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HL Paper 1

The following diagram shows a frame that is made from wire. The total length of wire is equal to $15 cm$ . The frame is made up of two identical sectors of a circle that are parallel to each other. The sectors have angle $θ$ radians and radius $r cm$ . They are connected by $1 cm$ lengths of wire perpendicular to the sectors. This is shown in the diagram below.

The faces of the frame are covered by paper to enclose a volume, $V$ .

Show that $r = \frac{6}{2 + θ}$ .

[2]

Find an expression for $V$ in terms of $θ$ .

[2]

b.i.

Find the expression $\frac{d V}{d θ}$ .

[3]

b.ii.

Solve algebraically $\frac{d V}{d θ} = 0$ to find the value of $θ$ that will maximize the volume, $V$ .

[2]

b.iii.

The position vector of a particle, $P$ , relative to a fixed origin $O$ at time $t$ is given by

$\vec{OP} = (\begin{matrix} \sin (t^{2}) \\ \cos (t^{2}) \end{matrix})$ .

Find the velocity vector of $P$ .

[2]

Show that the acceleration vector of $P$ is never parallel to the position vector of $P$ .

[5]

The graph of $y = - {x^3}$ is transformed onto the graph of $y = 33 - 0.08{x^3}$ by a translation of $a$ units vertically and a stretch parallel to the $x$ -axis of scale factor $b$ .

Write down the value of $a$ .

[1]

a.i.

Find the value of $b$ .

[2]

a.ii.

The outer dome of a large cathedral has the shape of a hemisphere of diameter 32 m, supported by vertical walls of height 17 m. It is also supported by an inner dome which can be modelled by rotating the curve $y = 33 - 0.08{x^3}$ through 360° about the $y$ -axis between $y$ = 0 and $y$ = 33, as indicated in the diagram.

Find the volume of the space between the two domes.

[5]

A function $f$ is of the form $f (t) = p e^{q \cos (r t)}, p, q, r \in ℝ^{+}$ . Part of the graph of $f$ is shown.

The points $A$ and $B$ have coordinates $A (0, 6.5)$ and $B (5.2, 0.2)$ , and lie on $f$ .

The point $A$ is a local maximum and the point $B$ is a local minimum.

Find the value of $p$ , of $q$ and of $r$ .

The production of oil $(P)$ , in barrels per day, from an oil field satisfies the differential equation $\frac{d P}{d t} = \frac{1000}{2 + t}$ where $t$ is measured in days from the start of production.

The production of oil at $t = 0$ is $20, 000$ barrels per day.

Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .

[1]

a.i.

State in context what this value represents.

[1]

a.ii.

Find an expression for $P$ in terms of $t$ .

[4]

Determine $\int_{0}^{365} P (t) d t$ and state what it represents.

[2]

A tank of water initially contains $400$ litres. Water is leaking from the tank such that after $10$ minutes there are $324$ litres remaining in the tank.

The volume of water, $V$ litres, remaining in the tank after $t$ minutes, can be modelled by the differential equation

$\frac{d V}{d t} = - k \sqrt{V}$ , where $k$ is a constant.

Show that $V = {(20 - \frac{t}{5})}^{2}$ .

[6]

Find the time taken for the tank to empty.

[2]

Consider the function $f (x) = \sqrt{- a x^{2} + x + a}, a \in ℝ^{+}$ .

For $a > 0$ the curve $y = f (x)$ has a single local maximum.

Find $f' (x)$ .

[2]

Find in terms of $a$ the value of $x$ at which the maximum occurs.

[2]

Hence find the value of $a$ for which $y$ has the smallest possible maximum value.

[4]

Let $f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0$ .

The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.

[5]

Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .

[5]

b.i.

The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.

[3]

b.ii.

Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.

[4]

Juri skis from the top of a hill to a finishing point at the bottom of the hill. She takes the shortest route, heading directly to the finishing point $(F)$ .

Let $h (x)$ define the height of the hill above $F$ at a horizontal distance $x$ from the starting point at the top of the hill.

The graph of the derivative of $h (x)$ is shown below. The graph of $h' (x)$ has local minima and maxima when $x$ is equal to $a, c$ and $e$ . The graph of $h' (x)$ intersects the $x$ -axis when $x$ is equal to $b, d$ , and $f$ .

