1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

1. In 1997 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(190 \mathrm {~g} / \mathrm { km }\).

In 2005 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK had fallen to \(169 \mathrm {~g} / \mathrm { km }\).
Given \(\mathrm { Ag } / \mathrm { km }\) is the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK \(n\) years after 1997 and using a linear model,

1. form an equation linking \(A\) with \(n\). In 2016 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(120 \mathrm {~g} / \mathrm { km }\).
2. Comment on the suitability of your model in light of this information.

1. The distance a particular car can travel in a journey starting with a full tank of fuel was investigated.

• From a full tank of fuel, 40 litres remained in the car's fuel tank after the car had travelled 80 km
• From a full tank of fuel, 25 litres remained in the car's fuel tank after the car had travelled 200 km

Using a linear model, with \(V\) litres being the volume of fuel remaining in the car's fuel tank and \(d \mathrm {~km}\) being the distance the car had travelled,

1. find an equation linking \(V\) with \(d\). Given that, on a particular journey
   • the fuel tank of the car was initially full
   • the car continued until it ran out of fuel
     find, according to the model,
   • 1. the initial volume of fuel that was in the fuel tank of the car,
     2. the distance that the car travelled on this journey.
   In fact the car travelled 320 km on this journey.
2. Evaluate the model in light of this information.

10. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(y = \frac { 3 } { 5 } x + 6\) The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(B ( 8,0 )\), as shown in the sketch in Figure 4.

1. Show that an equation for line \(l _ { 2 }\) is $$5 x + 3 y = 40$$ Given that

15. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of a circle \(C\) with centre \(N ( 7,4 )\) The line \(l\) with equation \(y = \frac { 1 } { 3 } x\) is a tangent to \(C\) at the point \(P\).
Find

1. the equation of line \(P N\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
2. an equation for \(C\). The line with equation \(y = \frac { 1 } { 3 } x + k\), where \(k\) is a non-zero constant, is also a tangent to \(C\).
3. Find the value of \(k\).

1. A tank, which contained water, started to leak from a hole in its base.

The volume of water in the tank 24 minutes after the leak started was \(4 \mathrm {~m} ^ { 3 }\) The volume of water in the tank 60 minutes after the leak started was \(2.8 \mathrm {~m} ^ { 3 }\) The volume of water, \(V \mathrm {~m} ^ { 3 }\), in the tank \(t\) minutes after the leak started, can be described by a linear model between \(V\) and \(t\).

1. Find an equation linking \(V\) with \(t\). Use this model to find
2. 1. the initial volume of water in the tank,
   2. the time taken for the tank to empty.
3. Suggest a reason why this linear model may not be suitable.

6. \begin{figure}[h] \end{figure} The circle \(C\) has centre \(A\) with coordinates \(( - 3,1 )\).
The line \(l _ { 1 }\) with equation \(y = - 4 x + 6\), is the tangent to \(C\) at the point \(Q\), as shown in Figure 3.
a. Find the equation of the line \(A Q\) in the form \(a x + b y = c\).
b. Show that the equation of the circle \(C\) is \(( x + 3 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 17\) The line \(l _ { 2 }\) with equation \(y = - 4 x + k , k \neq 6\), is also a tangent to \(C\).
c. Find the value of the constant \(k\).

1. a. Sketch the graph of the function with equation

$$y = 11 - 2 | 2 - x |$$ Stating the coordinates of the maximum point and any points where the graph cuts the \(y\)-axis.
b. Solve the equation $$4 x = 11 - 2 | 2 - x |$$ A straight line \(l\) has equation \(y = k x + 13\), where \(k\) is a constant.
Given that \(l\) does not meet or intersect \(y = 11 - 2 | 2 - x |\) c. find the range of possible value of \(k\).

6. \begin{figure}[h] \end{figure} Not to scale The circle \(C\) has centre \(A\) with coordinates (7,5).
The line \(l\), with equation \(y = 2 x + 1\), is the tangent to \(C\) at the point \(P\), as shown in Figure 3 .

1. Show that an equation of the line \(P A\) is \(2 y + x = 17\)
2. Find an equation for \(C\). The line with equation \(y = 2 x + k , \quad k \neq 1\) is also a tangent to \(C\).
3. Find the value of the constant \(k\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a rectangle \(A B C D\).
The point \(A\) lies on the \(y\)-axis and the points \(B\) and \(D\) lie on the \(x\)-axis as shown in Figure 1. Given that the straight line through the points \(A\) and \(B\) has equation \(5 y + 2 x = 10\)

1. show that the straight line through the points \(A\) and \(D\) has equation \(2 y - 5 x = 4\)
2. find the area of the rectangle \(A B C D\).

