Questions - AQA C1 - A-Level Maths

Given that $x + 3$ is a factor of $\mathrm { p } ( x )$, show that $$3 c - d = 8$$
The remainder when $\mathrm { p } ( x )$ is divided by $x - 2$ is 65 . Obtain a further equation in $c$ and $d$.
Use the equations from parts (a) and (b) to find the value of $c$ and the value of $d$. [3 marks]

AQA C1 2014 June Q6

7 marks

6 The diagram shows a curve and a line which intersect at the points $A , B$ and $C$.
\includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation $y = x ^ { 3 } - x ^ { 2 } - 5 x + 7$ and the straight line has equation $y = x + 7$. The point $B$ has coordinates ( 0,7 ).

1. Show that the $x$-coordinates of the points $A$ and $C$ satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
2. Find the coordinates of the points $A$ and $C$.
Find $\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x$.
Find the area of the shaded region $R$ bounded by the curve and the line segment $A B$.
[0pt] [4 marks]
$7 \quad$ A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0$. The point $A ( 3 , - 2 )$ lies on the circle.

AQA C1 2014 June Q8

4 marks

8 Solve the following inequalities:

$\quad 3 ( 1 - 2 x ) - 5 ( 3 x + 2 ) > 0$
$\quad 6 x ^ { 2 } \leqslant x + 12$
[0pt] [4 marks]

AQA C1 2015 June Q1

1 The line $A B$ has equation $3 x + 5 y = 7$.

Find the gradient of $A B$.
Find an equation of the line that is perpendicular to the line $A B$ and which passes through the point $( - 2 , - 3 )$. Express your answer in the form $p x + q y + r = 0$, where $p , q$ and $r$ are integers.
The line $A C$ has equation $2 x - 3 y = 30$. Find the coordinates of $A$.

AQA C1 2015 June Q2

5 marks

2 The point $P$ has coordinates $( \sqrt { 3 } , 2 \sqrt { 3 } )$ and the point $Q$ has coordinates $( \sqrt { 5 } , 4 \sqrt { 5 } )$. Show that the gradient of $P Q$ can be expressed as $n + \sqrt { 15 }$, stating the value of the integer $n$.
[0pt] [5 marks]

AQA C1 2015 June Q3

5 marks

3 The diagram shows a sketch of a curve and a line.
\includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-06_520_588_351_742} The curve has equation $y = x ^ { 4 } + 3 x ^ { 2 } + 2$. The points $A ( - 1,6 )$ and $B ( 2,30 )$ lie on the curve.

Find an equation of the tangent to the curve at the point $A$.
1. Find $\int _ { - 1 } ^ { 2 } \left( x ^ { 4 } + 3 x ^ { 2 } + 2 \right) \mathrm { d } x$.
2. Calculate the area of the shaded region bounded by the curve and the line $A B$.
  [0pt] [3 marks] $4 \quad$ A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } + 2 x - 6 y - 40 = 0$.

AQA C1 2015 June Q5

Express $x ^ { 2 } + 3 x + 2$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are rational numbers.
A curve has equation $y = x ^ { 2 } + 3 x + 2$.
1. Use the result from part (a) to write down the coordinates of the vertex of the curve.
2. State the equation of the line of symmetry of the curve.
The curve with equation $y = x ^ { 2 } + 3 x + 2$ is translated by the vector $\left[ \begin{array} { l } 2
4 \end{array} \right]$. Find the equation of the resulting curve in the form $y = x ^ { 2 } + b x + c$.

AQA C1 2015 June Q6

4 marks

6 The diagram shows a cylindrical container of radius $r \mathrm {~cm}$ and height $h \mathrm {~cm}$. The container has an open top and a circular base.
\includegraphics[max width=\textwidth, alt={}, center]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-12_389_426_404_751} The external surface area of the container's curved surface and base is $48 \pi \mathrm {~cm} ^ { 2 }$.
When the radius of the base is $r \mathrm {~cm}$, the volume of the container is $V \mathrm {~cm} ^ { 3 }$.

1. Find an expression for $h$ in terms of $r$.
2. Show that $V = 24 \pi r - \frac { \pi } { 2 } r ^ { 3 }$.
1. Find $\frac { \mathrm { d } V } { \mathrm {~d} r }$.
2. Find the positive value of $r$ for which $V$ is stationary, and determine whether this stationary value is a maximum value or a minimum value.
  [0pt] [4 marks]

AQA C1 2015 June Q7

3 marks

Sketch the curve with equation $y = x ^ { 2 } ( x - 3 )$.
The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 2 } ( x - 3 ) + 20$.
1. Find the remainder when $\mathrm { p } ( x )$ is divided by $x - 4$.
2. Use the Factor Theorem to show that $x + 2$ is a factor of $\mathrm { p } ( x )$.
3. Express $\mathrm { p } ( x )$ in the form $( x + 2 ) \left( x ^ { 2 } + b x + c \right)$, where $b$ and $c$ are integers.
4. Hence show that the equation $\mathrm { p } ( x ) = 0$ has exactly one real root and state its value.
  [0pt] [3 marks]

AQA C1 2015 June Q8

8 A curve has equation $y = x ^ { 2 } + ( 3 k - 4 ) x + 13$ and a line has equation $y = 2 x + k$, where $k$ is a constant.

