Questions - AQA C2 - A-Level Maths

1. Verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 2$.
  (l mark)
2. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 2$.
3. Hence state whether $P$ is a maximum point or a minimum point, giving a reason for your answer.
Find the equation of the curve.

AQA C2 2012 June Q7

7 It is given that $( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0$.

Find the possible values of $\tan \theta$.
Hence solve the equation $( \tan \theta + 1 ) \left( \sin ^ { 2 } \theta - 3 \cos ^ { 2 } \theta \right) = 0$, giving all solutions for $\theta$, in degrees, in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.

AQA C2 2012 June Q8

Sketch the curve with equation $y = 7 ^ { x }$, indicating the coordinates of any point where the curve intersects the coordinate axes.
The curve $C _ { 1 }$ has equation $y = 7 ^ { x }$. The curve $C _ { 2 }$ has equation $y = 7 ^ { 2 x } - 12$.
1. By forming and solving a quadratic equation, prove that the curves $C _ { 1 }$ and $C _ { 2 }$ intersect at exactly one point. State the $y$-coordinate of this point.
2. Use logarithms to find the $x$-coordinate of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$, giving your answer to three significant figures.
  (2 marks)

AQA C2 2012 June Q9

9 The diagram shows part of a curve whose equation is $y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)$.
\includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-5_355_451_367_799}

Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 1 } \log _ { 10 } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer to three significant figures.
The graph of $y = 2 \log _ { 10 } x$ can be transformed into the graph of $y = 1 + 2 \log _ { 10 } x$ by means of a translation. Write down the vector of the translation.
1. Show that $\log _ { 10 } \left( 10 x ^ { 2 } \right) = 1 + 2 \log _ { 10 } x$.
2. Show that the graph of $y = 2 \log _ { 10 } x$ can also be transformed into the graph of $y = 1 + 2 \log _ { 10 } x$ by means of a stretch, and describe the stretch.
3. The curve with equation $y = 1 + 2 \log _ { 10 } x$ intersects the curve $y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)$ at the point $P$. Given that the $x$-coordinate of $P$ is positive, find the gradient of the line $O P$, where $O$ is the origin. Give your answer in the form $\log _ { 10 } \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers.

AQA C2 2013 June Q1

1 A geometric series has first term 80 and common ratio $\frac { 1 } { 2 }$.

Find the third term of the series.
Find the sum to infinity of the series.
Find the sum of the first 12 terms of the series, giving your answer to two decimal places.

AQA C2 2013 June Q2

2 The diagram shows a sector $O A B$ of a circle with centre $O$.
\includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_341_371_968_815} The radius of the circle is 20 cm and the angle $A O B = 0.8$ radians.

Find the length of the arc $A B$.
Find the area of the sector $O A B$.
A line from $B$ meets the radius $O A$ at the point $D$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_344_371_1747_815} The length of $B D$ is 15 cm . Find the size of the obtuse angle $O D B$, in radians, giving your answer to three significant figures.

AQA C2 2013 June Q3

1. Using the binomial expansion, or otherwise, express $( 2 + y ) ^ { 3 }$ in the form $a + b y + c y ^ { 2 } + y ^ { 3 }$, where $a , b$ and $c$ are integers.
2. Hence show that $\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }$ can be expressed in the form $p + q x ^ { - 4 }$, where $p$ and $q$ are integers.
1. Hence find $\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x$.
2. Hence find the value of $\int _ { 1 } ^ { 2 } \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x$.

AQA C2 2013 June Q4

Sketch the graph of $y = 9 ^ { x }$, indicating the value of the intercept on the $y$-axis.
(2 marks)
Use logarithms to solve the equation $9 ^ { x } = 15$, giving your value of $x$ to three significant figures.
The curve $y = 9 ^ { x }$ is reflected in the $y$-axis to give the curve with equation $y = \mathrm { f } ( x )$. Write down an expression for $\mathrm { f } ( x )$.
(l mark)

AQA C2 2013 June Q5

Use the trapezium rule with five ordinates (four strips) to find an approximate value for $\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x$, giving your answer to three significant figures.
Describe the single transformation that maps the graph of $y = \sqrt { 8 x ^ { 3 } + 1 }$ onto the graph of $y = \sqrt { x ^ { 3 } + 1 }$.
The curve with equation $y = \sqrt { x ^ { 3 } + 1 }$ is translated by $\left[ \begin{array} { c } 2
- 0.7 \end{array} \right]$ to give the curve with equation $y = \mathrm { g } ( x )$. Find the value of $\mathrm { g } ( 4 )$.
(3 marks)

