Questions C4 - A-Level Maths

AQA C4 2008 June Q8

8

The number of fish in a lake is decreasing. After $t$ years, there are $x$ fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.
1. Formulate a differential equation, in the variables $x$ and $t$ and a constant of proportionality $k$, where $k > 0$, to model the rate at which the number of fish in the lake is decreasing.
2. At a certain time, there were 20000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of $k$.
The equation $$P = 2000 - A \mathrm { e } ^ { - 0.05 t }$$ is proposed as a model for the number of fish, $P$, in another lake, where $t$ is the time in years and $A$ is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake.
1. Taking 1 January 2008 as $t = 0$, find the value of $A$.
2. Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900.

AQA C4 2009 June Q1

1

Use the Remainder Theorem to find the remainder when $3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5$ is divided by $3 x - 1$.
Express $\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }$ in the form $a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }$, where $a , b$ and $c$ are integers.

AQA C4 2009 June Q2

2 A curve is defined by the parametric equations $$x = \frac { 1 } { t } , \quad y = t + \frac { 1 } { 2 t }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Find an equation of the normal to the curve at the point where $t = 1$.
Show that the cartesian equation of the curve can be written in the form $$x ^ { 2 } - 2 x y + k = 0$$ where $k$ is an integer.

AQA C4 2009 June Q3

3

Find the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
1. Express $\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }$ in the form $\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }$, where $A$ and $B$ are integers.
2. Find the binomial expansion of $\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }$ up to and including the term in $x ^ { 2 }$.
Find the range of values of $x$ for which the binomial expansion of $\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }$ is valid.

AQA C4 2009 June Q4

4 A car depreciates in value according to the model $$V = A k ^ { t }$$ where $\pounds V$ is the value of the car $t$ months from when it was new, and $A$ and $k$ are constants. Its value when new was $\pounds 12499$ and 36 months later its value was $\pounds 7000$.

1. Write down the value of $A$.
2. Show that the value of $k$ is 0.984025 , correct to six decimal places.
The value of this car first dropped below $\pounds 5000$ during the $n$th month from new. Find the value of $n$.

AQA C4 2009 June Q5

5 A curve is defined by the equation $4 x ^ { 2 } + y ^ { 2 } = 4 + 3 x y$.
Find the gradient at the point ( 1,3 ) on this curve.

AQA C4 2009 June Q6

6

1. Show that the equation $3 \cos 2 x + 7 \cos x + 5 = 0$ can be written in the form $a \cos ^ { 2 } x + b \cos x + c = 0$, where $a , b$ and $c$ are integers.
2. Hence find the possible values of $\cos x$.
1. Express $7 \sin \theta + 3 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $\alpha$ is an acute angle. Give your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $7 \sin \theta + 3 \cos \theta = 4$ for all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$, giving $\theta$ to the nearest $0.1 ^ { \circ }$.
1. Given that $\beta$ is an acute angle and that $\tan \beta = 2 \sqrt { 2 }$, show that $\cos \beta = \frac { 1 } { 3 }$.
2. Hence show that $\sin 2 \beta = p \sqrt { 2 }$, where $p$ is a rational number.

AQA C4 2009 June Q7

7 The points $A$ and $B$ have coordinates ( $3 , - 2,5$ ) and ( $4,0,1$ ) respectively. The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 6
- 1
5 \end{array} \right] + \lambda \left[ \begin{array} { r } 2
- 1
4 \end{array} \right]$.

Find the distance between the points $A$ and $B$.
Verify that $B$ lies on $l _ { 1 }$.
(2 marks)
The line $l _ { 2 }$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3
- 2
5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1
3
- 8 \end{array} \right]$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $C$. Show that the points $A , B$ and $C$ form an isosceles triangle.
(6 marks)

AQA C4 2009 June Q8

8

Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ given that $x = 20$ when $t = \frac { \pi } { 4 }$, giving your solution in the form $x ^ { 2 } = \mathrm { f } ( t )$. (6 marks)
The oscillations of a 'baby bouncy cradle' are modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 150 \cos 2 t } { x }$$ where $x \mathrm {~cm}$ is the height of the cradle above its base $t$ seconds after the cradle begins to oscillate. Given that the cradle is 20 cm above its base at time $t = \frac { \pi } { 4 }$ seconds, find:
1. the height of the cradle above its base 13 seconds after it starts oscillating, giving your answer to the nearest millimetre;
2. the time at which the cradle will first be 11 cm above its base, giving your answer to the nearest tenth of a second.
  (2 marks)

OCR C4 Q3

3 The line $L _ { 1 }$ passes through the points $( 2 , - 3,1 )$ and $( - 1 , - 2 , - 4 )$. The line $L _ { 2 }$ passes through the point $( 3,2 , - 9 )$ and is parallel to the vector $4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }$.

Find an equation for $L _ { 1 }$ in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.
Prove that $L _ { 1 }$ and $L _ { 2 }$ are skew.

OCR C4 Q4

4

Show that the substitution $x = \tan \theta$ transforms $\int \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$ to $\int \cos ^ { 2 } \theta \mathrm {~d} \theta$.
Hence find the exact value of $\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$.
$5 A B C D$ is a parallelogram. The position vectors of $A , B$ and $C$ are given respectively by $$\mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { c } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } .$$
Find the position vector of $D$.
Determine, to the nearest degree, the angle $A B C$. 6 The equation of a curve is $x y ^ { 2 } = 2 x + 3 y$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - y ^ { 2 } } { 2 x y - 3 }$.
Show that the curve has no tangents which are parallel to the $y$-axis. 7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent at the point $P \left( 4 , - \frac { 1 } { 2 } \right)$ is $$x - 16 y = 12$$
Find the value of the parameter at the point where the tangent at $P$ meets the curve again. June 2005
8
Given that $\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { 1 + x } + \frac { B } { 2 + x } + \frac { C } { ( 2 + x ) ^ { 2 } }$, find $A , B$ and $C$.
Hence or otherwise expand $\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
State the set of values of $x$ for which the expansion in part (ii) is valid. 9 Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of $20 ^ { \circ } \mathrm { C }$. At time $t$ minutes later, the temperature of the liquid is $\theta ^ { \circ } \mathrm { C }$.
Explain how the information above leads to the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$ where $k$ is a positive constant.
The liquid is initially at a temperature of $100 ^ { \circ } \mathrm { C }$. It takes 5 minutes for the liquid to cool from $100 ^ { \circ } \mathrm { C }$ to $68 ^ { \circ } \mathrm { C }$. Show that $$\theta = 20 + 80 \mathrm { e } ^ { - \left( \frac { 1 } { 5 } \ln \frac { 5 } { 3 } \right) t }$$
Calculate how much longer it takes for the liquid to cool by a further $32 ^ { \circ } \mathrm { C }$. 1 Simplify $\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }$. 2 Given that $\sin y = x y + x ^ { 2 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. 3
Find the quotient and the remainder when $3 x ^ { 3 } - 2 x ^ { 2 } + x + 7$ is divided by $x ^ { 2 } - 2 x + 5$.
Hence, or otherwise, determine the values of the constants $a$ and $b$ such that, when $3 x ^ { 3 } - 2 x ^ { 2 } + a x + b$ is divided by $x ^ { 2 } - 2 x + 5$, there is no remainder. 4
Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.
Hence find $\int x \tan ^ { 2 } x \mathrm {~d} x$.

