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Date	November 2019	Marks available	6	Reference code	19N.2.AHL.TZ0.H_11
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Show that	Question number	H_11	Adapted from	N/A

Question

The following diagram shows part of the graph of $2{x^2} = {\text{si}}{{\text{n}}^3}\,y$ for $0 \leqslant y \leqslant \pi$ .

The shaded region $R$ is the area bounded by the curve, the $y$ -axis and the lines $y = 0$ and $y = \pi$ .

Using implicit differentiation, find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .

[4]

a.i.

Find the equation of the tangent to the curve at the point $\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)$ .

[4]

a.ii.

Find the area of $R$ .

[3]

The region $R$ is now rotated about the $y$ -axis, through $2\pi$ radians, to form a solid.

By writing ${{\text{si}}{{\text{n}}^3}\,y}$ as $\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y$ , show that the volume of the solid formed is $\frac{{2\pi }}{3}$ .

[6]

Markscheme

valid attempt to differentiate implicitly (M1)

$4x = 3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y\frac{{{\text{d}}y}}{{{\text{d}}x}}$ A1A1

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4x}}{{3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y}}$ A1

[4 marks]

a.i.

at $\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right){\text{, }}\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4x}}{{3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y}} = \frac{1}{{3{{\left( {\frac{1}{2}} \right)}^2}\left( { - \frac{{\sqrt 3 }}{2}} \right)}}$ (M1)

$\Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{8}{{3\sqrt 3 }}\left( { = - 1.54} \right)$ A1

hence equation of tangent is

$y - \frac{{5\pi }}{6} = - 1.54\left( {x - \frac{1}{4}} \right)$ OR $y = - 1.54x + 3.00$ (M1)A1

Note: Accept $y = - 1.54x + 3$ .

[4 marks]

a.ii.

$x = \sqrt {\frac{1}{2}{\text{si}}{{\text{n}}^3}\,y}$ (M1)

$\int_0^\pi {\sqrt {\frac{1}{2}{\text{si}}{{\text{n}}^3}\,y\,{\text{d}}y} }$ (A1)

$= 1.24$ A1

[3 marks]

use of volume $= \int {\pi {x^2}} \,{\text{d}}y$ (M1)

$= \int_0^\pi {\frac{1}{2}} \pi \,{\text{si}}{{\text{n}}^3}\,y\,{\text{d}}y$ A1

$= \frac{1}{2}\pi \int_0^\pi {\left( {{\text{sin}}\,y - {\text{sin}}\,y\,{\text{co}}{{\text{s}}^2}\,y} \right){\text{d}}y}$

Note: Condone absence of limits up to this point.

reasonable attempt to integrate (M1)

$= \frac{1}{2}\pi \left[ { - {\text{cos}}\,y + \frac{1}{3}{\text{co}}{{\text{s}}^3}\,y} \right]_0^\pi$ A1A1

Note: Award A1 for correct limits (not to be awarded if previous M1 has not been awarded) and A1 for correct integrand.

$= \frac{1}{2}\pi \left( {1 - \frac{1}{3}} \right) - \frac{1}{2}\pi \left( { - 1 + \frac{1}{3}} \right)$ A1

$= \frac{{2\pi }}{3}$ AG

Note: Do not accept decimal answer equivalent to $\frac{{2\pi }}{3}$ .

[6 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

Syllabus sections

Topic 5 —Calculus » SL 5.5—Integration introduction, areas between curve and x axis

