Questions - AQA C1 - A-Level Maths

1. State the coordinates of the point $B$ and hence find the area of the triangle $A O B$, where $O$ is the origin.
2. Find $\int \left( 3 x ^ { 5 } + 2 x + 5 \right) \mathrm { d } x$.
3. Hence find the area of the shaded region bounded by the curve and the line $A B$.
1. Find the gradient of the curve with equation $y = 3 x ^ { 5 } + 2 x + 5$ at the point $A ( - 1,0 )$.
2. Hence find an equation of the tangent to the curve at the point $A$.

AQA C1 2007 January Q7

7 The quadratic equation $( k + 1 ) x ^ { 2 } + 12 x + ( k - 4 ) = 0$ has real roots.

Show that $k ^ { 2 } - 3 k - 40 \leqslant 0$.
Hence find the possible values of $k$.

AQA C1 2008 January Q1

1 The triangle $A B C$ has vertices $A ( - 2,3 ) , B ( 4,1 )$ and $C ( 2 , - 5 )$.

Find the coordinates of the mid-point of $B C$.
1. Find the gradient of $A B$, in its simplest form.
2. Hence find an equation of the line $A B$, giving your answer in the form $x + q y = r$, where $q$ and $r$ are integers.
3. Find an equation of the line passing through $C$ which is parallel to $A B$.
Prove that angle $A B C$ is a right angle.

AQA C1 2008 January Q2

2 The curve with equation $y = x ^ { 4 } - 32 x + 5$ has a single stationary point, $M$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the $x$-coordinate of $M$.
1. Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
2. Hence, or otherwise, determine whether $M$ is a maximum or a minimum point.
Determine whether the curve is increasing or decreasing at the point on the curve where $x = 0$.

AQA C1 2008 January Q3

Express $5 \sqrt { 8 } + \frac { 6 } { \sqrt { 2 } }$ in the form $n \sqrt { 2 }$, where $n$ is an integer.
Express $\frac { \sqrt { 2 } + 2 } { 3 \sqrt { 2 } - 4 }$ in the form $c \sqrt { 2 } + d$, where $c$ and $d$ are integers.

AQA C1 2008 January Q4

4 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0$.

By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
Write down:
1. the coordinates of $C$;
2. the radius of the circle, leaving your answer in surd form.
A line has equation $y = 2 x$.
1. Show that the $x$-coordinate of any point of intersection of the line and the circle satisfies the equation $x ^ { 2 } - 4 x + 4 = 0$.
2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, $P$.
Prove that the point $Q ( - 1,4 )$ lies inside the circle.

AQA C1 2008 January Q5

Factorise $9 - 8 x - x ^ { 2 }$.
Show that $25 - ( x + 4 ) ^ { 2 }$ can be written as $9 - 8 x - x ^ { 2 }$.
A curve has equation $y = 9 - 8 x - x ^ { 2 }$.
1. Write down the equation of its line of symmetry.
2. Find the coordinates of its vertex.
3. Sketch the curve, indicating the values of the intercepts on the $x$-axis and the $y$-axis.

AQA C1 2008 January Q6

The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6$.
1. Use the Factor Theorem to show that $x + 1$ is a factor of $\mathrm { p } ( x )$.
2. Express $\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6$ as the product of three linear factors.
The curve with equation $y = x ^ { 3 } - 7 x - 6$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the $x$-axis at the point $A$ and the points $B ( - 1,0 )$ and $C ( 3,0 )$.
1. State the coordinates of the point $A$.
2. Find $\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x$.
3. Hence find the area of the shaded region bounded by the curve $y = x ^ { 3 } - 7 x - 6$ and the $x$-axis between $B$ and $C$.
4. Find the gradient of the curve $y = x ^ { 3 } - 7 x - 6$ at the point $B$.
5. Hence find an equation of the normal to the curve at the point $B$.

AQA C1 2008 January Q7

7 The curve $C$ has equation $y = x ^ { 2 } + 7$. The line $L$ has equation $y = k ( 3 x + 1 )$, where $k$ is a constant.

