Level 2

[Maximum mark: 8]

AB Technologies produces DC motors of which $5\%$ are defective. The motors are packaged into boxes, each box containing $60$ motors. A box is selected at random.

1. Find the probability that the box contains:

   1. exactly one defective motor;

   2. at least two defective motors. [4]
      

2. Given that there are at least two defective motors in the box, find the probability that there are at most four defective motors. [2]
   

James orders 5 boxes of motors for his company.

3. Find the probability that James will receive at least one but no more than fifteen defective motors in his order. [2]
   

[Maximum mark: 5]

The functions $f$ and $g$ are defined such that $f(x) = \dfrac{x-2}{3}$ and $g(x) = 12x+4$.

1. Show that $(g\circ f)(x) = 4x-4$. [2]
   

2. Given that $(g\circ f)^{-1}(a) = 10$, find the value of $a$. [3]
   

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.

[Maximum mark: 8]

In a triangle ABC, $\text{AB} = 3$ cm, $\text{BC} = 5$ cm and $\text{A\^{C}B} = \dfrac{\pi}{6}$.

1. Find, to three significant figures, the two possible lengths of [AC]. [5]
   

2. Find the difference between the areas of the two possible triangles ABC.[3]
   

[Maximum mark: 8]

A function $f(x)$ has derivative $f'(x) = 6x^2-24x$. The graph of $f$ has an $x$-intercept at $x = 1$.

1. Find $f(x)$. [4]
   

2. The graph of $f$ has a point of inflexion at $x = k$. Find $k$. [2]
   

3. Find the values of $x$ for which the graph of $f$ is concave-up. [2]
   

[Maximum mark: 5]

Consider the vectors $\mathbf{a} = 2\textbf{i} + \textbf{j} - 4\textbf{k}$ and $\mathbf{b} = \textbf{i} - 2\textbf{j} + 5\textbf{k}$.

1. Find $\mathbf{a} \times \mathbf{b}$. [2]
   

2. Hence find the Cartesian equation of the plane containing the vectors $\textbf{a}$ and $\textbf{b}$ and passing through the point P$(2,-1,0)$. [3]
   

[Maximum mark: 6]

The following diagram shows the graph of $y = f(x)$, for $-1 \leq x \leq 2$.

1. Write down the value of:

   1. $f(1)$;

   2. $f^{-1}(-2)$. [2]
      

2. Find $(f\circ f)(1)$. [2]
   

3. Sketch the graph of $y = f(-x)$ on the same grid above. [2]
   

[Maximum mark: 5]

A function $f$ is defined by $f(x) = -x^5 + e^{-x} + 2$, $x \in \mathbb{R}$. By considering $f'(x)$, determine whether $f$ is a one-to-one or many-to-one function.

[Maximum mark: 6]

Let $f(x) = p - \dfrac{16}{x}$, for $x \neq 0$, where $p$ is a constant.

The line $y = x - p$ intersects the graph of $y = f(x)$ at two distinct points.

Find the possible values of $p$.

[Maximum mark: 5]

Consider two events $A$ and $B$ such that $\mathrm{P}(A)=k$, $\mathrm{P}(B)=2k$, $\mathrm{P}(A \cap B) = 4k^2-0.11$ and $\mathrm{P}(A \cup B) = 0.65$.

Find the value of $k$.