Show that the $x$-coordinate of any point of intersection of the line and curve satisfies the equation $$x ^ { 2 } + 3 ( k - 2 ) x + 13 - k = 0$$
Given that the line and the curve do not intersect:
1. show that $9 k ^ { 2 } - 32 k - 16 < 0$;
2. find the possible values of $k$.
  \includegraphics[max width=\textwidth, alt={}]{c7f38f7e-75aa-4b41-96fd-f38f968c225c-18_1657_1714_1050_153}

AQA C1 2016 June Q1

2 marks

1 The line $A B$ has equation $5 x + 3 y + 3 = 0$.

The line $A B$ is parallel to the line with equation $y = m x + 7$. Find the value of $m$.
The line $A B$ intersects the line with equation $3 x - 2 y + 17 = 0$ at the point $B$. Find the coordinates of $B$.
The point with coordinates $( 2 k + 3,4 - 3 k )$ lies on the line $A B$. Find the value of $k$.
[0pt] [2 marks]

AQA C1 2016 June Q2

4 marks

Simplify $( 3 \sqrt { 5 } ) ^ { 2 }$.
Express $\frac { ( 3 \sqrt { 5 } ) ^ { 2 } + \sqrt { 5 } } { 7 + 3 \sqrt { 5 } }$ in the form $m + n \sqrt { 5 }$, where $m$ and $n$ are integers.
[0pt] [4 marks]

AQA C1 2016 June Q3

3 marks

1. Express $x ^ { 2 } - 7 x + 2$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are rational numbers.
2. Hence write down the minimum value of $x ^ { 2 } - 7 x + 2$.
Describe the geometrical transformation which maps the graph of $y = x ^ { 2 } - 7 x + 2$ onto the graph of $y = ( x - 4 ) ^ { 2 }$.
[0pt] [3 marks]

AQA C1 2016 June Q4

3 marks

4 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } - 8 x + 48$.

1. Use the Factor Theorem to show that $x + 3$ is a factor of $\mathrm { p } ( x )$.
2. Express $\mathrm { p } ( x )$ as a product of three linear factors.
1. Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x - 2$.
2. Express $\mathrm { p } ( x )$ in the form $( x - 2 ) \left( x ^ { 2 } + b x + c \right) + r$, where $b , c$ and $r$ are integers. [3 marks]

AQA C1 2016 June Q5

5 A circle with centre $C ( 5 , - 3 )$ passes through the point $A ( - 2,1 )$.

Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
Given that $A B$ is a diameter of the circle, find the coordinates of the point $B$.
Find an equation of the tangent to the circle at the point $A$, giving your answer in the form $p x + q y + r = 0$, where $p , q$ and $r$ are integers.
The point $T$ lies on the tangent to the circle at $A$ such that $A T = 4$. Find the length of $C T$.

AQA C1 2016 June Q6

3 marks

A curve has equation $y = 8 - 4 x - 2 x ^ { 2 }$.
1. Find the values of $x$ where the curve crosses the $x$-axis, giving your answer in the form $m \pm \sqrt { n }$, where $m$ and $n$ are integers.
2. Sketch the curve, giving the value of the $y$-intercept.
A line has equation $y = k ( x + 4 )$, where $k$ is a constant.
1. Show that the $x$-coordinates of any points of intersection of the line with the curve $y = 8 - 4 x - 2 x ^ { 2 }$ satisfy the equation $$2 x ^ { 2 } + ( k + 4 ) x + 4 ( k - 2 ) = 0$$
2. Find the values of $k$ for which the line is a tangent to the curve $y = 8 - 4 x - 2 x ^ { 2 }$.
  [0pt] [3 marks]

AQA C1 2016 June Q7

3 marks

7 The diagram shows the sketch of a curve and the tangent to the curve at $P$.
\includegraphics[max width=\textwidth, alt={}, center]{0d5b9235-af2b-4fd5-8fcf-b2b45e3c0a3c-14_519_817_356_614} The curve has equation $y = 4 - x ^ { 2 } - 3 x ^ { 3 }$ and the point $P ( - 2,24 )$ lies on the curve. The tangent at $P$ crosses the $x$-axis at $Q$.