9 In this question you must show detailed reasoning.
Find the equation of the straight line with positive gradient that passes through \(( 0,2 )\) and is a tangent to the curve \(y = x ^ { 2 } - x + 6\).

3 Sam invested in a shares scheme. The value, \(\pounds V\), of Sam's shares was reported \(t\) months after investment.

• Exactly 6 months after investment, the value of Sam's shares was \(\pounds 2375\).
• Exactly 1 year after investment, the value of Sam's shares was \(\pounds 2825\).
  1. Using a straight-line model, determine an equation for \(V\) in terms of \(t\).

Sam's original investment in the scheme was \(\pounds 1900\).

Explain whether or not this fact supports the use of the straight-line model in part (a).

7 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-05_848_1049_260_242} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 9 y + 19 = 0\) and centre \(C\).

1. Find the following.
   The tangent to the circle at \(D\) meets the \(x\)-axis at the point \(A \left( \frac { 55 } { 4 } , 0 \right)\) and the \(y\)-axis at the point \(B ( 0 , - 11 )\).
2. Determine the area of triangle \(O B D\).

6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.

10 A triangle has vertices \(A ( 1,4 ) , B ( 7,0 )\) and \(C ( - 4 , - 1 )\).

1. Show that the equation of the line AC is \(\mathrm { y } = \mathrm { x } + 3\). M is the midpoint of AB . The line AC intersects the \(x\)-axis at D .
2. Determine the angle DMA.

5 The graph shows displacement \(s m\) against time \(t \mathrm {~s}\) for a model of the motion of a bead moving along a straight wire. The points \(( 0,4 ) , ( 2,7 ) , ( 5,7 )\) and \(( 9 , - 7 )\) are the endpoints of the line segments. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-4_741_1301_404_239}

1. Find an expression for the displacement of the bead for \(0 \leqslant t \leqslant 2\).
2. Sketch the velocity-time graph for this model.
3. Explain why the model may not be suitable at \(t = 2\) and \(t = 5\).

5 In this question you must show detailed reasoning.

1. Show that the gradient of the curve \(\mathrm { y } = \sqrt { \mathrm { x } } \left( \frac { 1 } { \mathrm { x } ^ { 2 } } - 2 \mathrm { x } \right)\) at the point \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) is \(- \frac { 99 } { 2 }\).
2. Find the equation of the tangent to the curve at \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

8 The point A has coordinates \(( - 1 , - 2 )\) and the point B has coordinates (7,4). The perpendicular bisector of \(A B\) intersects the line \(y + 2 x = k\) at \(P\). Determine the coordinates of P in terms of \(k\).

9

1. Sketch both of the following on the axes provided in the Printed Answer Booklet.
   1. The curve \(\mathrm { y } = \frac { 12 } { \mathrm { x } }\), stating the coordinates of at least one point on the curve.
   2. The line \(y = 2 x + 8\), stating the coordinates of the points at which the line crosses the axes.
2. In this question you must show detailed reasoning. Determine the exact coordinates of the points of intersection of the curve and the line.

11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).

1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
2. State the rate of increase in calls according to model 1.
3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .

6 Determine the equation of the line which passes through the point \(( 4 , - 1 )\) and is perpendicular to the line with equation \(2 x + 3 y = 6\). Give your answer in the form \(y = m x + c\), where \(m\) is a fraction in its lowest terms and \(c\) is an integer.

12 Fig. 12 shows the circle \(( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 25\), the line \(4 y = 3 x - 32\) and the tangent to the circle at the point \(\mathrm { A } ( 5,2 )\). D is the point of intersection of the line \(4 y = 3 x - 32\) and the tangent at A . \begin{figure}[h] \end{figure}

1. Write down the coordinates of C , the centre of the circle.
2. (A) Show that the line \(4 y = 3 x - 32\) is a tangent to the circle.
   (B) Find the coordinates of B , the point where the line \(4 y = 3 x - 32\) touches the circle.
3. Prove that ADBC is a square.
4. The point E is the lowest point on the circle. Find the area of the sector ECB .

2 Show that the line which passes through the points \(( 2 , - 4 )\) and \(( - 1,5 )\) does not intersect the line \(3 x + y = 10\).

7 Determine the exact distance between the two points at which the line through ( 4,5 ) and ( \(6 , - 1\) ) meets the curve \(y = 2 x ^ { 2 } - 7 x + 1\).

11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h] \end{figure}

1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
2. Determine the area of the triangle AOB .

8 The curves \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) and \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) crosses the \(x\)-axis at D and the \(y\)-axis at C . \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-7_826_819_520_255}

1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
2. Determine the coordinates of A, B, C and D.
3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(\mathrm { y } = \mathrm { mx } + c\), where \(m\) and \(c\) are constants to be determined.
4. Points A , B , C and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\), \(b\) and \(r ^ { 2 }\) are constants to be determined.