AQA C2 2013 June Q6

6 A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$

Express $\frac { 12 + x ^ { 2 } \sqrt { x } } { x }$ in the form $12 x ^ { p } + x ^ { q }$.
1. Hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Find an equation of the normal to the curve at the point on the curve where $x = 4$.
3. The curve has a stationary point $P$. Show that the $x$-coordinate of $P$ can be written in the form $2 ^ { k }$, where $k$ is a rational number.

AQA C2 2013 June Q7

7 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 two terms of the sequence are given by $u _ { 1 } = 96$ and $u _ { 2 } = 72$.
The limit of $u _ { n }$ as $n$ tends to infinity is 24 .

Show that $p = \frac { 2 } { 3 }$.
Find the value of $u _ { 3 }$.

AQA C2 2013 June Q8

Given that $\log _ { a } b = c$, express $b$ in terms of $a$ and $c$.
By forming a quadratic equation, show that there is only one value of $x$ which satisfies the equation $2 \log _ { 2 } ( x + 7 ) - \log _ { 2 } ( x + 5 ) = 3$.

AQA C2 2013 June Q9

1. On the axes given below, sketch the graph of $y = \tan x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
2. Solve the equation $\tan x = - 1$, giving all values of $x$ in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
1. Given that $6 \tan \theta \sin \theta = 5$, show that $6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0$.
2. Hence solve the equation $6 \tan 3 x \sin 3 x = 5$, giving all values of $x$ to the nearest degree in the interval $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
  \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}

AQA C2 2014 June Q1

3 marks

1 The diagram shows a triangle $A B C$. The size of angle $B A C$ is $47 ^ { \circ }$ and the lengths of $A B$ and $A C$ are 5 cm and 12 cm respectively.

Calculate the area of the triangle $A B C$, giving your answer to the nearest $\mathrm { cm } ^ { 2 }$.
Calculate the length of $B C$, giving your answer, in cm , to one decimal place.
[0pt] [3 marks]

AQA C2 2014 June Q2

Find $\int \left( 1 + 3 x ^ { \frac { 1 } { 2 } } + x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$.
1. The expression $( 1 + y ) ^ { 3 }$ can be written in the form $1 + 3 y + n y ^ { 2 } + y ^ { 3 }$. Write down the value of the constant $n$.
2. Hence, or otherwise, expand $( 1 + \sqrt { x } ) ^ { 3 }$.
Hence find the exact value of $\int _ { 0 } ^ { 1 } ( 1 + \sqrt { x } ) ^ { 3 } \mathrm {~d} x$.

AQA C2 2014 June Q3

3 marks

3 The first term of a geometric series is 54 and the common ratio of the series is $\frac { 8 } { 9 }$.

Find the sum to infinity of the series.
Find the second term of the series.
Show that the 12th term of the series can be written in the form $\frac { 2 ^ { p } } { 3 ^ { q } }$, where $p$ and $q$ are integers.
[0pt] [3 marks]

AQA C2 2014 June Q4

5 marks

4 A curve has equation $y = \frac { 1 } { x ^ { 2 } } + 4 x$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
The point $P ( - 1 , - 3 )$ lies on the curve. Find an equation of the normal to the curve at the point $P$.
Find an equation of the tangent to the curve that is parallel to the line $y = - 12 x$.
[0pt] [5 marks]

AQA C2 2014 June Q5

6 marks

5 The diagram shows a sector $O A B$ of a circle with centre $O$ and radius $r \mathrm {~cm}$.
\includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-10_346_360_360_824} The angle $A O B$ is $\theta$ radians.
The area of the sector is $12 \mathrm {~cm} ^ { 2 }$.
The perimeter of the sector is four times the length of the $\operatorname { arc } A B$.
Find the value of $r$.
[0pt] [6 marks]

\includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-10_1533_1712_1174_150}

AQA C2 2014 June Q6

2 marks

Sketch, on the axes given below, the graph of $y = \sin x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Describe the geometrical transformation that maps the graph of $y = \sin x$ onto the graph of $y = \sin 5 x$.
Describe the single geometrical transformation that maps the graph of $y = \sin 5 x$ onto the graph of $y = \sin \left( 5 x + 10 ^ { \circ } \right)$.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-12_675_1417_906_370}

