1.02q Use intersection points: of graphs to solve equations

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = x ^ { 2 } - 5 x + 13\) The point \(M\) is the minimum point of \(C\). The straight line \(l\) passes through the origin \(O\) and intersects \(C\) at the points \(M\) and \(N\) as shown. Find, showing your working,

1. the coordinates of \(M\),
2. the coordinates of \(N\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_531_561_1793_680} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the curve \(C\) and the line \(l\). The finite region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
3. Use inequalities to define the region \(R\).

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

Solutions relying on calculator technology are not acceptable.

1. Write \(\frac { 8 - \sqrt { 15 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(a \sqrt { 3 } + b \sqrt { 5 }\) where \(a\) and \(b\) are integers to be found.
2. Hence, or otherwise, solve $$( x + 5 \sqrt { 3 } ) \sqrt { 5 } = 40 - 2 x \sqrt { 3 }$$ giving your answer in simplest form.

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

\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation $$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$ The curve \(C _ { 2 }\) has equation $$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$

1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet when $$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at points \(P\) and \(Q\)
2. find, using algebra, the exact distance \(P Q\)

13. (a) On separate axes sketch the graphs of

1. \(y = c ^ { 2 } - x ^ { 2 }\)
2. \(y = x ^ { 2 } ( x - 3 c )\) where \(c\) is a positive constant.
   Show clearly the coordinates of the points where each graph crosses or meets the \(x\)-axis and the \(y\)-axis.
   (b) Prove that the \(x\) coordinate of any point of intersection of $$y = c ^ { 2 } - x ^ { 2 } \text { and } y = x ^ { 2 } ( x - 3 c )$$ where \(c\) is a positive constant, is given by a solution of the equation $$x ^ { 3 } + ( 1 - 3 c ) x ^ { 2 } - c ^ { 2 } = 0$$ Given that the graphs meet when \(x = 2\) (c) find the exact value of \(c\), writing your answer as a fully simplified surd.

15. \begin{figure}[h] \end{figure} The straight line \(l\) with equation \(y = 5 - 3 x\) cuts the curve \(C\), with equation \(y = 20 x - 12 x ^ { 2 }\), at the points \(P\) and \(Q\), as shown in Figure 3.

1. Use algebra to find the exact coordinates of the points \(P\) and \(Q\). The finite region \(R\), shown shaded in Figure 3, is bounded by the line \(l\), the \(x\)-axis and the curve \(C\).
2. Use calculus to find the exact area of \(R\).

10. (a) On the same axes sketch the graphs of the curves with equations

1. \(y = x ^ { 2 } ( x - 2 )\),
2. \(y = x ( 6 - x )\),
   and indicate on your sketches the coordinates of all the points where the curves cross the \(x\)-axis.
   (b) Use algebra to find the coordinates of the points where the graphs intersect.

8. The point \(P ( 1 , a )\) lies on the curve with equation \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\).

1. Find the value of \(a\).
2. On the axes below sketch the curves with the following equations:
   1. \(y = ( x + 1 ) ^ { 2 } ( 2 - x )\),
   2. \(y = \frac { 2 } { x }\). On your diagram show clearly the coordinates of any points at which the curves meet the axes.
3. With reference to your diagram in part (b) state the number of real solutions to the equation $$( x + 1 ) ^ { 2 } ( 2 - x ) = \frac { 2 } { x } .$$
   \includegraphics[max width=\textwidth, alt={}]{871f5957-180d-4379-88ce-186432f57bad-10_1347_1344_1245_297}

10. (a) On the axes below, sketch the graphs of

1. \(y = x ( x + 2 ) ( 3 - x )\)
2. \(y = - \frac { 2 } { x }\) showing clearly the coordinates of all the points where the curves cross the coordinate axes.
   (b) Using your sketch state, giving a reason, the number of real solutions to the equation $$x ( x + 2 ) ( 3 - x ) + \frac { 2 } { x } = 0$$ \includegraphics[max width=\textwidth, alt={}, center]{95e11fd7-765c-477d-800b-7574bc1af81f-13_994_997_1270_479}

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y = 2 - \frac { 1 } { x } , \quad x \neq 0$$ The curve crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\).
2. Show that the equation of the normal to \(C\) at \(A\) can be written as $$2 x + 8 y - 1 = 0$$ The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 2 .
3. Find the coordinates of \(B\).

