1.03b - OCR Spec

AQA C1 2010 June Q5

11 marks Moderate -0.8

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 2011 June Q1

9 marks Moderate -0.8

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 $B$.

AQA C1 2011 June Q8

13 marks Moderate -0.3

8 A circle has centre $C ( 3 , - 8 )$ and radius 10 .

Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
Find the $x$-coordinates of the points where the circle crosses the $x$-axis.
The tangent to the circle at the point $A$ has gradient $\frac { 5 } { 2 }$. Find an equation of the line $C A$, giving your answer in the form $r x + s y + t = 0$, where $r , s$ and $t$ are integers.
The line with equation $y = 2 x + 1$ intersects the circle.
1. Show that the $x$-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + 6 x - 2 = 0$$
2. Hence show that the $x$-coordinates of the points of intersection are of the form $m \pm \sqrt { n }$, where $m$ and $n$ are integers.

AQA C1 2012 June Q2

10 marks Moderate -0.8

2 The line $A B$ has equation $4 x - 3 y = 7$.

1. Find the gradient of $A B$.
2. Find an equation of the straight line that is parallel to $A B$ and which passes through the point $C ( 3 , - 5 )$, giving your answer in the form $p x + q y = r$, where $p , q$ and $r$ are integers.
The line $A B$ intersects the line with equation $3 x - 2 y = 4$ at the point $D$. Find the coordinates of $D$.
The point $E$ with coordinates $( k - 2,2 k - 3 )$ lies on the line $A B$. Find the value of the constant $k$.

AQA C1 2012 June Q7

15 marks Moderate -0.8

7 The gradient, $\frac { \mathrm { d } y } { \mathrm {~d} x }$, of a curve $C$ at the point $( x , y )$ is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 20 x - 6 x ^ { 2 } - 16$$

1. Show that $y$ is increasing when $3 x ^ { 2 } - 10 x + 8 < 0$.
2. Solve the inequality $3 x ^ { 2 } - 10 x + 8 < 0$.
The curve $C$ passes through the point $P ( 2,3 )$.
1. Verify that the tangent to the curve at $P$ is parallel to the $x$-axis.
2. The point $Q ( 3 , - 1 )$ also lies on the curve. The normal to the curve at $Q$ and the tangent to the curve at $P$ intersect at the point $R$. Find the coordinates of $R$.
  (7 marks)

AQA C1 2013 June Q1

10 marks Moderate -0.3

1 The line $A B$ has equation $3 x - 4 y + 5 = 0$.

The point with coordinates $( p , p + 2 )$ lies on the line $A B$. Find the value of the constant $p$.
Find the gradient of $A B$.
The point $A$ has coordinates ( 1,2 ). The point $C ( - 5 , k )$ is such that $A C$ is perpendicular to $A B$. Find the value of $k$.
The line $A B$ intersects the line with equation $2 x - 5 y = 6$ at the point $D$. Find the coordinates of $D$.

AQA C1 2014 June Q1

13 marks Standard +0.3

1 The point $A$ has coordinates $( - 1,2 )$ and the point $B$ has coordinates $( 3 , - 5 )$.

1. Find the gradient of $A B$.
2. Hence find an equation of the line $A B$, giving your answer in the form $p x + q y = r$, where $p , q$ and $r$ are integers.
The midpoint of $A B$ is $M$.
1. Find the coordinates of $M$.
2. Find an equation of the line which passes through $M$ and which is perpendicular to $A B$. [3 marks]
The point $C$ has coordinates $( k , 2 k + 3 )$. Given that the distance from $A$ to $C$ is $\sqrt { 13 }$, find the two possible values of the constant $k$.
[0pt] [4 marks]

AQA C1 2015 June Q1

8 marks Moderate -0.8

1 The line $A B$ has equation $3 x + 5 y = 7$.

Find the gradient of $A B$.
Find an equation of the line that is perpendicular to the line $A B$ and which passes through the point $( - 2 , - 3 )$. Express your answer in the form $p x + q y + r = 0$, where $p , q$ and $r$ are integers.
The line $A C$ has equation $2 x - 3 y = 30$. Find the coordinates of $A$.

