IB Questionbank

User interface language: English | Español

HL Paper 1

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]

The graph of $y = f (x)$ is given on the following set of axes. The graph passes through the points $(- 2, 6)$ and $(0, 1)$ , and has a horizontal asymptote at $y = 0$ .

Let $g (x) = 2 f (x - 2) + 4$ .

Find $g (0)$ .

[2]

On the same set of axes draw the graph of $y = g (x)$ , showing any intercepts and asymptotes.

[2]

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$ , which have a magnitude of at least $M$ . For a particular region the equation is

$\log_{10} N = a - M$ , for some $a \in ℝ$ .

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$ .

The equation for this region can also be written as $N = \frac{b}{10^{M}}$ .

Within this region the most severe earthquake recorded had a magnitude of $7.2$ .

The number of earthquakes in a given year with a magnitude of at least $7.2$ can be modelled by a Poisson distribution, with mean $N$ . The number of earthquakes in one year is independent of the number of earthquakes in any other year.

Let $Y$ be the number of years between the earthquake of magnitude $7.2$ and the next earthquake of at least this magnitude.

Find the value of $a$ .

[2]

Find the value of $b$ .

[2]

Find the average number of earthquakes in a year with a magnitude of at least $7.2$ .

[1]

Find $P (Y > 100)$ .

[3]

The rate, $A$ , of a chemical reaction at a fixed temperature is related to the concentration of two compounds, $B$ and $C$ , by the equation

$A = k{B^x}{C^y}$ , where $x$ , $y$ , $k \in \mathbb{R}$ .

A scientist measures the three variables three times during the reaction and obtains the following values.

Find $x$ , $y$ and $k$ .

The function $f$ is defined by $f\left( x \right) = \frac{{ax + b}}{{cx + d}}$ , for $x \in \mathbb{R},\,\,x \ne - \frac{d}{c}$ .

The function $g$ is defined by $g\left( x \right) = \frac{{2x - 3}}{{x - 2}},\,\,x \in \mathbb{R},\,\,x \ne 2$

Find the inverse function ${f^{ - 1}}$ , stating its domain.

[5]

Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.

[2]

b.i.

Sketch the graph of $y = g\left( x \right)$ . State the equations of any asymptotes and the coordinates of any intercepts with the axes.

[3]

b.ii.

The function $h$ is defined by $h\left( x \right) = \sqrt x$ , for $x$ ≥ 0.

State the domain and range of $h \circ g$ .

[4]

The following table shows the time, in days, from December $1 st$ and the percentage of Christmas trees in stock at a shop on the beginning of that day.

The following table shows the natural logarithm of both $d$ and $x$ on these days to $2$ decimal places.

Use the data in the second table to find the value of $m$ and the value of $b$ for the regression line, $\ln x = m (\ln d) + b$ .

[2]

Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when $d = 25$ .

[3]

The function $f$ is defined by $f(x) = 2{x^3} + 5,{\text{ }} - 2 \leqslant x \leqslant 2$ .

Write down the range of $f$ .

[2]

Find an expression for ${f^{ - 1}}(x)$ .

[2]

Write down the domain and range of ${f^{ - 1}}$ .

[2]

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]

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]

Roger buys a new laptop for himself at a cost of $£ 495$ . At the same time, he buys his daughter Chloe a higher specification laptop at a cost of $£ 2200$ .

It is anticipated that Roger’s laptop will depreciate at a rate of $10 %$ per year, whereas Chloe’s laptop will depreciate at a rate of $15 %$ per year.

Roger and Chloe’s laptops will have the same value $k$ years after they were purchased.

Estimate the value of Roger’s laptop after $5$ years.

[2]

Find the value of $k$ .

[2]

Comment on the validity of your answer to part (b).

[1]

Consider the function $g\left( x \right) = 4\,{\text{cos}}\,x + 1$ , $a \leqslant x \leqslant \frac{\pi }{2}$ where $a < \frac{\pi }{2}$ .

For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.

[3]

Write down the least value of $a$ such that $g$ has an inverse.

[1]

For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .

[1]

c.i.

For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .

[2]

c.ii.

The function $f (x) = \ln (\frac{1}{x - 2})$ is defined for $x > 2, x \in ℝ$ .

Find an expression for $f^{- 1} (x)$ . You are not required to state a domain.

[3]

Solve $f (x) = f^{- 1} (x)$ .

[2]

It is believed that the power $P$ of a signal at a point $d$ km from an antenna is inversely proportional to $d^{n}$ where $n \in ℤ^{+}$ .

The value of $P$ is recorded at distances of $1 m$ to $5 m$ and the values of $\log_{10} d$ and $\log_{10} P$ are plotted on the graph below.

The values of $\log_{10} d$ and $\log_{10} P$ are shown in the table below.

Explain why this graph indicates that $P$ is inversely proportional to $d^{n}$ .

[2]

Find the equation of the least squares regression line of $\log_{10} P$ against $\log_{10} d$ .

[2]

Use your answer to part (b) to write down the value of $n$ to the nearest integer.

[1]

c.i.

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

[2]

c.ii.

The function $p\left( x \right)$ is defined by $p\left( x \right) = {x^3} + 3{x^2} + 8x - 24$ where $x \in \mathbb{R}$ .

Find the remainder when $p\left( x \right)$ is divided by $\left( {x - 2} \right)$ .

[2]

a.i.

Find the remainder when $p\left( x \right)$ is divided by $\left( {x - 3} \right)$ .

[1]

a.ii.

