Questions - AQA C2 - A-Level Maths

Show that the common ratio, $r$, of the geometric series is $\frac { 3 } { 4 }$.
The first term of the geometric series is 48 . Find the sum of the first 10 terms of the series, giving your answer to four decimal places.
The $n$th term of the geometric series is $u _ { n }$ and the ( $2 n$ )th term of the series is $u _ { 2 n }$.
1. Write $u _ { n }$ and $u _ { 2 n }$ in terms of $n$.
2. Hence show that $\log _ { 10 } \left( u _ { n } \right) - \log _ { 10 } \left( u _ { 2 n } \right) = n \log _ { 10 } \left( \frac { 4 } { 3 } \right)$.
3. Hence show that the value of $$\log _ { 10 } \left( \frac { u _ { 100 } } { u _ { 200 } } \right)$$ is 12.5 correct to three significant figures.

AQA C2 2005 June Q6

Using the binomial expansion, or otherwise, express $( 1 + x ) ^ { 4 }$ in ascending powers of $x$.
1. Hence show that $( 1 + \sqrt { 5 } ) ^ { 4 } = 56 + 24 \sqrt { 5 }$.
2. Hence show that $\log _ { 2 } ( 1 + \sqrt { 5 } ) ^ { 4 } = k + \log _ { 2 } ( 7 + 3 \sqrt { 5 } )$, where $k$ is an integer.

AQA C2 2005 June Q7

7 A curve is defined, for $x > 0$, by the equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$

Express $\frac { x ^ { 8 } - 1 } { x ^ { 3 } }$ in the form $x ^ { p } - x ^ { q }$, where $p$ and $q$ are integers.
1. Hence differentiate $\mathrm { f } ( x )$ to find $\mathrm { f } ^ { \prime } ( x )$.
2. Hence show that f is an increasing function.
Find the gradient of the normal to the curve at the point $( 1,0 )$.

AQA C2 2005 June Q8

1. Show that the equation $$4 \tan \theta \sin \theta = 15$$ can be written as $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ (1 mark)
2. Use an appropriate identity to show that the equation $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ can be written as $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$
1. Solve the equation $4 c ^ { 2 } + 15 c - 4 = 0$.
2. Hence explain why the only value of $\cos \theta$ which satisfies the equation $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$ is $\cos \theta = \frac { 1 } { 4 }$.
3. Hence solve the equation $4 \tan \theta \sin \theta = 15$ giving all solutions to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
Write down all the values of $x$ in the interval $0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }$ for which $$4 \tan 4 x \sin 4 x = 15$$ giving your answers to the nearest degree.

AQA C2 2006 June Q1

1 The diagram shows a sector of a circle of radius 5 cm and angle $\theta$ radians.
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_327_358_571_842} The area of the sector is $8.1 \mathrm {~cm} ^ { 2 }$.

Show that $\theta = 0.648$.
Find the perimeter of the sector.

AQA C2 2006 June Q2

2 The diagram shows a triangle $A B C$.
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-2_757_558_1409_726} The lengths of $A C$ and $B C$ are 4.8 cm and 12 cm respectively.
The size of the angle $B A C$ is $100 ^ { \circ }$.

Show that angle $A B C = 23.2 ^ { \circ }$, correct to the nearest $0.1 ^ { \circ }$.
Calculate the area of triangle $A B C$, giving your answer in $\mathrm { cm } ^ { 2 }$ to three significant figures.

AQA C2 2006 June Q3

3 The first term of an arithmetic series is 1 . The common difference of the series is 6 .

Find the tenth term of the series.
The sum of the first $n$ terms of the series is 7400 .
1. Show that $3 n ^ { 2 } - 2 n - 7400 = 0$.
2. Find the value of $n$.

AQA C2 2006 June Q4

The expression $( 1 - 2 x ) ^ { 4 }$ can be written in the form $$1 + p x + q x ^ { 2 } - 32 x ^ { 3 } + 16 x ^ { 4 }$$ By using the binomial expansion, or otherwise, find the values of the integers $p$ and $q$.
Find the coefficient of $x$ in the expansion of $( 2 + x ) ^ { 9 }$.
Find the coefficient of $x$ in the expansion of $( 1 - 2 x ) ^ { 4 } ( 2 + x ) ^ { 9 }$.