6. Figure 1 shows a sketch of the curve with equation \(y = \frac { 2 } { x } , x \neq 0\) The curve \(C\) has equation \(y = \frac { 2 } { x } - 5 , x \neq 0\), and the line \(l\) has equation \(y = 4 x + 2\)

1. Sketch and clearly label the graphs of \(C\) and \(l\) on a single diagram. On your diagram, show clearly the coordinates of the points where \(C\) and \(l\) cross the coordinate axes.
2. Write down the equations of the asymptotes of the curve \(C\).
3. Find the coordinates of the points of intersection of \(y = \frac { 2 } { x } - 5\) and \(y = 4 x + 2\)

6. The curve \(C\) has equation \(y = \frac { 3 } { x }\) and the line \(l\) has equation \(y = 2 x + 5\).

1. On the axes below, sketch the graphs of \(C\) and \(l\), indicating clearly the coordinates of any intersections with the axes.
2. Find the coordinates of the points of intersection of \(C\) and \(l\). \includegraphics[max width=\textwidth, alt={}, center]{9451ec48-d955-44a8-9988-68f7c0fb9821-07_1137_1141_1046_397}

10. (a) On the axes below sketch the graphs of

1. \(y = x ( 4 - x )\)
2. \(y = x ^ { 2 } ( 7 - x )\) showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\)-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 4 \right) = 0\) The point \(A\) lies on both of the curves and the \(x\) and \(y\) coordinates of \(A\) are both positive.
   (c) Find the exact coordinates of \(A\), leaving your answer in the form ( \(p + q \sqrt { } 3 , r + s \sqrt { } 3\) ), where \(p , q , r\) and \(s\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}

9. The line \(L _ { 1 }\) has equation \(4 y + 3 = 2 x\) The point \(A ( p , 4 )\) lies on \(L _ { 1 }\)

1. Find the value of the constant \(p\). The line \(L _ { 2 }\) passes through the point \(C ( 2,4 )\) and is perpendicular to \(L _ { 1 }\)
2. Find an equation for \(L _ { 2 }\) giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The line \(L _ { 1 }\) and the line \(L _ { 2 }\) intersect at the point \(D\).
3. Find the coordinates of the point \(D\).
4. Show that the length of \(C D\) is \(\frac { 3 } { 2 } \sqrt { } 5\) A point \(B\) lies on \(L _ { 1 }\) and the length of \(A B = \sqrt { } ( 80 )\) The point \(E\) lies on \(L _ { 2 }\) such that the length of the line \(C D E = 3\) times the length of \(C D\).
5. Find the area of the quadrilateral \(A C B E\).

9. (a) On separate axes sketch the graphs of

1. \(y = - 3 x + c\), where \(c\) is a positive constant,
2. \(y = \frac { 1 } { x } + 5\) On each sketch show the coordinates of any point at which the graph crosses the \(y\)-axis and the equation of any horizontal asymptote. Given that \(y = - 3 x + c\), where \(c\) is a positive constant, meets the curve \(y = \frac { 1 } { x } + 5\) at two distinct points,
   (b) show that \(( 5 - c ) ^ { 2 } > 12\) (c) Hence find the range of possible values for \(c\).

9. \begin{figure}[h] \end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 3.
2. Use calculus to find the exact area of \(R\).

5. \begin{figure}[h] \end{figure} Figure 2 shows the line with equation \(y = 10 - x\) and the curve with equation \(y = 10 x - x ^ { 2 } - 8\) The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). The shaded area \(R\) is bounded by the line and the curve, as shown in Figure 2.
2. Calculate the exact area of \(R\).

6. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 2 x - 5 | + 3 \quad x \geqslant 0$$ The vertex of the graph is at point \(P\) as shown.

1. State the coordinates of \(P\).
2. Solve the equation \(\mathrm { f } ( x ) = 3 x - 2\) Given that the equation $$f ( x ) = k x + 2$$ where \(k\) is a constant, has exactly two roots,
3. find the range of values of \(k\).
   

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | 3 x + a | + a$$ and where \(a\) is a positive constant. The graph has a vertex at the point \(P\), as shown in Figure 2 .

