Questions - A-Level Maths

AQA C4 2008 June Q4

7 marks Standard +0.3

4

1. Obtain the binomial expansion of $( 1 - x ) ^ { \frac { 1 } { 4 } }$ up to and including the term in $x ^ { 2 }$.
2. Hence show that $( 81 - 16 x ) ^ { \frac { 1 } { 4 } } \approx 3 - \frac { 4 } { 27 } x - \frac { 8 } { 729 } x ^ { 2 }$ for small values of $x$.
Use the result from part (a)(ii) to find an approximation for $\sqrt [ 4 ] { 80 }$, giving your answer to seven decimal places.

AQA C4 2008 June Q5

10 marks Moderate -0.3

5

The angle $\alpha$ is acute and $\sin \alpha = \frac { 4 } { 5 }$.
1. Find the value of $\cos \alpha$.
2. Express $\cos ( \alpha - \beta )$ in terms of $\sin \beta$ and $\cos \beta$.
3. Given also that the angle $\beta$ is acute and $\cos \beta = \frac { 5 } { 13 }$, find the exact value of $\cos ( \alpha - \beta )$.
1. Given that $\tan 2 x = 1$, show that $\tan ^ { 2 } x + 2 \tan x - 1 = 0$.
2. Hence, given that $\tan 45 ^ { \circ } = 1$, show that $\tan 22 \frac { 1 } { 2 } ^ { \circ } = \sqrt { 2 } - 1$.

AQA C4 2008 June Q6

10 marks Moderate -0.3

6

Express $\frac { 2 } { x ^ { 2 } - 1 }$ in the form $\frac { A } { x - 1 } + \frac { B } { x + 1 }$.
Hence find $\int \frac { 2 } { x ^ { 2 } - 1 } \mathrm {~d} x$.
Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y } { 3 \left( x ^ { 2 } - 1 \right) }$, given that $y = 1$ when $x = 3$. Show that the solution can be written as $y ^ { 3 } = \frac { 2 ( x - 1 ) } { x + 1 }$.

AQA C4 2008 June Q7

12 marks Standard +0.8

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

Find the distance between $A$ and $B$.
Find the acute angle between the lines $A B$ and $l$. Give your answer to the nearest degree.
The points $B$ and $C$ lie on $l$ such that the distance $A C$ is equal to the distance $A B$. Find the coordinates of $C$.

AQA C4 2008 June Q8

9 marks Moderate -0.5

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

5 marks Moderate -0.8

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

11 marks Standard +0.3

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

13 marks Moderate -0.3

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

6 marks Moderate -0.3

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 marks Standard +0.3

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

15 marks Standard +0.3

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

10 marks Moderate -0.3

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

10 marks Moderate -0.3

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

7 marks Standard +0.3

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 Q7

10 marks Standard +0.8

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 MEI C4 2006 January Q1

5 marks Moderate -0.3

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

OCR MEI C4 2006 January Q2

5 marks Moderate -0.8

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

6 marks Moderate -0.3

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

6 marks Moderate -0.3

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

7 marks Moderate -0.3

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

7 marks Standard +0.3

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

17 marks Standard +0.3

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

19 marks Standard +0.3

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}
For Examiner's Use
Qu. Mark
1
2
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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

8 marks Moderate -0.3

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

11 marks Standard +0.3

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 $\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }$.

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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

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

For Examiner's Use
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