Questions - AQA C1 - A-Level Maths

1. Find the gradient of $A B$.
2. Hence show that angle $A B C$ is a right angle.
1. Find the coordinates of $M$, the mid-point of $A C$.
2. Show that the lengths of $A B$ and $B C$ are equal.
3. Hence find an equation of the line of symmetry of the triangle $A B C$.

AQA C1 2010 January Q3

3 The depth of water, $y$ metres, in a tank after time $t$ hours is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - 2 t ^ { 2 } + 4 t , \quad 0 \leqslant t \leqslant 4$$

Find:
1. $\frac { \mathrm { d } y } { \mathrm {~d} t }$;
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$.
Verify that $y$ has a stationary value when $t = 2$ and determine whether it is a maximum value or a minimum value.
1. Find the rate of change of the depth of water, in metres per hour, when $t = 1$.
2. Hence determine, with a reason, whether the depth of water is increasing or decreasing when $t = 1$.

AQA C1 2010 January Q4

Show that $\frac { \sqrt { 50 } + \sqrt { 18 } } { \sqrt { 8 } }$ is an integer and find its value.
(3 marks)
Express $\frac { 2 \sqrt { 7 } - 1 } { 2 \sqrt { 7 } + 5 }$ in the form $m + n \sqrt { 7 }$, where $m$ and $n$ are integers.
(4 marks)

AQA C1 2010 January Q5

Express $( x - 5 ) ( x - 3 ) + 2$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
(3 marks)
1. Sketch the graph of $y = ( x - 5 ) ( x - 3 ) + 2$, stating the coordinates of the minimum point and the point where the graph crosses the $y$-axis.
2. Write down an equation of the tangent to the graph of $y = ( x - 5 ) ( x - 3 ) + 2$ at its vertex.
Describe the geometrical transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = ( x - 5 ) ( x - 3 ) + 2$.

AQA C1 2010 January Q6

6 The curve with equation $y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{2f7a8e95-4994-4732-a9a4-306c7b6cad92-3_444_819_1434_609} The curve crosses the $x$-axis at the origin $O$, and the point $A ( 2 , - 6 )$ lies on the curve.

1. Find the gradient of the curve with equation $y = 12 x ^ { 2 } - 19 x - 2 x ^ { 3 }$ at the point $A$.
2. Hence find the equation of the normal to the curve at the point $A$, giving your answer in the form $x + p y + q = 0$, where $p$ and $q$ are integers.
1. Find the value of $\int _ { 0 } ^ { 2 } \left( 12 x ^ { 2 } - 19 x - 2 x ^ { 3 } \right) \mathrm { d } x$.
2. Hence determine the area of the shaded region bounded by the curve and the line $O A$.

AQA C1 2010 January Q7

7 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 4 x + 12 y + 15 = 0$.

Find:
1. the coordinates of $C$;
2. the radius of the circle.
Explain why the circle lies entirely below the $x$-axis.
The point $P$ with coordinates $( 5 , k )$ lies outside the circle.
1. Show that $P C ^ { 2 } = k ^ { 2 } + 12 k + 45$.
2. Hence show that $k ^ { 2 } + 12 k + 20 > 0$.
3. Find the possible values of $k$.

AQA C1 2011 January Q1

1 The curve with equation $y = 13 + 18 x + 3 x ^ { 2 } - 4 x ^ { 3 }$ passes through the point $P$ where $x = - 1$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Show that the point $P$ is a stationary point of the curve and find the other value of $x$ where the curve has a stationary point.
1. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $P$.
2. Hence, or otherwise, determine whether $P$ is a maximum point or a minimum point.
  (l mark)

AQA C1 2011 January Q2

Simplify $( 3 \sqrt { 3 } ) ^ { 2 }$.
Express $\frac { 4 \sqrt { 3 } + 3 \sqrt { 7 } } { 3 \sqrt { 3 } + \sqrt { 7 } }$ in the form $\frac { m + \sqrt { 21 } } { n }$, where $m$ and $n$ are integers.

AQA C1 2011 January Q3

3 The line $A B$ has equation $3 x + 2 y = 7$. The point $C$ has coordinates $( 2 , - 7 )$.

1. Find the gradient of $A B$.
2. The line which passes through $C$ and which is parallel to $A B$ crosses the $y$-axis at the point $D$. Find the $y$-coordinate of $D$.
The line with equation $y = 1 - 4 x$ intersects the line $A B$ at the point $A$. Find the coordinates of $A$.
The point $E$ has coordinates $( 5 , k )$. Given that $C E$ has length 5 , find the two possible values of the constant $k$.

AQA C1 2011 January Q4

4 The curve sketched below passes through the point $A ( - 2,0 )$.
\includegraphics[max width=\textwidth, alt={}, center]{889639d6-0a31-4569-8370-1e72291a0c47-3_538_734_365_662} The curve has equation $y = 14 - x - x ^ { 4 }$ and the point $P ( 1,12 )$ lies on the curve.

