IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	May 2018	Marks available	5	Reference code	18M.2.AHL.TZ1.H_9
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	H_9	Adapted from	N/A

Question

The following graph shows the two parts of the curve defined by the equation ${x^2}y = 5 - {y^4}$ , and the normal to the curve at the point P(2 , 1).

Show that there are exactly two points on the curve where the gradient is zero.

[7]

a.

Find the equation of the normal to the curve at the point P.

[5]

b.

The normal at P cuts the curve again at the point Q. Find the $x$ -coordinate of Q.

[3]

c.

The shaded region is rotated by 2 $\pi$ about the $y$ -axis. Find the volume of the solid formed.

[7]

d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

differentiating implicitly: M1

$2xy + {x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 4{y^3}\frac{{{\text{d}}y}}{{{\text{d}}x}}$ A1A1

Note: Award A1 for each side.

if $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ then either $x = 0$ or $y = 0$ M1A1

$x = 0 \Rightarrow$ two solutions for $y\left( {y = \pm \sqrt[4]{5}} \right)$ R1

$y = 0$ not possible (as 0 ≠ 5) R1

hence exactly two points AG

Note: For a solution that only refers to the graph giving two solutions at $x = 0$ and no solutions for $y = 0$ award R1 only.

[7 marks]

a.

at (2, 1) $4 + 4\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 4\frac{{{\text{d}}y}}{{{\text{d}}x}}$ M1

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{1}{2}$ (A1)

gradient of normal is 2 M1

1 = 4 + c (M1)

equation of normal is $y = 2x - 3$ A1

[5 marks]

b.

substituting (M1)

${x^2}\left( {2x - 3} \right) = 5 - {\left( {2x - 3} \right)^4}$ or ${\left( {\frac{{y + 3}}{2}} \right)^2}\,y = 5 - {y^4}$ (A1)

$x = 0.724$ A1

[3 marks]

c.

recognition of two volumes (M1)

volume $1 = \pi \int_1^{\sqrt[4]{5}} {\frac{{5 - {y^4}}}{y}} {\text{d}}y\left( { = 101\pi = 3.178 \ldots } \right)$ M1A1A1

Note: Award M1 for attempt to use $\pi \int {{x^2}} {\text{d}}y$ , A1 for limits, A1 for ${\frac{{5 - {y^4}}}{y}}$ Condone omission of $\pi$ at this stage.

volume 2

EITHER

$= \frac{1}{3}\pi \times {2^2} \times 4\left( { = 16.75 \ldots } \right)$ (M1)(A1)

OR

$= \pi \int_{ - 3}^1 {{{\left( {\frac{{y + 3}}{2}} \right)}^2}} {\text{d}}y\left( { = \frac{{16\pi }}{3} = 16.75 \ldots } \right)$ (M1)(A1)

THEN

total volume = 19.9 A1

[7 marks]

d.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

Syllabus sections

Topic 5—Calculus » SL 5.1—Introduction of differential calculus

Show 80 related questions

18N.2.SL.TZ0.T_6a:
Calculate the area of cloth, in cm², needed to make Haruka’s bag.
18N.2.SL.TZ0.T_6e:
Use your answers to parts (c) and (d) to show that

$A = 3{x^2} + \frac{{10368}}{x}$ .
19M.1.AHL.TZ1.H_5:
A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at $t$ = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.

A car travels along the road at a speed of 24 ms⁻¹. Let the position of the car be X and let OĈX = θ.

Find $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ , the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
17M.2.SL.TZ1.T_6d.i:
Find $g’( - 1)$ .
22M.2.AHL.TZ2.4b:
Given that $\lim_{T \to \infty} k = A$ and $\lim_{T \to 0} k = 0$ , sketch the graph of $k$ against $T$ .
22M.3.AHL.TZ1.1e:
Determine the value of $a$ and of $b$ . Give your answers correct to one decimal place.
17N.2.AHL.TZ0.H_11c:
Solve the equation $f(x) = 0$ , giving your answers in the form $\arctan k$ where $k \in \mathbb{Z}$ .
21N.1.AHL.TZ0.8b:
Juri starts at a height of $60$ metres and finishes at $F$ , where $x = f$ .