Show that the $x$-coordinates of any points of intersection of the line $L$ with the curve $C$ satisfy the equation $$x ^ { 2 } - 3 k x + 7 - k = 0$$
The curve $C$ and the line $L$ intersect in two distinct points. Show that $$9 k ^ { 2 } + 4 k - 28 > 0$$
Solve the inequality $9 k ^ { 2 } + 4 k - 28 > 0$.

AQA C1 2007 June Q1

1 The points $A$ and $B$ have coordinates $( 6 , - 1 )$ and $( 2,5 )$ respectively.

1. Show that the gradient of $A B$ is $- \frac { 3 } { 2 }$.
2. Hence find an equation of the line $A B$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
1. Find an equation of the line which passes through $B$ and which is perpendicular to the line $A B$.
2. The point $C$ has coordinates ( $k , 7$ ) and angle $A B C$ is a right angle. Find the value of the constant $k$.

AQA C1 2007 June Q2

Express $\frac { \sqrt { 63 } } { 3 } + \frac { 14 } { \sqrt { 7 } }$ in the form $n \sqrt { 7 }$, where $n$ is an integer.
Express $\frac { \sqrt { 7 } + 1 } { \sqrt { 7 } - 2 }$ in the form $p \sqrt { 7 } + q$, where $p$ and $q$ are integers.

AQA C1 2007 June Q3

1. Express $x ^ { 2 } + 10 x + 19$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
2. Write down the coordinates of the vertex (minimum point) of the curve with equation $y = x ^ { 2 } + 10 x + 19$.
3. Write down the equation of the line of symmetry of the curve $y = x ^ { 2 } + 10 x + 19$.
4. Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 10 x + 19$.
Determine the coordinates of the points of intersection of the line $y = x + 11$ and the curve $y = x ^ { 2 } + 10 x + 19$.

AQA C1 2007 June Q4

4 A model helicopter takes off from a point $O$ at time $t = 0$ and moves vertically so that its height, $y \mathrm {~cm}$, above $O$ after time $t$ seconds is given by $$y = \frac { 1 } { 4 } t ^ { 4 } - 26 t ^ { 2 } + 96 t , \quad 0 \leqslant t \leqslant 4$$

Find:
1. $\frac { \mathrm { d } y } { \mathrm {~d} t }$;
  (3 marks)
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$.
  (2 marks)
Verify that $y$ has a stationary value when $t = 2$ and determine whether this stationary value is a maximum value or a minimum value.
(4 marks)
Find the rate of change of $y$ with respect to $t$ when $t = 1$.
Determine whether the height of the helicopter above $O$ is increasing or decreasing at the instant when $t = 3$.

AQA C1 2007 June Q5

5 A circle with centre $C$ has equation $( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25$.

Write down:
1. the coordinates of $C$;
2. the radius of the circle.
1. Verify that the point $N ( 0 , - 2 )$ lies on the circle.
2. Sketch the circle.
3. Find an equation of the normal to the circle at the point $N$.
The point $P$ has coordinates (2, 6).
1. Find the distance $P C$, leaving your answer in surd form.
2. Find the length of a tangent drawn from $P$ to the circle.

AQA C1 2007 June Q6

The polynomial $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5$.
1. Use the Factor Theorem to show that $x - 1$ is a factor of $\mathrm { f } ( x )$.
2. Express $\mathrm { f } ( x )$ in the form $( x - 1 ) \left( x ^ { 2 } + p x + q \right)$, where $p$ and $q$ are integers.
3. Hence show that the equation $\mathrm { f } ( x ) = 0$ has exactly one real root and state its value.
The curve with equation $y = x ^ { 3 } + 4 x - 5$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the $x$-axis at the point $A ( 1,0 )$ and the point $B ( 2,11 )$ lies on the curve.
1. Find $\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x$.
2. Hence find the area of the shaded region bounded by the curve and the line $A B$.

AQA C1 2007 June Q7

7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where $k$ is a constant, has real roots.