Prove that $p\left( x \right)$ has only one real zero.

[4]

Write down the transformation that will transform the graph of $y = p\left( x \right)$ onto the graph of $y = 8{x^3} + 12{x^2} + 16x - 24$ .

[2]

The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6{\text{P}}\left( {X = 3} \right) = 3{\text{P}}\left( {X = 2} \right) - 2{\text{P}}\left( {X = 1} \right) + 3{\text{P}}\left( {X = 0} \right)$ .

Find the value of $\lambda$ .

[6]

It is believed that two variables, $m$ and $p$ are related. Experimental values of $m$ and $p$ are obtained. A graph of ${\text{ln}}\,m$ against $p$ shows a straight line passing through (2.1, 7.3) and (5.6, 2.4).

Hence, find

Find the equation of the straight line, giving your answer in the form ${\text{ln}}\,m = ap + b$ , where $a{\text{,}}\,\,b \in \mathbb{R}$ .

[3]

a formula for $m$ in terms of $p$ .

[1]

b.i.

the value of $m$ when $p = 0$ .

[1]

b.ii.

A function is defined by $f (x) = 2 - \frac{12}{x + 5}$ for $- 7 \leq x \leq 7, x \neq - 5$ .

Find the range of $f$ .

[3]

Find an expression for the inverse function $f^{- 1} (x)$ . The domain is not required.

[3]

Write down the range of $f^{- 1} (x)$ .

[1]

Consider the functions $f$ and $g$ defined by $f\left( x \right) = {\text{ln}}\left| x \right|$ , $x \in \mathbb{R}$ \ $\left\{ 0 \right\}$ , and $g\left( x \right) = {\text{ln}}\left| {x + k} \right|$ , $x \in \mathbb{R}$ \ $\left\{ { - k} \right\}$ , where $k \in \mathbb{R}$ , $k > 2$ .

The graphs of $f$ and $g$ intersect at the point P .

Describe the transformation by which $f\left( x \right)$ is transformed to $g\left( x \right)$ .

[1]

State the range of $g$ .

[1]

Sketch the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$ on the same axes, clearly stating the points of intersection with any axes.

[6]

Find the coordinates of P.

[2]

The tangent to $y = f\left( x \right)$ at P passes through the origin (0, 0).

Determine the value of $k$ .

[7]

The graph of the function $f (x) = \ln x$ is translated by $(\begin{matrix} a \\ b \end{matrix})$ so that it then passes through the points $(0, 1)$ and $(e^{3}, 1 + \ln 2)$ .

Find the value of $a$ and the value of $b$ .

Adesh wants to model the cooling of a metal rod. He heats the rod and records its temperature as it cools.

He believes the temperature can be modeled by $T\left( t \right) = a{{\text{e}}^{bt}} + 25$ , where $a{\text{,}}\,\,b \in \mathbb{R}$ .

Hence

Show that ${\text{ln}}\left( {T - 25} \right) = bt + {\text{ln}}\,a$ .

[2]

Find the equation of the regression line of ${\text{ln}}\left( {T - 25} \right)$ on $t$ .

[3]

find the value of $a$ and of $b$ .

[3]

c.i.

predict the temperature of the metal rod after 3 minutes.

[2]

c.ii.

Consider the function $f\left( x \right) = {x^4} - 6{x^2} - 2x + 4$ , $x \in \mathbb{R}$ .

The graph of $f$ is translated two units to the left to form the function $g\left( x \right)$ .

Express $g\left( x \right)$ in the form $a{x^4} + b{x^3} + c{x^2} + dx + e$ where $a$ , $b$ , $c$ , $d$ , $e \in \mathbb{Z}$ .

Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$ , showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

N17/5/MATHL/HP1/ENG/TZ0/06.a

[4]

Hence or otherwise, solve the inequality $\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2$ .

[5]

Consider the graphs of $y = \left| x \right|$ and $y = - \left| x \right| + b$ , where $b \in {\mathbb{Z}^ + }$ .

Sketch the graphs on the same set of axes.

[2]

Given that the graphs enclose a region of area 18 square units, find the value of b.

[3]

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]

A rational function is defined by $f(x) = a + \frac{b}{{x - c}}$ where the parameters $a,{\text{ }}b,{\text{ }}c \in \mathbb{Z}$ and $x \in \mathbb{R}\backslash \{ c\}$ . The following diagram represents the graph of $y = f(x)$ .

N16/5/MATHL/HP1/ENG/TZ0/03

Using the information on the graph,

state the value of $a$ and the value of $c$ ;

[2]

find the value of $b$ .

[2]

Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following axes.

[3]

Solve the equation $\frac{x}{2} + 1 = \left| {x - 2} \right|$ .

[4]

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 $r$ .

It is believed that two variables, $v$ and $w$ are related by the equation $v = k{w^n}$ , where $k{\text{,}}\,\,n \in \mathbb{R}$ . Experimental values of $v$ and $w$ are obtained. A graph of ${\text{ln}}\,v$ against ${\text{ln}}\,w$ shows a straight line passing through (−1.7, 4.3) and (7.1, 17.5).

Find the value of $k$ and of $n$ .

Let $f (x) = a \cos (b (x - c)), a, b, c \in ℝ^{+}$ .

Part of the graph of $y = f (x)$ is shown below. Point $A$ is a local maximum and has coordinates $(1, 3)$ and point $B$ is a local minimum with coordinates $(2, - 3)$ .

Write down a sequence of transformations that will transform the graph of $y = \cos x$ onto the graph of $y = f (x)$ .