1.02q Use intersection points: of graphs to solve equations

1 Find the set of values of \(m\) for which the line with equation \(y = m x + 1\) and the curve with equation \(y = 3 x ^ { 2 } + 2 x + 4\) intersect at two distinct points.

7 \includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-12_631_1031_267_534} The diagram shows the curve with equation \(y = ( 3 x - 2 ) ^ { \frac { 1 } { 2 } }\) and the line \(y = \frac { 1 } { 2 } x + 1\). The curve and the line intersect at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\).
2. Hence find the area of the region enclosed between the curve and the line.

9 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 26 = 0\).

1. Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the lowest point on the circle.
2. Find the set of values of the constant \(k\) for which the line with equation \(y = k x - 5\) intersects the circle at two distinct points.

5 The equation of a curve is \(y = 4 x ^ { 2 } - k x + \frac { 1 } { 2 } k ^ { 2 }\) and the equation of a line is \(y = x - a\), where \(k\) and \(a\) are constants.

1. Given that the curve and the line intersect at the points with \(x\)-coordinates 0 and \(\frac { 3 } { 4 }\), find the values of \(k\) and \(a\).
2. Given instead that \(a = - \frac { 7 } { 2 }\), find the values of \(k\) for which the line is a tangent to the curve. [5]

7 \includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-10_887_1003_258_571} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20\) and with centre \(C\). The point \(B\) has coordinates \(( 0,2 )\) and the line segment \(B C\) intersects the circle at \(P\).

1. Find the equation of \(B C\).
2. Hence find the coordinates of \(P\), giving your answer in exact form.

7 A curve has equation \(y = 2 + 3 \sin \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant 4 \pi\).

1. State greatest and least values of \(y\).
2. Sketch the curve. \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-09_1127_1219_904_495}
3. State the number of solutions of the equation $$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$ for \(0 \leqslant x \leqslant 4 \pi\).

5 A circle has equation \(( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40\). A line with equation \(y = x - 9\) intersects the circle at points \(A\) and \(B\).

1. Find the coordinates of the two points of intersection.
2. Find an equation of the circle with diameter \(A B\).

3 The equation of a curve is \(y = 2 x ^ { 2 } + m ( 2 x + 1 )\), where \(m\) is a constant, and the equation of a line is \(y = 6 x + 4\). Show that, for all values of \(m\), the line intersects the curve at two distinct points.

11 A curve has equation \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).

1. State the greatest and least values of \(y\).
2. Sketch the graph of \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
3. By considering the straight line \(y = k x\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2 x + 2 = k x\) for \(0 \leqslant x \leqslant \pi\) in each of the following cases.
   1. \(k = - 3\)
   2. \(k = 1\)
   3. \(k = 3\) Functions \(\mathrm { f } , \mathrm { g }\) and h are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } ( x ) = 3 \cos 2 x + 2 \\ & \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4 \\ & \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right) \end{aligned}$$
4. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { g } ( x )\).
5. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { h } ( x )\). [2]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A curve has equation \(y = 3 x ^ { 2 } - 4 x + 4\) and a straight line has equation \(y = m x + m - 1\), where \(m\) is a constant. Find the set of values of \(m\) for which the curve and the line have two distinct points of intersection.

2 A curve has equation \(y = k x ^ { 2 } + 2 x - k\) and a line has equation \(y = k x - 2\), where \(k\) is a constant. Find the set of values of \(k\) for which the curve and line do not intersect.

5 \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-07_778_878_255_630} The diagram shows part of the graph of \(y = a \cos ( b x ) + c\).

1. Find the values of the positive integers \(a , b\) and \(c\).
2. For these values of \(a\), \(b\) and \(c\), use the given diagram to determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations.
   1. \(a \cos ( b x ) + c = \frac { 6 } { \pi } x\)
   2. \(a \cos ( b x ) + c = 6 - \frac { 6 } { \pi } x\) The diagram shows a metal plate \(A B C\) in which the sides are the straight line \(A B\) and the arcs \(A C\) and \(B C\). The line \(A B\) has length 6 cm . The arc \(A C\) is part of a circle with centre \(B\) and radius 6 cm , and the arc \(B C\) is part of a circle with centre \(A\) and radius 6 cm .

