IB Questionbank

User interface language: English | Español

Date	May 2022	Marks available	5	Reference code	22M.1.SL.TZ2.8
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Find	Question number	8	Adapted from	N/A

Question

Consider the functions $f (x) = \frac{1}{x - 4} + 1$ , for $x \neq 4$ , and $g (x) = x - 3$ for $x \in ℝ$ .

The following diagram shows the graphs of $f$ and $g$ .

The graphs of $f$ and $g$ intersect at points $A$ and $B$ . The coordinates of $A$ are $(3, 0)$ .

In the following diagram, the shaded region is enclosed by the graph of $f$ , the graph of $g$ , the $x$ -axis, and the line $x = k$ , where $k \in ℤ$ .

The area of the shaded region can be written as $\ln (p) + 8$ , where $p \in ℤ$ .

Find the coordinates of $B$ .

[5]

Find the value of $k$ and the value of $p$ .

[10]

Markscheme

$\frac{1}{x - 4} + 1 = x - 3$ (M1)

$x^{2} - 8 x + 15 = 0$ OR ${(x - 4)}^{2} = 1$ (A1)

valid attempt to solve their quadratic (M1)

$(x - 3) (x - 5) = 0$ OR $x = \frac{8 \pm \sqrt{8^{2} - 4 (1) (15)}}{2 (1)}$ OR $(x - 4) = \pm 1$

$x = 5 (x = 3, x = 5)$ (may be seen in answer) A1

$B (5, 2)$ (accept $x = 5, y = 2$ ) A1

[5 marks]

recognizing two correct regions from $x = 3$ to $x = 5$ and from $x = 5$ to $x = k$ (R1)

triangle $+ \int_{5}^{k} f (x) d x$ OR $\int_{3}^{5} g (x) d x + \int_{5}^{k} f (x) d x$ OR $\int_{3}^{5} (x - 3) d x + \int_{5}^{k} (\frac{1}{x - 4} + 1) d x$

area of triangle is $2$ OR $\frac{2 \cdot 2}{2}$ OR $(\frac{5^{2}}{2} - 3 (5)) - (\frac{3^{2}}{2} - 3 (3))$ (A1)

correct integration (A1)(A1)

$\int (\frac{1}{x - 4} + 1) d x = \ln (x - 4) + x (+ C)$

Note: Award A1 for $\ln (x - 4)$ and A1 for $x$ .
Note: The first three A marks may be awarded independently of the R mark.

substitution of their limits (for $x$ ) into their integrated function (in terms of $x$ ) (M1)

$\ln (k - 4) + k - (\ln 1 + 5)$

${[\ln (x - 4) + x]}_{5}^{k} = \ln (k - 4) + k - 5$ A1

adding their two areas (in terms of $k$ ) and equating to $\ln p + 8$ (M1)

$2 + \ln (k - 4) + k - 5 = \ln p + 8$

equating their non-log terms to $8$ (equation must be in terms of $k$ ) (M1)

$k - 3 = 8$

$k = 11$ A1

$11 - 4 = p$

$p = 7$ A1

[10 marks]

Examiners report

Nearly all candidates knew to set up an equation with $f (x) = g (x)$ in order to find the intersection of the two graphs, and most were able to solve the resulting quadratic equation. Candidates were not as successful in part (b), however. While some candidates recognized that there were two regions to be added together, very few were able to determine the correct boundaries of these regions, with many candidates integrating one or both functions from $x = 3$ to $x = k$ . While a good number of candidates were able to correctly integrate the function(s), without the correct bounds the values of $k$ and $p$ were unattainable.

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology

Show 100 related questions

22M.1.SL.TZ1.7d:
Sketch the graph of $f ″$ , the second derivative of $f$ . Indicate clearly the $x$ -intercept and the $y$ -intercept.
22M.1.SL.TZ2.4a.ii:
Write down the equation of the horizontal asymptote.
22M.1.SL.TZ2.4a.i:
Write down the equation of the vertical asymptote.
22M.1.SL.TZ2.4b:
On the set of axes below, sketch the graph of $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
22M.1.AHL.TZ2.3a.i:
Write down the equation of the vertical asymptote.
22M.2.SL.TZ2.8a.ii:
Plant $A$ correct to three significant figures.
18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
19M.2.SL.TZ1.T_4a:
Find the value of $f\left( x \right)$ when $x = \frac{1}{2}$ .
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
19N.2.SL.TZ0.S_7b:
Find the difference between the greatest possible expected value and the least possible expected value.
22M.3.AHL.TZ2.1b.ii:
By considering each curve from part (a), identify two key features that would distinguish one curve from the other.
22M.3.AHL.TZ2.1f.ii:
Hence, find the coordinates of the rational point $Q$ where this tangent intersects $C$ , expressing each coordinate as a fraction.
22M.1.SL.TZ2.4c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.1.AHL.TZ2.3a.ii:
Write down the equation of the horizontal asymptote.
22M.1.AHL.TZ2.3c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.2.SL.TZ1.8c.i:
Find the value of $p$ and the value of $q$ .
EXN.1.AHL.TZ0.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.
22M.2.AHL.TZ2.10a.ii:
Plant $A$ correct to three significant figures.
22M.2.AHL.TZ2.10b:
Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .
21M.2.AHL.TZ1.7b:
Find the value of $k$ .
20N.2.SL.TZ0.S_1b:
Solve $f (x) = 0$ .
17M.2.SL.TZ2.S_10c:
Jill plays the game nine times. Find the probability that she wins exactly two prizes.
21M.2.AHL.TZ1.11d:
Sketch the graph of $y = f (x)$ for $- 3 \leq x \leq 3$ , showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
21M.2.SL.TZ2.2a:
Sketch the graph of $f$ on the grid below.
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
20N.2.SL.TZ0.S_1c:
The graph of $f$ has a local minimum at point $A$ .