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18M.2.SL.TZ2.S_3b:
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19M.2.SL.TZ2.S_2b:
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19N.1.SL.TZ0.S_8d.ii:
Justify your answer.
17N.2.SL.TZ0.S_9d:
Find the total distance travelled by P when its velocity is increasing.
19N.1.SL.TZ0.S_8c:
Find $\frac{{{\text{d}}V}}{{{\text{d}}x}}$ .
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
SPM.2.AHL.TZ0.11c:
Sketch the graph of $x$ versus $t$ for 0 ≤ $t$ ≤ 60 and hence find the maximum amount of salt in the tank and the value of $t$ at which this occurs.
19M.2.SL.TZ2.S_2a:
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21M.2.SL.TZ1.1a:
Find $\int (6 x + 7) d x$ .
17M.2.SL.TZ1.S_10a.i:
Write down the value of $q$ ;
22M.2.SL.TZ2.2:
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21N.1.SL.TZ0.7c:
Find the total distance travelled by $P$ .
21N.1.SL.TZ0.7a.ii:
Show that the distance of $P$ from $O$ at this time is $\frac{88}{27}$ metres.
22M.1.AHL.TZ1.1:
Find the value of $\int_{1}^{9} (\frac{3 \sqrt{x} - 5}{\sqrt{x}}) d x$ .
18M.1.SL.TZ2.T_7a:
State the number of boys who answered questions in Portuguese.
19M.1.SL.TZ1.T_12a:
Find the probability that both spins are yellow.
17N.1.SL.TZ0.T_7b:
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16N.2.SL.TZ0.T_2d:
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19M.1.SL.TZ2.T_5c:
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Find the probability that this student likes kiwi fruit smoothies given that they like mango smoothies.
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State, with a reason, whether you would reject the null hypothesis.
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Determine whether the events $S$ and $M$ are independent.
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
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Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
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Find the value of $t$ when particle A first reaches point P.
17N.2.SL.TZ0.S_9c:
Find an expression for the velocity of P at time $t$ .
18M.2.SL.TZ1.T_2h:
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Find the probability that each of these flights travelled at least 5000 km.
SPM.2.AHL.TZ0.11e:
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Find the amount of salt that left the tank during the first 60 minutes.
18M.2.SL.TZ1.S_4c:
Find the area of the region enclosed by the graphs of f and g.
17M.2.SL.TZ2.S_7:
Note: In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity $v$ of the particle, at time $t$ , is given by $v(t) = \frac{{2{t^2} - 4t}}{{{t^2} - 2t + 2}}$ , for $0 \leqslant t \leqslant 5$ . The following diagram shows the graph of $v$

There are $t$ -intercepts at $(0,{\text{ }}0)$ and $(2,{\text{ }}0)$ .

Find the maximum distance of the particle from A during the time $0 \leqslant t \leqslant 5$ and justify your answer.
17M.1.SL.TZ1.S_10a:
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20N.1.AHL.TZ0.H_12a:
State the equation of the vertical asymptote on the graph of $y = f (x)$ .
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19N.1.AHL.TZ0.H_10c:
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19M.1.SL.TZ2.S_10b:
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20N.1.AHL.TZ0.H_12e:
The region bounded by the $x$ -axis, the curve $y = f (x)$ , and the lines $x = 5$ and $x = 7$ is rotated through $2 π$ about the $x$ -axis. Find the volume of the solid generated, giving your answer in the form $π (a + b \ln 2)$ , where $a, b \in ℤ$ .
17M.2.SL.TZ1.S_7b:
A second particle Q also moves along a straight line. Its velocity, ${v_{\text{Q}}}{\text{ m}}\,{{\text{s}}^{ - 1}}$ after $t$ seconds is given by ${v_{\text{Q}}} = \sqrt t$ for $0 \leqslant t \leqslant 8$ . After $k$ seconds Q has travelled the same total distance as P.

Find $k$ .
16N.2.SL.TZ0.S_4a:
Find the value of $p$ and of $q$ .
19N.2.AHL.TZ0.H_11b:
Find the area of $R$ .
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Find the probability that this student is taught in Spanish, given that the student studies Biology.
19N.1.AHL.TZ0.H_10d:
The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .
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Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.
16N.2.SL.TZ0.S_9d:
(i) Find the total distance travelled by P between $t = 1$ and $t = p$ .

