H2 Math Question Bank

A child’s toy has $36$ slots, numbered from $1$ to $36$. A child puts a ball into the toy, the ball falls into one of the $36$ slots and the child’s score is the number of that slot. The ball is equally likely to fall into any one of the slots.

Sadiq is investigating four possible events, $A$, $B$, $C$, and $D$, which are defined as follows.

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $A$: The score is odd.
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $B$: The score is even.
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $C$: The score is a multiple of $3$.
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $D$: The score is a multiple of $6$.

1. 1. State which pairs of the events, if any, are mutually exclusive.

      [1]

   2. Show that $A$ and $C$ are independent events, and state another pair of independent events.

      [2]

Sadiq notices that a ball has become stuck in the slot labelled $36$, and so balls put into the toy are now falling, with equal probability, into one of only $35$ slots, and the score can only be from $1$ to $35$.

2. 1. State which pairs of the four events, if any, are now mutually exclusive.

      [1]

   2. Determine whether $A$ and $C$ are now independent events.

      [1]

A bag contains $r$ red counters and $b$ blue counters, where $r > 12$ and $b > 12$. Mei randomly removes $12$ counters from the bag. The probability that there are $4$ red counters among Mei’s $12$ counters is the same as the probability that there are $3$ red counters.

1. Show that $9r+5=4b$.

   [2]

The probability that there are $3$ red counters among Mei’s $12$ counters is $\frac{5}{3}$ times the probability that there are $2$ red counters.

2. Derive an equation similar to the equation in part (a) and hence find the probability that just one of the $12$ counters removed is red.

   [6]

A bag contains $3$ blue discs, $2$ red discs and $y$ yellow discs. Li chooses $3$ discs at random from the bag, without replacement.

1. Show that the probability Li chooses $1$ blue disc, $1$ red disc and $1$ yellow disc is $\frac{36y}{\left( y+5 \right)\left( y+4 \right)\left( y+3 \right)}$.

   [1]

Li’s discs are replaced in the bag.
Darvina chooses $3$ discs at random from the bag, without replacement. The random variable $S$ is the sum of the number of blue discs chosen and twice the number of red discs chosen.

2. Find an expression in terms of $y$ for $\text{P}\left( S=3 \right)$.

   [2]

3. Given that $\text{P}\left( S=3 \right)=\frac{7}{20}$, calculate the value of $y$. Hence find the probability distribution of $S$.

   [6]

When performing a trick a magician says the word ABRACADABRA. The $11$ letters of this word are arranged in a row

1. Find the number of different arrangements that can be made.

   [2]

2. Find the number of different arrangements in which the $2$ B’s are next to each other, the $2$ R’s are next to each other, exactly $4$ of the A’s are next to each other, and the C is next to the D.

   [3]

3. Given that the $11$ letters are arranged randomly, find the probability that all $5$ A’s are together.

   [3]

Show that the differential equation $y\left( \frac{\text{d}y}{\text{d}x}+2y \right)=\frac{x}{{{\text{e}}^{4x+{{x}^{2}}}}}$ can be reduced by the substitution $z={{y}^{2}}{{\text{e}}^{4x}}$ to $\frac{\text{d}z}{\text{d}x}=\frac{2x}{{{\text{e}}^{{{x}^{2}}}}}$. Hence, find the general solution in the form ${{y}^{2}}=\text{f}\left( x \right)$.

[4]

It is given that $y={{e}^{-x}}$, $y={{e}^{2x}}$ and $y={{e}^{-3x}}$ are solutions to the differential equation

$\frac{{{\text{d}}^{3}}y}{\text{d}{{x}^{3}}}+a\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+b\frac{\text{d}y}{\text{d}x}+cy=0$.

Find the values of the constants $a$, $b$ and $c$.

It is given that $\left( 2xy\frac{\text{d}y}{\text{d}x}+{{y}^{2}} \right)\cos x=\frac{1}{x{{y}^{2}}}$, where $0 < x < \frac{\pi }{2}$ and $y\ne 0$.

1. Using the substitution $u=x{{y}^{2}}$, show that the differential equation can be reduced to $\frac{\text{d}u}{\text{d}x}=\frac{\sec x}{u}$.

   [3]

2. Hence find the general solution to the differential equation.

   [3]

There are three distinct points $A$, $B$ and $P$. The point $P$ lies on the circle with centre $O$ and diameter $AB$. Relative to the origin $O$, the position vectors of points $A$ and $P$ are $\mathbf{a}$ and $\mathbf{p}$ respectively.

1. By expressing $\overrightarrow{AP}$ and $\overrightarrow{BP}$ in terms of $\mathbf{a}$ and $\mathbf{p}$, show that $\overrightarrow{AP}\cdot \overrightarrow{BP}=0$.

   [3]

The point $C$ with position vector $\mathbf{c}$ lies on $AB$ produced such that $AC:AB=\lambda :1$.

2. Find $\mathbf{c}$ in terms of $\lambda $ and $\mathbf{a}$.

   [2]

3. It is given that $PC$ is a tangent to the circle and that angle $AOP={{120}^{{}^\circ }}$. Using a suitable scalar product, find the value of $\lambda $.

   [4]

It is given that $y={{e}^{{{\tan }^{-1}}4x}}$.

1. Show that $\left( 1+16{{x}^{2}} \right)\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+k\frac{\text{d}y}{\text{d}x}\left( 8x-1 \right)=0$, where $k$ is a constant to be found.

   [3]

2. By further differentiation of the result in part (a), find the Maclaurin series for $y$ up to and including the term in ${{x}^{3}}$.

   [3]

3. The first three non-zero terms of the Maclaurin expansion of ${{e}^{{{\tan }^{-1}}4x}}$ are equal to the first three non-zero terms of the series expansion of $\ln \left( \frac{a}{1-bx} \right)$, where $a$ and $b$ are constants. Using standard series from the List of Formulae (MF27), find the exact values of $a$ and $b$.

   [2]

The region bounded by the axes and the curve $y=\cos x$ from $x=0$ to $x=\frac{1}{2}\pi $ is divided into two parts, of areas ${{A}_{1}}$ and ${{A}_{2}}$, by the curve $y=\sin x$ (see diagram). Prove that ${{A}_{2}}=\left( \sqrt{2} \right){{A}_{1}}$.

[6]

The two curves meet at $P$. The line through $P$parallel to the $x$-axis meets the $y$-axis at $Q$. The region $OPQ$, bounded by the arc $OP$ and the lines $PQ$ and $QO$, is rotated through $4$ right angles about the $y$-axis to form a solid of revolution of volume $V$. It is given that

$C=\pi \int_{0}^{\frac{1}{\sqrt{2}}}{{{\left( {{\sin }^{-1}}y \right)}^{2}}}\text{d}y$.

1. By substituting $u={{\sin }^{-1}}y$, show that $V=\pi \int_{0}^{\frac{1}{4}\pi }{{{u}^{2}}\cos u\,\text{d}u}$.

   [2]

2. Show that $\frac{\text{d}}{\text{d}u}\left( {{u}^{2}}\sin u+2u\cos u-2\sin u \right)={{u}^{2}}\cos u$.

   [2]

3. Hence find the exact value of $V$.

   [2]