Simultaneous equations

Solve the simultaneous equations $$y = x - 2,$$ $$y^2 + x^2 = 10.$$ [7]

1 Solve the simultaneous equations $$\begin{aligned} x ^ { 2 } + y ^ { 2 } & = 5 \\ y & = 2 x \end{aligned}$$

11. \begin{figure}[h] \end{figure} The line \(y = x + 2\) meets the curve \(x ^ { 2 } + 4 y ^ { 2 } - 2 x = 35\) at the points \(A\) and \(B\) as shown in Figure 2.

1. Find the coordinates of \(A\) and the coordinates of \(B\).
2. Find the distance \(A B\) in the form \(r \sqrt { 2 }\) where \(r\) is a rational number.

Solve the simultaneous equations $$x + y = 2$$ $$x^2 + 2y = 12.$$ [6]

Simplify

1. \(( 2 \sqrt { } 5 ) ^ { 2 }\)
2. \(\frac { \sqrt { } 2 } { 2 \sqrt { } 5 - 3 \sqrt { } 2 }\) giving your answer in the form \(a + \sqrt { } b\), where \(a\) and \(b\) are integers.

7 In this question you must show detailed reasoning.
Fig. 7 shows the curve \(y = 5 x - x ^ { 2 }\). \begin{figure}[h] \end{figure} The line \(y = 4 - k x\) crosses the curve \(y = 5 x - x ^ { 2 }\) on the \(x\)-axis and at one other point.
Determine the coordinates of this other point.

6 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - 2 x y + y ^ { 2 } = 9 .$$

4 Find, algebraically, the coordinates of the point of intersection of the lines \(y = 2 x - 5\) and \(6 x + 2 y = 7\).

6 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - 2 x y + y ^ { 2 } = 9 .$$

1 Solve the equation \(\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3\).

1 The line \(x + 2 y = 9\) intersects the curve \(x y + 18 = 0\) at the points \(A\) and \(B\). Find the coordinates of \(A\) and \(B\).

3 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).

1. Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]

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 \(k\).

7 The straight line \(\mathrm { y } = \mathrm { x } + 5\) meets the curve \(2 \mathrm { x } ^ { 2 } + 3 \mathrm { y } ^ { 2 } = \mathrm { k }\) at a single point \(P\).

1. Find the value of the constant \(k\).
2. Find the coordinates of \(P\).

1. A circle \(C\) with radius \(r\)

• lies only in the 1st quadrant
• touches the \(x\)-axis and touches the \(y\)-axis

The line \(l\) has equation \(2 x + y = 12\)

1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$ Given also that \(l\) is a tangent to \(C\),
2. find the two possible values of \(r\), giving your answers as fully simplified surds.

Solve the equation \(\frac{4x}{x+1} - \frac{3}{2x+1} = 1\). [5]

1. Given that \(3^x = 9^{y-1}\), show that \(x = 2y - 2\). [2]
2. Solve the simultaneous equations \begin{align} x &= 2y - 2,
   x^2 &= y^2 + 7. \end{align} [6]

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation $$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$ The curve \(C _ { 2 }\) has equation $$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$

1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet when $$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at points \(P\) and \(Q\)
2. find, using algebra, the exact distance \(P Q\)

3 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).

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

Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]

1 Show that the system of equations $$\begin{aligned} 14 x - 4 y + 6 z & = 5 \\ x + y + k z & = 3 \\ - 21 x + 6 y - 9 z & = 14 \end{aligned}$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.

4. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0 \\ & \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$

6. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows

• the line \(l\) with equation \(y - 5 x = 75\)
• the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)

The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).

1. The curve \(C _ { 1 }\) has equation

$$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)

1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
2. Hence find the coordinates of \(P\)

1. A tree was planted.

Exactly 3 years after it was planted, the height of the tree was 2 m . Exactly 5 years after it was planted, the height of the tree was 2.4 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted, can be modelled by the equation $$H ^ { 3 } = p t ^ { 2 } + q$$ where \(p\) and \(q\) are constants,

1. find, to 3 significant figures where necessary, the value of \(p\) and the value of \(q\). Exactly \(T\) years after the tree was planted, its height was 5 m .
2. Find the value of \(T\) according to the model, giving your answer to one decimal place.

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

The equation of a line is \(2y + x = k\), where \(k\) is a constant, and the equation of a curve is \(xy = 6\).

1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(AB\). [6]
2. Find the set of values of \(k\) for which the line \(2y + x = k\) intersects the curve \(xy = 6\) at two distinct points. [3]

1. Solve the following simultaneous equations:

$$\begin{aligned} & y = 4 x ^ { 2 } + 2 x - 5 \\ & y = | 4 x + 1 | \end{aligned}$$