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Date	November 2020	Marks available	2	Reference code	20N.1.SL.TZ0.S_10
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find	Question number	S_10	Adapted from	N/A

Question

The following diagram shows part of the graph of $f (x) = \frac{k}{x}$ , for $x > 0, k > 0$ .

Let $P (p, \frac{k}{p})$ be any point on the graph of $f$ . Line $L_{1}$ is the tangent to the graph of $f$ at $P$ .

Line $L_{1}$ intersects the $x$ -axis at point $A (2 p, 0)$ and the $y$ -axis at point B.

Find $f' (p)$ in terms of $k$ and $p$ .

[2]

a.i.

Show that the equation of $L_{1}$ is $k x + p^{2} y - 2 p k = 0$ .

[2]

a.ii.

Find the area of triangle $AOB$ in terms of $k$ .

[5]

The graph of $f$ is translated by $(\begin{matrix} 4 \\ 3 \end{matrix})$ to give the graph of $g$ .
In the following diagram:

point $Q$ lies on the graph of $g$
points $C$ , $D$ and $E$ lie on the vertical asymptote of $g$
points $D$ and $F$ lie on the horizontal asymptote of $g$
point $G$ lies on the $x$ -axis such that $FG$ is parallel to $DC$ .

Line $L_{2}$ is the tangent to the graph of $g$ at $Q$ , and passes through $E$ and $F$ .

Given that triangle $EDF$ and rectangle $CDFG$ have equal areas, find the gradient of $L_{2}$ in terms of $p$ .

[6]

Markscheme

$f' (x) = - k x^{- 2}$ (A1)

$f' (p) = - k p^{- 2} (= - \frac{k}{p^{2}})$ A1 N2

[2 marks]

a.i.

attempt to use point and gradient to find equation of $L_{1}$ M1

eg $y - \frac{k}{p} = - k p^{- 2} (x - p), \frac{k}{p} = - \frac{k}{p^{2}} (p) + b$

correct working leading to answer A1

eg $p^{2} y - k p = - k x + k p, y - \frac{k}{p} = - \frac{k}{p^{2}} x + \frac{k}{p}, y = - \frac{k}{p^{2}} x + \frac{2 k}{p}$

$k x + p^{2} y - 2 p k = 0$ AG N0

[2 marks]

a.ii.

METHOD 1 – area of a triangle

recognizing $x = 0$ at $B$ (M1)

correct working to find $y$ -coordinate of null (A1)

eg $p^{2} y - 2 p k = 0$

$y$ -coordinate of null at $y = \frac{2 k}{p}$ (may be seen in area formula) A1

correct substitution to find area of triangle (A1)

eg $\frac{1}{2} (2 p) (\frac{2 k}{p}), p \times (\frac{2 k}{p})$

area of triangle $AOB = 2 k$ A1 N3

METHOD 2 – integration

recognizing to integrate $L_{1}$ between $0$ and $2 p$ (M1)

eg $\int_{0}^{2 p} L_{1} d x, \int_{0}^{2 p} - \frac{k}{p^{2}} x + \frac{2 k}{p}$

correct integration of both terms A1

eg $- \frac{k x^{2}}{2 p^{2}} + \frac{2 k x}{p}, - \frac{k}{2 p^{2}} x^{2} + \frac{2 k}{p} x + c, {[- \frac{k}{2 p^{2}} x^{2} + \frac{2 k}{p} x]}_{0}^{2 p}$

substituting limits into their integrated function and subtracting (in either order) (M1)

eg $- \frac{k {(2 p)}^{2}}{2 p^{2}} + \frac{2 k (2 p)}{p} - (0), - \frac{4 k p^{2}}{2 p^{2}} + \frac{4 k p}{p}$

correct working (A1)

eg $- 2 k + 4 k$

area of triangle $AOB = 2 k$ A1 N3

[5 marks]

Note: In this question, the second M mark may be awarded independently of the other marks, so it is possible to award (M0)(A0)M1(A0)(A0)A0.

recognizing use of transformation (M1)

eg area of triangle $AOB$ = area of triangle $DEF$ , $g (x) = \frac{k}{x - 4} + 3,$ gradient of $L_{2} =$ gradient of $L_{1}$ , $D (4, 3), 2p+4,$ one correct shift

correct working (A1)

eg area of triangle $DEF = 2 k, CD = 3, DF = 2 p, CG = 2 p, E (4, \frac{2 k}{p} + 3), F (2 p + 4, 3), Q (p + 4, \frac{k}{p} + 3),$

gradient of $L_{2} = - \frac{k}{p^{2}}, g' (x) = - \frac{k}{{(x - 4)}^{2}},$ area of rectangle $CDFG = 2 k$

valid approach (M1)