Sketch a possible diagram of the hill on the following pair of coordinate axes.
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
19M.2.SL.TZ2.T_5f:
Find the equation of the tangent line to the graph of $y = f\left( x \right)$ at $x = 2$ . Give the equation in the form $ax + by + d = 0$ where, $a$ , $b$ , and $d \in \mathbb{Z}$ .
17N.1.SL.TZ0.T_14b:
Find the point on the graph of $f$ at which the gradient of the tangent is equal to 6.
18N.2.SL.TZ0.T_6b:
Calculate the volume, in cm³, of the bag.
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
18M.2.SL.TZ2.T_6f:
Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.
18M.1.SL.TZ1.T_5a:
Write down the coordinates of C, the midpoint of line segment AB.
18N.2.SL.TZ0.T_6h:
The cloth used to make Nanako’s bag costs 4 Japanese Yen (JPY) per cm².

Find the cost of the cloth used to make Nanako’s bag.
18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
18M.2.SL.TZ2.T_6b:
A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
18M.2.AHL.TZ2.H_11b.i:
Find the coordinates of P and Q.
17N.1.SL.TZ0.T_14a:
Write down the derivative of $f$ .
17M.2.SL.TZ1.T_6d.ii:
Hence justify that $g$ is decreasing at $x = - 1$ .
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
18N.2.SL.TZ0.T_6f:
Find $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ .
18M.1.SL.TZ2.T_14c:
Find the x-coordinate of the point at which the normal to the graph of f has gradient ${ - \frac{1}{8}}$ .
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.
17N.2.AHL.TZ0.H_11b.i:
Express $\sin x$ in terms of $\mu$ .
18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
18M.1.SL.TZ2.T_14a:
Find f'(x)
17N.2.AHL.TZ0.H_11a.iii:
Find the $x$ -coordinate(s) of the point(s) of inflexion of the graph of $y = f(x)$ , labelling these clearly on the graph of $y = f’(x)$ .
16N.1.AHL.TZ0.H_9b:
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 $y = - x$ .
17N.2.AHL.TZ0.H_11b.iii:
Hence show that $f(x) = 0$ can be expressed as ${u^3} - 7{u^2} + 15u - 9 = 0$ .
18M.1.SL.TZ1.T_5c:
Find the equation of the line DC. Write your answer in the form ax + by + d = 0 where a , b and d are integers.
17N.2.AHL.TZ0.H_10a.i:
Show that the $x$ -coordinate of the minimum point on the curve $y = f(x)$ satisfies the equation $\tan x = 2x$ .
17N.2.AHL.TZ0.H_11a.ii:
Sketch a graph of $y = f’(x)$ for $0 \leqslant x < \frac{\pi }{2}$ .
19M.2.SL.TZ1.T_6a:
Show that $k = - 6$ .
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 $y = mx + c$ .
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.
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 $\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)$ .
17N.2.AHL.TZ0.H_11a.i:
Determine an expression for $f’(x)$ in terms of $x$ .
17N.2.AHL.TZ0.H_10c:
Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y = - x$ .
17N.2.AHL.TZ0.H_10b:
Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
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.
17N.2.AHL.TZ0.H_10a.ii:
Determine the values of $x$ for which $f(x)$ is a decreasing function.
18M.2.AHL.TZ1.H_9a:
Show that there are exactly two points on the curve where the gradient is zero.
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
17M.2.AHL.TZ1.H_2a:
Find $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .
17M.2.SL.TZ1.T_6b.i:
Show that $k = 6$ .
17M.2.SL.TZ1.T_6a:
Find $g'(x)$ .
18M.2.AHL.TZ1.H_9d:
The shaded region is rotated by 2 $\pi$ about the $y$ -axis. Find the volume of the solid formed.
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .
19M.2.SL.TZ1.T_6b:
Find the coordinates of the local minimum.
18M.1.SL.TZ1.T_5b:
Find the gradient of the line DC.
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 $\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)$
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 $y = mx + c$ .
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 $y = g(x)$ .
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
18M.2.AHL.TZ1.H_9c:
The normal at P cuts the curve again at the point Q. Find the $x$ -coordinate of Q.
19M.2.SL.TZ1.T_6c:
Write down the interval where the gradient of the graph of $f\left( x \right)$ is negative.
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
17N.2.AHL.TZ0.H_11b.ii:
Express $\sin 2x$ in terms of $u$ .
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .
21N.1.AHL.TZ0.8a.i:
Identify the $x$ value of the point where $| h' (x) |$ has its maximum value.
21N.1.AHL.TZ0.8a.ii:
Interpret this point in the given context.

Hide related questions

Topic 5—Calculus » SL 5.4—Tangents and normal

Topic 5—Calculus » SL 5.6—Stationary points, local max and min

Topic 5—Calculus » SL 5.7—Optimisation

Topic 5—Calculus » AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y

Topic 5—Calculus