1.02c Simultaneous equations: two variables by elimination and substitution

8 Find the coordinates of the point of intersection of the lines \(x + 2 y = 5\) and \(y = 5 x - 1\).

12 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

5 You are given that \(\mathrm { f } ( x ) = x ^ { 2 } + k x + c\).
Given also that \(\mathrm { f } ( 2 ) = 0\) and \(\mathrm { f } ( - 3 ) = 35\), find the values of the constants \(k\) and \(c\).

8 Rearrange the equation \(5 c + 9 t = a ( 2 c + t )\) to make \(c\) the subject.

11

1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.

5 Make \(a\) the subject of \(3 ( a + 4 ) = a c + 5 f\).

2 Find the coordinates of the point of intersection of the lines \(2 x + 3 y = 12\) and \(y = 7 - 3 x\).

4 You are given that \(a = \frac { 3 c + 2 a } { 2 c - 5 }\). Express \(a\) in terms of \(c\).

8

1. Use logarithms to solve the equation \(7 ^ { w - 3 } - 4 = 180\), giving your answer correct to 3 significant figures.
2. Solve the simultaneous equations $$\log _ { 10 } x + \log _ { 10 } y = \log _ { 10 } 3 , \quad \log _ { 10 } ( 3 x + y ) = 1$$

5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).

1. Find the centre and radius of the circle.
2. Find the coordinates of any points where the line \(y = 2 x - 3\) meets the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
3. State what can be deduced from the answer to part (ii) about the line \(y = 2 x - 3\) and the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).

2 Alex is comparing the cost of mobile phone contracts. Contract \(\boldsymbol { A }\) has a set-up cost of \(\pounds 40\) and then costs 4 p per minute. Contract \(\boldsymbol { B }\) has no set-up cost, does not charge for the first 100 minutes and then costs 6 p per minute.

1. Find an expression for the cost of each of the contracts in terms of \(m\), where \(m\) is the number of minutes for which the phone is used and \(m > 100\).
2. Hence find the value of \(m\) for which both contracts would cost the same.

3

1. In this question you must show detailed reasoning.
   Find the coordinates of the points of intersection of the curves with equations \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\).
2. The diagram shows the curves \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\). This diagram is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-5_377_542_603_322} On the diagram in the Printed Answer Booklet, draw the line \(y = 2 x - 2\).
3. Show on your diagram in the Printed Answer Booklet the region of the \(x - y\) plane within which all three of the following inequalities are satisfied. \(y \geqslant x ^ { 2 } - 2 x + 1 \quad y \leqslant - x ^ { 2 } + 6 x - 5 \quad y \leqslant 2 x - 2\) You should indicate the region for which all the inequalities hold by labelling the region \(R\).[1]

12. Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\).
2. [diagram]
   The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).

11. \begin{figure}[h] \end{figure} The figure 5 shows part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { c c } C _ { 1 } : y = x ^ { 3 } - 2 x ^ { 2 } & x > 0 \\ C _ { 2 } : y = 9 - \frac { 5 } { 2 } ( x - 3 ) ^ { 2 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\).
a. Verify that the point \(Q\) has coordinates \(( 3,9 )\) b. Use algebra to find the coordinates of the point \(P\).

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 2 x ^ { 3 } + 10 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 42 x - 15 x ^ { 2 } - 7 \quad x > 0$$

1. Verify that the curves intersect at \(x = \frac { 1 } { 2 }\) The curves intersect again at the point \(P\)
2. Using algebra and showing all stages of working, find the exact \(x\) coordinate of \(P\)

1. A small factory makes bars of soap.

On any day, the total cost to the factory, \(\pounds y\), of making \(x\) bars of soap is modelled to be the sum of two separate elements:

• a fixed cost
• a cost that is proportional to the number of bars of soap that are made that day
  1. Write down a general equation linking \(y\) with \(x\), for this model.

The bars of soap are sold for \(\pounds 2\) each.
On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of \(\pounds 80\) Using the above information,

show that \(y = 0.84 x + 428\)

With reference to the model, interpret the significance of the value 0.84 in the equation. Assuming that each bar of soap is sold on the day it is made,

find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.

1. The line \(l\) has equation

$$3 x - 2 y = k$$ where \(k\) is a real constant.
Given that the line \(l\) intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for \(k\).

7 Determine the points of intersection of the curve \(3 x y + x ^ { 2 } + 14 = 0\) and the line \(x + 2 y = 4\).

7

1. In this question you must show detailed reasoning. Find the range of values of the constant \(m\) for which the simultaneous equations \(y = m x\) and \(x ^ { 2 } + y ^ { 2 } - 6 x - 2 y + 5 = 0\) have real solutions.
2. Give a geometrical interpretation of the solution in the case where \(m = 2\).

5 In this question you must show detailed reasoning.
The line \(x + 5 y = k\) is a tangent to the curve \(x ^ { 2 } - 4 y = 10\). Find the value of the constant \(k\).

8 A team of volunteers donates cakes for sale at a charity stall. The number of cakes that can be sold depends on the price. A model for this is \(\mathrm { y } = 190 - 70 \mathrm { x }\), where \(y\) cakes can be sold when the price of a cake is \(\pounds\) x.

1. Find how many cakes could be given away for free according to this model. The number of volunteers who are willing to donate cakes goes up as the price goes up. If the cakes sell for \(\pounds 1.20\) they will donate 50 cakes, but if they sell for \(\pounds 2.40\) they will donate 140 cakes. They use the linear model \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) to relate the number of cakes donated, \(y\), to the price of a cake, \(\pounds x\).
2. Find the values of the constants \(m\) and \(c\) for which this linear model fits the two data points.
3. Explain why the model is not suitable for very low prices.
4. The team would like to sell all the cakes that they donate. Find the set of possible prices that the cakes could have to achieve this.

8 In this question you must show detailed reasoning.

1. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 4 y - 20 = 0\).
2. Find the points of intersection of the circle with the line \(x + 3 y - 10 = 0\).

1 Find the coordinates of the point of intersection of the lines \(y = 3 x - 2\) and \(x + 2 y = 10\).

15 A family is planning a holiday in Europe. They need to buy some euros before they go. The exchange rate, \(y\), is the number of euros they can buy per pound. They believe that the exchange rate may be modelled by the formula \(y = a t ^ { 2 } + b t + c\),
where \(t\) is the time in days from when they first check the exchange rate.
Initially, when \(t = 0\), the exchange rate is 1.14 .

1. Write down the value of \(c\). When \(t = 2 , y = 1.20\) and when \(t = 4 , y = 1.25\).
2. Calculate the values of \(a\) and \(b\). The family will only buy their euros when their model predicts an exchange rate of at least 1.29 .
3. Determine the range of values of \(t\) for which, according to their model, they will buy their euros.
4. Explain why the family's model is not viable in the long run.

4 The vectors \(\mathbf { v } _ { 1 }\) and \(\mathbf { v } _ { 2 }\) are defined by \(\mathbf { v } _ { 1 } = 2 \mathrm { a } \mathbf { i } + \mathrm { bj }\) and \(\mathbf { v } _ { 2 } = b \mathbf { i } - 3 \mathbf { j }\) where \(a\) and \(b\) are constants. Given that \(3 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } = 22 \mathbf { i } - 9 \mathbf { j }\), find the values of \(a\) and \(b\).