1.02c Simultaneous equations: two variables by elimination and substitution

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\)

• has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic function
• cuts the \(x\)-axis at the origin and at \(x = 4\)
• has a minimum turning point at ( \(2 , - 4.8\) )
  1. find \(\mathrm { f } ( x )\)

Given that \(C _ { 2 }\)

The curves \(C _ { 1 }\) and \(C _ { 2 }\) meet in the first quadrant at the point \(P\), shown in Figure 1.

Use algebra to find the coordinates of \(P\).

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

\section*{Solutions relying on calculator technology are not acceptable.}

1. Sketch the curve \(C\) with equation $$y = \frac { 1 } { 2 - x } \quad x \neq 2$$ State on your sketch
   • the equation of the vertical asymptote
   • the coordinates of the intersection of \(C\) with the \(y\)-axis
   The straight line \(l\) has equation \(y = k x - 4\), where \(k\) is a constant.
   Given that \(l\) cuts \(C\) at least once,
2. 1. show that $$k ^ { 2 } - 5 k + 4 \geqslant 0$$
   2. find the range of possible values for \(k\).

3 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - x y + y ^ { 2 } = 7 .$$

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

2 Solve the following simultaneous equations. $$x ^ { 2 } + 2 y ^ { 2 } = 36 \quad x + 2 y = 10$$

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

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

Show that the curve with equation \(x^2 - 3xy - 40 = 0\) and the line with equation \(3x + y + k = 0\) meet for all values of the constant \(k\). [5]

The equation of a curve is \(y = \frac{1}{3}k^2x^2 - 2kx + 2\) and the equation of a line is \(y = kx + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).

1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left(\frac{6}{5}, \frac{3}{5}\right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection. [7]
2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\). [3]

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]

Variables \(u\), \(x\) and \(y\) are such that \(u = 2x(y - x)\) and \(x + 3y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\). [5]

The point \(C\) lies on the perpendicular bisector of the line joining the points \(A(4, 6)\) and \(B(10, 2)\). \(C\) also lies on the line parallel to \(AB\) through \((3, 11)\).

1. Find the equation of the perpendicular bisector of \(AB\). [4]
2. Calculate the coordinates of \(C\). [3]

The coordinates of two points \(A\) and \(B\) are \((1, 3)\) and \((9, -1)\) respectively and \(D\) is the mid-point of \(AB\). A point \(C\) has coordinates \((x, y)\), where \(x\) and \(y\) are variables.

1. State the coordinates of \(D\). [1]
2. It is given that \(CD^2 = 20\). Write down an equation relating \(x\) and \(y\). [1]
3. It is given that \(AC\) and \(BC\) are equal in length. Find an equation relating \(x\) and \(y\) and show that it can be simplified to \(y = 2x - 9\). [3]
4. Using the results from parts (ii) and (iii), and showing all necessary working, find the possible coordinates of \(C\). [4]

The equation of a curve is \(y = 2x + \frac{12}{x}\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the line does not meet the curve. [3]

In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]
2. Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\). [3]

The variables \(x\) and \(y\) satisfy the equation \(a^{2y} = e^{3x+k}\), where \(a\) and \(k\) are constants. The graph of \(y\) against \(x\) is a straight line.

1. Use logarithms to show that the gradient of the straight line is \(\frac{3}{2\ln a}\). [1]
2. Given that the straight line passes through the points \((0.4, 0.95)\) and \((3.3, 3.80)\), find the values of \(a\) and \(k\). [4]

Solve the simultaneous equations $$y + 4x + 1 = 0$$ $$y^2 + 5x^2 + 2y = 0$$ [6]

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

Solve the simultaneous equations $$x - 2y = 1,$$ $$x^2 + y^2 = 29.$$ [6]

The line \(l_1\) passes through the points \(P(-1, 2)\) and \(Q(11, 8)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [4]

The line \(l_2\) passes through the point \(R(10, 0)\) and is perpendicular to \(l_1\). The lines \(l_1\) and \(l_2\) intersect at the point \(S\).

1. Calculate the coordinates of \(S\). [5]
2. Show that the length of \(RS\) is \(3\sqrt{5}\). [2]
3. Hence, or otherwise, find the exact area of triangle \(PQR\). [4]

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

The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).

1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]

The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Calculate, as exact fractions, the coordinates of \(A\). [3]
2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

Solve the simultaneous equations \begin{align} x - 3y + 1 &= 0,
x^2 - 3xy + y^2 &= 11. \end{align} [7]

\includegraphics{figure_2} Figure 2 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\). [1]
2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
3. Show that \(\angle PAQ\) is a right angle. [4]