Questions C2 - A-Level Maths

Show that $\theta = 0.430$ correct to three significant figures.
Use the cosine rule to calculate the length of $A B$, giving your answer to two significant figures.
The point $D$ lies on $C B$ such that $A D$ is an arc of a circle centre $C$ and radius 8 cm . The region bounded by the arc $A D$ and the straight lines $D B$ and $A B$ is shaded in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555} Calculate, to two significant figures:
1. the length of the $\operatorname { arc } A D$;
2. the area of the shaded region.

AQA C2 2006 January Q5

9 marks Moderate -0.3

5 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 $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$

Show that $p = 0.6$ and find the value of $q$.
Find the value of $u _ { 4 }$.
The limit of $u _ { n }$ as $n$ tends to infinity is $L$. Write down an equation for $L$ and hence find the value of $L$.

AQA C2 2006 January Q6

12 marks Moderate -0.8

Describe the geometrical transformation that maps the curve with equation $y = \sin x$ onto the curve with equation:
1. $y = 2 \sin x$;
2. $y = - \sin x$;
3. $y = \sin \left( x - 30 ^ { \circ } \right)$.
Solve the equation $\sin \left( \theta - 30 ^ { \circ } \right) = 0.7$, giving your answers to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
Prove that $( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2$.

AQA C2 2006 January Q7

5 marks Standard +0.3

7 It is given that $n$ satisfies the equation $$2 \log _ { a } n - \log _ { a } ( 5 n - 24 ) = \log _ { a } 4$$

Show that $n ^ { 2 } - 20 n + 96 = 0$.
Hence find the possible values of $n$.

AQA C2 2006 January Q8

18 marks Standard +0.3

8 A curve, drawn from the origin $O$, crosses the $x$-axis at the point $A ( 9,0 )$. Tangents to the curve at $O$ and $A$ meet at the point $P$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-5_778_901_461_571} The curve, defined for $x \geqslant 0$, has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
1. Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point $O$ and hence write down an equation of the tangent at $O$.
2. Show that the equation of the tangent at $A ( 9,0 )$ is $2 y = 3 x - 27$.
3. Hence find the coordinates of the point $P$ where the two tangents meet.
Find $\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x$.
Calculate the area of the shaded region bounded by the curve and the tangents $O P$ and $A P$.

AQA C2 2008 January Q1

6 marks Moderate -0.8

1 The diagrams show a rectangle of length 6 cm and width 3 cm , and a sector of a circle of radius 6 cm and angle $\theta$ radians.
\includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-2_266_1128_589_424} The area of the rectangle is twice the area of the sector.

Show that $\theta = 0.5$.
Find the perimeter of the sector.

AQA C2 2008 January Q2

5 marks Moderate -0.8

2 The arithmetic series $$51 + 58 + 65 + 72 + \ldots + 1444$$ has 200 terms.

Write down the common difference of the series.
Find the 101st term of the series.
Find the sum of the last 100 terms of the series.

AQA C2 2008 January Q3

6 marks Easy -1.2

3 The diagram shows a triangle $A B C$. The length of $A C$ is 18.7 cm , and the sizes of angles $B A C$ and $A B C$ are $72 ^ { \circ }$ and $50 ^ { \circ }$ respectively.

Show that the length of $B C = 23.2 \mathrm {~cm}$, correct to the nearest 0.1 cm .
Calculate the area of triangle $A B C$, giving your answer to the nearest $\mathrm { cm } ^ { 2 }$.

AQA C2 2008 January Q4

4 marks Moderate -0.8

4 Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 0 } ^ { 3 } \sqrt { x ^ { 2 } + 3 } d x$$ giving your answer to three decimal places.

AQA C2 2008 January Q5

20 marks Moderate -0.8

5 A curve, drawn from the origin $O$, crosses the $x$-axis at the point $P ( 4,0 )$.
The normal to the curve at $P$ meets the $y$-axis at the point $Q$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for $x \geqslant 0$, has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$

1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
  (3 marks)
2. Show that the gradient of the curve at $P ( 4,0 )$ is - 2 .
3. Find an equation of the normal to the curve at $P ( 4,0 )$.
4. Find the $y$-coordinate of $Q$ and hence find the area of triangle $O P Q$.
5. The curve has a maximum point $M$. Find the $x$-coordinate of $M$.
1. Find $\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$.
2. Find the total area of the region bounded by the curve and the lines $P Q$ and $Q O$.