1. Find the equation of the tangent to the curve at the point $P$, giving your answer in the form $y = m x + c$.
2. Hence find the $x$-coordinate of $Q$.
1. Find $\int _ { - 2 } ^ { 1 } \left( 4 - x ^ { 2 } - 3 x ^ { 3 } \right) \mathrm { d } x$.
2. The point $R ( 1,0 )$ lies on the curve. Calculate the area of the shaded region bounded by the curve and the lines $P Q$ and $Q R$.
  [0pt] [3 marks]

AQA C1 2016 June Q8

8 The gradient, $\frac { \mathrm { d } y } { \mathrm {~d} x }$, at the point $( x , y )$ on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
2. The curve passes through the point $P \left( - 1 \frac { 1 } { 2 } , 4 \right)$. Verify that the curve has a minimum point at $P$.
1. Show that at the points on the curve where $y$ is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
2. Solve the inequality $2 x ^ { 2 } - 9 x - 18 > 0$.

AQA C1 2014 June Q7

Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
1. Write down the coordinates of $C$.
2. Show that the circle has radius $n \sqrt { 5 }$, where $n$ is an integer.
Find the equation of the tangent to the circle at the point $A$, giving your answer in the form $x + p y = q$, where $p$ and $q$ are integers.
The point $B$ lies on the tangent to the circle at $A$ and the length of $B C$ is 6. Find the length of $A B$.
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}

AQA C1 2015 June Q4

Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
1. State the coordinates of $C$.
2. Find the radius of the circle, giving your answer in the form $n \sqrt { 2 }$.
The point $P$ with coordinates $( 4 , k )$ lies on the circle. Find the possible values of $k$.
The points $Q$ and $R$ also lie on the circle, and the length of the chord $Q R$ is 2 . Calculate the shortest distance from $C$ to the chord $Q R$.
[0pt] [2 marks]

AQA C1 2007 January Q1

1 The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where $k$ is a constant.

1. Given that $x + 2$ is a factor of $\mathrm { p } ( x )$, show that $k = 10$.
2. Express $\mathrm { p } ( x )$ as the product of three linear factors.
Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x - 3$.
Sketch the curve with equation $y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10$, indicating the values where the curve crosses the $x$-axis and the $y$-axis. (You are not required to find the coordinates of the stationary points.)

AQA C1 2007 January Q2

2 The line $A B$ has equation $3 x + 5 y = 8$ and the point $A$ has coordinates (6, -2).

1. Find the gradient of $A B$.
2. Hence find an equation of the straight line which is perpendicular to $A B$ and which passes through $A$.
The line $A B$ intersects the line with equation $2 x + 3 y = 3$ at the point $B$. Find the coordinates of $B$.
The point $C$ has coordinates $( 2 , k )$ and the distance from $A$ to $C$ is 5 . Find the two possible values of the constant $k$.

AQA C1 2007 January Q3

Express $\frac { \sqrt { 5 } + 3 } { \sqrt { 5 } - 2 }$ in the form $p \sqrt { 5 } + q$, where $p$ and $q$ are integers.
1. Express $\sqrt { 45 }$ in the form $n \sqrt { 5 }$, where $n$ is an integer.
2. Solve the equation $$x \sqrt { 20 } = 7 \sqrt { 5 } - \sqrt { 45 }$$ giving your answer in its simplest form.

AQA C1 2007 January Q4

4 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0$.

By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
Write down:
1. the coordinates of $C$;
2. the radius of the circle.
Show that the circle does not intersect the $x$-axis.
The line with equation $x + y = 4$ intersects the circle at the points $P$ and $Q$.
1. Show that the $x$-coordinates of $P$ and $Q$ satisfy the equation $$x ^ { 2 } + 3 x - 10 = 0$$
2. Given that $P$ has coordinates (2,2), find the coordinates of $Q$.
3. Hence find the coordinates of the midpoint of $P Q$.

AQA C1 2007 January Q5

5 The diagram shows an open-topped water tank with a horizontal rectangular base and four vertical faces. The base has width $x$ metres and length $2 x$ metres, and the height of the tank is $h$ metres.
\includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-4_403_410_477_792} The combined internal surface area of the base and four vertical faces is $54 \mathrm {~m} ^ { 2 }$.

1. Show that $x ^ { 2 } + 3 x h = 27$.
2. Hence express $h$ in terms of $x$.
3. Hence show that the volume of water, $V \mathrm {~m} ^ { 3 }$, that the tank can hold when full is given by $$V = 18 x - \frac { 2 x ^ { 3 } } { 3 }$$
1. Find $\frac { \mathrm { d } V } { \mathrm {~d} x }$.
2. Verify that $V$ has a stationary value when $x = 3$.
Find $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$ and hence determine whether $V$ has a maximum value or a minimum value when $x = 3$.
(2 marks)

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