OCR C4 Q7

7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent at the point $P \left( 4 , - \frac { 1 } { 2 } \right)$ is $$x - 16 y = 12$$
Find the value of the parameter at the point where the tangent at $P$ meets the curve again. June 2005

OCR C4 Q9

9 Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of $20 ^ { \circ } \mathrm { C }$. At time $t$ minutes later, the temperature of the liquid is $\theta ^ { \circ } \mathrm { C }$.

Explain how the information above leads to the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$ where $k$ is a positive constant.
The liquid is initially at a temperature of $100 ^ { \circ } \mathrm { C }$. It takes 5 minutes for the liquid to cool from $100 ^ { \circ } \mathrm { C }$ to $68 ^ { \circ } \mathrm { C }$. Show that $$\theta = 20 + 80 \mathrm { e } ^ { - \left( \frac { 1 } { 5 } \ln \frac { 5 } { 3 } \right) t }$$
Calculate how much longer it takes for the liquid to cool by a further $32 ^ { \circ } \mathrm { C }$. 1 Simplify $\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }$. 2 Given that $\sin y = x y + x ^ { 2 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. 3
Find the quotient and the remainder when $3 x ^ { 3 } - 2 x ^ { 2 } + x + 7$ is divided by $x ^ { 2 } - 2 x + 5$.
Hence, or otherwise, determine the values of the constants $a$ and $b$ such that, when $3 x ^ { 3 } - 2 x ^ { 2 } + a x + b$ is divided by $x ^ { 2 } - 2 x + 5$, there is no remainder. 4
Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.
Hence find $\int x \tan ^ { 2 } x \mathrm {~d} x$. 5 A curve is given parametrically by the equations $x = t ^ { 2 } , y = 2 t$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent to the curve at $\left( p ^ { 2 } , 2 p \right)$ is $$p y = x + p ^ { 2 } .$$
Find the coordinates of the point where the tangent at $( 9,6 )$ meets the tangent at $( 25 , - 10 )$. 6
Show that the substitution $x = \sin ^ { 2 } \theta$ transforms $\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x$ to $\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta$.
Hence find $\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x$. 7 The expression $\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }$ is denoted by $\mathrm { f } ( x )$.
Express $\mathrm { f } ( x )$ in the form $\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }$, where $A , B$ and $C$ are constants.
Given that $| x | < 1$, find the first 3 terms in the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$. 8
Solve the differential equation $$\frac { d y } { d x } = \frac { 2 - x } { y - 3 }$$ giving the particular solution that satisfies the condition $y = 4$ when $x = 5$.
Show that this particular solution can be expressed in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where the values of the constants $a , b$ and $k$ are to be stated.
Hence sketch the graph of the particular solution, indicating clearly its main features. 9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4
2
- 6 \end{array} \right) + t \left( \begin{array} { r } - 8
1
- 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2
a
- 2 \end{array} \right) + s \left( \begin{array} { r } - 9
2
- 5 \end{array} \right) ,$$ where $a$ is a constant.
Calculate the acute angle between the lines.
Given that these two lines intersect, find $a$ and the point of intersection. \section*{June 2006} 1 Find the gradient of the curve $4 x ^ { 2 } + 2 x y + y ^ { 2 } = 12$ at the point $( 1,2 )$. 2
Expand $( 1 - 3 x ) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
Find the coefficient of $x ^ { 2 }$ in the expansion of $\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }$ in ascending powers of $x$. 3
Express $\frac { 3 - 2 x } { x ( 3 - x ) }$ in partial fractions.
Show that $\int _ { 1 } ^ { 2 } \frac { 3 - 2 x } { x ( 3 - x ) } \mathrm { d } x = 0$.
What does the result of part (ii) indicate about the graph of $y = \frac { 3 - 2 x } { x ( 3 - x ) }$ between $x = 1$ and $x = 2$ ? 4 The position vectors of three points $A , B$ and $C$ relative to an origin $O$ are given respectively by and $$\begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k } ,
& \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }
& \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } . \end{aligned}$$
Find the angle between $A B$ and $A C$.
Find the area of triangle $A B C$. 5 A forest is burning so that, $t$ hours after the start of the fire, the area burnt is $A$ hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to $A ^ { 2 }$.
Write down a differential equation which models this situation.
After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt. 6
Show that the substitution $u = \mathrm { e } ^ { x } + 1$ transforms $\int \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x$ to $\int \frac { u - 1 } { u } \mathrm {~d} u$.
Hence show that $\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x = \mathrm { e } - 1 - \ln \left( \frac { \mathrm { e } + 1 } { 2 } \right)$. \section*{June 2006} 7 Two lines have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } + \lambda ( 3 \mathbf { i } + \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) ,$$ where $a$ is a constant.
Given that the lines are skew, find the value that $a$ cannot take.
Given instead that the lines intersect, find the point of intersection. 8
Show that $\int \cos ^ { 2 } 6 x \mathrm {~d} x = \frac { 1 } { 2 } x + \frac { 1 } { 24 } \sin 12 x + c$.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } x \cos ^ { 2 } 6 x \mathrm {~d} x$. 9 A curve is given parametrically by the equations $$x = 4 \cos t , \quad y = 3 \sin t$$ where $0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Show that the equation of the tangent at the point $P$, where $t = p$, is $$3 x \cos p + 4 y \sin p = 12$$
The tangent at $P$ meets the $x$-axis at $R$ and the $y$-axis at $S$. $O$ is the origin. Show that the area of triangle $O R S$ is $\frac { 12 } { \sin 2 p }$.
Write down the least possible value of the area of triangle $O R S$, and give the corresponding value of $p$. Jan 2007
1 It is given that $$f ( x ) = \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - 4 x } \quad \text { for } x \neq 0 , x \neq 4$$ Express $\mathrm { f } ( x )$ in its simplest form. 2 Find the exact value of $\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x$. 3 The points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to an origin $O$, where $\mathbf { a } = 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$ and $\mathbf { b } = - 7 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k }$.
Find the length of $A B$.
Use a scalar product to find angle $O A B$. 4 Use the substitution $u = 2 x - 5$ to show that $\int _ { \frac { 5 } { 2 } } ^ { 3 } ( 4 x - 8 ) ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x = \frac { 17 } { 72 }$.
Expand $( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 - 3 \left( x + x ^ { 3 } \right) \right) ^ { - \frac { 1 } { 3 } }$. 6
Express $\frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } }$ in the form $\frac { A } { x - 3 } + \frac { B } { ( x - 3 ) ^ { 2 } }$, where $A$ and $B$ are constants.
Hence find the exact value of $\int _ { 4 } ^ { 10 } \frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x$, giving your answer in the form $a + b \ln c$, where $a , b$ and $c$ are integers. 7 The equation of a curve is $2 x ^ { 2 } + x y + y ^ { 2 } = 14$. Show that there are two stationary points on the curve and find their coordinates. 8 The parametric equations of a curve are $x = 2 t ^ { 2 } , y = 4 t$. Two points on the curve are $P \left( 2 p ^ { 2 } , 4 p \right)$ and $Q \left( 2 q ^ { 2 } , 4 q \right)$.
Show that the gradient of the normal to the curve at $P$ is $- p$.
Show that the gradient of the chord joining the points $P$ and $Q$ is $\frac { 2 } { p + q }$.
The chord $P Q$ is the normal to the curve at $P$. Show that $p ^ { 2 } + p q + 2 = 0$.
The normal at the point $R ( 8,8 )$ meets the curve again at $S$. The normal at $S$ meets the curve again at $T$. Find the coordinates of $T$. 9
Find the general solution of the differential equation $$\frac { \sec ^ { 2 } y } { \cos ^ { 2 } ( 2 x ) } \frac { d y } { d x } = 2$$
For the particular solution in which $y = \frac { 1 } { 4 } \pi$ when $x = 0$, find the value of $y$ when $x = \frac { 1 } { 6 } \pi$.

