Questions C1 - A-Level Maths

Find:
1. $\frac { \mathrm { d } V } { \mathrm {~d} t }$;
  (3 marks)
2. $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }$.
  (2 marks)
Find the rate of change of the volume of water in the tank, in $\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }$, when $t = 2$.
(2 marks)
1. Verify that $V$ has a stationary value when $t = 1$.
  (2 marks)
2. Determine whether this is a maximum or minimum value.
  (2 marks)

AQA C1 2006 January Q8

8 The diagram shows the curve with equation $y = 3 x ^ { 2 } - x ^ { 3 }$ and the line $L$.
\includegraphics[max width=\textwidth, alt={}, center]{81f6fc30-982b-47b5-bab3-076cc0cc6563-5_479_816_406_596} The points $A$ and $B$ have coordinates $( - 1,0 )$ and $( 2,0 )$ respectively. The curve touches the $x$-axis at the origin $O$ and crosses the $x$-axis at the point $( 3,0 )$. The line $L$ cuts the curve at the point $D$ where $x = - 1$ and touches the curve at $C$ where $x = 2$.

Find the area of the rectangle $A B C D$.
1. Find $\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x$.
2. Hence find the area of the shaded region bounded by the curve and the line $L$.
For the curve above with equation $y = 3 x ^ { 2 } - x ^ { 3 }$ :
1. find $\frac { \mathrm { d } y } { \mathrm {~d} x }$;
2. hence find an equation of the tangent at the point on the curve where $x = 1$;
3. show that $y$ is decreasing when $x ^ { 2 } - 2 x > 0$.
Solve the inequality $x ^ { 2 } - 2 x > 0$.

AQA C1 2009 January Q1

1 The points $A$ and $B$ have coordinates $( 1,6 )$ and $( 5 , - 2 )$ respectively. The mid-point of $A B$ is $M$.

Find the coordinates of $M$.
Find the gradient of $A B$, giving your answer in its simplest form.
A straight line passes through $M$ and is perpendicular to $A B$.
1. Show that this line has equation $x - 2 y + 1 = 0$.
2. Given that this line passes through the point $( k , k + 5 )$, find the value of the constant $k$.

AQA C1 2009 January Q2

Factorise $2 x ^ { 2 } - 5 x + 3$.
Hence, or otherwise, solve the inequality $2 x ^ { 2 } - 5 x + 3 < 0$.

AQA C1 2009 January Q3

Express $\frac { 7 + \sqrt { 5 } } { 3 + \sqrt { 5 } }$ in the form $m + n \sqrt { 5 }$, where $m$ and $n$ are integers.
Express $\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }$ in the form $k \sqrt { 5 }$, where $k$ is an integer.

AQA C1 2009 January Q4

1. Express $x ^ { 2 } + 2 x + 5$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
2. Hence show that $x ^ { 2 } + 2 x + 5$ is always positive.
A curve has equation $y = x ^ { 2 } + 2 x + 5$.
1. Write down the coordinates of the minimum point of the curve.
2. Sketch the curve, showing the value of the intercept on the $y$-axis.
Describe the geometrical transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 2 x + 5$.

AQA C1 2009 January Q5

5 A model car moves so that its distance, $x$ centimetres, from a fixed point $O$ after time $t$ seconds is given by $$x = \frac { 1 } { 2 } t ^ { 4 } - 20 t ^ { 2 } + 66 t , \quad 0 \leqslant t \leqslant 4$$

Find:
1. $\frac { \mathrm { d } x } { \mathrm {~d} t }$;
2. $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } }$.
Verify that $x$ has a stationary value when $t = 3$, and determine whether this stationary value is a maximum value or a minimum value.
Find the rate of change of $x$ with respect to $t$ when $t = 1$.
Determine whether the distance of the car from $O$ is increasing or decreasing at the instant when $t = 2$.

AQA C1 2009 January Q6

The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } + x - 10$.
1. Use the Factor Theorem to show that $x - 2$ is a factor of $\mathrm { p } ( x )$.
2. Express $\mathrm { p } ( x )$ in the form $( x - 2 ) \left( x ^ { 2 } + a x + b \right)$, where $a$ and $b$ are constants.
The curve $C$ with equation $y = x ^ { 3 } + x - 10$, sketched below, crosses the $x$-axis at the point $Q ( 2,0 )$.
\includegraphics[max width=\textwidth, alt={}, center]{22c93dd5-d96a-4e31-8507-9c802e386231-3_444_547_1781_756}
1. Find the gradient of the curve $C$ at the point $Q$.
2. Hence find an equation of the tangent to the curve $C$ at the point $Q$.
3. Find $\int \left( x ^ { 3 } + x - 10 \right) \mathrm { d } x$.
4. Hence find the area of the shaded region bounded by the curve $C$ and the coordinate axes.

AQA C1 2009 January Q7

7 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$.

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.
The point $D$ has coordinates (7, -2).
1. Verify that the point $D$ lies on the circle.
2. Find an equation of the normal to the circle at the point $D$, giving your answer in the form $m x + n y = p$, where $m , n$ and $p$ are integers.
1. A line has equation $y = k x$. Show that the $x$-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
2. Find the values of $k$ for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
3. Describe the geometrical relationship between the line and the circle when $k$ takes either of the values found in part (d)(ii).

AQA C1 2010 January Q1

1 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 13 x - 12$.

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.

AQA C1 2010 January Q2

2 The triangle $A B C$ has vertices $A ( 1,3 ) , B ( 3,7 )$ and $C ( - 1,9 )$.

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.

Questions C1 (1442 questions)

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