\[Maximum mark: 4\]

Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your
answer.





\[Maximum mark: 6\]

A bag contains $7$ blue and $5$ red marbles. Two marbles are selected at
random without replacement.




1.  Complete the tree diagram below. :marks[3]

 

:::center




![865fd050d3db84e24416b401670caec65ade51d0.svg](question:95276d16-56a8-4a45-a61b-ffdf60c8d8ac)

:::




2.  Find the probability that exactly one of the selected
    marbles is blue. :marks[3]






\[Maximum mark: 6\]

The following diagram shows part of a circle with centre O and radius
$5$ cm.


:::center




![d3a02f29895c6ce0b991606f98b7b99a2c21d836.svg](question:67d70f28-0bff-466b-91cd-942f091eff72)

:::


Points A, B lie on the circle, chord AB has a length of $8$ cm and
$\text{A\^{O}B} = \theta$. 



1.  Find the value of $\theta$, giving your answer in radians.
    :marks[3]

2.  Find the area of the shaded region. :marks[3]






\[Maximum mark: 6\]

A school basketball team of $5$ students is selected from $8$ boys and
$4$ girls. 



1.  Determine how many possible teams can be chosen. :marks[2]

2.  Determine how many teams can be formed consisting of $3$ boys and
    $2$ girls? :marks[2]

3.  Determine how many teams can be formed consisting of at most $3$
    girls? :marks[2]






\[Maximum mark: 6\]

Let $f(x) = p x^3 - qx$. At $x = 0$, the gradient of the curve of $f$ is
$2$. Given that\
$f^{-1} (12) = 2$, find the value of $p$ and $q$.





\[Maximum mark: 6\]

The function $f$ is of the form $f(x) = \dfrac{ax+b}{2x+c}$, for
$x \neq -\dfrac{c}{2}$, where $a,b,c \in \mathbb{Z}$.
Given

that the graph of $y = f(x)$ has asymptotes $x = -5$ and $y = 2$, and
that the point 

P$\left(1,-\dfrac{1}{12}\right)$ lies on the graph, find the values of
$a, b$ and $c$.





\[Maximum mark: 6\]

Given that $\log_b 3 = 10$. 



1.  Find the exact value of $\log_b 81$. :marks[2]

2.  Find the exact value of $\log_{b^2} 3$. :marks[2]

3.  Find the value of $b$, giving your answer correct to $3$ significant
    figures. :marks[2]






\[Maximum mark: 7\]

In a geometric sequence, $u_2 = 6$, $u_5 = 20.25$.




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

2.  Find $u_1$. :marks[2]

3.  Find the greatest value of $n$ such that $u_n < 200$. :marks[3]






\[Maximum mark: 5\]

Let $f(x) = 9 - 2\ln(x^2 + 4)$, for $x \in \mathbb{R}$. The graph of $f$
passes through the point $(p,3)$, where
$p > 0$. 



1.  Find the value of $p$. :marks[2]


The following diagram shows part of the graph of $f$.


:::center




![6016001b65155100f0dc6eb98ff244252ca74b02.svg](question:a3dfe8f1-2f42-41c8-a72c-77591e2cdb87)

:::


The region enclosed by the graph of $f$, the $x$-axis and the lines
$x = -p$ and $x = p$ is rotated $360^\circ$ about the $x$-axis.




2.  Find the volume of the solid formed. :marks[3]






\[Maximum mark: 6\]

A particle moves in a straight line with velocity
$v(t) = 2 t - 0.3 t^3 + 2$, for $t \geq 0$, where $v$ is in ms$^{-1}$
and $t$ in seconds. 



1.  Find the acceleration of the particle after $2.2$ seconds.
    :marks[3]

2.  
    1.  Find the time when the acceleration is zero.

    2.  Find the velocity when the acceleration is zero. :marks[3]

    






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