Find the coordinates of $A$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
21M.2.AHL.TZ2.11d.ii:
State the cross-sectional radius of the bowl at this point.
21M.2.AHL.TZ2.11d.i:
By sketching the graph of a suitable derivative of $f$ , find where the cross-sectional radius of the bowl is decreasing most rapidly.
21M.2.AHL.TZ2.12f:
Sketch the graph of $y = g^{- 1} (x)$ , clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.
20N.1.SL.TZ0.T_4a.i:
Write down the zero of $f (x)$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
EXN.2.SL.TZ0.9e:
Find the time taken for the water to have a temperature of $50 ° C$ . Give your answer correct to the nearest second.
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
20N.2.SL.TZ0.S_1a:
Find $f (1)$ .
EXN.2.SL.TZ0.6a:
Find the $x$ -coordinates of the points of intersection of the two curves.
EXN.2.SL.TZ0.9d:
Sketch the graph of $T$ versus $t$ , clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.
21N.2.SL.TZ0.5b:
Hence determine the value of $θ$ .
21N.2.SL.TZ0.7b.iii:
Write down the minimum value of $T$ .
21N.2.SL.TZ0.7c.ii:
Write down the minimum total cost for this journey.
21N.2.SL.TZ0.8a:
Show that $b = \frac{π}{6}$ .
21N.2.SL.TZ0.8c:
Find the value of $d$ .
21N.2.SL.TZ0.8d:
Find the smallest possible value of $c$ .
22M.2.AHL.TZ1.10a:
Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.
22M.2.AHL.TZ2.10a.i:
Plant $B$ .
22M.3.AHL.TZ2.1g:
The point $S (- 1, 1)$ also lies on $C$ . The line $[QS]$ intersects $C$ at a further point. Determine the coordinates of this point.
22M.3.AHL.TZ2.1c:
By varying the value of $b$ , suggest two key features common to these curves.
22M.2.SL.TZ2.8a.i:
Plant $B$ .
21N.2.SL.TZ0.7b.i:
Find an expression, in terms of $x$ for the travel time $T$ , from $A$ to $B$ , passing through $D$ .
21N.2.SL.TZ0.8b:
Find the value of $a$ .
21N.2.AHL.TZ0.9a:
Show that $b = \frac{π}{6}$ .
21N.2.AHL.TZ0.9b:
Find the value of $a$ .
21N.2.AHL.TZ0.9d:
Find the smallest possible value of $c$ .
19M.2.SL.TZ2.T_6a:
Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ2.T_6c:
After $m$ complete months Tommaso will, for the first time, have enough money in his account to buy the bicycle.

Find the value of $m$ .
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
20N.1.SL.TZ0.T_4b:
Consider the function $g (x) = 3 - x$ .

Solve $f (x) = g (x)$ .
21N.2.SL.TZ0.5a:
Given that the areas of the two shaded regions are equal, show that $θ = 2 \sin θ$ .
21N.2.SL.TZ0.7a.ii:
the boat travels directly to $B$ .
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
19N.2.SL.TZ0.S_7a:
Show that $b = 0.3 - a$ .
21M.2.SL.TZ1.9a:
Find the $x$ -coordinate of $P$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
21M.3.AHL.TZ1.1f:
Consider the function $g (x) = x^{3} - 3 c x + d$ for $x \in ℝ$ and where $c, d \in ℝ$ .

Find all conditions on $c$ and $d$ such that the graph of $y = g (x)$ has exactly one $x$ -axis intercept, explaining your reasoning.
21M.2.AHL.TZ1.12d:
Find the value of $k$ , giving your answer correct to four significant figures.
EXN.2.SL.TZ0.9f:
The model for the temperature of the water can also be expressed in the form $T = T_{0} a^{\frac{t}{10}}$ for $t \geq 0$ and $a$ is a positive constant.

Find the exact value of $a$ .
21M.2.AHL.TZ1.11f:
By considering the graph of $y = g (x) - f (x)$ , or otherwise, solve $f (x) < g (x)$ for $x \in ℝ$ .
21M.2.SL.TZ2.2b:
Find the value of $x$ for which $f' (x) = 0$ .
21N.2.AHL.TZ0.9e:
Find the height of the water at $12 : 00$ .
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.2.SL.TZ2.T_6b:
Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.
19M.2.SL.TZ1.T_4c:
Find the coordinates of the bridges relative to the centre of Orangeton.
20N.1.SL.TZ0.T_4a.ii:
Write down the coordinates of the local minimum point.
19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
21N.2.SL.TZ0.7a.i:
the boat is taken from $A$ to $P$ , and the bus from $P$ to $B$ .
21N.2.SL.TZ0.7b.ii:
Find the value of $x$ so that $T$ is a minimum.
21N.2.SL.TZ0.7c.i:
Find the new value of $x$ so that the total cost $C$ to travel from $A$ to $B$ via $D$ is a minimum.
21N.2.SL.TZ0.8e:
Find the height of the water at $12 : 00$ .
21N.2.SL.TZ0.8f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.7a:
Determine the value of $m$ .
21N.2.AHL.TZ0.7b:
Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .
21N.2.AHL.TZ0.9c:
Find the value of $d$ .
21N.2.AHL.TZ0.9f:
Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.
21N.2.AHL.TZ0.9g:
A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.
19M.2.SL.TZ1.T_4b.i:
Find the function, $P\left( x \right)$ , that would define this footpath on the map.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .

Hide related questions

Topic 2—Functions » SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots

Topic 2—Functions » SL 2.10—Solving equations graphically and analytically

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options