Mock Exam Set 1 - Paper 1

[Maximum mark: 5]

The random variable $X$ is normally distributed with a mean of $120$. The following diagram shows the normal curve for $X$.

Let $R$ be the shaded region under the curve between $105$ and $135$. The area of $R$ is $0.4$.

1. Write down $\mathrm{P}(105 < X < 135)$. [1]
   

2. Find $\mathrm{P}(X < 135)$. [2]
   

3. Find $\mathrm{P}(X > 105\hspace{0.25em}|\hspace{0.25em} X < 135)$. [2]
   

[Maximum mark: 6]

Prove by contradiction that the equation $3x^3-7x^2+5=0$ has no integer roots.

[Maximum mark: 6]

The diagram below shows the graph of a quadratic function $f(x) = 2x^2 + bx + c$.

1. Write down the value of $c$. [1]
   

2. Find the value of $b$ and write down $f(x)$. [3]
   

3. Calculate the coordinates of the vertex of the graph of $f$. [2]
   

[Maximum mark: 7]

A real estate company keeps a register of the monthly cost of rent, $R$, of their apartments and their corresponding area, $A$, in m$^2$.

The areas of the apartments registered are summarised in the following box and whisker diagram.

1. Find the smallest area $A$ that would not be considered an outlier. [3]
   

The regression line $A$ on $R$ is $A=\dfrac{5}{12}R-50$.

Meanwhile, the regression line $R$ on $A$ is $R=\dfrac{5}{2}A+100$.

2. One of the apartments has a monthly rent of $\$480$. Estimate the area of the rental. [2]
   

3. Find the mean rental cost of all the real estate company's apartments. [2]
   

[Maximum mark: 5]

Using the substitution $u=\sqrt{x}-1$, find the value of $\displaystyle\int_1^4 \dfrac{2\sqrt{\sqrt{x}-1}}{\sqrt{x}}\,\,\mathrm{d}x$

[Maximum mark: 7]

Consider the functions $f(x)=3\cos(x)+\dfrac{9}{2}$ and $g(x)=3\cos\left(x+\dfrac{\pi}{3}\right)+A$, where $x\in \mathbb{R}$ and $A< \dfrac{9}{2}$.

1. Describe a sequence of two transformations that transforms the graph of $f$ to the graph of $g$. [3]
   

The $y$-intercept of the graph $g$ is at the point $\left(0\,,\dfrac{9}{2}\right)$

2. Find the range of $g$. [4]
   

[Maximum mark: 7]

Points A and B represent the complex numbers $z_1 = \sqrt{3} - {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_2 = -3 - 3{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ as shown on the Argand diagram below.

1. Find the angle AOB. [3]
   

2. Find the argument of $z_1z_2$. [1]
   

3. Given that the real powers of $pz_1z_2$, for $p > 0$, all lie on a unit circle centred at the origin, find the exact value of $p$. [3]
   

[Maximum mark: 6]

Consider the curve $y=(kx-1)\ln(2x)$ where $k\in \mathbb{R}$ and $x>0$.

The tangent to the curve at $x=2$ is perpendicular to the line $y=\dfrac{2}{5+4\hspace{0.15em}\ln 4}x$.

Find the value of $k$.

[Maximum mark: 8]

Consider the function $f(x)=\sin\left(\dfrac{\pi}{12}-\dfrac{x}{4}\right)$ for $x \in \mathbb{R}$.

1. Show that the $y$-intercept of $f(x)$ is $\dfrac{\sqrt{6}-\sqrt{2}}{4}$ [3]
   

2. Find the least positive value of $x$ for which $f(x) = \dfrac{\sqrt{3}}{2}$. [5]
   

[Maximum mark: 18]

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

First consider the case in which the series is geometric.

1. 1. Find the possible values of $q$.

   2. Hence or otherwise, show that the series is convergent. [3]
      

2. Given that $q>0$ and $S_\infty=8\ln{3}$, find the value of $x$. [3]
   

Now suppose that the series is arithmetic.

3. 1. Show that $q=\dfrac{5}{4}$.

   2. Write down the common difference in the form $m\ln x$, where $m \in \mathbb{Q}$. [4]
      

4. Given that the sum of the first $n$ terms of the sequence is $\ln \sqrt{x^5}$, find the value of $n$. [8]
   

[Maximum mark: 15]

Consider the planes $\Pi_1$, $\Pi_2$, $\Pi_3$ given by the following equations:

1. Show that the three planes do not intersect. [4]
   

It is given that the point Q$(-1,-1,0)$ lies on both $\Pi_1$ and $\Pi_2$.

Let $\ell$ be the line of intersection of $\Pi_1$ and $\Pi_2$.

2. Find a vector expression for $\ell$. [4]
   

3. Show that $\ell$ is parallel to plane $\Pi_3$. [2]
   

4. Hence or otherwise, find the distance between $\ell$ and $\Pi_3$ Express your answer in the form $\dfrac{p}{\sqrt{q}}$, where $p$, $q\in \mathbb{Z}$. [5]
   

[Maximum mark: 20]

Consider the function defined by $f(x) = \dfrac{7}{x^2+6x-7}$ for $x\in \mathbb{R}$, $x\neq -7$, $x\neq 1$.

1. Sketch the graph of $y=f(x)$, showing the values of any axes intercepts, the coordinates of any local maxima and minima, and the graphs of any asymptotes. [6]
   

Next, consider the function $g$ defined by $g(x)=\dfrac{7}{x^2+6x-7}$ for $x\in \mathbb{R}$, $x> 1$.

2. Show that $g^{-1}(x)=-3 + \sqrt{\dfrac{16x+7}{x}}$. [6]
   

3. State the domain of $g^{-1}$. [1]
   

Now, consider the function $h$ defined by $h(x)=\arccos\left(\dfrac{x}{7}\right)$.

4. Given that $\left(h\circ g\right)(a) = \dfrac{\pi}{3}$, find the value of $a$. Give your answer in the form $p+q\sqrt{2}$ where $p$, $q\in \mathbb{Z}$. [7]