1. Find the gradient of the curve at the point $P$.
2. Hence find the equation of the tangent to the curve at the point $P$, giving your answer in the form $y = m x + c$.
1. Find $\int _ { - 2 } ^ { 1 } \left( 14 - x - x ^ { 4 } \right) \mathrm { d } x$.
2. Hence find the area of the shaded region bounded by the curve $y = 14 - x - x ^ { 4 }$ and the line $A P$.
  (2 marks)

AQA C1 2011 January Q5

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

AQA C1 2011 January Q6

6 A circle has centre $C ( - 3,1 )$ and radius $\sqrt { 13 }$.

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Hence find the equation of the circle in the form $$x ^ { 2 } + y ^ { 2 } + m x + n y + p = 0$$ where $m , n$ and $p$ are integers.
The circle cuts the $y$-axis at the points $A$ and $B$. Find the distance $A B$.
1. Verify that the point $D ( - 5 , - 2 )$ lies on the circle.
2. Find the gradient of $C D$.
3. Hence find an equation of the tangent to the circle at the point $D$.

AQA C1 2011 January Q7

1. Express $4 - 10 x - x ^ { 2 }$ in the form $p - ( x + q ) ^ { 2 }$.
2. Hence write down the equation of the line of symmetry of the curve with equation $y = 4 - 10 x - x ^ { 2 }$.
The curve $C$ has equation $y = 4 - 10 x - x ^ { 2 }$ and the line $L$ has equation $y = k ( 4 x - 13 )$, where $k$ is a constant.
1. Show that the $x$-coordinates of any points of intersection of the curve $C$ with the line $L$ satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
2. Given that the curve $C$ and the line $L$ intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
3. Solve the inequality $4 k ^ { 2 } + 33 k + 29 > 0$.

AQA C1 2012 January Q1

1 The point $A$ has coordinates (6, -4) and the point $B$ has coordinates (-2, 7).

Given that the point $O$ has coordinates $( 0,0 )$, show that the length of $O A$ is less than the length of $O B$.
1. Find the gradient of $A B$.
2. Find an equation of the line $A B$ in the form $p x + q y = r$, where $p , q$ and $r$ are integers.
The point $C$ has coordinates $( k , 0 )$. The line $A C$ is perpendicular to the line $A B$. Find the value of the constant $k$.

AQA C1 2012 January Q2

Factorise $x ^ { 2 } - 4 x - 12$.
Sketch the graph with equation $y = x ^ { 2 } - 4 x - 12$, stating the values where the curve crosses the coordinate axes.
1. Express $x ^ { 2 } - 4 x - 12$ in the form $( x - p ) ^ { 2 } - q$, where $p$ and $q$ are positive integers.
2. Hence find the minimum value of $x ^ { 2 } - 4 x - 12$.
The curve with equation $y = x ^ { 2 } - 4 x - 12$ is translated by the vector $\left[ \begin{array} { r } - 3
2 \end{array} \right]$. Find an equation of the new curve. You need not simplify your answer.

AQA C1 2012 January Q3

1. Simplify $( 3 \sqrt { 2 } ) ^ { 2 }$.
2. Show that $( 3 \sqrt { 2 } - 1 ) ^ { 2 } + ( 3 + \sqrt { 2 } ) ^ { 2 }$ is an integer and find its value.
Express $\frac { 4 \sqrt { 5 } - 7 \sqrt { 2 } } { 2 \sqrt { 5 } + \sqrt { 2 } }$ in the form $m - \sqrt { n }$, where $m$ and $n$ are integers.

AQA C1 2012 January Q4

4 The curve with equation $y = x ^ { 5 } - 3 x ^ { 2 } + x + 5$ is sketched below. The point $O$ is at the origin and the curve passes through the points $A ( - 1,0 )$ and $B ( 1,4 )$.
\includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_447_752_438_653}

Given that $y = x ^ { 5 } - 3 x ^ { 2 } + x + 5$, find:
1. $\frac { \mathrm { d } y } { \mathrm {~d} x }$;
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Find an equation of the tangent to the curve at the point $A ( - 1,0 )$.
Verify that the point $B$, where $x = 1$, is a minimum point of the curve.
The curve with equation $y = x ^ { 5 } - 3 x ^ { 2 } + x + 5$ is sketched below. The point $O$ is at the origin and the curve passes through the points $A ( - 1,0 )$ and $B ( 1,4 )$.
\includegraphics[max width=\textwidth, alt={}, center]{91170a77-e266-4c81-89ee-1fc29a538485-3_451_757_1736_648}
1. Find $\int _ { - 1 } ^ { 1 } \left( x ^ { 5 } - 3 x ^ { 2 } + x + 5 \right) \mathrm { d } x$.
2. Hence find the area of the shaded region bounded by the curve between $A$ and $B$ and the line segments $A O$ and $O B$.

