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Date	May 2021	Marks available	2	Reference code	21M.2.AHL.TZ1.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	7	Adapted from	N/A

Question

A continuous random variable $X$ has the probability density function $f$ given by

$f (x) = \{\begin{cases} \frac{x}{\sqrt{{(x^{2} + k)}^{3}}} 0 \leq x \leq 4 \\ 0 otherwise \end{cases}$

where $k \in ℝ^{+}$ .

Show that $\sqrt{16 + k} - \sqrt{k} = \sqrt{k} \sqrt{16 + k}$ .

[5]

Find the value of $k$ .

[2]

Markscheme

recognition of the need to integrate $\frac{x}{\sqrt{{(x^{2} + k)}^{3}}}$ (M1)

$\int \frac{x}{\sqrt{{(x^{2} + k)}^{3}}} d x (= 1)$

EITHER

$u = x^{2} + k \Rightarrow \frac{d u}{d x} = 2 x$ (or equivalent) (A1)

$\int \frac{x}{\sqrt{{(x^{2} + k)}^{3}}} d x = \frac{1}{2} \int u^{- \frac{3}{2}} d u$

$= - u^{- \frac{1}{2}} (+ c) (= - {(x^{2} + k)}^{- \frac{1}{2}} (+ c))$ A1

$\int \frac{x}{\sqrt{{(x^{2} + k)}^{3}}} d x = \frac{1}{2} \int \frac{2 x}{\sqrt{{(x^{2} + k)}^{3}}} d x$ (A1)

$= - {(x^{2} + k)}^{- \frac{1}{2}} (+ c)$ A1

THEN

attempt to use correct limits for their integrand and set equal to $1$ M1

${[- u^{- \frac{1}{2}}]}_{k}^{16 + k} = 1$ OR ${[- {(x^{2} + k)}^{- \frac{1}{2}}]}_{0}^{4} = 1$

$- {(16 + k)}^{- \frac{1}{2}} + k^{- \frac{1}{2}} = 1 (\Rightarrow \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{16 + k}} = 1)$ A1

$\sqrt{16 + k} - \sqrt{k} = \sqrt{k} \sqrt{16 + k}$ AG

[5 marks]

attempt to solve $\sqrt{16 + k} - \sqrt{k} = \sqrt{k} \sqrt{16 + k}$ (M1)

$k = 0.645038 \dots$

$= 0.645$ A1

[2 marks]

Examiners report

[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.
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Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
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Write down the number of possible solutions to the equation $f(x) = 5$ .
17M.2.AHL.TZ1.H_12c:
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19M.2.SL.TZ1.T_4a:
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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 $h_{A} (t) = h_{B} (t)$ .
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:
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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.1.SL.TZ2.8a:
Find the coordinates of $B$ .
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 $f(x)$ .

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