1. Find, in terms of \(a\), the coordinates of \(P\).
2. Sketch the graph with equation \(y = g ( x )\), where $$g ( x ) = | x + 5 a |$$ On your sketch, show the coordinates, in terms of \(a\), of each point where the graph cuts or meets the coordinate axes. The graph with equation \(y = \mathrm { g } ( x )\) intersects the graph with equation \(y = \mathrm { f } ( x )\) at two points.
3. Find, in terms of \(a\), the coordinates of the two points. \includegraphics[max width=\textwidth, alt={}, center]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-11_2255_50_314_34}
   

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7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph of \(C _ { 1 }\) with equation $$y = 5 - | 3 x - 22 |$$

1. Write down the coordinates of
   1. the vertex of \(C _ { 1 }\)
   2. the intersection of \(C _ { 1 }\) with the \(y\)-axis.
2. Find the \(x\) coordinates of the intersections of \(C _ { 1 }\) with the \(x\)-axis. Diagram 1, shown on page 21, is a copy of Figure 3.
3. On Diagram 1, sketch the curve \(C _ { 2 }\) with equation $$y = \frac { 1 } { 9 } x ^ { 2 } - 9$$ Identify clearly the coordinates of any points of intersection of \(C _ { 2 }\) with the coordinate axes.
4. Find the coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying entirely on calculator technology are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-21_629_803_1137_573} \section*{Diagram 1} Solutions relying entirely on calculator technology are not acceptable.

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curves \(C _ { 1 }\) and \(C _ { 2 }\) $$\begin{aligned} & C _ { 1 } \text { has equation } y = 3 + \mathrm { e } ^ { x + 1 } \quad x \in \mathbb { R } \\ & C _ { 2 } \text { has equation } y = 10 - \mathrm { e } ^ { x } \quad x \in \mathbb { R } \end{aligned}$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) cut the \(y\)-axis at the points \(P\) and \(Q\) respectively,

1. find the exact distance \(P Q\). \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point \(R\).
2. Find the exact coordinates of \(R\).

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6. \begin{figure}[h] \end{figure} Figure 1 is a sketch showing part of the curve with equation \(y = 2 ^ { x + 1 } - 3\) and part of the line with equation \(y = 17 - x\). The curve and the line intersect at the point \(A\).

1. Show that the \(x\) coordinate of \(A\) satisfies the equation $$x = \frac { \ln ( 20 - x ) } { \ln 2 } - 1$$
2. Use the iterative formula $$x _ { n + 1 } = \frac { \ln \left( 20 - x _ { n } \right) } { \ln 2 } - 1 , \quad x _ { 0 } = 3$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
3. Use your answer to part (b) to deduce the coordinates of the point \(A\), giving your answers to one decimal place.

1. The parabola \(C\) has equation \(y ^ { 2 } = 20 x\)

The point \(P\) on \(C\) has coordinates ( \(5 p ^ { 2 } , 10 p\) ) where \(p\) is a non-zero constant.

1. Use calculus to show that the tangent to \(C\) at \(P\) has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
2. Write down the coordinates of \(A\). The point \(S\) is the focus of \(C\).
3. Write down the coordinates of \(S\). The straight line \(l _ { 1 }\) passes through \(A\) and \(S\).
   The straight line \(l _ { 2 }\) passes through \(O\) and \(P\), where \(O\) is the origin. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\),
4. show that the coordinates of \(B\) satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where c is a positive constant. The point \(A\) on \(H\) has \(x\)-coordinate \(3 c\).

1. Write down the \(y\)-coordinate of \(A\).
2. Show that an equation of the normal to \(H\) at \(A\) is $$3 y = 27 x - 80 c$$ The normal to \(H\) at \(A\) meets \(H\) again at the point \(B\).
3. Find, in terms of \(c\), the coordinates of \(B\).

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has parametric equations $$x = \sin t - 3 \cos ^ { 2 } t \quad y = 3 \sin t + 2 \cos t \quad 0 \leqslant t \leqslant 5$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) where \(t = \pi\) The point \(P\) lies on \(C\) where \(t = \pi\)
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. Given that the tangent to the curve at \(P\) cuts \(C\) at the point \(Q\)
3. show that the value of \(t\) at point \(Q\) satisfies the equation $$9 \cos ^ { 2 } t + 2 \cos t - 7 = 0$$
4. Hence find the exact value of the \(y\) coordinate of \(Q\)

6. (a) On the same diagram, sketch the graphs of \(y = \left| x ^ { 2 } - 4 \right|\) and \(y = | 2 x - 1 |\), showing the coordinates of the points where the graphs meet the axes.
(b) Solve \(\left| x ^ { 2 } - 4 \right| = | 2 x - 1 |\), giving your answers in surd form where appropriate.
(c) Hence, or otherwise, find the set of values of \(x\) for which \(\left| x ^ { 2 } - 4 \right| > | 2 x - 1 |\).
(3)(Total 12 marks)