OCR C4 Q10

3 marks

10 The position vectors of the points $P$ and $Q$ with respect to an origin $O$ are $5 \mathbf { i } + 2 \mathbf { j } - 9 \mathbf { k }$ and $4 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k }$ respectively.

Find a vector equation for the line $P Q$. The position vector of the point $T$ is $\mathbf { i } + 2 \mathbf { j } - \mathbf { k }$.
Write down a vector equation for the line $O T$ and show that $O T$ is perpendicular to $P Q$. It is given that $O T$ intersects $P Q$.
Find the position vector of the point of intersection of $O T$ and $P Q$.
Hence find the perpendicular distance from $O$ to $P Q$, giving your answer in an exact form. 1 The equation of a curve is $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }$.
Express $\mathrm { f } ( x )$ in partial fractions.
Hence find $\mathrm { f } ^ { \prime } ( x )$ and deduce that the gradient of the curve is negative at all points on the curve. 2 Find the exact value of $\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$. 3 Find the exact volume generated when the region enclosed between the $x$-axis and the portion of the curve $y = \sin x$ between $x = 0$ and $x = \pi$ is rotated completely about the $x$-axis. 4
Expand $( 2 + x ) ^ { - 2 }$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, and state the set of values of $x$ for which the expansion is valid.
Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }$. 5 A curve $C$ has parametric equations $$x = \cos t , \quad y = 3 + 2 \cos 2 t , \quad \text { where } 0 \leqslant t \leqslant \pi$$
Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$ and hence show that the gradient at any point on $C$ cannot exceed 8 .
Show that all points on $C$ satisfy the cartesian equation $y = 4 x ^ { 2 } + 1$.
Sketch the curve $y = 4 x ^ { 2 } + 1$ and indicate on your sketch the part which represents $C$. 6 The equation of a curve is $x ^ { 2 } + 3 x y + 4 y ^ { 2 } = 58$. Find the equation of the normal at the point $( 2,3 )$ on the curve, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. 7
Find the quotient and the remainder when $2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12$ is divided by $x ^ { 2 } + 4$.
Hence express $\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }$ in the form $A x + B + \frac { C x + D } { x ^ { 2 } + 4 }$, where the values of the constants $A , B , C$ and $D$ are to be stated.
Use the result of part (ii) to find the exact value of $\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x$. \section*{June 2007} 8 The height, $h$ metres, of a shrub $t$ years after planting is given by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 6 - h } { 20 }$$ A shrub is planted when its height is 1 m .
Show by integration that $t = 20 \ln \left( \frac { 5 } { 6 - h } \right)$.
How long after planting will the shrub reach a height of 2 m ?
Find the height of the shrub 10 years after planting.
State the maximum possible height of the shrub. 9 Lines $L _ { 1 } , L _ { 2 }$ and $L _ { 3 }$ have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) ,
& L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) ,
& L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$
Calculate the acute angle between $L _ { 1 }$ and $L _ { 2 }$.
Given that $L _ { 1 }$ and $L _ { 3 }$ are parallel, find the value of $c$.
Given instead that $L _ { 2 }$ and $L _ { 3 }$ intersect, find the value of $c$. 1 Find the angle between the vectors $\mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }$ and $2 \mathbf { i } + \mathbf { j } + \mathbf { k }$. 2
Express $\frac { x } { ( x + 1 ) ( x + 2 ) }$ in partial fractions.
Hence find $\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x$. 3 When $x ^ { 4 } - 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x + a$ is divided by $x ^ { 2 } + 2 x - 1$, the quotient is $x ^ { 2 } + b x + 2$ and the remainder is $c x + 7$. Find the values of the constants $a , b$ and $c$. 4 Find the equation of the normal to the curve $$x ^ { 3 } + 4 x ^ { 2 } y + y ^ { 3 } = 6$$ at the point $( 1,1 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. 5 The vector equations of two lines are $$\mathbf { r } = ( 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) + s ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = ( 2 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) + t ( 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k } )$$ Prove that the two lines are
perpendicular,
skew. 6
Expand $( 1 + a x ) ^ { - 4 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
The coefficients of $x$ and $x ^ { 2 }$ in the expansion of $( 1 + b x ) ( 1 + a x ) ^ { - 4 }$ are 1 and - 2 respectively. Given that $a > 0$, find the values of $a$ and $b$. 7
Given that $$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$ find the values of the constants $A$ and $B$.
Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } d \theta$$ giving your answer in the form $a \pi - \ln b$. 8 Water flows out of a tank through a hole in the bottom and, at time $t$ minutes, the depth of water in the tank is $x$ metres. At any instant, the rate at which the depth of water in the tank is decreasing is proportional to the square root of the depth of water in the tank.
Write down a differential equation which models this situation.
When $t = 0 , x = 2$; when $t = 5 , x = 1$. Find $t$ when $x = 0.5$, giving your answer correct to 1 decimal place. 9 The parametric equations of a curve are $x = t ^ { 3 } , y = t ^ { 2 }$.
Show that the equation of the tangent at the point $P$ where $t = p$ is $$3 p y - 2 x = p ^ { 3 }$$
Given that this tangent passes through the point ( $- 10,7$ ), find the coordinates of each of the three possible positions of $P$. 10
