1.02q Use intersection points: of graphs to solve equations

16. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\) The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\) b. Show that an equation for \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$ c. The cartesian equation for the curve \(C\) can be written in the form $$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found. The straight line with equation $$y = \frac { 1 } { 3 } x + k$$ where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).

1. A curve \(C\) has parametric equations

$$x = 3 + 2 \sin t , \quad y = 4 + 2 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$

1. Show that all points on \(C\) satisfy \(y = 6 - ( x - 3 ) ^ { 2 }\)
2. 1. Sketch the curve \(C\).
   2. Explain briefly why \(C\) does not include all points of \(y = 6 - ( x - 3 ) ^ { 2 } , \quad x \in \mathbb { R }\) The line with equation \(x + y = k\), where \(k\) is a constant, intersects \(C\) at two distinct points.
3. State the range of values of \(k\), writing your answer in set notation.

3 The diagram in the Printed Answer Booklet shows part of the graph of \(y = x ^ { 2 } - 4 x + 3\).

1. It is required to solve the equation \(x ^ { 2 } - 3 x + 1 = 0\) graphically by drawing a straight line with equation \(y = m x + c\) on the diagram, where \(m\) and \(c\) are constants. Find the values of \(m\) and \(c\).
2. Use the graph to find approximate values of the roots of the equation \(x ^ { 2 } - 3 x + 1 = 0\).
3. By shading, or otherwise, indicate clearly the regions where all of the following inequalities are satisfied. You should use the values of \(m\) and \(c\) found in part (a). \(x \geqslant 0\) \(x \leqslant 4\) \(y \leqslant x ^ { 2 } - 4 x + 3\) \(y \geqslant m x + c\)

9

1. Sketch both of the following on the axes provided in the Printed Answer Booklet.
   1. The curve \(\mathrm { y } = \frac { 12 } { \mathrm { x } }\), stating the coordinates of at least one point on the curve.
   2. The line \(y = 2 x + 8\), stating the coordinates of the points at which the line crosses the axes.
2. In this question you must show detailed reasoning. Determine the exact coordinates of the points of intersection of the curve and the line.

10

1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.

11 In this question you must show detailed reasoning.
Determine for what values of \(k\) the graphs \(y = 2 x ^ { 2 } - k x\) and \(y = x ^ { 2 } - k\) intersect.

7

1. On the pair of axes in the Printed Answer Booklet, sketch the graphs of

3

1. Sketch the graph of \(\mathrm { y } = \arctan \mathrm { x }\) where \(x\) is in radians.
2. In this question you must show detailed reasoning. Find all points of intersection of the curves \(\mathrm { y } = 3 \sin \mathrm { xcos } \mathrm { x }\) and \(\mathrm { y } = \cos ^ { 2 } \mathrm { x }\) for \(- \pi \leqslant x \leqslant \pi\).

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

7 In this question you must show detailed reasoning.
The points \(\mathrm { A } ( - 1,4 )\) and \(\mathrm { B } ( 7 , - 2 )\) are at opposite ends of a diameter of a circle.

1. Find the equation of the circle.
2. Find the coordinates of the points of intersection of the circle and the line \(y = 2 x + 5\).
3. Q is the point of intersection with the larger \(y\)-coordinate. Calculate the area of the triangle ABQ .

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 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).

1. 1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
   2. Find the coordinates of the points \(A\) and \(C\).
2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
   [0pt] [4 marks] \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.

10. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).

1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
2. Find an equation for the tangent to the curve at \(P\).
3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).

8. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

8

1. Sketch the curve with equation \(y = 7 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. The curve \(C _ { 1 }\) has equation \(y = 7 ^ { x }\). The curve \(C _ { 2 }\) has equation \(y = 7 ^ { 2 x } - 12\).
   1. By forming and solving a quadratic equation, prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at exactly one point. State the \(y\)-coordinate of this point.
   2. Use logarithms to find the \(x\)-coordinate of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\), giving your answer to three significant figures.
      (2 marks)

7

1. Sketch the graph of \(y = \tan ^ { - 1 } x\).
2. 1. By drawing a suitable straight line on your sketch, show that the equation \(\tan ^ { - 1 } x = 2 x - 1\) has only one root.
   2. Given that the root of this equation is \(\alpha\), show that \(0.8 < \alpha < 0.9\).
3. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \left( \tan ^ { - 1 } x _ { n } + 1 \right)\) with \(x _ { 1 } = 0.8\) to find the value of \(x _ { 3 }\), giving your answer to two significant figures.

\(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 } { x ^ { 2 } + x - 6 }\).