AQA C2 2008 January Q6

10 marks Easy -1.8

Using the binomial expansion, or otherwise:
1. express $( 1 + x ) ^ { 3 }$ in ascending powers of $x$;
2. express $( 1 + x ) ^ { 4 }$ in ascending powers of $x$.
Hence, or otherwise:
1. express $( 1 + 4 x ) ^ { 3 }$ in ascending powers of $x$;
2. express $( 1 + 3 x ) ^ { 4 }$ in ascending powers of $x$.
Show that the expansion of $$( 1 + 3 x ) ^ { 4 } - ( 1 + 4 x ) ^ { 3 }$$ can be written in the form $$p x ^ { 2 } + q x ^ { 3 } + r x ^ { 4 }$$ where $p , q$ and $r$ are integers.

AQA C2 2008 January Q7

4 marks Easy -1.2

Given that $$\log _ { a } x = \log _ { a } 16 - \log _ { a } 2$$ write down the value of $x$.
Given that $$\log _ { a } y = 2 \log _ { a } 3 + \log _ { a } 4 + 1$$ express $y$ in terms of $a$, giving your answer in a form not involving logarithms.

AQA C2 2008 January Q8

12 marks Moderate -0.8

Sketch the graph of $y = 3 ^ { x }$, stating the coordinates of the point where the graph crosses the $y$-axis.
Describe a single geometrical transformation that maps the graph of $y = 3 ^ { x }$ :
1. onto the graph of $y = 3 ^ { 2 x }$;
2. onto the graph of $y = 3 ^ { x + 1 }$.
1. Using the substitution $Y = 3 ^ { x }$, show that the equation $$9 ^ { x } - 3 ^ { x + 1 } + 2 = 0$$ can be written as $$( Y - 1 ) ( Y - 2 ) = 0$$
2. Hence show that the equation $9 ^ { x } - 3 ^ { x + 1 } + 2 = 0$ has a solution $x = 0$ and, by using logarithms, find the other solution, giving your answer to four decimal places.
  (4 marks)

AQA C2 2008 January Q9

8 marks Standard +0.3

Given that $$\frac { 3 + \sin ^ { 2 } \theta } { \cos \theta - 2 } = 3 \cos \theta$$ show that $$\cos \theta = - \frac { 1 } { 2 }$$
Hence solve the equation $$\frac { 3 + \sin ^ { 2 } 3 x } { \cos 3 x - 2 } = 3 \cos 3 x$$ giving all solutions in degrees in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.

AQA C2 2009 January Q1

7 marks Easy -1.2

1 The diagram shows a sector $O A B$ of a circle with centre $O$ and radius 10 cm .
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-2_371_378_555_824} The angle $A O B$ is 0.8 radians.

Find the area of the sector.
1. Find the perimeter of the sector $O A B$.
2. The perimeter of the sector $O A B$ is equal to the perimeter of a square. Find the area of the square.

AQA C2 2009 January Q2

5 marks Moderate -0.3

Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 1.5 } ^ { 6 } x ^ { 2 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
State how you could obtain a better approximation to the value of the integral using the trapezium rule.
(1 mark)

AQA C2 2009 January Q3

7 marks Moderate -0.8

3 The diagram shows a triangle $A B C$. The size of angle $A$ is $63 ^ { \circ }$, and the lengths of $A B$ and $A C$ are 7.4 m and 5.26 m respectively.

Calculate the area of triangle $A B C$, giving your answer in $\mathrm { m } ^ { 2 }$ to three significant figures.
Show that the length of $B C$ is 6.86 m , correct to three significant figures.
Find the value of $\sin \boldsymbol { B }$ to two significant figures.

AQA C2 2009 January Q4

14 marks Moderate -0.3

4 The diagram shows a sketch of the curves with equations $y = 2 x ^ { \frac { 3 } { 2 } }$ and $y = 8 x ^ { \frac { 1 } { 2 } }$.
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point $A$, where $x = 4$.

