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OCR FP3 2014 June Q6

8 marks Standard +0.8

6 The line $l$ has equations $\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 7 } { 5 }$. The plane $\Pi$ has equation $4 x - y - z = 8$.

Show that $l$ is parallel to $\Pi$ but does not lie in $\Pi$.
The point $A ( 1 , - 2,7 )$ is on $l$. Write down a vector equation of the line through $A$ which is perpendicular to $\Pi$. Hence find the position vector of the point on $\Pi$ which is closest to $A$.
Hence write down a vector equation of the line in $\Pi$ which is parallel to $l$ and closest to it.

OCR FP3 2014 June Q7

8 marks Challenging +1.2

By expressing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$, show that $$\sin ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta ) .$$
Hence solve the equation $$\sin 5 \theta + 4 \sin \theta = 5 \sin 3 \theta$$ for $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. 8 consists of the set of matrices of the form $\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)$, where $a$ and $b$ are real and $a ^ { 2 } + b ^ { 2 } \neq 0$, combined under the operation of matrix multiplication.

Prove that $G$ is a group. You may assume that matrix multiplication is associative.
Determine whether $G$ is commutative.
Find the order of $\left( \begin{array} { c c } 0 & - 1 \\ 1 & 0 \end{array} \right)$.

OCR FP3 2015 June Q1

8 marks Standard +0.8

1 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 13 y = \sin x$$

OCR FP3 2015 June Q2

8 marks Standard +0.8

2 The elements of a group $G$ are polynomials of the form $a + b x + c x ^ { 2 }$, where $a , b , c \in \{ 0,1,2,3,4 \}$. The group operation is addition, where the coefficients are added modulo 5 .

State the identity element.
State the inverse of $3 + 2 x + x ^ { 2 }$.
State the order of $G$. The proper subgroup $H$ contains $2 + x$ and $1 + x$.
Find the order of $H$, justifying your answer.

OCR FP3 2015 June Q3

11 marks Standard +0.3

3 The plane $\Pi$ passes through the points $( 1,2,1 ) , ( 2,3,6 )$ and $( 4 , - 1,2 )$.

Find a cartesian equation of the plane $\Pi$. The line $l$ has equation $\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)$.
Find the coordinates of the point of intersection of $\Pi$ and $l$.
Find the acute angle between $\Pi$ and $l$.

OCR FP3 2015 June Q4

9 marks Standard +0.8

4 In an Argand diagram, the complex numbers $0 , z$ and $z \mathrm { e } ^ { \frac { 1 } { 6 } \mathrm { i } \pi }$ are represented by the points $O , A$ and $B$ respectively.

Sketch a possible Argand diagram showing the triangle $O A B$. Show that the triangle is isosceles and state the size of angle $A O B$. The complex numbers $1 + \mathrm { i }$ and $5 + 2 \mathrm { i }$ are represented by the points $C$ and $D$ respectively. The complex number $w$ is represented by the point $E$, such that $C D = C E$ and angle $D C E = \frac { 1 } { 6 } \pi$.
Calculate the possible values of $w$, giving your answers exactly in the form $a + b \mathrm { i }$.

Use de Moivre's theorem to show that $\tan 4 \theta \equiv \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$.
Hence find the exact roots of $t ^ { 4 } + 4 \sqrt { 3 } t ^ { 3 } - 6 t ^ { 2 } - 4 \sqrt { 3 } t + 1 = 0$.

OCR FP3 2015 June Q8

12 marks Challenging +1.8

8 Let $G$ be any multiplicative group. $H$ is a subset of $G$. $H$ consists of all elements $h$ such that $h g = g h$ for every element $g$ in $G$.

Prove that $H$ is a subgroup of $G$. Now consider the case where $G$ is given by the following table:

	$e$	$p$	$q$	$r$	$s$	$t$
$e$	$e$	$p$	$q$	$r$	$s$	$t$
$p$	$p$	$q$	$e$	$s$	$t$	$r$
$q$	$q$	$e$	$p$	$t$	$r$	$s$
$r$	$r$	$t$	$s$	$e$	$q$	$p$
$s$	$s$	$r$	$t$	$p$	$e$	$q$
$t$	$t$	$s$	$r$	$q$	$p$	$e$

Show that $H$ consists of just the identity element.

