Level 2

[Maximum mark: 6]

Consider the infinite geometric sequence $4480$, $-3360$, $2520$, $-1890,\dots$

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

2. Find the $20$th term. [2]
   

3. Find the exact sum of the infinite sequence. [2]
   

[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 a right-angled triangle ABC and a sector ABD of a circle with centre A. The point E lies on [AC] such that [BE] is perpendicular to [AC]. The region $R$ is bounded by [DE], [BE] and arc BFD.

$\text{AB} = 5$ cm, $\text{AC} = 13$ cm, $\text{BC} = 12$ cm.

1. Find BÂC, giving your answer in radians. [2]
   

2. Find the area of $R$. [4]
   

[Maximum mark: 6]

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

1. Find $f(3)$. [2]
   

2. Find $(g\circ f)(3)$. [2]
   

3. Find $g^{-1}(x)$. [2]
   

[Maximum mark: 5]

Let $f(x) = a(x-h)^2 + k$, for $x \in \mathbb{R}$.

The vertex of the graph of $f$ is at P$(3,4)$ and the graph passes through Q$(1,-4)$.

1. Write down the values of $h$ and $k$. [2]
   

2. Find the value of $a$. [3]
   

[Maximum mark: 6]

A random variable $X$ has a probability distribution given in the following table.

1. Find the value of $q$. [3]
   

2. Find $\mathrm{E}[X]$. [3]
   

[Maximum mark: 5]

Consider the expansion of $(x-3)^8$.

1. Write down the number of terms in this expansion. [1]
   

2. Find the coefficient of the term in $x^6$. [4]
   

[Maximum mark: 5]

Let $f(x)=2x^2-5x+7$. The line $L$ intersects $f$ at P$(3,10)$ and is perpendicular to the tangent to the curve of $f$ at P. Find the equation of $L$ in the form $y = mx + c$.

[Maximum mark: 7]

Given that $\sin x = \dfrac{3}{5}$, where $x$ is an acute angle, find:

1. $\cos x$; [3]
   

2. $\sin 2x$; [2]
   

3. $\cos 2x$. [2]
   

[Maximum mark: 6]

Let $f(x) = 3x^2 - 4x$. The graph of $f$ is shown in the following diagram.

1. Find $\displaystyle \int (3x^2 - 4x)\,\mathrm{d}x$. [2]
   

2. Find the area of the region enclosed by the graph of $f$, the $x$-axis and the lines $x = 2$ and $x = 4$. [4]