1. For the curve $y = 2 x ^ { \frac { 3 } { 2 } }$, find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = 4$.
  (2 marks)
2. Find an equation of the normal to the curve $y = 2 x ^ { \frac { 3 } { 2 } }$ at the point $A$.
1. Find $\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x$.
2. Find the area of the shaded region bounded by the two curves.
Describe a single geometrical transformation that maps the graph of $y = 2 x ^ { \frac { 3 } { 2 } }$ onto the graph of $y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }$.
(2 marks)

AQA C2 2009 January Q5

11 marks Moderate -0.5

By using the binomial expansion, or otherwise, express $( 1 + 2 x ) ^ { 4 }$ in the form $$1 + a x + b x ^ { 2 } + c x ^ { 3 } + 16 x ^ { 4 }$$ where $a$, $b$ and $c$ are integers.
Hence show that $( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 } = 2 + 48 x ^ { 2 } + 32 x ^ { 4 }$.
Hence show that the curve with equation $$y = ( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 }$$ has just one stationary point and state its coordinates.

AQA C2 2009 January Q6

9 marks Moderate -0.8

Write each of the following in the form $\log _ { a } k$, where $k$ is an integer:
1. $\log _ { a } 4 + \log _ { a } 10$;
2. $\log _ { a } 16 - \log _ { a } 2$;
3. $3 \log _ { a } 5$.
Use logarithms to solve the equation $( 1.5 ) ^ { 3 x } = 7.5$, giving your value of $x$ to three decimal places.
Given that $\log _ { 2 } p = m$ and $\log _ { 8 } q = n$, express $p q$ in the form $2 ^ { y }$, where $y$ is an expression in $m$ and $n$.

AQA C2 2009 January Q7

13 marks Moderate -0.8

Solve the equation $\sin x = 0.8$ in the interval $0 \leqslant x \leqslant 2 \pi$, giving your answers in radians to three significant figures.
The diagram shows the graph of the curve $y = \sin x , 0 \leqslant x \leqslant 2 \pi$ and the lines $y = k$ and $y = - k$.
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689} The line $y = k$ intersects the curve at the points $P$ and $Q$, and the line $y = - k$ intersects the curve at the points $R$ and $S$. The point $M$ is the minimum point of the curve.
1. Write down the coordinates of the point $M$.
2. The $x$-coordinate of $P$ is $\alpha$. Write down the $x$-coordinate of the point $Q$ in terms of $\pi$ and $\alpha$.
3. Find the length of $R S$ in terms of $\pi$ and $\alpha$, giving your answer in its simplest form.
Sketch the graph of $y = \sin 2 x$ for $0 \leqslant x \leqslant 2 \pi$, indicating the coordinates of points where the graph intersects the $x$-axis and the coordinates of any maximum points.

AQA C2 2009 January Q8

9 marks Standard +0.3

8 The 25th term of an arithmetic series is 38 .
The sum of the first 40 terms of the series is 1250 .

Show that the common difference of this series is 1.5 .
Find the number of terms in the series which are less than 100 .

AQA C2 2010 January Q1

9 marks Moderate -0.3

1 The diagram shows a sector $O A B$ of a circle with centre $O$.
\includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_444_373_541_804} The radius of the circle is 15 cm and angle $A O B = 1.2$ radians.

1. Show that the area of the sector is $135 \mathrm {~cm} ^ { 2 }$.
2. Calculate the length of the arc $A B$.
The point $P$ lies on the radius $O B$ such that $O P = 10 \mathrm {~cm}$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_449_378_1436_799} Calculate the perimeter of the shaded region bounded by $A P , P B$ and the arc $A B$, giving your answer to three significant figures.
(5 marks)

AQA C2 2010 January Q2

7 marks Moderate -0.8

2 At the point $( x , y )$ on a curve, where $x > 0$, the gradient is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$

Write $\sqrt { x ^ { 5 } }$ in the form $x ^ { k }$, where $k$ is a fraction.
Find $\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x$.
Hence find the equation of the curve, given that the curve passes through the point $( 1,3 )$.

AQA C2 2010 January Q3

7 marks Moderate -0.8

Find the value of $x$ in each of the following:
1. $\quad \log _ { 9 } x = 0$;
2. $\quad \log _ { 9 } x = \frac { 1 } { 2 }$.
Given that $$2 \log _ { a } n = \log _ { a } 18 + \log _ { a } ( n - 4 )$$ find the possible values of $n$.

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