Identify the $x$ value of the point where $| h' (x) |$ has its maximum value.

[1]

a.i.

Interpret this point in the given context.

[1]

a.ii.

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.

[3]

The slope field for the differential equation $\frac{d y}{d x} = e^{- x^{2}} - y$ is shown in the following two graphs.

On the second graph,

Calculate the value of $\frac{d y}{d x}$ at the point $(0, 1)$ .

[1]

Sketch, on the first graph, a curve that represents the points where $\frac{d y}{d x} = 0$ .

[2]

(i) sketch the solution curve that passes through the point $(0, 0)$ .

(ii) sketch the solution curve that passes through the point $(0, 0.75)$ .

[4]

The region bounded by $y = \frac{1}{(x + 2)} + 1, x = 0, x = 2$ and the $x$ -axis is rotated through $2 π$ about the $x$ -axis to form a solid.

Expand ${(\frac{1}{u} + 1)}^{2}$ .

[1]

a.i.

Find $\int {(\frac{1}{(x + 2)} + 1)}^{2} d x$ .

[3]

a.ii.

Find the volume of the solid formed. Give your answer in the form $\frac{π}{4} (a + b \ln (c))$ , where $a, b, c \in ℤ$ .

[4]

Consider the curve $y = 2 x (4 - e^{x})$ .

Find $\frac{d y}{d x}$ .

[2]

a.i.

Find $\frac{d^{2} y}{d x^{2}}$ .

[2]

a.ii.

The curve has a point of inflexion at $(a, b)$ .

Find the value of $a$ .

[2]

The rates of change of the area covered by two types of fungi, X and Y, on a particular tree are given by the following equations, where $x$ is the area covered by X and $y$ is the area covered by Y.

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = 3x - 2y$

$\frac{{{\text{d}}y}}{{{\text{d}}t}} = 2x - 2y$

The matrix $\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 2&{ - 2} \end{array}} \right)$ has eigenvalues of 2 and −1 with corresponding eigenvectors $\left( {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right)$ and $\left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right)$ .

Initially $x$ = 8 cm² and $y$ = 10 cm².

Find the value of $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ when $t = 0$ .

[2]

On the following axes, sketch a possible trajectory for the growth of the two fungi, making clear any asymptotic behaviour.

[4]

Consider the function $f(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1$ .

Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$ .

[1]

a.i.

Factorize ${x^2} + 3x + 2$ .

[1]

a.ii.

Sketch the graph of $f(x)$ , indicating on it the equations of the asymptotes, the coordinates of the $y$ -intercept and the local maximum.

[5]

Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$ .

[1]

Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$ .

[4]

Sketch the graph of $y = f\left( {\left| x \right|} \right)$ .

[2]

Determine the area of the region enclosed between the graph of $y = f\left( {\left| x \right|} \right)$ , the $x$ -axis and the lines with equations $x = - 1$ and $x = 1$ .

[3]

A particle, A, moves so that its velocity ( $\nu$ ms⁻¹) at time $t$ is given by $\nu$ = 2 sin $t$ , $t$ ≥ 0.

The kinetic energy ( $E$ ) of the particle A is measured in joules (J) and is given by $E$ = 5 $\nu$ ².

Write down an expression for $E$ as a function of time.

[1]

Hence find $\frac{{{\text{d}}E}}{{{\text{d}}t}}$ .

[2]

Hence or otherwise find the first time at which the kinetic energy is changing at a rate of 5 J s⁻¹.

[2]

The graph of $y = f'\left( x \right)$ , 0 ≤ $x$ ≤ 5 is shown in the following diagram. The curve intercepts the $x$ -axis at (1, 0) and (4, 0) and has a local minimum at (3, −1).

The shaded area enclosed by the curve $y = f'\left( x \right)$ , the $x$ -axis and the $y$ -axis is 0.5. Given that $f\left( 0 \right) = 3$ ,

The area enclosed by the curve $y = f'\left( x \right)$ and the $x$ -axis between $x = 1$ and $x = 4$ is 2.5 .

Write down the $x$ -coordinate of the point of inflexion on the graph of $y = f\left( x \right)$ .

[1]

find the value of $f\left( 1 \right)$ .

[3]

find the value of $f\left( 4 \right)$ .