1. Using algebraic division, show that $$f ( x ) = x ^ { 2 } + A + \frac { B } { x + C }$$ where \(A , B\) and \(C\) are integers to be found.
2. By sketching two suitable graphs on the same set of axes, show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root.
3. Use the iterative formula $$x _ { n + 1 } = 2 + \frac { 1 } { x _ { n } ^ { 2 } + 1 } ,$$ with a suitable starting value to find the root of the equation \(\mathrm { f } ( x ) = 0\) correct to 3 significant figures and justify the accuracy of your answer.

8 A curve has equation \(y ^ { 2 } = 12 x\).

1. Sketch the curve.
2. 1. The curve is translated by 2 units in the positive \(y\) direction. Write down the equation of the curve after this translation.
   2. The original curve is reflected in the line \(y = x\). Write down the equation of the curve after this reflection.
3. 1. Show that if the straight line \(y = x + c\), where \(c\) is a constant, intersects the curve \(y ^ { 2 } = 12 x\), then the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
   2. Hence find the value of \(c\) for which the straight line is a tangent to the curve.
   3. Using this value of \(c\), find the coordinates of the point where the line touches the curve.
   4. In the case where \(c = 4\), determine whether the line intersects the curve or not.

8 A curve \(C\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 9 } = 1$$

1. Find the \(y\)-coordinates of the points on \(C\) for which \(x = 10\), giving each answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
2. Sketch the curve \(C\), indicating the coordinates of any points where the curve intersects the coordinate axes.
3. Write down the equation of the tangent to \(C\) at the point where \(C\) intersects the positive \(x\)-axis.
4. 1. Show that, if the line \(y = x - 4\) intersects \(C\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$16 x ^ { 2 } - 200 x + 625 = 0$$
   2. Solve this equation and hence state the relationship between the line \(y = x - 4\) and the curve \(C\).

9 A hyperbola \(H\) has equation $$x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$$

1. Find the equations of the two asymptotes of \(H\), giving each answer in the form \(y = m x\).
2. Draw a sketch of the two asymptotes of \(H\), using roughly equal scales on the two coordinate axes. Using the same axes, sketch the hyperbola \(H\).
3. 1. Show that, if the line \(y = x + c\) intersects \(H\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$x ^ { 2 } - 2 c x - \left( c ^ { 2 } + 2 \right) = 0$$
   2. Hence show that the line \(y = x + c\) intersects \(H\) in two distinct points, whatever the value of \(c\).
   3. Find, in terms of \(c\), the \(y\)-coordinates of these two points.

7 A graph has equation $$y = \frac { x - 4 } { x ^ { 2 } + 9 }$$

1. Explain why the graph has no vertical asymptote and give the equation of the horizontal asymptote.
2. Show that, if the line \(y = k\) intersects the graph, the \(x\)-coordinates of the points of intersection of the line with the graph must satisfy the equation $$k x ^ { 2 } - x + ( 9 k + 4 ) = 0$$
3. Show that this equation has real roots if \(- \frac { 1 } { 2 } \leqslant k \leqslant \frac { 1 } { 18 }\).
4. Hence find the coordinates of the two stationary points on the curve.
   (No credit will be given for methods involving differentiation.)

7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$

1. Find the equations of the asymptotes of \(H\).
2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3 \\ 0 \end{array} \right]\).
   1. Write down the equation of the translated curve.
   2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}

9 A curve has equation $$y = \frac { x } { x - 1 }$$

1. Find the equations of the asymptotes of this curve.
2. Given that the line \(y = - 4 x + c\) intersects the curve, show that the \(x\)-coordinates of the points of intersection must satisfy the equation $$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
3. It is given that the line \(y = - 4 x + c\) is a tangent to the curve.
   1. Find the two possible values of \(c\).
      (No credit will be given for methods involving differentiation.)
   2. For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.

9 An ellipse is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cf9337b9-b766-4ce5-967c-5d7522e2aa42-5_453_633_365_699} The ellipse intersects the \(x\)-axis at the points \(A\) and \(B\). The equation of the ellipse is $$\frac { ( x - 4 ) ^ { 2 } } { 4 } + y ^ { 2 } = 1$$

1. Find the \(x\)-coordinates of \(A\) and \(B\).
2. The line \(y = m x ( m > 0 )\) is a tangent to the ellipse, with point of contact \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$\left( 1 + 4 m ^ { 2 } \right) x ^ { 2 } - 8 x + 12 = 0$$
   2. Hence find the exact value of \(m\).
   3. Find the coordinates of \(P\).

7

1. 1. Write down the equations of the two asymptotes of the curve \(y = \frac { 1 } { x - 3 }\).
   2. Sketch the curve \(y = \frac { 1 } { x - 3 }\), showing the coordinates of any points of intersection with the coordinate axes.
   3. On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line \(y = 2 x - 5\).
2. 1. Solve the equation $$\frac { 1 } { x - 3 } = 2 x - 5$$
   2. Find the solution of the inequality $$\frac { 1 } { x - 3 } < 2 x - 5$$ □ \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}