1.02q Use intersection points: of graphs to solve equations

The ellipse \(E\) has equation $$x^2 + \frac{y^2}{9} = 1$$ The line with equation \(y = mx + 4\) is a tangent to \(E\) Without using differentiation show that \(m = \pm\sqrt{7}\) [4 marks]

The points \(A(-2, 4)\) and \(B(6, 10)\) are such that \(AB\) is the diameter of a circle.

1. Show that the centre of the circle has coordinates \((2, 7)\). [1]
2. The equation of the circle is \(x^2 + y^2 + ax + by + c = 0\). Determine the values of \(a\), \(b\), \(c\). [3]

A straight line, with equation \(y = x + 6\), passes through the point \(A\) and cuts the circle again at the point \(C\).

1. Find the coordinates of \(C\). [5]
2. Calculate the exact area of the triangle \(ABC\). [3]

The line \(L_1\) passes through the points \(A(0, 5)\) and \(B(3, -1)\).

1. Find the equation of the line \(L_1\). [3]

The line \(L_2\) is perpendicular to \(L_1\) and passes through the origin \(O\).

1. Write down the equation of \(L_2\). [1]

The lines \(L_1\) and \(L_2\) intersect at the point \(C\).

1. Calculate the area of triangle \(OAC\). [4]
2. Find the equation of the line \(L_3\) which is parallel to \(L_1\) and passes through the point \(D(4, 2)\). [2]
3. The line \(L_3\) intersects the \(y\)-axis at the point \(E\). Find the area of triangle \(ODE\). [1]

The curve \(C_1\) has equation \(y = -x^2 + 2x + 3\) and the curve \(C_2\) has equation \(y = x^2 - x - 6\). The two curves intersect at the points \(A\) and \(B\).

1. Determine the coordinates of \(A\) and \(B\). [4]
2. On the same set of axes, sketch the graphs of \(C_1\) and \(C_2\). Clearly label the points where the two curves intersect. [3]
3. In the diagram drawn in part (b), shade the region satisfying the following inequalities: [2] $$x > 0,$$ $$y < -x^2 + 2x + 3,$$ $$y > x^2 - x - 6.$$

A circle \(C\) has centre \(A\) and equation \(x^2 + y^2 - 4x - 6y = 3\).

1. Find the coordinates of \(A\) and the radius of \(C\). [3]

The line \(L\) with equation \(y = x + 5\) intersects \(C\) at the points \(P\) and \(Q\).

1. Determine the coordinates of \(P\) and \(Q\). [4]
2. The point \(B\) is on \(PQ\) and is such that \(AB\) is perpendicular to \(PQ\). Find the length of \(PB\). [2]
3. Show that the area of the smaller segment enclosed by \(C\) and \(L\) is \(4\pi - 8\). [2]

By drawing suitable graphs, show that \(x - 1 = \cos x\) has only one root. Starting with \(x_0 = 1\), use the Newton-Raphson method to find the value of this root correct to two decimal places. [6]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3x - 2\sqrt{x}\), \(x \geqslant 0\) and the line \(l\) with equation \(y = 8x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\). [5]
2. \includegraphics{figure_4} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\). [3]

\includegraphics{figure_1} Figure 1 shows part of the graph of \(y = f(x)\), \(x \in \mathbb{R}\). The graph consists of two line segments that meet at the point \((1, a)\), \(a < 0\). One line meets the \(x\)-axis at \((3, 0)\). The other line meets the \(x\)-axis at \((-1, 0)\) and the \(y\)-axis at \((0, b)\), \(b < 0\). In separate diagrams, sketch the graph with equation

1. \(y = f(x + 1)\), [2]
2. \(y = f(|x|)\). [2]

Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(f(x) = |x - 1| - 2\), find

1. the value of \(a\) and the value of \(b\), [2]
2. the value of \(x\) for which \(f(x) = 5x\). [3]

Let \(f(x) = x^2(x - 2)\) and \(g(x) = 2x - 1\) for all real \(x\).

1. Sketch the graph of \(y = f(x)\) and explain briefly why the function f has no inverse. [2]
2. Write down \(g^{-1}(x)\). [1]
3. On the same diagram, sketch the graphs of \(y = f(x - 1) - 3\) and \(y = g^{-1}(x)\) and state the number of real roots of the equation \(f(x - 1) - 3 = g^{-1}(x)\). [3]