9 The line \(y = 2 x + 5\) intersects the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\) at \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\) in surd form and hence find the exact length of the chord \(A B\).
   A straight line through the point \(( 10,0 )\) with gradient \(m\) is a tangent to the circle.
2. Find the two possible values of \(m\).

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 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-2_629_659_715_740} The diagram shows the curve \(y = 3 \sqrt { } x\) and the line \(y = x\) intersecting at \(O\) and \(P\). Find

1. the coordinates of \(P\),
2. the area of the shaded region.

6

1. Sketch the graph of the curve \(y = 3 \sin x\), for \(- \pi \leqslant x \leqslant \pi\). The straight line \(y = k x\), where \(k\) is a constant, passes through the maximum point of this curve for \(- \pi \leqslant x \leqslant \pi\).
2. Find the value of \(k\) in terms of \(\pi\).
3. State the coordinates of the other point, apart from the origin, where the line and the curve intersect.

6 The curve \(y = 9 - \frac { 6 } { x }\) and the line \(y + x = 8\) intersect at two points. Find

1. the coordinates of the two points,
2. the equation of the perpendicular bisector of the line joining the two points.

2 Find the set of values of \(k\) for which the line \(y = k x - 4\) intersects the curve \(y = x ^ { 2 } - 2 x\) at two distinct points.

7 A curve has equation \(y = x ^ { 2 } - 4 x + 4\) and a line has equation \(y = m x\), where \(m\) is a constant.

1. For the case where \(m = 1\), the curve and the line intersect at the points \(A\) and \(B\). Find the coordinates of the mid-point of \(A B\).
2. Find the non-zero value of \(m\) for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.

5

1. Sketch, on the same diagram, the curves \(y = \sin 2 x\) and \(y = \cos x - 1\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Hence state the number of solutions, in the interval \(0 \leqslant x \leqslant 2 \pi\), of the equations
   1. \(2 \sin 2 x + 1 = 0\),
   2. \(\sin 2 x - \cos x + 1 = 0\).

6 The equation of a curve is \(y = 3 \cos 2 x\) and the equation of a line is \(2 y + \frac { 3 x } { \pi } = 5\).

1. State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leqslant x \leqslant 2 \pi\).
2. Sketch, on the same diagram, the graphs of \(y = 3 \cos 2 x\) and \(2 y + \frac { 3 x } { \pi } = 5\) for \(0 \leqslant x \leqslant 2 \pi\).
3. State the number of solutions of the equation \(6 \cos 2 x = 5 - \frac { 3 x } { \pi }\) for \(0 \leqslant x \leqslant 2 \pi\).

4

1. Sketch and label, on the same diagram, the graphs of \(y = 2 \sin x\) and \(y = \cos 2 x\), for the interval \(0 \leqslant x \leqslant \pi\).
2. Hence state the number of solutions of the equation \(2 \sin x = \cos 2 x\) in the interval \(0 \leqslant x \leqslant \pi\).

5 The equation of a curve is \(y = x ^ { 2 } - 4 x + 7\) and the equation of a line is \(y + 3 x = 9\). The curve and the line intersect at the points \(A\) and \(B\).

1. The mid-point of \(A B\) is \(M\). Show that the coordinates of \(M\) are \(\left( \frac { 1 } { 2 } , 7 \frac { 1 } { 2 } \right)\).
2. Find the coordinates of the point \(Q\) on the curve at which the tangent is parallel to the line \(y + 3 x = 9\).
3. Find the distance \(M Q\).

1 Determine the set of values of the constant \(k\) for which the line \(y = 4 x + k\) does not intersect the curve \(y = x ^ { 2 }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_554_723_1557_712} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 0 , - 2 )\) and \(C\) is \(( 12,14 )\). The diagonal \(B D\) is parallel to the \(x\)-axis.

1. Explain why the \(y\)-coordinate of \(D\) is 6 . The \(x\)-coordinate of \(D\) is \(h\).
2. Express the gradients of \(A D\) and \(C D\) in terms of \(h\).
3. Calculate the \(x\)-coordinates of \(D\) and \(B\).
4. Calculate the area of the rectangle \(A B C D\).