AQA C2 2006 June Q5

Given that $$\log _ { a } x = 2 \log _ { a } 6 - \log _ { a } 3$$ show that $x = 12$.
Given that $$\log _ { a } y + \log _ { a } 5 = 7$$ express $y$ in terms of $a$, giving your answer in a form not involving logarithms.
(3 marks)

AQA C2 2006 June Q6

6 The diagram shows a sketch of the curve with equation $y = 27 - 3 ^ { x }$.
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484} The curve $y = 27 - 3 ^ { x }$ intersects the $y$-axis at the point $A$ and the $x$-axis at the point $B$.

1. Find the $y$-coordinate of point $A$.
2. Verify that the $x$-coordinate of point $B$ is 3 .
The region, $R$, bounded by the curve $y = 27 - 3 ^ { x }$ and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of $R$.
1. Use logarithms to solve the equation $3 ^ { x } = 13$, giving your answer to four decimal places.
2. The line $y = k$ intersects the curve $y = 27 - 3 ^ { x }$ at the point where $3 ^ { x } = 13$. Find the value of $k$.
1. Describe the single geometrical transformation by which the curve with equation $y = - 3 ^ { x }$ can be obtained from the curve $y = 27 - 3 ^ { x }$.
2. Sketch the curve $y = - 3 ^ { x }$.

AQA C2 2006 June Q7

7 At the point $( x , y )$, where $x > 0$, the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$

1. Verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 4$.
  (1 mark)
2. Write $\frac { 16 } { x ^ { 2 } }$ in the form $16 x ^ { k }$, where $k$ is an integer.
3. Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
4. Hence determine whether the point where $x = 4$ is a maximum or a minimum, giving a reason for your answer.
The point $P ( 1,8 )$ lies on the curve.
1. Show that the gradient of the curve at the point $P$ is 12 .
2. Find an equation of the normal to the curve at $P$.
1. Find $\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x$.
2. Hence find the equation of the curve which passes through the point $P ( 1,8 )$.

AQA C2 2006 June Q8

Describe the single geometrical transformation by which the curve with equation $y = \tan \frac { 1 } { 2 } x$ can be obtained from the curve $y = \tan x$.
Solve the equation $\tan \frac { 1 } { 2 } x = 3$ in the interval $\mathbf { 0 } < \boldsymbol { x } < \mathbf { 4 } \boldsymbol { \pi }$, giving your answers in radians to three significant figures.
Solve the equation $$\cos \theta ( \sin \theta - 3 \cos \theta ) = 0$$ in the interval $0 < \theta < 2 \pi$, giving your answers in radians to three significant figures.
(5 marks)

AQA C2 2008 June Q1

Write $\sqrt { x ^ { 3 } }$ in the form $x ^ { k }$, where $k$ is a fraction.
(1 mark)
A curve, defined for $x \geqslant 0$, has equation $$y = x ^ { 2 } - \sqrt { x ^ { 3 } }$$
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Find the equation of the tangent to the curve at the point where $x = 4$, giving your answer in the form $y = m x + c$.

AQA C2 2008 June Q2

2 The diagram shows a shaded segment of a circle with centre $O$ and radius 14 cm , where $P Q$ is a chord of the circle.
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-2_423_551_1270_740} In triangle $O P Q$, angle $P O Q = \frac { 3 \pi } { 7 }$ radians and angle $O P Q = \alpha$ radians.

Find the length of the arc $P Q$, giving your answer as a multiple of $\pi$.
Find $\alpha$ in terms of $\pi$.
Find the perimeter of the shaded segment, giving your answer to three significant figures.

AQA C2 2008 June Q3

3 A geometric series begins $$20 + 16 + 12.8 + 10.24 + \ldots$$

Find the common ratio of the series.
Find the sum to infinity of the series.
Find the sum of the first 20 terms of the series, giving your answer to three decimal places.
Prove that the $n$th term of the series is $25 \times 0.8 ^ { n }$.

AQA C2 2008 June Q4

4 The diagram shows a triangle $A B C$.
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-3_394_522_1062_751} The size of angle $B A C$ is $65 ^ { \circ }$, and the lengths of $A B$ and $A C$ are 7.6 m and 8.3 m respectively.

Show that the length of $B C$ is 8.56 m , correct to three significant figures.
Calculate the area of triangle $A B C$, giving your answer in $\mathrm { m } ^ { 2 }$ to three significant figures.
The perpendicular from $A$ to $B C$ meets $B C$ at the point $D$. Calculate the length of $A D$, giving your answer to the nearest 0.1 m .