OCR MEI FP2 2012 June Q1

18 marks Standard +0.3

1. Differentiate the equation $\sin y = x$ with respect to $x$, and hence show that the derivative of $\arcsin x$ is $\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$.
2. Evaluate the following integrals, giving your answers in exact form.
  (A) $\int _ { - 1 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x$ (B) $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - 2 x ^ { 2 } } } \mathrm {~d} x$
A curve has polar equation $r = \tan \theta , 0 \leqslant \theta < \frac { 1 } { 2 } \pi$. The points on the curve have cartesian coordinates $( x , y )$. A sketch of the curve is given in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99f0c663-bb5b-4456-854c-df177f5d8349-2_493_796_1123_605} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Show that $x = \sin \theta$ and that $r ^ { 2 } = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }$.
Hence show that the cartesian equation of the curve is $$y = \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } .$$ Give the cartesian equation of the asymptote of the curve.

OCR MEI FP2 2012 June Q2

18 marks Standard +0.8

1. Given that $z = \cos \theta + \mathrm { j } \sin \theta$, express $z ^ { n } + \frac { 1 } { z ^ { n } }$ and $z ^ { n } - \frac { 1 } { z ^ { n } }$ in simplified trigonometric form.
2. Beginning with an expression for $\left( z + \frac { 1 } { z } \right) ^ { 4 }$, find the constants $A , B , C$ in the identity $$\cos ^ { 4 } \theta \equiv A + B \cos 2 \theta + C \cos 4 \theta$$
3. Use the identity in part (ii) to obtain an expression for $\cos 4 \theta$ as a polynomial in $\cos \theta$.
1. Given that $z = 4 \mathrm { e } ^ { \mathrm { j } \pi / 3 }$ and that $w ^ { 2 } = z$, write down the possible values of $w$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$. Show $z$ and the possible values of $w$ in an Argand diagram.
2. Find the least positive integer $n$ for which $z ^ { n }$ is real. Show that there is no positive integer $n$ for which $z ^ { n }$ is imaginary.
  For each possible value of $w$, find the value of $w ^ { 3 }$ in the form $a + \mathrm { j } b$ where $a$ and $b$ are real.

OCR MEI FP2 2012 June Q3

18 marks Challenging +1.2

Find the value of $a$ for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & a & 4 \\ 3 & - 2 & 2 \end{array} \right)$$ does not have an inverse.
Assuming that $a$ does not have this value, find the inverse of $\mathbf { M }$ in terms of $a$.
Hence solve the following system of equations. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
Find the value of $b$ for which the following system of equations has a solution. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 6 y + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = b \end{aligned}$$ Find the general solution in this case and describe the solution geometrically.

OCR MEI FP2 2012 June Q4

18 marks Challenging +1.2

Prove, from definitions involving exponential functions, that $$\cosh 2 u = 2 \sinh ^ { 2 } u + 1$$
Prove that, if $y \geqslant 0$ and $\cosh y = u$, then $y = \ln \left( u + \sqrt { } \left( u ^ { 2 } - 1 \right) \right)$.
Using the substitution $2 x = \cosh u$, show that $$\int \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x = a x \sqrt { 4 x ^ { 2 } - 1 } - b \operatorname { arcosh } 2 x + c$$ where $a$ and $b$ are constants to be determined and $c$ is an arbitrary constant.
Find $\int _ { \frac { 1 } { 2 } } ^ { 1 } \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x$, expressing your answer in an exact form involving logarithms.

OCR MEI FP2 2012 June Q5

18 marks Challenging +1.2

5 This question concerns curves with polar equation $r = \sec \theta + a$, where $a$ is a constant.

State the set of values of $\theta$ between 0 and $2 \pi$ for which $r$ is undefined. For the rest of the question you should assume that $\theta$ takes all values between 0 and $2 \pi$ for which $r$ is defined.
Use your graphical calculator to obtain a sketch of the curve in the case $a = 0$. Confirm the shape of the curve by writing the equation in cartesian form.
Sketch the curve in the case $a = 1$. Now consider the curve in the case $a = - 1$. What do you notice?
By considering both curves for $0 < \theta < \pi$ and $\pi < \theta < 2 \pi$ separately, describe the relationship between the cases $a = 1$ and $a = - 1$.
What feature does the curve exhibit for values of $a$ greater than 1 ? Sketch a typical case.
Show that a cartesian equation of the curve $r = \sec \theta + a$ is $\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }$.

OCR MEI FP2 2013 June Q3

18 marks Standard +0.8

3 You are given the matrix $\mathbf { A } = \left( \begin{array} { r r r } k & - 7 & 4 \\ 2 & - 2 & 3 \\ 1 & - 3 & - 2 \end{array} \right)$.