Use the substitution $x = \sin \theta$ to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
Find the exact value of $$\int _ { 1 } ^ { 3 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ \section*{June 2008} 1
(a) Simplify $\frac { \left( 2 x ^ { 2 } - 7 x - 4 \right) ( x + 1 ) } { \left( 3 x ^ { 2 } + x - 2 \right) ( x - 4 ) }$.
(b) Find the quotient and remainder when $x ^ { 3 } + 2 x ^ { 2 } - 6 x - 5$ is divided by $x ^ { 2 } + 4 x + 1$. 2 Find the exact value of $\int _ { 1 } ^ { \mathrm { e } } x ^ { 4 } \ln x \mathrm {~d} x$. 3 The equation of a curve is $x ^ { 2 } y - x y ^ { 2 } = 2$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - 2 x y } { x ^ { 2 } - 2 x y }$.
(a) Show that, if $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$, then $y = 2 x$.
(b) Hence find the coordinates of the point on the curve where the tangent is parallel to the $x$-axis. 4 Relative to an origin $O$, the points $A$ and $B$ have position vectors $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ and $\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ respectively.
Find a vector equation of the line passing through $A$ and $B$.
Find the position vector of the point $P$ on $A B$ such that $O P$ is perpendicular to $A B$. 5
Show that $\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }$, for $| x | < 1$.
By taking $x = \frac { 2 } { 7 }$, show that $\sqrt { 5 } \approx \frac { 111 } { 49 }$. 6 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 1
0
- 5 \end{array} \right) + t \left( \begin{array} { l } 2
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 12
0
5 \end{array} \right) + s \left( \begin{array} { r } 1
- 4
- 2 \end{array} \right) .$$
Show that the lines intersect.
Find the angle between the lines. 7
Show that, if $y = \operatorname { cosec } x$, then $\frac { \mathrm { d } y } { \mathrm {~d} x }$ can be expressed as $- \operatorname { cosec } x \cot x$.
Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that $x = \frac { 1 } { 6 } \pi$ when $t = \frac { 1 } { 2 } \pi$. 8
Given that $\frac { 2 t } { ( t + 1 ) ^ { 2 } }$ can be expressed in the form $\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }$, find the values of the constants $A$ and $B$.
Show that the substitution $t = \sqrt { 2 x - 1 }$ transforms $\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x$ to $\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t$.
Hence find the exact value of $\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x$. 9 The parametric equations of a curve are $$x = 2 \theta + \sin 2 \theta , \quad y = 4 \sin \theta ,$$ and part of its graph is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{0eb9fe7c-7fc2-48ed-b08b-d8ff1c26f4e3-15_630_1131_1059_507}
Find the value of $\theta$ at $A$ and the value of $\theta$ at $B$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec \theta$.
At the point $C$ on the curve, the gradient is 2 . Find the coordinates of $C$, giving your answer in an exact form. 1 Simplify $\frac { 20 - 5 x } { 6 x ^ { 2 } - 24 x }$. 2 Find $\int x \sec ^ { 2 } x \mathrm {~d} x$. 3
Expand $( 1 + 2 x ) ^ { \frac { 1 } { 2 } }$ as a series in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Hence find the expansion of $\frac { ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } } { ( 1 + x ) ^ { 3 } }$ as a series in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
State the set of values of $x$ for which the expansion in part (ii) is valid. 4 Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 1 + \sin x ) ^ { 2 } \mathrm {~d} x$. 5
Show that the substitution $u = \sqrt { x }$ transforms $\int \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x$ to $\int \frac { 2 } { u ( 1 + u ) } \mathrm { d } u$.
Hence find the exact value of $\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x$. 6 A curve has parametric equations $$x = t ^ { 2 } - 6 t + 4 , \quad y = t - 3$$ Find
the coordinates of the point where the curve meets the $x$-axis,
the equation of the curve in cartesian form, giving your answer in a simple form without brackets,
the equation of the tangent to the curve at the point where $t = 2$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. 7
Show that the straight line with equation $\mathbf { r } = \left( \begin{array} { r } 2
- 3
5 \end{array} \right) + t \left( \begin{array} { r } 1
4
- 2 \end{array} \right)$ meets the line passing through ( $9,7,5$ ) and ( $7,8,2$ ), and find the point of intersection of these lines.
Find the acute angle between these lines. Jan 2009
8 The equation of a curve is $x ^ { 3 } + y ^ { 3 } = 6 x y$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Show that the point $\left( 2 ^ { \frac { 4 } { 3 } } , 2 ^ { \frac { 5 } { 3 } } \right)$ lies on the curve and that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ at this point.
The point $( a , a )$, where $a > 0$, lies on the curve. Find the value of $a$ and the gradient of the curve at this point. 9 A liquid is being heated in an oven maintained at a constant temperature of $160 ^ { \circ } \mathrm { C }$. It may be assumed that the rate of increase of the temperature of the liquid at any particular time $t$ minutes is proportional to $160 - \theta$, where $\theta ^ { \circ } \mathrm { C }$ is the temperature of the liquid at that time.
Write down a differential equation connecting $\theta$ and $t$. When the liquid was placed in the oven, its temperature was $20 ^ { \circ } \mathrm { C }$ and 5 minutes later its temperature had risen to $65 ^ { \circ } \mathrm { C }$.
Find the temperature of the liquid, correct to the nearest degree, after another 5 minutes. June 2009