AQA C2 2014 June Q7

4 marks

Given that $\frac { \cos ^ { 2 } x + 4 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 7$, show that $\tan ^ { 2 } x = \frac { 3 } { 2 }$.
Hence solve the equation $\frac { \cos ^ { 2 } 2 \theta + 4 \sin ^ { 2 } 2 \theta } { 1 - \sin ^ { 2 } 2 \theta } = 7$ in the interval $0 ^ { \circ } < \theta < 180 ^ { \circ }$, giving your values of $\theta$ to the nearest degree.
[0pt] [4 marks]

AQA C2 2014 June Q8

5 marks

8 An arithmetic series has first term $a$ and common difference $d$. The sum of the first 5 terms of the series is 575 .

Show that $a + 2 d = 115$.
Given also that the 10th term of the series is 87, find the value of $d$.
The $n$th term of the series is $u _ { n }$. Given that $u _ { k } > 0$ and $u _ { k + 1 } < 0$, find the value of $\sum _ { n = 1 } ^ { k } u _ { n }$.
[0pt] [5 marks]

AQA C2 2014 June Q9

5 marks

9 A curve has equation $y = 3 \times 12 ^ { x }$.

The point ( $k , 6$ ) lies on the curve $y = 3 \times 12 ^ { x }$. Use logarithms to find the value of $k$, giving your answer to three significant figures.
Use the trapezium rule with four ordinates (three strips) to find an approximate value for $\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x$, giving your answer to two significant figures.
The curve $y = 3 \times 12 ^ { x }$ is translated by the vector $\left[ \begin{array} { l } 1
p \end{array} \right]$ to give the curve $y = \mathrm { f } ( x )$. Given that the curve $y = \mathrm { f } ( x )$ passes through the origin ( 0,0 ), find the value of the constant $p$.
The curve with equation $y = 2 ^ { 2 - x }$ intersects the curve $y = 3 \times 12 ^ { x }$ at the point $T$. Show that the $x$-coordinate of $T$ can be written in the form $\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }$, where $q$ is an integer. State the value of $q$.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-20_2288_1707_221_153}

AQA C2 2015 June Q1

4 marks

1 The diagram shows a sector $O A B$ of a circle with centre $O$ and radius 5 cm .
\includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-02_378_451_648_790} The angle $A O B$ is $\theta$ radians and the area of the sector is $15 \mathrm {~cm} ^ { 2 }$.
Find the perimeter of the sector.
[0pt] [4 marks]

AQA C2 2015 June Q2

4 marks

2 The diagram shows a triangle $A B C$. The size of angle $B A C$ is $72 ^ { \circ }$ and the size of angle $A B C$ is $48 ^ { \circ }$. The length of $B C$ is 20 cm .

Show that the length of $A C$ is 15.6 cm , correct to three significant figures.
The midpoint of $B C$ is $M$. Calculate the length of $A M$, giving your answer, in cm , to three significant figures.
[0pt] [4 marks]

AQA C2 2015 June Q3

3 The first term of an infinite geometric series is 48 . The common ratio of the series is 0.6 .

Find the third term of the series.
Find the sum to infinity of the series.
The $n$th term of the series is $u _ { n }$. Find the value of $\sum _ { n = 4 } ^ { \infty } u _ { n }$.

Questions — AQA C2 (184 questions)

Browse by module

Browse by board

AQA C2 2012 June Q6

AQA C2 2012 June Q7

AQA C2 2012 June Q8

AQA C2 2012 June Q9

AQA C2 2013 June Q1

AQA C2 2013 June Q2

AQA C2 2013 June Q3

AQA C2 2013 June Q4

AQA C2 2013 June Q5

AQA C2 2013 June Q6

AQA C2 2013 June Q7

AQA C2 2013 June Q8

AQA C2 2013 June Q9

AQA C2 2014 June Q1

AQA C2 2014 June Q2

AQA C2 2014 June Q3

AQA C2 2014 June Q4

AQA C2 2014 June Q5

AQA C2 2014 June Q6

AQA C2 2014 June Q7

AQA C2 2014 June Q8

AQA C2 2014 June Q9

AQA C2 2015 June Q1

AQA C2 2015 June Q2

AQA C2 2015 June Q3