Questions - AQA C1 - A-Level Maths

Find the $y$-coordinates of the points where the circle crosses the $y$-axis.
Find the radius of the circle.
The point $P ( 2,5 )$ lies outside the circle.
1. Find the length of $C P$, giving your answer in the form $\sqrt { n }$, where $n$ is an integer.
2. The point $Q$ lies on the circle so that $P Q$ is a tangent to the circle. Find the length of $P Q$.

AQA C1 2013 January Q8

8 A curve has equation $y = 2 x ^ { 2 } - x - 1$ and a line has equation $y = k ( 2 x - 3 )$, where $k$ is a constant.

Show that the $x$-coordinate of any point of intersection of the curve and the line satisfies the equation $$2 x ^ { 2 } - ( 2 k + 1 ) x + 3 k - 1 = 0$$
The curve and the line intersect at two distinct points.
1. Show that $4 k ^ { 2 } - 20 k + 9 > 0$.
2. Find the possible values of $k$.

AQA C1 2005 June Q1

1 The point $A$ has coordinates $( 6,5 )$ and the point $B$ has coordinates $( 2 , - 1 )$.

Find the coordinates of the midpoint of $A B$.
Show that $A B$ has length $k \sqrt { 13 }$, where $k$ is an integer.
1. Find the gradient of the line $A B$.
2. Hence, or otherwise, show that the line $A B$ has equation $3 x - 2 y = 8$.
The line $A B$ intersects the line with equation $2 x + y = 10$ at the point $C$. Find the coordinates of $C$.

AQA C1 2005 June Q2

Express $x ^ { 2 } - 6 x + 16$ in the form $( x - p ) ^ { 2 } + q$.
A curve has equation $y = x ^ { 2 } - 6 x + 16$. Using your answer from part (a), or otherwise:
1. find the coordinates of the vertex (minimum point) of the curve;
2. sketch the curve, indicating the value where the curve crosses the $y$-axis;
3. state the equation of the line of symmetry of the curve.
Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } - 6 x + 16$.

AQA C1 2005 June Q3

3 A circle has centre $C ( 2 , - 1 )$ and radius 5 . The point $P$ has coordinates $( 6,2 )$.

Write down an equation of the circle.
Verify that the point $P$ lies on the circle.
Find the gradient of the line $C P$.
1. Find the gradient of a line which is perpendicular to $C P$.
2. Hence find an equation for the tangent to the circle at the point $P$.

AQA C1 2005 June Q4

4 The curve with equation $y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{3729de55-7139-4f41-8584-640f173c0e09-3_444_588_411_717} The curve touches the $x$-axis at the point $A ( 1,0 )$ and cuts the $x$-axis at the point $B$.

1. Use the factor theorem to show that $x - 3$ is a factor of $$\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$$
2. Hence find the coordinates of $B$.
The point $M$, shown on the diagram, is a minimum point of the curve with equation $y = x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Hence determine the $x$-coordinate of $M$.
Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 1$.
1. Find $\int \left( x ^ { 3 } - 5 x ^ { 2 } + 7 x - 3 \right) \mathrm { d } x$.
2. Hence determine the area of the shaded region bounded by the curve and the coordinate axes.

AQA C1 2005 June Q5

5 Express each of the following in the form $m + n \sqrt { 3 }$, where $m$ and $n$ are integers:

$( \sqrt { 3 } + 1 ) ^ { 2 }$;
$\frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }$.

AQA C1 2005 June Q6

6 The cubic polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = ( x - 2 ) \left( x ^ { 2 } + x + 3 \right)$.

Show that $\mathrm { p } ( x )$ can be written in the form $x ^ { 3 } + a x ^ { 2 } + b x - 6$, where $a$ and $b$ are constants whose values are to be found.
Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x + 1$.
(2 marks)
Prove that the equation $( x - 2 ) \left( x ^ { 2 } + x + 3 \right) = 0$ has only one real root and state its value.
(3 marks)

AQA C1 2005 June Q7

7 Solve each of the following inequalities:

$3 ( x - 1 ) > 3 - 5 ( x + 6 )$;
$\quad x ^ { 2 } - x - 6 < 0$.

