Questions - AQA 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 $t = 2$.
1. Verify that $V$ has a stationary value when $t = 1$.
2. Determine whether this is a maximum or minimum value.

AQA C1 Q8

25 marks

8 The diagram shows the curve with equation $y = 3 x ^ { 2 } - x ^ { 3 }$ and the line $L$.
\includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} 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 2005 January Q1

1 The point $A$ has coordinates $( 11,2 )$ and the point $B$ has coordinates $( - 1 , - 1 )$.

1. Find the gradient of $A B$.
2. Hence, or otherwise, show that the line $A B$ has equation $$x - 4 y = 3$$
The line with equation $3 x + 5 y = 26$ intersects the line $A B$ at the point $C$. Find the coordinates of $C$.

AQA C1 2005 January Q2

2 A curve has equation $y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
The point $P$ on the curve has coordinates $( 2,3 )$.
1. Show that the gradient of the curve at $P$ is 5 .
2. Hence find an equation of the normal to the curve at $P$, expressing your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
Determine whether $y$ is increasing or decreasing when $x = 1$.

AQA C1 2005 January Q3

3 A circle has equation $x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0$.

By completing the square, express the equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
Write down:
1. the coordinates of the centre of the circle;
2. the radius of the circle.
The line with equation $y = x + 4$ intersects the circle at the points $P$ and $Q$.
1. Show that the $x$-coordinates of $P$ and $Q$ satisfy the equation $$x ^ { 2 } - 5 x + 6 = 0$$
2. Find the coordinates of $P$ and $Q$.

AQA C1 2005 January Q4

The function f is defined for all values of $x$ by $\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8$.
1. Find the remainder when $\mathrm { f } ( x )$ is divided by $x + 1$.
2. Given that $\mathrm { f } ( 1 ) = 0$ and $\mathrm { f } ( - 2 ) = 0$, write down two linear factors of $\mathrm { f } ( x )$.
3. Hence express $x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8$ as the product of three linear factors.
The curve with equation $y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
1. The curve intersects the $y$-axis at the point $A$. Find the $y$-coordinate of $A$.
2. The curve crosses the $x$-axis when $x = - 2$, when $x = 1$ and also at the point $B$. Use the results from part (a) to find the $x$-coordinate of $B$.
1. Find $\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x$.
2. Hence find the area of the shaded region bounded by the curve and the $x$-axis.

AQA C1 2005 January Q5

Simplify $( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )$.
Express $\sqrt { 12 }$ in the form $m \sqrt { 3 }$, where $m$ is an integer.
Express $\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }$ in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.

AQA C1 2005 January Q6

6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm .
\includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side $x \mathrm {~cm}$ is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height $x \mathrm {~cm}$.

Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
1. Find $\frac { \mathrm { d } V } { \mathrm {~d} x }$.
2. Show that any stationary values of $V$ must occur when $x ^ { 2 } - 11 x + 18 = 0$.
3. Solve the equation $x ^ { 2 } - 11 x + 18 = 0$.
4. Explain why there is only one value of $x$ for which $V$ is stationary.
1. Find $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$.
2. Hence determine whether the stationary value is a maximum or minimum.

AQA C1 2005 January Q7

Simplify $( k + 5 ) ^ { 2 } - 12 k ( k + 2 )$.
The quadratic equation $3 ( k + 2 ) x ^ { 2 } + ( k + 5 ) x + k = 0$ has real roots.
1. Show that $( k - 1 ) ( 11 k + 25 ) \leqslant 0$.
2. Hence find the possible values of $k$.

AQA C1 2006 January Q1

Simplify $( \sqrt { 5 } + 2 ) ( \sqrt { 5 } - 2 )$.
Express $\sqrt { 8 } + \sqrt { 18 }$ in the form $n \sqrt { 2 }$, where $n$ is an integer.

AQA C1 2006 January Q2

2 The point $A$ has coordinates $( 1,1 )$ and the point $B$ has coordinates $( 5 , k )$. The line $A B$ has equation $3 x + 4 y = 7$.

1. Show that $k = - 2$.
2. Hence find the coordinates of the mid-point of $A B$.
Find the gradient of $A B$.
The line $A C$ is perpendicular to the line $A B$.
1. Find the gradient of $A C$.
2. Hence find an equation of the line $A C$.
3. Given that the point $C$ lies on the $x$-axis, find its $x$-coordinate.

AQA C1 2006 January Q3

1. Express $x ^ { 2 } - 4 x + 9$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
2. Hence, or otherwise, state the coordinates of the minimum point of the curve with equation $y = x ^ { 2 } - 4 x + 9$.
The line $L$ has equation $y + 2 x = 12$ and the curve $C$ has equation $y = x ^ { 2 } - 4 x + 9$.
1. Show that the $x$-coordinates of the points of intersection of $L$ and $C$ satisfy the equation $$x ^ { 2 } - 2 x - 3 = 0$$
2. Hence find the coordinates of the points of intersection of $L$ and $C$.

AQA C1 2006 January Q4

4 The quadratic equation $x ^ { 2 } + ( m + 4 ) x + ( 4 m + 1 ) = 0$, where $m$ is a constant, has equal roots.

Show that $m ^ { 2 } - 8 m + 12 = 0$.
Hence find the possible values of $m$.

AQA C1 2006 January Q5

5 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11$.

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.
The point $O$ has coordinates $( 0,0 )$.
1. Find the length of CO .
2. Hence determine whether the point $O$ lies inside or outside the circle, giving a reason for your answer.

AQA C1 2006 January Q6

6 The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$

1. Using the factor theorem, show that $x - 2$ is a factor of $\mathrm { p } ( x )$.
2. Hence express $\mathrm { p } ( x )$ as the product of three linear factors.
Sketch the curve with equation $y = x ^ { 3 } + x ^ { 2 } - 10 x + 8$, showing the coordinates of the points where the curve cuts the axes.
(You are not required to calculate the coordinates of the stationary points.)

AQA C1 2006 January Q7

7 The volume, $V \mathrm {~m} ^ { 3 }$, of water in a tank at time $t$ seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

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.

Questions — AQA C1 (156 questions)

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