1 Find the quotient and the remainder when $3 x ^ { 4 } - x ^ { 3 } - 3 x ^ { 2 } - 14 x - 8$ is divided by $x ^ { 2 } + x + 2$. 2 Use the substitution $x = \tan \theta$ to find the exact value of $$\int _ { 1 } ^ { \sqrt { 3 } } \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \mathrm {~d} x$$ 3
Expand $( a + x ) ^ { - 2 }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
When $( 1 - x ) ( a + x ) ^ { - 2 }$ is expanded, the coefficient of $x ^ { 2 }$ is 0 . Find the value of $a$. 4
Differentiate $\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )$, simplifying your answer.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x$. 5 A curve has parametric equations $$x = 2 t + t ^ { 2 } , \quad y = 2 t ^ { 2 } + t ^ { 3 }$$
Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$ and find the gradient of the curve at the point $( 3 , - 9 )$.
By considering $\frac { y } { x }$, find a cartesian equation of the curve, giving your answer in a form not involving fractions. 6 The expression $\frac { 4 x } { ( x - 5 ) ( x - 3 ) ^ { 2 } }$ is denoted by $\mathrm { f } ( x )$.
Express $\mathrm { f } ( x )$ in the form $\frac { A } { x - 5 } + \frac { B } { x - 3 } + \frac { C } { ( x - 3 ) ^ { 2 } }$, where $A , B$ and $C$ are constants.
Hence find the exact value of $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$. 7
The vector $\mathbf { u } = \frac { 3 } { 13 } \mathbf { i } + b \mathbf { j } + c \mathbf { k }$ is perpendicular to the vector $4 \mathbf { i } + \mathbf { k }$ and to the vector $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$. Find the values of $b$ and $c$, and show that $\mathbf { u }$ is a unit vector.
Calculate, to the nearest degree, the angle between the vectors $4 \mathbf { i } + \mathbf { k }$ and $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$. June 2009
8
Given that $14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 28 x - 7 y } { 7 x - 2 y }$.
The points $L$ and $M$ on the curve $14 x ^ { 2 } - 7 x y + y ^ { 2 } = 2$ each have $x$-coordinate 1 . The tangents to the curve at $L$ and $M$ meet at $N$. Find the coordinates of $N$. 9 A tank contains water which is heated by an electric water heater working under the action of a thermostat. The temperature of the water, $\theta ^ { \circ } \mathrm { C }$, may be modelled as follows. When the water heater is first switched on, $\theta = 40$. The heater causes the temperature to increase at a rate $k _ { 1 } { } ^ { \circ } \mathrm { C }$ per second, where $k _ { 1 }$ is a constant, until $\theta = 60$. The heater then switches off.
Write down, in terms of $k _ { 1 }$, how long it takes for the temperature to increase from $40 ^ { \circ } \mathrm { C }$ to $60 ^ { \circ } \mathrm { C }$. The temperature of the water then immediately starts to decrease at a variable rate $k _ { 2 } ( \theta - 20 ) ^ { \circ } \mathrm { C }$ per second, where $k _ { 2 }$ is a constant, until $\theta = 40$.
Write down a differential equation to represent the situation as the temperature is decreasing.
Find the total length of time for the temperature to increase from $40 ^ { \circ } \mathrm { C }$ to $60 ^ { \circ } \mathrm { C }$ and then decrease to $40 ^ { \circ } \mathrm { C }$. Give your answer in terms of $k _ { 1 }$ and $k _ { 2 }$. 1 Find the quotient and the remainder when $x ^ { 4 } + 11 x ^ { 3 } + 28 x ^ { 2 } + 3 x + 1$ is divided by $x ^ { 2 } + 5 x + 2$. 2 Points $A , B$ and $C$ have position vectors $- 5 \mathbf { i } - 10 \mathbf { j } + 12 \mathbf { k } , \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }$ and $3 \mathbf { i } + 6 \mathbf { j } + p \mathbf { k }$ respectively, where $p$ is a constant.
Given that angle $A B C = 90 ^ { \circ }$, find the value of $p$.
Given instead that $A B C$ is a straight line, find the value of $p$. 3 By expressing $\cos 2 x$ in terms of $\cos x$, find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \cos 2 x } { \cos ^ { 2 } x } \mathrm {~d} x$. 4 Use the substitution $u = 2 + \ln t$ to find the exact value of $$\int _ { 1 } ^ { \mathrm { e } } \frac { 1 } { t ( 2 + \ln t ) ^ { 2 } } \mathrm {~d} t$$ 5
Expand $( 1 + x ) ^ { \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
(a) Hence, or otherwise, expand $( 8 + 16 x ) ^ { \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
(b) State the set of values of $x$ for which the expansion in part (ii) (a) is valid. 6 A curve has parametric equations $$x = 9 t - \ln ( 9 t ) , \quad y = t ^ { 3 } - \ln \left( t ^ { 3 } \right)$$ Show that there is only one value of $t$ for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3$ and state that value. 7 Find the equation of the normal to the curve $x ^ { 3 } + 2 x ^ { 2 } y = y ^ { 3 } + 15$ at the point $( 2,1 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. 8
State the derivative of $\mathrm { e } ^ { \cos x }$.
Hence use integration by parts to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$ Jan 2010
9 The equation of a straight line $l$ is $\mathbf { r } = \left( \begin{array} { l } 3
1
1 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
2 \end{array} \right) . O$ is the origin.
The point $P$ on $l$ is given by $t = 1$. Calculate the acute angle between $O P$ and $l$.
Find the position vector of the point $Q$ on $l$ such that $O Q$ is perpendicular to $l$.
Find the length of $O Q$. 10
Express $\frac { 1 } { ( 3 - x ) ( 6 - x ) }$ in partial fractions.
In a chemical reaction, the amount $x$ grams of a substance at time $t$ seconds is related to the rate at which $x$ is changing by the equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 3 - x ) ( 6 - x )$$ where $k$ is a constant. When $t = 0 , x = 0$ and when $t = 1 , x = 1$.
(a) Show that $k = \frac { 1 } { 3 } \ln \frac { 5 } { 4 }$.