AQA C1 2016 June Q1

7 marks Moderate -0.8

1 The line $A B$ has equation $5 x + 3 y + 3 = 0$.

The line $A B$ is parallel to the line with equation $y = m x + 7$. Find the value of $m$.
The line $A B$ intersects the line with equation $3 x - 2 y + 17 = 0$ at the point $B$. Find the coordinates of $B$.
The point with coordinates $( 2 k + 3,4 - 3 k )$ lies on the line $A B$. Find the value of $k$.
[0pt] [2 marks]

Edexcel C1 Q5

10 marks Moderate -0.8

5. The points $A$ and $B$ have coordinates $( 4,6 )$ and $( 12,2 )$ respectively. The straight line $l _ { 1 }$ passes through $A$ and $B$.

Find an equation for $l _ { 1 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integers. The straight line $l _ { 2 }$ passes through the origin and has gradient - 4 .
Write down an equation for $l _ { 2 }$. The lines $l _ { 1 }$ and $l _ { 2 }$ intercept at the point $C$.
Find the exact coordinates of the mid-point of $A C$.

Edexcel C1 Q9

12 marks Standard +0.3

9. The points $A ( - 1 , - 2 ) , B ( 7,2 )$ and $C ( k , 4 )$, where $k$ is a constant, are the vertices of $\triangle A B C$. Angle $A B C$ is a right angle.

Find the gradient of $A B$.
Calculate the value of $k$.
Show that the length of $A B$ may be written in the form $p \sqrt { 5 }$, where $p$ is an integer to be found.
Find the exact value of the area of $\triangle A B C$.
Find an equation for the straight line $l$ passing through $B$ and $C$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C1 Q8

12 marks Standard +0.3

8. The points $A ( - 1 , - 2 ) , B ( 7,2 )$ and $C ( k , 4 )$, where $k$ is a constant, are the vertices of $\triangle A B C$. Angle $A B C$ is a right angle.

Find the gradient of $A B$.
Calculate the value of $k$.
Show that the length of $A B$ may be written in the form $p \sqrt { 5 }$, where $p$ is an integer to be found.
Find the exact value of the area of $\triangle A B C$.
Find an equation for the straight line $l$ passing through $B$ and $C$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C1 Q7

13 marks Moderate -0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8bae58f7-c53a-43ed-9a1d-2f718bd1e539-3_563_570_785_561} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A ( - 3 , - 2 )$ and $B ( 8,4 )$ are at the ends of a diameter of the circle shown in Fig. 1.

Find the coordinates of the centre of the circle.
Find an equation of the diameter $A B$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Find an equation of tangent to the circle at $B$. The line $l$ passes through $A$ and the origin.
Find the coordinates of the point at which $l$ intersects the tangent to the circle at $B$, giving your answer as exact fractions.

Edexcel C1 Q7

8 marks Moderate -0.8

7. The straight line $l _ { 1 }$ has gradient 2 and passes through the point with coordinates $( 4 , - 5 )$.

Find an equation for $l _ { 1 }$ in the form $y = m x + c$. The straight line $l _ { 2 }$ is perpendicular to the line with equation $3 x - y = 4$ and passes through the point with coordinates $( 3,0 )$.
Find an equation for $l _ { 2 }$.
Find the coordinates of the point where $l _ { 1 }$ and $l _ { 2 }$ intersect.

Edexcel C1 Q8

11 marks Standard +0.3

8. The straight line $l _ { 1 }$ passes through the point $A ( - 2,5 )$ and the point $B ( 4,1 )$.

Find an equation for $l _ { 1 }$ in the form $a x + b y = c$, where $a , b$ and $c$ are integers. The straight line $l _ { 2 }$ passes through $B$ and is perpendicular to $l _ { 1 }$.
Find an equation for $l _ { 2 }$. Given that $l _ { 2 }$ meets the $y$-axis at the point $C$,
show that triangle $A B C$ is isosceles.

Edexcel C1 Q10

13 marks Standard +0.3

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5e8f154b-c232-49ee-a798-f61ff08ca0b9-4_663_1113_950_402} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the parallelogram $A B C D$.
The points $A$ and $B$ have coordinates $( - 1,3 )$ and $( 3,4 )$ respectively and lie on the straight line $l _ { 1 }$.

Find an equation for $l _ { 1 }$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The points $C$ and $D$ lie on the straight line $l _ { 2 }$ which has the equation $x - 4 y - 21 = 0$.
Show that the distance between $l _ { 1 }$ and $l _ { 2 }$ is $k \sqrt { 17 }$, where $k$ is an integer to be found.
Find the area of parallelogram $A B C D$.

Edexcel C1 Q1

4 marks Easy -1.2

The points $A , B$ and $C$ have coordinates $( - 3,0 ) , ( 5 , - 2 )$ and $( 4,1 )$ respectively.