[2]

Sketch the curve $y = f\left( x \right)$ , 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

[3]

It is given that ${\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0$ .

Show that ${\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x$ where $r,\,x \in {\mathbb{R}^ + }$ .

[2]

Express $y$ in terms of $x$ . Give your answer in the form $y = p{x^q}$ , where p , q are constants.

[5]

The region R, is bounded by the graph of the function found in part (b), the x-axis, and the lines $x = 1$ and $x = \alpha$ where $\alpha > 1$ . The area of R is $\sqrt 2$ .

Find the value of $\alpha$ .

[5]

Consider the second order differential equation

$\ddot{x} + 4 {(\dot{x})}^{2} - 2 t = 0$

where $x$ is the displacement of a particle for $t \geq 0$ .

Write the differential equation as a system of coupled first order differential equations.

[2]

When $t = 0$ , $x = \dot{x} = 0$

Use Euler’s method with a step length of $0.1$ to find an estimate for the value of the displacement and velocity of the particle when $t = 1$ .

[4]

The wind chill index $W$ is a measure of the temperature, in $° C$ , felt when taking into account the effect of the wind.

When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind $v$ in kilometres per hour $({km h}^{- 1})$ is given by the equation

$W = 19.34 - 7.405 v^{0.16}$

Find an expression for $\frac{d W}{d v}$ .

[2]

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5 {km h}^{- 1} {minute}^{- 1}$ .

Find the rate of change of $W$ at this time.

[5]

A slope field for the differential equation $\frac{d y}{d x} = x^{2} + \frac{y}{2}$ is shown.

Some of the solutions to the differential equation have a local maximum point and a local minimum point.

Write down the equation of the curve on which all these maximum and minimum points lie.

[1]

a.i.

Sketch this curve on the slope field.

[1]

a.ii.

The solution to the differential equation that passes through the point $(0, - 2)$ has both a local maximum point and a local minimum point.

On the slope field, sketch the solution to the differential equation that passes through $(0, - 2)$ .

[2]

The sides of a bowl are formed by rotating the curve $y = 6 \ln x, 0 \leq y \leq 9$ , about the $y$ -axis, where $x$ and $y$ are measured in centimetres. The bowl contains water to a height of $h cm$ .

Show that the volume of water, $V$ , in terms of $h$ is $V = 3 π (e^{\frac{h}{3}} - 1)$ .

[5]

Hence find the maximum capacity of the bowl in ${cm}^{3}$ .

[2]

The shape of a vase is formed by rotating a curve about the $y$ -axis.

The vase is $10 cm$ high. The internal radius of the vase is measured at $2 cm$ intervals along the height:

Use the trapezoidal rule to estimate the volume of water that the vase can hold.

Let $y = {\text{si}}{{\text{n}}^2}\theta ,\,\,0 \leqslant \theta \leqslant \pi$ .

Find $\frac{{{\text{d}}y}}{{{\text{d}}\theta }}$

[2]

Hence find the values of θ for which $\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2y$ .

[5]

A window is made in the shape of a rectangle with a semicircle of radius $r$ metres on top, as shown in the diagram. The perimeter of the window is a constant P metres.

M17/5/MATHL/HP1/ENG/TZ2/10

Find the area of the window in terms of P and $r$ .

[4]

a.i.

Find the width of the window in terms of P when the area is a maximum, justifying that this is a maximum.

[5]

a.ii.

Show that in this case the height of the rectangle is equal to the radius of the semicircle.

[2]

A particle moves along a straight line. Its displacement, $s$ metres, at time $t$ seconds is given by $s = t + \cos 2t,{\text{ }}t \geqslant 0$ . The first two times when the particle is at rest are denoted by ${t_1}$ and ${t_2}$ , where ${t_1} < {t_2}$ .

Find ${t_1}$ and ${t_2}$ .

[5]

Find the displacement of the particle when $t = {t_1}$

[2]

A particle moves in a straight line such that at time $t$ seconds $(t \geqslant 0)$ , its velocity $v$ , in ${\text{m}}{{\text{s}}^{ - 1}}$ , is given by $v = 10t{{\text{e}}^{ - 2t}}$ . Find the exact distance travelled by the particle in the first half-second.

