IB Mathematics AA HL  Mock Exams
Mock Exam Set 1  Paper 1
Trial Examinations for IB Mathematics AA HL
Paper 1
12 Questions
120 mins
110 marks
Paper
Difficulty
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 Section A 
Question 1
[Maximum mark: 5]
The random variable $X$ is normally distributed with a mean of $120$. The following diagram shows the normal curve for $X$.
Let $R$ be the shaded region under the curve between $105$ and $135$. The area of $R$ is $0.4$.

Write down $\mathrm{P}(105 < X < 135)$. [1]

Find $\mathrm{P}(X < 135)$. [2]

Find $\mathrm{P}(X > 105\hspace{0.25em}\hspace{0.25em} X < 135)$. [2]
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Question 2
[Maximum mark: 6]
Prove by contradiction that the equation $3x^37x^2+5=0$ has no integer roots.
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Question 3
[Maximum mark: 6]
The diagram below shows the graph of a quadratic function $f(x) = 2x^2 + bx + c$.

Write down the value of $c$. [1]

Find the value of $b$ and write down $f(x)$. [3]

Calculate the coordinates of the vertex of the graph of $f$. [2]
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Question 4
[Maximum mark: 7]
A real estate company keeps a register of the monthly cost of rent, $R$, of their apartments and their corresponding area, $A$, in m$^2$.
The areas of the apartments registered are summarised in the following box and whisker diagram.
 Find the smallest area $A$ that would not be considered an outlier. [3]
The regression line $A$ on $R$ is $A=\dfrac{5}{12}R50$.
Meanwhile, the regression line $R$ on $A$ is $R=\dfrac{5}{2}A+100$.

One of the apartments has a monthly rent of $\$480$. Estimate the area of the rental. [2]

Find the mean rental cost of all the real estate company's apartments. [2]
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Question 5
[Maximum mark: 5]
Using the substitution $u=\sqrt{x}1$, find the value of $\displaystyle\int_1^4 \dfrac{2\sqrt{\sqrt{x}1}}{\sqrt{x}}\,\,\mathrm{d}x$
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Question 6
[Maximum mark: 7]
Consider the functions $f(x)=3\cos(x)+\dfrac{9}{2}$ and $g(x)=3\cos\left(x+\dfrac{\pi}{3}\right)+A$, where $x\in \mathbb{R}$ and $A< \dfrac{9}{2}$.
 Describe a sequence of two transformations that transforms the graph of $f$ to the graph of $g$. [3]
The $y$intercept of the graph $g$ is at the point $\left(0\,,\dfrac{9}{2}\right)$
 Find the range of $g$. [4]
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Question 7
[Maximum mark: 7]
Points A and B represent the complex numbers $z_1 = \sqrt{3}  {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_2 = 3  3{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ as shown on the Argand diagram below.

Find the angle AOB. [3]

Find the argument of $z_1z_2$. [1]

Given that the real powers of $pz_1z_2$, for $p > 0$, all lie on a unit circle centred at the origin, find the exact value of $p$. [3]
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Question 8
[Maximum mark: 6]
Consider the curve $y=(kx1)\ln(2x)$ where $k\in \mathbb{R}$ and $x>0$.
The tangent to the curve at $x=2$ is perpendicular to the line $y=\dfrac{2}{5+4\hspace{0.15em}\ln 4}x$.
Find the value of $k$.
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Question 9
[Maximum mark: 8]
Consider the function $f(x)=\sin\left(\dfrac{\pi}{12}\dfrac{x}{4}\right)$ for $x \in \mathbb{R}$.

Show that the $y$intercept of $f(x)$ is $\dfrac{\sqrt{6}\sqrt{2}}{4}$ [3]

Find the least positive value of $x$ for which $f(x) = \dfrac{\sqrt{3}}{2}$. [5]
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 Section B 
Question 10
[Maximum mark: 18]
The first three terms of an infinite sequence, in order, are
$2\ln x,\,\, q\ln x,\,\, \ln \sqrt{x}\,\,\, \text{ where $\ x > 0$.}$
First consider the case in which the series is geometric.


Find the possible values of $q$.

Hence or otherwise, show that the series is convergent. [3]


Given that $q>0$ and $S_\infty=8\ln{3}$, find the value of $x$. [3]
Now suppose that the series is arithmetic.


Show that $q=\dfrac{5}{4}$.

Write down the common difference in the form $m\ln x$, where $m \in \mathbb{Q}$. [4]


Given that the sum of the first $n$ terms of the sequence is $\ln \sqrt{x^5}$, find the value of $n$. [8]
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Question 11
[Maximum mark: 15]
Consider the planes $\Pi_1$, $\Pi_2$, $\Pi_3$ given by the following equations:
 Show that the three planes do not intersect. [4]
It is given that the point Q$(1,1,0)$ lies on both $\Pi_1$ and $\Pi_2$.
Let $\ell$ be the line of intersection of $\Pi_1$ and $\Pi_2$.

Find a vector expression for $\ell$. [4]

Show that $\ell$ is parallel to plane $\Pi_3$. [2]

Hence or otherwise, find the distance between $\ell$ and $\Pi_3$ Express your answer in the form $\dfrac{p}{\sqrt{q}}$, where $p$, $q\in \mathbb{Z}$. [5]
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Question 12
[Maximum mark: 20]
Consider the function defined by $f(x) = \dfrac{7}{x^2+6x7}$ for $x\in \mathbb{R}$, $x\neq 7$, $x\neq 1$.
 Sketch the graph of $y=f(x)$, showing the values of any axes intercepts, the coordinates of any local maxima and minima, and the graphs of any asymptotes. [6]
Next, consider the function $g$ defined by $g(x)=\dfrac{7}{x^2+6x7}$ for $x\in \mathbb{R}$, $x> 1$.

Show that $g^{1}(x)=3 + \sqrt{\dfrac{16x+7}{x}}$. [6]

State the domain of $g^{1}$. [1]
Now, consider the function $h$ defined by $h(x)=\arccos\left(\dfrac{x}{7}\right)$.
 Given that $\left(h\circ g\right)(a) = \dfrac{\pi}{3}$, find the value of $a$. Give your answer in the form $p+q\sqrt{2}$ where $p$, $q\in \mathbb{Z}$. [7]
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