Show that when $k = 5$ the determinant of $\mathbf { A }$ is zero. Obtain an expression for the inverse of $\mathbf { A }$ when $k \neq 5$.
Solve the matrix equation $$\left( \begin{array} { r r r }

OCR MEI FP2 2013 June Q4

18 marks Challenging +1.2

Prove, using exponential functions, that $\cosh ^ { 2 } u - \sinh ^ { 2 } u = 1$.
Given that $y = \operatorname { arsinh } x$, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$ and that $$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$
Show that $$\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + 9 x ^ { 2 } } } \mathrm {~d} x = \frac { 1 } { 3 } \ln ( 3 + \sqrt { 10 } )$$
Find, in exact logarithmic form, $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } } \operatorname { arsinh } x \mathrm {~d} x$$

CAIE S1 2020 Specimen Q1

5 marks Easy -1.2

1 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.

Females			Males
(4)	5200	20	3	(1)
(9)	988764000	21	007	(3)
(8)	87533100	22	004566	(6)
(6)	642100	23	002335677	(9)
(6)	754000	24	0112556889	(10)
(4)	9500	25	3457789	(7)
(2)	50	26	046	(3)

Key: 2 | 20 | 3 means \$20200for females and \$20300 for males.

Find the median and the quartiles of the females' salaries.
You are given that the median salary of the males is $\$ 24000$, the lower quartile is $\$ 22600$ and the upper quartile is $\$ 25300$.
Draw a pair of box-and-whisker plots in a single diagram on the grid below to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-02_994_1589_1736_310}

CAIE S1 2020 Specimen Q2

4 marks Easy -1.2

2 A summary of the speeds, $x$ kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \text { and } \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of $\Sigma x ^ { 2 }$.

CAIE S1 2020 Specimen Q3

7 marks Moderate -0.5

3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable $X$ represents the number of paperback books she chooses.

Show that the probability that she chooses exactly 2 paperback books is $\frac { 3 } { 14 }$.
Draw up the probability distribution table for $X$.
You are given that $\mathrm { E } ( X ) = 3$. Find $\operatorname { Var } ( X )$.

CAIE S1 2020 Specimen Q4

10 marks Moderate -0.5

4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.

Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
The daily sales at another petrol station are $X$ litres, where $X$ is normally distributed with mean $m$ and standard deviation 560. It is given that $\mathrm { P } ( X > 8000 ) = 0.122$.
Find the value of $m$.
Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.

CAIE S1 2020 Specimen Q5

7 marks Moderate -0.5

5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.

Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
Justify your use of the approximation in part (a).
On another occasion, the same die is thrown repeatedly until a 3 is obtained.
Find the probability that obtaining a 3 requires fewer than 7 throws.

CAIE S1 2020 Specimen Q6

7 marks Standard +0.3

6 A group of 8 friends travels to the airport in two taxis, $P$ and $Q$. Each taxi can take 4 passengers.

The 8 friends divide themselves into two groups of 4, one group for taxi $P$ and one group for taxi $Q$, with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_272_456_242_461} \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_281_455_233_1151} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi $P$ and Jon and Sarah sit in the back of taxi $P$ next to each other.
Find the number of different seating arrangements that are now possible for the 8 friends.

CAIE S1 2020 Specimen Q7

10 marks Standard +0.3

7 Bag $A$ contains 4 balls numbered 2, 4, 5, 8. Bag $B$ contains 5 balls numbered 1, 3, 6, 8, 8. Bag $C$ contains 7 balls numbered $2,7,8,8,8,8,9$. One ball is selected at random from each bag.

Event $X$ is 'exactly two of the selected balls have the same number'.
Event $Y$ is 'the ball selected from bag $A$ has number 4'.
1. Find $\mathrm { P } ( X )$.
2. Find $\mathrm { P } ( X \cap Y )$ and hence determine whether or not events $X$ and $Y$ are independent.
3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.

CAIE Further Paper 4 2020 Specimen Q1

7 marks Moderate -0.5

State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test. The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.
5.62 5.73 6.55 6.81 6.10 5.75 5.87 6.47 5.86 6.26 6.99 5.91
Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a $5\%$ significance level.
[0pt] [6]

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	\(e\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(e\)	\(e\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(p\)	\(p\)	\(q\)	\(e\)	\(s\)	\(t\)	\(r\)
\(q\)	\(q\)	\(e\)	\(p\)	\(t\)	\(r\)	\(s\)
\(r\)	\(r\)	\(t\)	\(s\)	\(e\)	\(q\)	\(p\)
\(s\)	\(s\)	\(r\)	\(t\)	\(p\)	\(e\)	\(q\)
\(t\)	\(t\)	\(s\)	\(r\)	\(q\)	\(p\)	\(e\)

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OCR FP3 2014 June Q6

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