eg $\frac{ED \times DF}{2} = CD \times DF, 2 p \cdot 3 = 2 k, ED = 2 CD, \int_{4}^{2 p + 4} L_{2} d x = 4 k$

correct working (A1)

eg $ED = 6, E (4, 9), k = 3 p, gradient = \frac{3 - (\frac{2 k}{p} + 3)}{(2 p + 4) - 4}, \frac{- 6}{(\frac{2 k}{3})}, - \frac{9}{k}$

correct expression for gradient (in terms of $p$ ) (A1)

eg $\frac{- 6}{2 p}, \frac{9 - 3}{4 - (2 p + 4)}, - \frac{3 p}{p^{2}}, \frac{3 - (\frac{2 (3 p)}{p} + 3)}{(2 p + 4) - 4}, - \frac{9}{3 p}$

gradient of $L_{2}$ is $- \frac{3}{p} (= - 3 p^{- 1})$ A1 N3

[6 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

Syllabus sections

Topic 5—Calculus » SL 5.3—Differentiating polynomials, n E Z

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17M.2.SL.TZ2.T_6a:
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16N.1.SL.TZ0.S_10c:
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17N.1.SL.TZ0.T_14a:
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17M.2.SL.TZ1.T_6d.ii:
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17N.1.AHL.TZ0.H_7:
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19N.2.SL.TZ0.T_6h:
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Find $f(2)$ .
19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
18N.2.SL.TZ0.T_6f:
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19N.1.SL.TZ0.T_14a:
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18M.1.SL.TZ2.T_14c:
Find the x-coordinate of the point at which the normal to the graph of f has gradient ${ - \frac{1}{8}}$ .
19N.2.SL.TZ0.T_6g:
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19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
18N.1.SL.TZ0.S_10b.ii:
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Find the coordinates of P.
16N.1.AHL.TZ0.H_9a:
Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
18N.2.SL.TZ0.T_6d:
Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.
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17M.2.SL.TZ1.S_6:
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Find the length of the cylinder for which $V$ is a maximum.
18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
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Show that $a = 8$ .
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Find $\frac{d V}{d r}$ .
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Find $f'\left( x \right)$ .
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Find f'(x)
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Find the equations of the tangents to this curve at the points where the curve intersects the line $x = 1$ .
18M.2.AHL.TZ2.H_11c:
Find the coordinates of the three points on C, nearest the origin, where the tangent is parallel to the line $y = - x$ .
18N.1.SL.TZ0.S_10c:
The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .
19N.2.SL.TZ0.T_6a:
Write down an expression for $V$ , the volume (cm³) of the speaker, in terms of $r$ , $l$ and $π$ .
17N.2.SL.TZ0.T_5b.i:
Expand the expression for $f(x)$ .
19M.2.SL.TZ1.T_6a:
Show that $k = - 6$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
17M.2.SL.TZ1.T_6b.ii:
Find the equation of the tangent to the graph of $y = g(x)$ at $x = 2$ . Give your answer in the form $y = mx + c$ .
19M.1.SL.TZ2.T_15b:
Differentiate $P\left( x \right)$ .
17N.2.SL.TZ0.T_5e:
Write down the coordinates of the point of intersection.
18M.2.AHL.TZ2.H_11b.ii:
Given that the gradients of the tangents to C at P and Q are m₁ and m₂ respectively, show that m₁ × m₂ = 1.
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
19N.1.SL.TZ0.T_14c:
At the point where $x = 2$ , the gradient of the tangent to the curve is $0.5$ .