AQA C1 2005 June Q8

8 A line has equation $y = m x - 1$, where $m$ is a constant.
A curve has equation $y = x ^ { 2 } - 5 x + 3$.

Show that the $x$-coordinate of any point of intersection of the line and the curve satisfies the equation $$x ^ { 2 } - ( 5 + m ) x + 4 = 0$$
Find the values of $m$ for which the equation $x ^ { 2 } - ( 5 + m ) x + 4 = 0$ has equal roots.
(4 marks)
Describe geometrically the situation when $m$ takes either of the values found in part (b).
(1 mark)

AQA C1 2006 June Q1

1 The point $A$ has coordinates $( 1,7 )$ and the point $B$ has coordinates $( 5,1 )$.

1. Find the gradient of the line $A B$.
2. Hence, or otherwise, show that the line $A B$ has equation $3 x + 2 y = 17$.
The line $A B$ intersects the line with equation $x - 4 y = 8$ at the point $C$. Find the coordinates of $C$.
Find an equation of the line through $A$ which is perpendicular to $A B$.

AQA C1 2006 June Q2

Express $x ^ { 2 } + 8 x + 19$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
Hence, or otherwise, show that the equation $x ^ { 2 } + 8 x + 19 = 0$ has no real solutions.
Sketch the graph of $y = x ^ { 2 } + 8 x + 19$, stating the coordinates of the minimum point and the point where the graph crosses the $y$-axis.
Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 8 x + 19$.

AQA C1 2006 June Q3

3 A curve has equation $y = 7 - 2 x ^ { 5 }$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find an equation for the tangent to the curve at the point where $x = 1$.
Determine whether $y$ is increasing or decreasing when $x = - 2$.

AQA C1 2006 June Q4

Express $( 4 \sqrt { 5 } - 1 ) ( \sqrt { 5 } + 3 )$ in the form $p + q \sqrt { 5 }$, where $p$ and $q$ are integers.
Show that $\frac { \sqrt { 75 } - \sqrt { 27 } } { \sqrt { 3 } }$ is an integer and find its value.

AQA C1 2006 June Q5

5 The curve with equation $y = x ^ { 3 } - 10 x ^ { 2 } + 28 x$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568} The curve crosses the $x$-axis at the origin $O$ and the point $A ( 3,21 )$ lies on the curve.

1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Hence show that the curve has a stationary point when $x = 2$ and find the $x$-coordinate of the other stationary point.
1. Find $\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x$.
2. Hence show that $\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }$.
3. Hence determine the area of the shaded region bounded by the curve and the line $O A$.

AQA C1 2006 June Q6

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

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

AQA C1 2006 June Q7

7 A circle has equation $x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0$.

Find:
1. the coordinates of the centre of the circle;
2. the radius of the circle in the form $p \sqrt { 2 }$, where $p$ is an integer.
A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
A line has equation $y = 2 k - x$, where $k$ is a constant.
1. Show that the $x$-coordinate of any point of intersection of the line and the circle satisfies the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
2. Find the values of $k$ for which the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$ has equal roots.
3. Describe the geometrical relationship between the line and the circle when $k$ takes either of the values found in part (c)(ii).

AQA C1 2010 June Q1

1 The trapezium $A B C D$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-2_298_591_557_737} The line $A B$ has equation $2 x + 3 y = 14$ and $D C$ is parallel to $A B$.

Find the gradient of $A B$.
The point $D$ has coordinates $( 3,7 )$.
1. Find an equation of the line $D C$.
2. The angle $B A D$ is a right angle. Find an equation of the line $A D$, giving your answer in the form $m x + n y + p = 0$, where $m , n$ and $p$ are integers.
The line $B C$ has equation $5 y - x = 6$. Find the coordinates of $B$.