AQA C1 2012 January Q5

5 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } + c x ^ { 2 } + d x - 12$, where $c$ and $d$ are constants.

When $\mathrm { p } ( x )$ is divided by $x + 2$, the remainder is - 150 . Show that $2 c - d + 65 = 0$.
Given that $x - 3$ is a factor of $\mathrm { p } ( x )$, find another equation involving $c$ and $d$.
By solving these two equations, find the value of $c$ and the value of $d$.

AQA C1 2012 January Q6

6 A rectangular garden is to have width $x$ metres and length $( x + 4 )$ metres.

The perimeter of the garden needs to be greater than 30 metres. Show that $2 x > 11$.
The area of the garden needs to be less than 96 square metres. Show that $x ^ { 2 } + 4 x - 96 < 0$.
Solve the inequality $x ^ { 2 } + 4 x - 96 < 0$.
Hence determine the possible values of the width of the garden.
$7 \quad$ A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } + 14 x - 10 y + 49 = 0$.
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.
Sketch the circle.
A line has equation $y = k x + 6$, where $k$ is a constant.
1. Show that the $x$-coordinates of any points of intersection of the line and the circle satisfy the equation $\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0$.
2. The equation $\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( k + 7 ) x + 25 = 0$ has equal roots. Show that $$12 k ^ { 2 } - 7 k - 12 = 0$$
3. Hence find the values of $k$ for which the line is a tangent to the circle.

AQA C1 2013 January Q1

1 The point $A$ has coordinates $( - 3,2 )$ and the point $B$ has coordinates $( 7 , k )$.
The line $A B$ has equation $3 x + 5 y = 1$.

1. Show that $k = - 4$.
2. Hence find the coordinates of the midpoint of $A B$.
Find the gradient of $A B$.
A line which passes through the point $A$ is perpendicular to the line $A B$. Find an equation of this line, giving your answer in the form $p x + q y + r = 0$, where $p , q$ and $r$ are integers.
The line $A B$, with equation $3 x + 5 y = 1$, intersects the line $5 x + 8 y = 4$ at the point $C$. Find the coordinates of $C$.

AQA C1 2013 January Q2

2 A bird flies from a tree. At time $t$ seconds, the bird's height, $y$ metres, above the horizontal ground is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - t ^ { 2 } + 5 , \quad 0 \leqslant t \leqslant 4$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} t }$.
1. Find the rate of change of height of the bird in metres per second when $t = 1$.
2. Determine, with a reason, whether the bird's height above the horizontal ground is increasing or decreasing when $t = 1$.
1. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$ when $t = 2$.
2. Given that $y$ has a stationary value when $t = 2$, state whether this is a maximum value or a minimum value.

AQA C1 2013 January Q3

1. Express $\sqrt { 18 }$ in the form $k \sqrt { 2 }$, where $k$ is an integer.
2. Simplify $\frac { \sqrt { 8 } } { \sqrt { 18 } + \sqrt { 32 } }$.
Express $\frac { 7 \sqrt { 2 } - \sqrt { 3 } } { 2 \sqrt { 2 } - \sqrt { 3 } }$ in the form $m + \sqrt { n }$, where $m$ and $n$ are integers.

AQA C1 2013 January Q4

1. Express $x ^ { 2 } - 6 x + 11$ in the form $( x - p ) ^ { 2 } + q$.
2. Use the result from part (a)(i) to show that the equation $x ^ { 2 } - 6 x + 11 = 0$ has no real solutions.
A curve has equation $y = x ^ { 2 } - 6 x + 11$.
1. Find the coordinates of the vertex of the curve.
2. Sketch the curve, indicating the value of $y$ where the curve crosses the $y$-axis.
3. Describe the geometrical transformation that maps the curve with equation $y = x ^ { 2 } - 6 x + 11$ onto the curve with equation $y = x ^ { 2 }$.

AQA C1 2013 January Q5

5 The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$

Use the Remainder Theorem to find the remainder when $\mathrm { p } ( x )$ is divided by $x + 1$.
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 linear factors.
Sketch the curve with equation $y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$, stating the values of $x$ where the curve meets the $x$-axis.

AQA C1 2013 January Q6

6 The gradient, $\frac { \mathrm { d } y } { \mathrm {~d} x }$, of a curve at the point $( x , y )$ is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 10 x ^ { 4 } - 6 x ^ { 2 } + 5$$ The curve passes through the point $P ( 1,4 )$.

Find the equation of the tangent to the curve at the point $P$, giving your answer in the form $y = m x + c$.
Find the equation of the curve.

Questions — AQA C1 (156 questions)

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