Consider the curve $y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}$ .

Find the x-coordinates of the points on the curve where the gradient is zero.

The cross-section of a beach is modelled by the equation $y = 0.02 x^{2}$ for $0 \leq x \leq 10$ where $y$ is the height of the beach (in metres) at a horizontal distance $x$ metres from an origin. $t$ is the time in hours after low tide.

At $t = 0$ the water is at the point $(0, 0)$ . The height of the water rises at a rate of $0.2$ metres per hour. The point $W (x (t), y (t))$ indicates where the water level meets the beach at time $t$ .

A snail is modelled as a single point. At $t = 0$ it is positioned at $(1, 0.02)$ . The snail travels away from the incoming water at a speed of $1$ metre per hour in the direction along the curve of the cross-section of the beach. The following diagram shows this for a value of $t$ , such that $t > 0$ .

When $W$ has an $x$ -coordinate equal to $1$ , find the horizontal component of the velocity of $W$ .

[3]

Find the time taken for the snail to reach the point $(10, 2)$ .

[4]

b.i.

Hence show that the snail reaches the point $(10, 2)$ before the water does.

[1]

b.ii.

A curve has equation $3x - 2{y^2}{{\text{e}}^{x - 1}} = 2$ .

Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ in terms of $x$ and $y$ .

[5]

Find the equations of the tangents to this curve at the points where the curve intersects the line $x = 1$ .

[4]

Find the coordinates of the points on the curve ${y^3} + 3x{y^2} - {x^3} = 27$ at which $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ .

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 .

Given that $\int_{ - 2}^2 {f\left( x \right){\text{d}}x = 10}$ and $\int_0^2 {f\left( x \right){\text{d}}x = 12}$ , find

$\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$ .

[4]

$\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$ .

[2]

A right circular cone of radius $r$ is inscribed in a sphere with centre O and radius $R$ as shown in the following diagram. The perpendicular height of the cone is $h$ , X denotes the centre of its base and B a point where the cone touches the sphere.

Show that the volume of the cone may be expressed by $V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)$ .

[4]

Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{{32\pi {R^3}}}{{81}}$ .

[4]

Let $y = {{\text{e}}^x}\sin x$ .

Consider the function $f$ defined by $f(x) = {{\text{e}}^x}\sin x,{\text{ }}0 \leqslant x \leqslant \pi$ .

The curvature at any point $(x,{\text{ }}y)$ on a graph is defined as $\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}$ .

Find an expression for $\frac{{{\text{d}}y}}{{{\text{d}}x}}$ .

[2]

Show that $\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^x}\cos x$ .

[2]

Show that the function $f$ has a local maximum value when $x = \frac{{3\pi }}{4}$ .

[2]

Find the $x$ -coordinate of the point of inflexion of the graph of $f$ .

[2]

Sketch the graph of $f$ , clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.

[3]

Find the area of the region enclosed by the graph of $f$ and the $x$ -axis.

[6]

Find the value of the curvature of the graph of $f$ at the local maximum point.

[3]

Find the value $\kappa$ for $x = \frac{\pi }{2}$ and comment on its meaning with respect to the shape of the graph.

[2]

The diagram shows the slope field for the differential equation

$\frac{d y}{d x} = \sin (x + y), - 4 \leq x \leq 5, 0 \leq y \leq 5$ .

The graphs of the two solutions to the differential equation that pass through points $(0, 1)$ and $(0, 3)$ are shown.

For the two solutions given, the local minimum points lie on the straight line $L_{1}$ .

Find the equation of $L_{1}$ , giving your answer in the form $y = m x + c$ .

[3]

For the two solutions given, the local maximum points lie on the straight line $L_{2}$ .

Find the equation of $L_{2}$ .

[2]

Consider the functions $f,\,\,g,$ defined for $x \in \mathbb{R}$ , given by $f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x$ and $g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x$ .

Find $f'\left( x \right)$ .

[2]

a.i.

Find $g'\left( x \right)$ .

[1]

a.ii.

Hence, or otherwise, find $\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$ .

[4]

The folium of Descartes is a curve defined by the equation ${x^3} + {y^3} - 3xy = 0$ , shown in the following diagram.

N17/5/MATHL/HP1/ENG/TZ0/07

Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the $y$ -axis.