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Paper 1

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Paper 2

\[Maximum mark: 6\]

Two events $A$ and $B$ are such that $\mathrm{P}(A) = 0.5$ and
$\mathrm{P}(A\cap B) = 0.1$. 



1.  Find $\mathrm{P}(A\cap B^{\prime})$. :marks[2]

2.  Given that
    $\mathrm{P}(A'\hspace{-0.075em}\cap B^{\prime}) = 0.2$,
    find : 

    

    1.  $\mathrm{P}(A\cup B)$;
    2.  $\mathrm{P}(B)$;

    3.  $\mathrm{P}(A\hspace{0.15em}|\hspace{0.15em}B^{\prime})$.
        :marks[4]

    




\[Maximum mark: 5\]

Two functions, $f$ and $g$, are defined in the following table.


:::center




![f16cb92d520a6fc6d08cc849fc6b256348e1335f.svg](question:72fb6a0c-ca68-40db-a765-15cc4604525e)

:::




1.  Write down the value of $f(2)$. :marks[1]

2.  Find the value of $(g\circ f)(2)$. :marks[2]

3.  Find the value of $g^{-1}(5)$. :marks[2]






\[Maximum mark: 6\]

The following diagram shows the curve
$y = a\cos\hspace{0.05em}(k(x - d)) + c$ where
$a, k, d$ and $c$ are all positive constants. The
curve has a minimum point at $(1.5,2)$ and
a $\text{maximum}$ point at $(3.5,7)$.


:::center




![3729611ba3a439e48776fd9a8da5e25aa14a08a8.svg](question:a3e8f6a8-2953-4ef7-8dc3-0a73047d48f4)

:::




1.  Write down the value of $a$ and the value of $c$. :marks[2]

2.  Find the value of $k$. :marks[2]

3.  Find the smallest possible value of $d$, given $d > 0$. :marks[2]






\[Maximum mark: 7\]



1.  Find $\displaystyle \int 6x^2e^{x^3+8}\,\mathrm{d}x$. :marks[4]

2.  Find $f(x)$, given that $f'(x) = 6x^2e^{x^3+8}$ and $f(-2) = 5$.
    :marks[3]






\[Maximum mark: 6\]

Solve the equation $\cos 2x - \sin^2 x = \cos^2 x + 3\cos x$, for
$0 \leq x \leq 2\pi$.





\[Maximum mark: 7\]

Consider $f(x) = \log_k(8x-2x^2)$, for $0 < x < 4$, where $k > 0$.


The equation $f(x) = 3$ has exactly one solution. Find the value of $k$.





\[Maximum mark: 15\]

Let $f(x) = 3x^2 + 12x + 9$, for $x \in \mathbb{R}$.




1.  For the graph of $f$, find: 
    1.  the $y$-intercept;

    2.  the $x$-intercepts. :marks[4]

    

 

The function $f$ can be written in the form $f(x) = a(x-h)^2+k$.




2.  Find the values of $a$, $h$ and $k$. :marks[5]

3.  For the graph of $f$, write down: 
    1.  the coordinates of the vertex;

    2.  the equation of the axis of symmetry. :marks[2]

    

 

The graph of a function $g$ is obtained from the graph of $f$ by a
reflection 

in the $x$-axis, followed by a translation by the vector
$\medmath{\begin{pmatrix} 0 \\ 4 \end{pmatrix}}$.




4.  Find $g(x)$, giving your answer in the form
    $g(x) = px^2 + qx + r$.:marks[4]




\[Maximum mark: 14\]

The first two terms of an infinite geometric sequence, in order, are

$$
3\log_3x,\,\, 2\log_3x,\,\, \text{where $x > 0$.}
$$




1.  Find the common ratio, $r$. :marks[2]

2.  Show that the sum of the infinite sequence is $9\log_3 x$.
    :marks[3]

 

The first three terms of an arithmetic sequence, in order, are

$$
\log_3x,\,\, \log_3 \dfrac{x}{3},\,\, \log_3\dfrac{x}{9},\,\, \text{where $x > 0$.}
$$




3.  Find the common difference $d$, giving your answer as an
    integer. :marks[3]

 

Let $S_6$ be the sum of the first $6$ terms of the arithmetic sequence.




4.  Show that $S_6 = 6\log_3 x - 15$. :marks[3]

5.  Given that $S_6$ is equal to one third of the sum of the
    infinite geometric\
    sequence, find $x$, giving your answer in the form
     $a^p$ where $a,p \in \mathbb{Z}$.
    :marks[3]




\[Maximum mark: 17\]

Consider a function $f$. The line $L_1$ with equation $y = 2x - 1$ is a
tangent to the graph of $f$ when $x = 3$. 



1.  
    1.  Write down $f'(3)$.

    2.  Find $f(3)$. :marks[4]

    

 

Let $g(x) = f(x^2 - 1)$ and P be the point on the graph of $g$ where
$x = 2$. 



2.  Show that the graph of $g$ has a gradient of $8$ at P.
    :marks[5]

 

Let $L_2$ be the tangent to the graph of $g$ at P. The line $L_1$
intersects $L_2$ at the point Q. 



3.  Find the $y$-coordinate of Q.:marks[8]







IB Math AA SL - Mock Exams

Mock Exam Set 1 - Paper 1

--- Section A ---

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

--- Section B ---

Question 7

Question 8

Question 9

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