Find an equation for the straight line which passes through $C$ and is parallel to $A B$.
Give your answer in the form $a x + b y = c$, where $a$, $b$ and $c$ are integers.

OCR C1 2007 January Q9

12 marks Moderate -0.3

Find the equation of the line through $A$ parallel to the line $y = 4 x - 5$, giving your answer in the form $y = m x + c$.
Calculate the length of $A B$, giving your answer in simplified surd form.
Find the equation of the line which passes through the mid-point of $A B$ and which is perpendicular to $A B$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

OCR C1 2009 June Q9

8 marks Moderate -0.8

Calculate the length of $A B$.
Find the coordinates of the mid-point of $A B$.
Find the equation of the line through $( 1,3 )$ which is parallel to $A B$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

OCR C1 Q9

10 marks Standard +0.3

Find an equation for the straight line $l$ which passes through $P$ and $Q$. Give your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers. The straight line $m$ has gradient 8 and passes through the origin, $O$.
Write down an equation for $m$. The lines $l$ and $m$ intersect at the point $R$.
Show that $O P = O R$.

OCR MEI AS Paper 2 2021 November Q10

6 marks Standard +0.3

Show that PQ is perpendicular to QR . A circle passes through $\mathrm { P } , \mathrm { Q }$ and R .
Determine the coordinates of the centre of the circle.

OCR H240/02 2021 November Q5

8 marks Standard +0.3

5 In this question you must show detailed reasoning. Points $A , B$ and $C$ have coordinates $( 0,6 ) , ( 7,5 )$ and $( 6 , - 2 )$ respectively.

Find an equation of the perpendicular bisector of $A B$.
Hence, or otherwise, find an equation of the circle that passes through points $A , B$ and $C$.

Edexcel C1 Q8

Moderate -0.8

8. The line $l _ { 1 }$ passes through the point $( 9 , - 4 )$ and has gradient $\frac { 1 } { 3 }$.

Find an equation for $l _ { 1 }$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The line $l _ { 2 }$ passes through the origin $O$ and has gradient - 2 . The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.
Calculate the coordinates of $P$. Given that $l _ { 1 }$ crosses the $y$-axis at the point $C$,
calculate the exact area of $\triangle O C P$. \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_59_2568_1882} \includegraphics[max width=\textwidth, alt={}, center]{307d6e38-b8ca-4473-9f1a-94c8660c0d9c-011_104_1829_2648_114}

Edexcel M2 Q3

11 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} The points $A ( 3,0 )$ and $B ( 0,4 )$ are two vertices of the rectangle $A B C D$, as shown in Fig. 2.

Write down the gradient of $A B$ and hence the gradient of $B C$. The point $C$ has coordinates $( 8 , k )$, where $k$ is a positive constant.
Find the length of $B C$ in terms of $k$. Given that the length of $B C$ is 10 and using your answer to part (b),
find the value of $k$,
find the coordinates of $D$.

Edexcel M2 Q16

13 marks Moderate -0.8

16. \section*{Figure 3}

\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-08_581_575_395_609}

The points $A ( - 3 , - 2 )$ and $B ( 8,4 )$ are at the ends of a diameter of the circle shown in Fig. 3.

Find the coordinates of the centre of the circle.
Find an equation of the diameter $A B$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Find an equation of tangent to the circle at $B$. The line $l$ passes through $A$ and the origin.
Find the coordinates of the point at which $l$ intersects the tangent to the circle at $B$, giving your answer as exact fractions.

1.03b Straight lines: parallel and perpendicular relationships

AQA C1 2010 June Q5

AQA C1 2011 June Q1

AQA C1 2011 June Q8

AQA C1 2012 June Q2

AQA C1 2012 June Q7

AQA C1 2013 June Q1

AQA C1 2014 June Q1

AQA C1 2015 June Q1

AQA C1 2016 June Q1

Edexcel C1 Q5

Edexcel C1 Q9

Edexcel C1 Q8

Edexcel C1 Q7

Edexcel C1 Q7

Edexcel C1 Q8

Edexcel C1 Q10

Edexcel C1 Q1

OCR C1 2007 January Q9

OCR C1 2009 June Q9

OCR C1 Q9

OCR MEI AS Paper 2 2021 November Q10

OCR H240/02 2021 November Q5

Edexcel C1 Q8

Edexcel M2 Q3

Edexcel M2 Q16