(b) Find the value of $x$ when $t = 2$. 1 Expand $( 1 + 3 x ) ^ { - \frac { 5 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. 2 Given that $y = \frac { \cos x } { 1 - \sin x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, simplifying your answer. 3 Express $\frac { x ^ { 2 } } { ( x - 1 ) ^ { 2 } ( x - 2 ) }$ in partial fractions. 4 Use the substitution $u = \sqrt { x + 2 }$ to find the exact value of $$\int _ { - 1 } ^ { 7 } \frac { x ^ { 2 } } { \sqrt { x + 2 } } \mathrm {~d} x$$ 5 Find the coordinates of the two stationary points on the curve with equation $$x ^ { 2 } + 4 x y + 2 y ^ { 2 } + 18 = 0$$ 6 Lines $l _ { 1 }$ and $l _ { 2 }$ have vector equations $$\mathbf { r } = \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + a \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - \mathbf { k } + s ( 2 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } )$$ respectively, where $t$ and $s$ are parameters and $a$ is a constant.
Given that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular, find the value of $a$.
Given instead that $l _ { 1 }$ and $l _ { 2 }$ intersect, find
(a) the value of $a$,
(b) the angle between the lines. 7 The parametric equations of a curve are $x = \frac { t + 2 } { t + 1 } , y = \frac { 2 } { t + 3 }$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 0$.
Find the cartesian equation of the curve, giving your answer in a form not involving fractions. 8
Find the quotient and the remainder when $x ^ { 2 } - 5 x + 6$ is divided by $x - 1$.
(a) Find the general solution of the differential equation $$\left( \frac { x - 1 } { x ^ { 2 } - 5 x + 6 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y - 5 .$$ (b) Given that $y = 7$ when $x = 8$, find $y$ when $x = 6$. 9
Find $\int ( x + \cos 2 x ) ^ { 2 } \mathrm {~d} x$.
\includegraphics[max width=\textwidth, alt={}, center]{0eb9fe7c-7fc2-48ed-b08b-d8ff1c26f4e3-23_547_940_376_644} The diagram shows the part of the curve $y = x + \cos 2 x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. The shaded region bounded by the curve, the axes and the line $x = \frac { 1 } { 2 } \pi$ is rotated completely about the $x$-axis to form a solid of revolution of volume $V$. Find $V$, giving your answer in an exact form. 1
Expand $( 1 - x ) ^ { \frac { 1 } { 2 } }$ in ascending powers of $x$ as far as the term in $x ^ { 2 }$.
Hence expand $\left( 1 - 2 y + 4 y ^ { 2 } \right) ^ { \frac { 1 } { 2 } }$ in ascending powers of $y$ as far as the term in $y ^ { 2 }$. 2
Express $\frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } }$ in the form $\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }$, where $A$ and $B$ are constants.
Hence find the exact value of $\int _ { 4 } ^ { 5 } \frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x$. 3
Show that the derivative of $\sec x$ can be written as $\sec x \tan x$.
Find $\int \frac { \tan x } { \sqrt { 1 + \cos 2 x } } \mathrm {~d} x$. 4 A curve has parametric equations $$x = 2 + t ^ { 2 } , \quad y = 4 t$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Find the equation of the normal at the point where $t = 4$, giving your answer in the form $y = m x + c$.
Find a cartesian equation of the curve. 5 In this question, $I$ denotes the definite integral $\int _ { 2 } ^ { 5 } \frac { 5 - x } { 2 + \sqrt { x - 1 } } \mathrm {~d} x$. The value of $I$ is to be found using two different methods.
Show that the substitution $u = \sqrt { x - 1 }$ transforms $I$ to $\int _ { 1 } ^ { 2 } \left( 4 u - 2 u ^ { 2 } \right) \mathrm { d } u$ and hence find the exact value of $I$.
(a) Simplify $( 2 + \sqrt { x - 1 } ) ( 2 - \sqrt { x - 1 } )$.
(b) By first multiplying the numerator and denominator of $\frac { 5 - x } { 2 + \sqrt { x - 1 } }$ by $2 - \sqrt { x - 1 }$, find the exact value of $I$. Jan 2011
$\mathbf { 6 }$ The line $l _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 3
0
- 2 \end{array} \right) + s \left( \begin{array} { r } 2
3
- 4 \end{array} \right)$. The line $l _ { 2 }$ has equation $\mathbf { r } = \left( \begin{array} { l } 5
3
2 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 2 \end{array} \right)$.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$.
Show that $l _ { 1 }$ and $l _ { 2 }$ are skew.
One of the numbers in the equation of line $l _ { 1 }$ is changed so that the equation becomes $\mathbf { r } = \left( \begin{array} { l } 3
0
a \end{array} \right) + s \left( \begin{array} { r } 2
3
- 4 \end{array} \right)$. Given that $l _ { 1 }$ and $l _ { 2 }$ now intersect, find $a$. 7 Show that $\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10$. 8 The points $P$ and $Q$ lie on the curve with equation $$2 x ^ { 2 } - 5 x y + y ^ { 2 } + 9 = 0$$ The tangents to the curve at $P$ and $Q$ are parallel, each having gradient $\frac { 3 } { 8 }$.
Show that the $x$ - and $y$-coordinates of $P$ and $Q$ are such that $x = 2 y$.
Hence find the coordinates of $P$ and $Q$. 9 Paraffin is stored in a tank with a horizontal base. At time $t$ minutes, the depth of paraffin in the tank is $x \mathrm {~cm}$. When $t = 0 , x = 72$. There is a tap in the side of the tank through which the paraffin can flow. When the tap is opened, the flow of the paraffin is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - 4 ( x - 8 ) ^ { \frac { 1 } { 3 } }$$
How long does it take for the level of paraffin to fall from a depth of 72 cm to a depth of 35 cm ?
The tank is filled again to its original depth of 72 cm of paraffin and the tap is then opened. The paraffin flows out until it stops. How long does this take? June 2011
1 Simplify $\frac { x ^ { 4 } - 10 x ^ { 2 } + 9 } { \left( x ^ { 2 } - 2 x - 3 \right) \left( x ^ { 2 } + 8 x + 15 \right) }$. 2 Find the unit vector in the direction of $\left( \begin{array} { c } 2