Show that $2 k ^ { 2 } - 5 k + 2 \leqslant 0$.
1. Factorise $2 k ^ { 2 } - 5 k + 2$.
2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$

AQA C1 2008 June Q1

1 The straight line $L$ has equation $y = 3 x - 1$ and the curve $C$ has equation $$y = ( x + 3 ) ( x - 1 )$$

Sketch on the same axes the line $L$ and the curve $C$, showing the values of the intercepts on the $x$-axis and the $y$-axis.
Show that the $x$-coordinates of the points of intersection of $L$ and $C$ satisfy the equation $x ^ { 2 } - x - 2 = 0$.
Hence find the coordinates of the points of intersection of $L$ and $C$.

AQA C1 2008 June Q2

2 It is given that $x = \sqrt { 3 }$ and $y = \sqrt { 12 }$.
Find, in the simplest form, the value of:

$x y$;
$\frac { y } { x }$;
$( x + y ) ^ { 2 }$.

AQA C1 2008 June Q3

3 Two numbers, $x$ and $y$, are such that $3 x + y = 9$, where $x \geqslant 0$ and $y \geqslant 0$. It is given that $V = x y ^ { 2 }$.

Show that $V = 81 x - 54 x ^ { 2 } + 9 x ^ { 3 }$.
1. Show that $\frac { \mathrm { d } V } { \mathrm {~d} x } = k \left( x ^ { 2 } - 4 x + 3 \right)$, and state the value of the integer $k$.
2. Hence find the two values of $x$ for which $\frac { \mathrm { d } V } { \mathrm {~d} x } = 0$.
Find $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$.
1. Find the value of $\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }$ for each of the two values of $x$ found in part (b)(ii).
2. Hence determine the value of $x$ for which $V$ has a maximum value.
3. Find the maximum value of $V$.

AQA C1 2008 June Q4

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

AQA C1 2008 June Q5

5 The curve with equation $y = 16 - x ^ { 4 }$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-3_435_663_824_685} The points $A ( - 2,0 ) , B ( 2,0 )$ and $C ( 1,15 )$ lie on the curve.

Find an equation of the straight line $A C$.
1. Find $\int _ { - 2 } ^ { 1 } \left( 16 - x ^ { 4 } \right) \mathrm { d } x$.
2. Hence calculate the area of the shaded region bounded by the curve and the line $A C$.

AQA C1 2008 June Q6

6 The polynomial $\mathrm { p } ( x )$ is given by $\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12$.

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 $x + 2$ is a factor of $\mathrm { p } ( x )$.
2. Express $\mathrm { p } ( x )$ as the product of linear factors.
1. The curve with equation $y = x ^ { 3 } + x ^ { 2 } - 8 x - 12$ passes through the point $( 0 , k )$. State the value of $k$.
2. Sketch the graph of $y = x ^ { 3 } + x ^ { 2 } - 8 x - 12$, indicating the values of $x$ where the curve touches or crosses the $x$-axis.

AQA C1 2008 June Q7

7 The circle $S$ has centre $C ( 8,13 )$ and touches the $x$-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}

Write down an equation for $S$, giving your answer in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
The point $P$ with coordinates $( 3,1 )$ lies on the circle.
1. Find the gradient of the straight line passing through $P$ and $C$.
2. Hence find an equation of the tangent to the circle $S$ at the point $P$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
3. The point $Q$ also lies on the circle $S$, and the length of $P Q$ is 10 . Calculate the shortest distance from $C$ to the chord $P Q$.

AQA C1 2008 June Q8

8 The quadratic equation $( k + 1 ) x ^ { 2 } + 4 k x + 9 = 0$ has real roots.

Show that $4 k ^ { 2 } - 9 k - 9 \geqslant 0$.
Hence find the possible values of $k$.

AQA C1 2009 June Q1

1 The line $A B$ has equation $3 x + 5 y = 11$.

1. Find the gradient of $A B$.
2. The point $A$ has coordinates (2,1). Find an equation of the line which passes through the point $A$ and which is perpendicular to $A B$.
The line $A B$ intersects the line with equation $2 x + 3 y = 8$ at the point $C$. Find the coordinates of $C$.

Questions — AQA C1 (156 questions)

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