AQA C1 2010 June Q2

Express $( 3 - \sqrt { 5 } ) ^ { 2 }$ in the form $m + n \sqrt { 5 }$, where $m$ and $n$ are integers.
Hence express $\frac { ( 3 - \sqrt { 5 } ) ^ { 2 } } { 1 + \sqrt { 5 } }$ in the form $p + q \sqrt { 5 }$, where $p$ and $q$ are integers.
(4 marks)

AQA C1 2010 June Q3

3 The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15$$

1. Use the Factor Theorem to show that $x + 3$ is a factor of $\mathrm { p } ( x )$.
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 $x - 2$.
1. Verify that $\mathrm { p } ( - 1 ) < \mathrm { p } ( 0 )$.
2. Sketch the curve with equation $y = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15$, indicating the values where the curve crosses the coordinate axes.

AQA C1 2010 June Q4

4 The curve with equation $y = x ^ { 4 } - 8 x + 9$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-5_410_609_383_721} The point $( 2,9 )$ lies on the curve.

1. Find $\int _ { 0 } ^ { 2 } \left( x ^ { 4 } - 8 x + 9 \right) \mathrm { d } x$.
2. Hence find the area of the shaded region bounded by the curve and the line $y = 9$.
The point $A ( 1,2 )$ lies on the curve with equation $y = x ^ { 4 } - 8 x + 9$.
1. Find the gradient of the curve at the point $A$.
2. Hence find an equation of the tangent to the curve at the point $A$.

AQA C1 2010 June Q5

5 A circle with centre $C ( - 5,6 )$ touches the $y$-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
1. Verify that the point $P ( - 2,2 )$ lies on the circle.
2. Find an equation of the normal to the circle at the point $P$.
3. The mid-point of $P C$ is $M$. Determine whether the point $P$ is closer to the point $M$ or to the origin $O$.

AQA C1 2010 June Q6

6 The diagram shows a block of wood in the shape of a prism with triangular cross-section. The end faces are right-angled triangles with sides of lengths $3 x \mathrm {~cm}$, $4 x \mathrm {~cm}$ and $5 x \mathrm {~cm}$, and the length of the prism is $y \mathrm {~cm}$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-7_394_825_459_548} The total surface area of the five faces is $144 \mathrm {~cm} ^ { 2 }$.

1. Show that $x y + x ^ { 2 } = 12$.
2. Hence show that the volume of the block, $V \mathrm {~cm} ^ { 3 }$, is given by $$V = 72 x - 6 x ^ { 3 }$$
1. Find $\frac { \mathrm { d } V } { \mathrm {~d} x }$.
2. Show that $V$ has a stationary value when $x = 2$.
Find $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$ and hence determine whether $V$ has a maximum value or a minimum value when $x = 2$.
(2 marks)

AQA C1 2010 June Q7

1. Express $2 x ^ { 2 } - 20 x + 53$ in the form $2 ( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
2. Use your result from part (a)(i) to explain why the equation $2 x ^ { 2 } - 20 x + 53 = 0$ has no real roots.
The quadratic equation $( 2 k - 1 ) x ^ { 2 } + ( k + 1 ) x + k = 0$ has real roots.
1. Show that $7 k ^ { 2 } - 6 k - 1 \leqslant 0$.
2. Hence find the possible values of $k$.

AQA C1 2011 June Q1

1 The line $A B$ has equation $7 x + 3 y = 13$.

Find the gradient of $A B$.
The point $C$ has coordinates $( - 1,3 )$.
1. Find an equation of the line which passes through the point $C$ and which is parallel to $A B$.
2. The point $\left( 1 \frac { 1 } { 2 } , - 1 \right)$ is the mid-point of $A C$. Find the coordinates of the point $A$.
The line $A B$ intersects the line with equation $3 x + 2 y = 12$ at the point $B$. Find the coordinates of $B$.

Questions — AQA C1 (156 questions)

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