1.02q Use intersection points: of graphs to solve equations

1

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-1_650_759_252_762} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
   (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (D) State what this limit represents.

10 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h] \end{figure}

1. Write down the coordinates of P .
2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{MEI STRUCTURED MATHEMATICS} Further Methods for Advanced Mathematics (FP2)
   Tuesday

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve \(C\) crosses the \(x\)-axis at the point \(A\).

1. State the \(x\) coordinate of the point \(A\). The curve \(D\) has equation \(y = x ^ { 2 } ( x - 2 )\), for all real values of \(x\).
2. A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve \(D\).
   Show on the sketch the coordinates of each point where the curve \(D\) crosses the coordinate axes.
3. Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}

9

1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).

10 A circle has equation \(( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).

1. Find the coordinates of the centre \(C\) and the length of the diameter.
2. Find the equation of the line which passes through \(C\) and the point \(P ( 7,2 )\).
3. Calculate the length of \(C P\) and hence determine whether \(P\) lies inside or outside the circle.
4. Determine algebraically whether the line with equation \(y = 2 x\) meets the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

13 \begin{figure}[h] \end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).

1. Find the exact coordinates of the points of intersection of this curve with the axes.
2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h] \end{figure}

1. Write down the coordinates of B .
2. Find the radius of the circle and hence write down the equation of the circle.
3. AD is a diameter of the circle. Find the coordinates of D .
4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).

1. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
2. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
3. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
4. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).

1. Use the intersections with both axes to express the equation of the curve in a factorised form.
2. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
3. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0 .$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. \section*{END OF QUESTION PAPER}

9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h] \end{figure}

1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
   (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
   (B) Find the equation of the translated curve, simplifying your answer.

11

1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}

11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.

4

1. Show by means of suitable sketch graphs that the equation $$( x - 2 ) ^ { 4 } = x + 16$$ has exactly 2 real roots.
2. State the value of the smaller root.
3. Use the iterative formula $$x _ { n + 1 } = 2 + \sqrt [ 4 ] { x _ { n } + 16 }$$ with a suitable starting value, to find the larger root correct to 3 decimal places.

5

1. It is given that \(k\) is a positive constant. By sketching the graphs of $$y = 14 - x ^ { 2 } \text { and } y = k \ln x$$ on a single diagram, show that the equation $$14 - x ^ { 2 } = k \ln x$$ has exactly one real root.
2. The real root of the equation \(14 - x ^ { 2 } = 3 \ln x\) is denoted by \(\alpha\).
   1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
   2. Use the iterative formula \(x _ { n + 1 } = \sqrt { 14 - 3 \ln x _ { n } }\), with a suitable starting value, to find \(\alpha\). Show the result of each iteration, and give \(\alpha\) correct to 2 decimal places.

6 \includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_524_720_246_676} The diagram shows the curve \(y = x ^ { 4 } - 8 x\).

1. By sketching a second curve on the copy of the diagram, show that the equation $$x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$$ has two real roots. State the equation of the second curve.
2. The larger root of the equation \(x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0\) is denoted by \(\alpha\).
   1. Show by calculation that \(2.1 < \alpha < 2.2\).
   2. Use an iterative process based on the equation $$x = \sqrt [ 4 ] { 9 + 8 x - x ^ { 2 } } ,$$ with a suitable starting value, to find \(\alpha\) correct to 3 decimal places. Give the result of each step of the iterative process.

8 The curve \(C _ { 1 }\) has equation \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are polynomials of degree 2 and 1 respectively. The asymptotes of the curve are \(x = - 2\) and \(y = \frac { 1 } { 2 } x + 1\), and the curve passes through the point \(\left( - 1 , \frac { 17 } { 2 } \right)\).

1. Express the equation of \(C _ { 1 }\) in the form \(y = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\).
2. For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
   (ii) For the curve \(C _ { 1 }\), find the range of values that \(y\) can take.
   Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
   (iii) It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of intersection. Another curve, \(C _ { 2 }\), has equation \(y ^ { 2 } = \frac { \mathrm { p } ( x ) } { \mathrm { q } ( x ) }\), where \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are the polynomials found in part (i).
   (iii) It is given that \(C _ { 2 }\) intersects the line \(y = \frac { 1 } { 2 } x + 1\) exactly once. Find the coordinates of the point of
   intersection. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + 4 }$$ where \(a\), \(b\) and \(c\) are constants. It is given that \(y = 2 x - 5\) is an asymptote of \(C\).

1. Find the values of \(a\) and \(b\).
2. Given also that \(C\) has a turning point at \(x = - 1\), find the value of \(c\).
3. Find the set of values of \(y\) for which there are no points on \(C\).
4. Draw a sketch of the curve with equation $$y = \frac { 2 ( x - 7 ) ^ { 2 } + 3 ( x - 7 ) - 2 } { x - 3 }$$ [You should state the equations of the asymptotes and the coordinates of the turning points.]

7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + p x + 1 } { x - 2 }\), where \(p\) is a constant. Given that \(C\) has two asymptotes, find the equation of each asymptote. Find the set of values of \(p\) for which \(C\) has two distinct turning points. Sketch \(C\) in the case \(p = - 1\). Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.

9
The diagram shows the graph of \(y = | 2 x - 3 |\).

1. State the coordinates of the points of intersection with the axes.
2. Given that the graphs of \(y = | 2 x - 3 |\) and \(y = a x + 2\) have two distinct points of intersection, determine
   1. the set of possible values of \(a\),
   2. the \(x\)-coordinates of the points of intersection of these graphs, giving your answers in terms of \(a\).

1. The curve \(C\) has equation

$$y = \frac { k ^ { 2 } } { x } + 1 \quad x \in \mathbb { R } , x \neq 0$$ where \(k\) is a constant.

1. Sketch \(C\) stating the equation of the horizontal asymptote. The line \(l\) has equation \(y = - 2 x + 5\)
2. Show that the \(x\) coordinate of any point of intersection of \(l\) with \(C\) is given by a solution of the equation $$2 x ^ { 2 } - 4 x + k ^ { 2 } = 0$$
3. Hence find the exact values of \(k\) for which \(l\) is a tangent to \(C\).

1. A company started mining tin in Riverdale on 1st January 2019.

A model to find the total mass of tin that will be mined by the company in Riverdale is given by the equation $$T = 1200 - 3 ( n - 20 ) ^ { 2 }$$ where \(T\) tonnes is the total mass of tin mined in the \(n\) years after the start of mining.
Using this model,

1. calculate the mass of tin that will be mined up to 1st January 2020,
2. deduce the maximum total mass of tin that could be mined,
3. calculate the mass of tin that will be mined in 2023.
4. State, giving reasons, the limitation on the values of \(n\).

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

$$y = \frac { 6 } { x } + 3$$

1. 1. Sketch \(C _ { 1 }\) stating the coordinates of any points where the curve cuts the coordinate axes.
   2. State the equations of any asymptotes to the curve \(C _ { 1 }\) The curve \(C _ { 2 }\) has equation $$y = 3 x ^ { 2 } - 4 x - 10$$
   (b) Show that \(C _ { 1 }\) and \(C _ { 2 }\) intersect when $$3 x ^ { 3 } - 4 x ^ { 2 } - 13 x - 6 = 0$$ Given that the \(x\) coordinate of one of the points of intersection is \(- \frac { 2 } { 3 }\)
2. use algebra to find the \(x\) coordinates of the other points of intersection between \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying on calculator technology are not acceptable.)