- 3
\sqrt { 12 } \end{array} \right)$. 3
Find the quotient when $3 x ^ { 3 } - x ^ { 2 } + 10 x - 3$ is divided by $x ^ { 2 } + 3$, and show that the remainder is $x$.
Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } - x ^ { 2 } + 10 x - 3 } { x ^ { 2 } + 3 } d x$$ 4 Use the substitution $x = \frac { 1 } { 3 } \sin \theta$ to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } } \frac { 1 } { \left( 1 - 9 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ 5 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\mathbf { r } = \left( \begin{array} { l } 4
6
4 \end{array} \right) + s \left( \begin{array} { l } 3
2
1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 1
0
0 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively.
Show that $l _ { 1 }$ and $l _ { 2 }$ are skew.
Find the acute angle between $l _ { 1 }$ and $l _ { 2 }$.
The point $A$ lies on $l _ { 1 }$ and $O A$ is perpendicular to $l _ { 1 }$, where $O$ is the origin. Find the position vector of $A$. 6 Find the coefficient of $x ^ { 2 }$ in the expansion in ascending powers of $x$ of $$\sqrt { \frac { 1 + a x } { 4 - x } } ,$$ giving your answer in terms of $a$. 7 The gradient of a curve at the point $( x , y )$, where $x > - 2$, is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 y ^ { 2 } ( x + 2 ) } .$$ The points $( 1,2 )$ and $( q , 1.5 )$ lie on the curve. Find the value of $q$, giving your answer correct to 3 significant figures. \section*{June 2011} 8 A curve has parametric equations $$x = \frac { 1 } { t + 1 } , \quad y = t - 1$$ The line $y = 3 x$ intersects the curve at two points.
Show that the value of $t$ at one of these points is - 2 and find the value of $t$ at the other point.
Find the equation of the normal to the curve at the point for which $t = - 2$.
Find the value of $t$ at the point where this normal meets the curve again.
Find a cartesian equation of the curve, giving your answer in the form $y = \mathrm { f } ( x )$. 9
Show that $\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x - x ) = \ln x$.
\includegraphics[max width=\textwidth, alt={}, center]{0eb9fe7c-7fc2-48ed-b08b-d8ff1c26f4e3-27_485_727_1062_751} In the diagram, $C$ is the curve $y = \ln x$. The region $R$ is bounded by $C$, the $x$-axis and the line $x = \mathrm { e }$.
(a) Find the exact volume of the solid of revolution formed by rotating $R$ completely about the $x$-axis.
(b) The region $R$ is rotated completely about the $y$-axis. Explain why the volume of the solid of revolution formed is given by $$\pi \mathrm { e } ^ { 2 } - \pi \int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 y } \mathrm {~d} y$$ and find this volume. 1 When the polynomial $\mathrm { f } ( x )$ is divided by $x ^ { 2 } + 1$, the quotient is $x ^ { 2 } + 4 x + 2$ and the remainder is $x - 1$. Find $\mathrm { f } ( x )$, simplifying your answer. 2
Find, in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$, an equation of the line $l$ through the points ( $4,2,7$ ) and ( $5 , - 4 , - 1$ ).
Find the acute angle between the line $l$ and a line in the direction of the vector $\left( \begin{array} { l } 1
2
3 \end{array} \right)$. 3 The equation of a curve $C$ is $( x + 3 ) ( y + 4 ) = x ^ { 2 } + y ^ { 2 }$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
The line $2 y = x + 3$ meets $C$ at two points. What can be said about the tangents to $C$ at these points? Justify your answer.
Find the equation of the tangent at the point ( 6,0 ), giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers. 4
Expand $( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
The term of lowest degree in the expansion of $$( 1 + a x ) \left( 1 + b x ^ { 2 } \right) ^ { 7 } - ( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$$ in ascending powers of $x$ is the term in $x ^ { 3 }$. Find the values of the constants $a$ and $b$. 5 Use the substitution $u = \cos x$ to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$ 6
\includegraphics[max width=\textwidth, alt={}, center]{0eb9fe7c-7fc2-48ed-b08b-d8ff1c26f4e3-29_606_848_251_612} The diagram shows the curves $y = \cos x$ and $y = \sin x$, for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. The region $R$ is bounded by the curves and the $x$-axis. Find the volume of the solid of revolution formed when $R$ is rotated completely about the $x$-axis, giving your answer in terms of $\pi$. 7 The equation of a straight line $l$ is $$\mathbf { r } = \left( \begin{array} { l } 1
0
2 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
0 \end{array} \right) .$$ $O$ is the origin.
Find the position vector of the point $P$ on $l$ such that $O P$ is perpendicular to $l$.
A point $Q$ on $l$ is such that the length of $O Q$ is 3 units. Find the two possible position vectors of $Q$. [3] 8 A curve is defined by the parametric equations $$x = \sin ^ { 2 } \theta , \quad y = 4 \sin \theta - \sin ^ { 3 } \theta ,$$ where $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 - 3 \sin ^ { 2 } \theta } { 2 \sin \theta }$.
Find the coordinates of the point on the curve at which the gradient is 2 .
Show that the curve has no stationary points.
Find a cartesian equation of the curve, giving your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$. 9 Find the exact value of $\int _ { 0 } ^ { 1 } \left( x ^ { 2 } + 1 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x$. 10
Write down the derivative of $\sqrt { y ^ { 2 } + 1 }$ with respect to $y$.
Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( x - 1 ) \sqrt { y ^ { 2 } + 1 } } { x y }$ and that $y = \sqrt { \mathrm { e } ^ { 2 } - 2 \mathrm { e } }$ when $x = \mathrm { e }$,
find a relationship between $x$ and $y$.

