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SL 2.10—Solving equations graphically and analytically - IB Questionbank | RevisionDojo

Practice SL 2.10—Solving equations graphically and analytically with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

A quadratic model $f(x)=ax^2+bx+c$ is fitted to three measured points $P(1,0.5+\ln 6)$ , $Q(3,1.5+\ln 9)$ , $R(6,\ln 5)$ .

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 $f$ . Give $h,k$ to 3 d.p.

[4]

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

[6]

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

[3]

Question 2

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+9),\quad x>-9,\ k>0.$ It is given that $(f\circ g)(1)=7$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and their maximal domains.

[4]

Solve $(x+9)^k=\ln(9+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

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

[4]

Question 3

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=7$ . It passes through the points $(0,5)$ and $(6,11)$ .

Show that the function can be written in the form $f(x)=7+\frac{k}{x-4}$ for some constant $k$ , and determine $k$ . Hence write $f(x)$ explicitly as $\dfrac{ax+b}{cx+d}$ with integer coefficients.

[4]

Determine all key features of $f$ : domain, range, intercepts, asymptotes, and the sign of $f(x)-7$ on $(- \infty,4)$ and $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to three decimal places and justify exactly one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe transformations mapping $y=1/x$ to $y=f(x)$ .

[2]

Question 4

SLPaper 2

A quadratic model $f(x)=ax^2+bx+c$ is fitted to three measured points $P(1,1+\ln 2)$ , $Q(3,e)$ , $R(6,\ln 2)$ .

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 $f$ . Give $h,k$ to 3 d.p.

[4]

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

[6]

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

[3]

Question 5

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+7),\quad x>-7,\ k>0.$ It is given that $(f\circ g)(0)=3$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and state maximal domains.

[4]

Solve $(x+7)^k=\ln(7+e^{kx})$ for $x\ge0$ (3 d.p.) and justify uniqueness.

[7]

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

[4]

Question 6

SLPaper 2

The basic function is $f(x)=\sqrt{x}$ for $x\ge 0$ . Define the transformed function $g(x)=3\sqrt{x-4}+1,\quad x\ge 4.$

Describe, step by step, the transformations that map $y=f(x)$ to $y=g(x)$ .

[4]

State the domain and range of $g$ .

[2]

Solve $3\sqrt{x-4}+1=x-2$ for $x\ge 4$ , giving your answers correct to 3 d.p.

[5]

Define $k(x)=|g(x)|,\ x\ge 4$ . Using technology, solve $k(x)=x-2$ for $4\le x\le 13$ . Give all solutions correct to 3 d.p.

[5]

Question 7

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]

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

[3]

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

[6]

Solve $q(x)\ge g(x)$ for $x\ge0$ (endpoints to 3 d.p.).

[3]

Question 8

SLPaper 2

Let $f(x) = x - 8$ , $g(x) = x^4 - 3$ , and $h(x) = f(g(x))$ .

Find $h\left( x \right)$ .

[2]

Let $D$ be a point on the graph of $h$ . The tangent to the graph of $h$ at $D$ is parallel to the graph of $f$ . Find the $x$ -coordinate of $D$ .

[5]

Question 9

SLPaper 2

Consider the exponential equation $9^x = 7x + 4$ .

Sketch the graph of $y = 9^x$ and $y = 7x + 4$ on the same axes.

[2]

Hence find the approximate solution to the equation $9^x = 7x + 4$ .

[2]

Hence, find the area between the curves within the interval of the intersection points

[2]

Question 10

SL & HLPaper 2

Consider the functions $f(x) = \sin x$ and $g(x) = \cos(2x)$ .

Graph the function $h(x) = f(x)g(x)$ for $x\in[0,\pi]$

[3]

Graph the function $(f \circ g)(x)$ on the same set of axes.

[3]

Find the intersection points of $f(x)g(x)$ and $(f \circ g)(x)$ .

[2]

SL 2.10—Solving equations graphically and analytically Questionbank