(ii) Hence or otherwise, find the displacement of P from A when $t = p$ .
21M.2.SL.TZ1.1b:
Given $f' (x) = 6 x + 7$ and $f (1.2) = 7.32$ , find $f (x)$ .
17M.2.SL.TZ1.S_10a.iii:
Write down the value of $k$ .
17M.2.SL.TZ1.S_10c:
Let $d$ be the vertical distance from a point on the graph of $h$ to the line $y = x$ . There is a point ${\text{P}}(a,{\text{ }}b)$ on the graph of $h$ where $d$ is a maximum.

Find the coordinates of P, where $0.111 < a < 3.31$ .
17M.2.SL.TZ1.S_10a.ii:
Write down the value of $h$ ;
17M.2.SL.TZ1.S_10b.ii:
Hence, find the area of the region enclosed by the graphs of $h$ and ${h^{ - 1}}$ .
18N.2.SL.TZ0.T_2a.i:
Find the number of students in the school that are taught in Spanish.
18N.2.SL.TZ0.T_2a.iii:
Find the number of students in the school that study both Biology and Mathematics.
19M.2.SL.TZ2.T_1d:
State whether or not H₀ should be rejected. Justify your statement.
17N.2.SL.TZ0.T_4c:
Copy and complete the tree diagram.
18M.2.SL.TZ1.T_2c:
Write down the number of degrees of freedom.
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18M.2.SL.TZ1.T_5d.ii:
A contestant is chosen at random. Find the probability that this contestant fell into a trap.
18N.2.SL.TZ0.T_2c.i:
Find the probability that this student studies Mathematics.
19M.2.SL.TZ2.T_1e.i:
Find the probability that the student does not take the Spanish class.
16N.2.SL.TZ0.T_2b:
Show that $x = 3$ .
18M.2.SL.TZ1.T_5d.i:
A contestant is chosen at random. Find the probability that this contestant fell into a trap while attempting to pass through a door in the second wall.
18M.1.SL.TZ2.T_7c:
Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.
19M.1.SL.TZ1.T_12b:
Find the probability that at least one of the spins is yellow.
17N.2.SL.TZ0.T_4e:
Find the probability that the liquid turns blue.
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
21N.1.SL.TZ0.7a.i:
Find the value of $t$ when $P$ reaches its maximum velocity.
16N.2.SL.TZ0.T_6a:
Write down a formula for $A$ , the surface area to be coated.
16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
18M.2.SL.TZ1.T_2f:
Write down the probability that this flight arrived on time.
19M.2.SL.TZ2.T_1e.ii:
Find the probability that neither of the two students take the Spanish class.
18M.2.SL.TZ2.T_1a.i:
Write down the value of a.
18M.2.SL.TZ2.T_1b.i:
Use the tree diagram to find the probability that an employee encountered traffic and was late for work.
18M.2.SL.TZ2.T_1b.ii:
Use the tree diagram to find the probability that an employee was late for work.
18M.2.SL.TZ2.T_1b.iii:
Use the tree diagram to find the probability that an employee encountered traffic given that they were late for work.
18M.2.SL.TZ2.T_1e:
Find $n\left( {\left( {C \cup B} \right) \cap P'} \right)$ .
16N.2.SL.TZ0.T_2a:
Draw a Venn diagram to represent the given information, using sets labelled $B$ , $C$ and $H$ .
16N.2.SL.TZ0.T_2c:
Write down the value of $n(B \cap C)$ .
18N.2.SL.TZ0.T_2c.ii:
Find the probability that this student studies neither Biology nor Mathematics.
17N.2.SL.TZ0.T_4b:
Find the probability that both people chosen are not allergic to nuts.
17N.2.SL.TZ0.T_4f:
Find the probability that the tested adult is allergic to nuts given that the liquid turned blue.
18M.2.SL.TZ1.T_5b:
Find the probability that only one of Ayako and Natsuko falls into a trap while attempting to pass through a door in the first wall.
19M.2.SL.TZ2.T_1b:
State the number of degrees of freedom.

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