Level 1

[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. [3]
   

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

[Maximum mark: 6]

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

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. [3]
   

2. Find the area of the shaded region. [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. [2]
   

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

3. Determine how many teams can be formed consisting of at most $3$ girls? [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 $\log_b 81$. [2]
   

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

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

[Maximum mark: 7]

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

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

2. Find $u_1$. [2]
   

3. Find the greatest value of $n$ such that $u_n < 200$. [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 $p$. [2]
   

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

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. [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. [3]
   

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

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