Find the value of $a$ .
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
18M.2.AHL.TZ2.H_11a:
Show that $\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)$ .
18N.2.SL.TZ0.T_6c:
Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.
20N.1.SL.TZ0.S_10a.ii:
Show that the equation of $L_{1}$ is $k x + p^{2} y - 2 p k = 0$ .
20N.1.SL.TZ0.T_13b:
Write down the gradient of this tangent.
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
17N.2.SL.TZ0.T_5c:
Use your answer to part (b)(ii) to find the values of $x$ for which $f$ is increasing.
18M.1.SL.TZ2.S_9b:
Show that $C = 20\pi {r^2} + \frac{{320\pi }}{r}$ .
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
17N.2.SL.TZ0.T_5a:
Find the exact value of each of the zeros of $f$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
SPM.2.SL.TZ0.4e:
Find the least number of boxes which must be sold each week in order to make a profit.
17M.2.AHL.TZ1.H_2a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
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 $g'(x) = 0$ ;
17M.2.SL.TZ1.T_6b.i:
Show that $k = 6$ .
17M.2.SL.TZ1.T_6a:
Find $g'(x)$ .
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .
20N.1.SL.TZ0.T_13c:
Find the value of $k$ .
19M.2.SL.TZ1.T_6b:
Find the coordinates of the local minimum.
SPM.2.SL.TZ0.4b:
Calculate the length AG.
17M.2.AHL.TZ1.H_8a:
Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms of $\theta$ .
19N.1.SL.TZ0.T_14b:
Find $\frac{d y}{d x}$ .
16N.1.SL.TZ0.T_14b:
Find the coordinates of P.
18M.1.SL.TZ2.T_14b:
Find the gradient of the graph of f at $x = - \frac{1}{2}$ .
17M.2.SL.TZ1.T_6e:
Find the $y$ -coordinate of the local minimum.
16N.2.SL.TZ0.T_6c:
Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
17M.2.AHL.TZ1.H_2b:
Determine the equation of the tangent to $C$ at the point $\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)$
17N.2.SL.TZ0.T_5d:
Draw the graph of $f$ for $- 3 \leqslant x \leqslant 3$ and $- 40 \leqslant y \leqslant 20$ . Use a scale of 2 cm to represent 1 unit on the $x$ -axis and 1 cm to represent 5 units on the $y$ -axis.
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).
19M.2.SL.TZ1.T_6d:
Determine the equation of the normal at $x = - 2$ in the form $y = mx + c$ .
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
17M.2.SL.TZ1.T_6c:
Use your answer to part (a) and the value of $k$ , to find the $x$ -coordinates of the stationary points of the graph of $y = g(x)$ .
21M.2.AHL.TZ1.2a.i:
Find $\frac{d y}{d x}$ .
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
SPM.2.SL.TZ0.4c:
Find the number of boxes that should be sold each week to maximize the profit.
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.T_6b:
Write down an equation for the surface area of the speaker in terms of $r$ , $l$ and $π$ .
20N.1.SL.TZ0.S_10c:
The graph of $f$ is translated by $(\begin{matrix} 4 \\ 3 \end{matrix})$ to give the graph of $g$ .
In the following diagram:
- point $Q$ lies on the graph of $g$
- points $C$ , $D$ and $E$ lie on the vertical asymptote of $g$
- points $D$ and $F$ lie on the horizontal asymptote of $g$
- point $G$ lies on the $x$ -axis such that $FG$ is parallel to $DC$ .
Line $L_{2}$ is the tangent to the graph of $g$ at $Q$ , and passes through $E$ and $F$ .

Given that triangle $EDF$ and rectangle $CDFG$ have equal areas, find the gradient of $L_{2}$ in terms of $p$ .
19M.2.SL.TZ1.T_6c:
Write down the interval where the gradient of the graph of $f\left( x \right)$ is negative.
20N.1.SL.TZ0.T_13a:
Write down $f' (x)$ .
20N.1.SL.TZ0.S_10b:
Find the area of triangle $AOB$ in terms of $k$ .
21M.2.SL.TZ2.5d:
Find an expression for $\frac{d V}{d x}$ .
19M.1.SL.TZ1.T_15a:
Write down an equation for the area, $A$ , of the curved surface in terms of $r$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
16N.1.SL.TZ0.T_14a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
17M.1.SL.TZ2.S_6a:
Find $h(1)$ .
EXN.1.SL.TZ0.7a:
Find an expression for $\frac{d y}{d x}$ .
18M.1.AHL.TZ1.H_2b:
Hence find the values of θ for which $\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2y$ .
18M.1.AHL.TZ1.H_2a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}\theta }}$
19M.1.AHL.TZ1.H_7:
Find the coordinates of the points on the curve ${y^3} + 3x{y^2} - {x^3} = 27$ at which $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ .
19M.1.SL.TZ1.T_15b:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18M.1.SL.TZ1.S_7:
Consider f(x), g(x) and h(x), for x∈ $\mathbb{R}$ where h(x) = $\left( {f \circ g} \right)$ (x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.
16N.1.SL.TZ0.S_10a:
(i) Find the first four derivatives of $f(x)$ .

(ii) Find ${f^{(19)}}(x)$ .
EXN.2.SL.TZ0.2e:
Hence or otherwise find the minimum length of ribbon required.
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .
EXN.2.SL.TZ0.2a:
Find an expression for the total length of the ribbon $L$ in terms of $x$ and $y$ .
EXN.2.SL.TZ0.2b:
Show that $L = 2 x + \frac{300}{x} + 22$
EXN.2.SL.TZ0.2c:
Find $\frac{d L}{d x}$
EXN.2.SL.TZ0.2d:
Solve $\frac{d L}{d x} = 0$
EXN.2.SL.TZ0.6d:
Use differentiation to show that $12 a + 4 b + c = 0$ .
21N.1.SL.TZ0.12a:
Find $S' (x)$ .
21N.1.SL.TZ0.12b.ii:
Interpret your answer to (b)(i) in context.

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