AQA C2 2008 June Q5

Write down the value of:
1. $\log _ { a } 1$;
2. $\log _ { a } a$.
Given that $$\log _ { a } x = \log _ { a } 5 + \log _ { a } 6 - \log _ { a } 1.5$$ find the value of $x$.

AQA C2 2008 June Q6

6 The $n$th term of a sequence is $u _ { n }$.
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where $p$ and $q$ are constants.
The first three terms of the sequence are given by $$u _ { 1 } = - 8 \quad u _ { 2 } = 8 \quad u _ { 3 } = 4$$

Show that $q = 6$ and find the value of $p$.
Find the value of $u _ { 4 }$.
The limit of $u _ { n }$ as $n$ tends to infinity is $L$.
1. Write down an equation for $L$.
2. Hence find the value of $L$.

AQA C2 2008 June Q7

The expression $\left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 }$ can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } + \frac { 64 } { x ^ { 6 } }$$ By using the binomial expansion, or otherwise, find the values of the integers $p$ and $q$.
1. Hence find $\int \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x$.
2. Hence find the value of $\int _ { 1 } ^ { 2 } \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x$.

AQA C2 2008 June Q8

8 The diagram shows a sketch of the curve with equation $y = 6 ^ { x }$.
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x$, giving your answer to three significant figures.
2. Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of $\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x$.
1. Describe a single geometrical transformation that maps the graph of $y = 6 ^ { x }$ onto the graph of $y = 6 ^ { 3 x }$.
2. The line $y = 84$ intersects the curve $y = 6 ^ { 3 x }$ at the point $A$. By using logarithms, find the $x$-coordinate of $A$, giving your answer to three decimal places.
  (4 marks)
The graph of $y = 6 ^ { x }$ is translated by $\left[ \begin{array} { c } 1
- 2 \end{array} \right]$ to give the graph of the curve with equation $y = \mathrm { f } ( x )$. Write down an expression for $\mathrm { f } ( x )$.

AQA C2 2008 June Q9

Solve the equation $\sin 2 x = \sin 48 ^ { \circ }$, giving the values of $x$ in the interval $0 ^ { \circ } \leqslant x < 360 ^ { \circ }$.
Solve the equation $2 \sin \theta - 3 \cos \theta = 0$ in the interval $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$, giving your answers to the nearest $0.1 ^ { \circ }$.

AQA C2 2009 June Q1

1 The triangle $A B C$, shown in the diagram, is such that $A B = 7 \mathrm {~cm} , A C = 5 \mathrm {~cm}$, $B C = 8 \mathrm {~cm}$ and angle $A B C = \theta$.

Show that $\theta = 38.2 ^ { \circ }$, correct to the nearest $0.1 ^ { \circ }$.
Calculate the area of triangle $A B C$, giving your answer, in $\mathrm { cm } ^ { 2 }$, to three significant figures.

AQA C2 2009 June Q2

Write down the value of $n$ given that $\frac { 1 } { x ^ { 4 } } = x ^ { n }$.
Expand $\left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 }$.
Hence find $\int \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x$.
Hence find the exact value of $\int _ { 1 } ^ { 3 } \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x$.

AQA C2 2009 June Q3

3 The $n$th term of a sequence is $u _ { n }$.
The sequence is defined by $$u _ { n + 1 } = k u _ { n } + 12$$ where $k$ is a constant.
The first two terms of the sequence are given by $$u _ { 1 } = 16 \quad u _ { 2 } = 24$$

Show that $k = 0.75$.
Find the value of $u _ { 3 }$ and the value of $u _ { 4 }$.
The limit of $u _ { n }$ as $n$ tends to infinity is $L$.
1. Write down an equation for $L$.
2. Hence find the value of $L$.

AQA C2 2009 June Q5

5 The diagram shows part of a curve with a maximum point $M$.
\includegraphics[max width=\textwidth, alt={}, center]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-06_472_791_358_630} The equation of the curve is $$y = 15 x ^ { \frac { 3 } { 2 } } - x ^ { \frac { 5 } { 2 } }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the coordinates of the maximum point $M$.
The point $P ( 1,14 )$ lies on the curve. Show that the equation of the tangent to the curve at $P$ is $y = 20 x - 6$.
The tangents to the curve at the points $P$ and $M$ intersect at the point $R$. Find the length of $R M$.

Questions — AQA C2 (184 questions)

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