OCR MEI C4 2006 January Q1

1 Solve the equation $\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3$.

OCR MEI C4 2006 January Q2

2 A curve is defined parametrically by the equations $$x = t - \ln t , \quad y = t + \ln t \quad ( t > 0 )$$ Find the gradient of the curve at the point where $t = 2$.

OCR MEI C4 2006 January Q3

3 A triangle ABC has vertices $\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )$ and $\mathrm { C } ( 4,8,3 )$. By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle.

OCR MEI C4 2006 January Q4

4 Solve the equation $2 \sin 2 \theta + \cos 2 \theta = 1$, for $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$.

OCR MEI C4 2006 January Q5

5

Find the cartesian equation of the plane through the point ( $2 , - 1,4$ ) with normal vector $$\mathbf { n } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right)$$
Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7
12
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
3
2 \end{array} \right)$$

OCR MEI C4 2006 January Q6

6

Find the first three non-zero terms of the binomial expansion of $\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }$ for $| x | < 2$.
Use this result to find an approximation for $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to
4 significant figures.
Given that $\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c$, evaluate $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to 4 significant figures.

OCR MEI C4 2006 January Q7

7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distance $y$ metres from the line TOA. Other distances and angles are as shown. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{897205bc-2f93-4628-8f21-2ec7fd3b3699-3_509_629_513_715} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Show that $\theta = \beta - \alpha$, and hence that $\tan \theta = \frac { 6 y } { 160 + y ^ { 2 } }$. Calculate the angle $\theta$ when $y = 6$.
By differentiating implicitly, show that $\frac { \mathrm { d } \theta } { \mathrm { d } y } = \frac { 6 \left( 160 - y ^ { 2 } \right) } { \left( 160 + y ^ { 2 } \right) ^ { 2 } } \cos ^ { 2 } \theta$.
Use this result to find the value of $y$ that maximises the angle $\theta$. Calculate this maximum value of $\theta$. [You need not verify that this value is indeed a maximum.]

OCR MEI C4 2006 January Q8

1 marks

8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population $x$, in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t } ,$$ where $t$ is the time in years, and $a$ and $k$ are constants. When $t = 0 , x = 2.5$.

Show that $\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }$.
Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate $a$ and $k$.
What is the long-term population of red squirrels predicted by this model? The population $y$, in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 } .$$ When $t = 0 , y = 1$.
Express $\frac { 1 } { 2 y - y ^ { 2 } }$ in partial fractions.
Hence show by integration that $\ln \left( \frac { y } { 2 - y } \right) = 2 t$. Show that $y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }$.
What is the long-term population of grey squirrels predicted by this model? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4)
Paper B: Comprehension
Monday
23 JANUARY 2006
Afternooon U Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2)
4754(B) \section*{Up to 1 hour
$$\text { o to } 1 \text { hour }$$} \section*{TIME} \section*{Up to 1 hour}
- Write your name, centre number and candidate number in the spaces at the top of this page.
- Answer all the questions.
- Write your answers in the spaces provided on the question paper.
- You are permitted to use a graphical calculator in this paper.
- Final answers should be given to a degree of accuracy appropriate to the context.
- The number of marks is given in brackets [ ] at the end of each question or part question.
- The insert contains the text for use with the questions.
- You may find it helpful to make notes and do some calculations as you read the passage.
- You are not required to hand in these notes with your question paper.
- You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
- The total number of marks for this section is 18.
For Examiner's Use
Qu. Mark
1
2
3
4
5
Total
1 Line 59 says "Again Party G just misses out; if there had been 7 seats G would have got the last one." Where is the evidence for this in the article? 26 parties, P, Q, R, S, T and U take part in an election for 7 seats. Their results are shown in the table below.
Party Votes (\%)
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Use the Trial-and-Improvement method, starting with values of $10 \%$ and $14 \%$, to find an acceptance percentage for 7 seats, and state the allocation of the seats.
Acceptance percentage, $\boldsymbol { a }$ \% 10\% 14\%
Party Votes (\%) Seats Seats Seats Seats Seats
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Total seats
Seat Allocation $\quad \mathrm { P } \ldots$... $\mathrm { Q } \ldots$ R ... S ... T ... $\mathrm { U } \ldots$.
Now apply the d'Hondt Formula to the same figures to find the allocation of the seats.
Round
Party 1 2 3 4 5 6 7 Residual
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Seat allocated to
Seat Allocation $\mathrm { P } \ldots$. Q … $\mathrm { R } \ldots$. S … T … $\mathrm { U } \ldots$. 3 In this question, use the figures for the example used in Table 5 in the article, the notation described in the section "Equivalence of the two methods" and the value of 11 found for $a$ in Table 4. Treating Party E as Party 5, verify that $\frac { V _ { 5 } } { N _ { 5 } + 1 } < a \leqslant \frac { V _ { 5 } } { N _ { 5 } }$.
4 Some of the intervals illustrated by the lines in the graph in Fig. 8 are given in this table.
Seats Interval Seats Interval
1 $22.2 < a \leqslant 27.0$ 5
2 $16.6 < a \leqslant 22.2$ 6 $10.6 < a \leqslant 11.1$
3 7
4
Describe briefly, giving an example, the relationship between the end-points of these intervals and the values in Table 5, which is reproduced below.

Complete the table above. \begin{table}[h]

	Round
Party	1	2	3	4	5	6	Residual
A	22.2	22.2	11.1	11.1	11.1	11.1	7.4
B	6.1	6.1	6.1	6.1	6.1	6.1	6.1
C	27.0	13.5	13.5	13.5	9.0	9.0	9.0
D	16.6	16.6	16.6	8.3	8.3	8.3	8.3
E	11.2	11.2	11.2	11.2	11.2	5.6	5.6
F	3.7	3.7	3.7	3.7	3.7	3.7	3.7
G	10.6	10.6	10.6	10.6	10.6	10.6	10.6
H	2.6	2.6	2.6	2.6	2.6	2.6	2.6
Seat allocated to	C	A	D	C	E	A

\captionsetup{labelformat=empty} \caption{Table 5}

\end{table} 5 The ends of the vertical lines in Fig. 8 are marked with circles. Those at the tops of the lines are filled in, e.g. • whereas those at the bottom are not, e.g. o.

Relate this distinction to the use of inequality signs.
Show that the inequality on line 102 can be rearranged to give $0 \leqslant V _ { k } - N _ { k } a < a$. [1]
Hence justify the use of the inequality signs in line 102.

OCR MEI C4 2006 June Q1

1 Fig. 1 shows part of the graph of $y = \sin x - \sqrt { 3 } \cos x$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-2_467_629_468_717} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Express $\sin x - \sqrt { 3 } \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi$.
Hence write down the exact coordinates of the turning point P .

OCR MEI C4 2006 June Q2

2

Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x } ,$$ where $A , B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$.
Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $( 1 + x ) ^ { - 2 }$ and $( 1 - 4 x ) ^ { - 1 }$. Hence find the first three terms of the expansion of $\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }$.

OCR MEI C4 2006 June Q3

3 Given that $\sin ( \theta + \alpha ) = 2 \sin \theta$, show that $\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }$. Hence solve the equation $\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

Questions C4 (1162 questions)

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