerically). Give your values correct to 3 significant figures.

[5]

Write $f (x)$ in vertex form $f (x) = a (x - h)^{2} + k$ . Hence state the vertex and the range of $f$ for $x \in R$ . Give $h, k$ to 3 d.p.

[4]

Let $g (x) = ln (x + 2) + 1$ . Solve $f (x) = g (x)$ for $x \geq 0$ , giving $x$ correct to 3 d.p., and justify the number of solutions.

[6]

Solve $f (x) \geq g (x)$ for $x \geq 0$ .

[3]

Question 3

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+2),\quad x>-2,\ k>0.$ It is given that $(f \circ g) (3) = 4$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and their maximal domains.

[4]

Solve $(x + 2)^{k} = ln (2 + e^{k x})$ for $x \geq 0$ (3 d.p.) and justify uniqueness.

[7]

Find $(f \circ g)^{- 1} (y)$ with domain and range.

[4]

Question 4

SLPaper 2

A line $L$ passes through $A (1, 4)$ with gradient $m$ . It meets the parabola $C:\ y=x^2-2x-3$ at two points $P, Q$ . The distance $P Q = 6$ .

Show that the $x$ -coordinates of $P, Q$ satisfy $x^2-(m+2)x+(m-7)=0.$

[4]

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

[2]

Show that $(x_{P} - x_{Q})^{2} = m^{2} + 32$ .

[4]

Using $P Q = 6$ , form an equation in $m$ and solve.

[5]

Question 5

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]

IB Mathematics Analysis and Approaches SL 2.4—key Features of Graphs, Intersections Using Technology Questionbank

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IB Mathematics Analysis and Approaches SL 2.4—key Features of Graphs, Intersections Using Technology Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 2.4—key Features of Graphs, Intersections Using Technology Questionbank

Unlock all 34 questions with a free account