Two Curves Intersection Area

8 \includegraphics[max width=\textwidth, alt={}, center]{cba7391d-2cf7-483c-9a21-bc9fd49860ee-12_570_961_260_591} The diagram shows the curves with equations \(y = x ^ { - \frac { 1 } { 2 } }\) and \(y = \frac { 5 } { 2 } - x ^ { \frac { 1 } { 2 } }\). The curves intersect at the points \(A \left( \frac { 1 } { 4 } , 2 \right)\) and \(B \left( 4 , \frac { 1 } { 2 } \right)\).

1. Find the area of the region between the two curves.
2. The normal to the curve \(y = x ^ { - \frac { 1 } { 2 } }\) at the point \(( 1,1 )\) intersects the \(y\)-axis at the point \(( 0 , p )\). Find the value of \(p\).

10 \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-16_942_933_262_605} Curves with equations \(y = 2 x ^ { \frac { 1 } { 2 } } + 1\) and \(y = \frac { 1 } { 2 } x ^ { 2 } - x + 1\) intersect at \(A ( 0,1 )\) and \(B ( 4,5 )\), as shown in the diagram.

1. Find the area of the region between the two curves.
   The acute angle between the two tangents at \(B\) is denoted by \(\alpha ^ { \circ }\), and the scales on the axes are the same.
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{0f59214c-df46-46a6-ae9e-b9c8f104e159-18_951_725_267_703} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\). Tangents touching the circle at points \(B\) and \(C\) pass through the point \(A ( 0,10 )\).
   1. By letting the equation of a tangent be \(y = m x + 10\), find the two possible values of \(m\).
   2. Find the coordinates of \(B\) and \(C\).
      The point \(D\) is where the circle crosses the positive \(x\)-axis.
   3. Find angle \(B D C\) in degrees.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-5_710_931_255_607} The diagram shows parts of the curves \(y = 9 - x ^ { 3 }\) and \(y = \frac { 8 } { x ^ { 3 } }\) and their points of intersection \(P\) and \(Q\). The \(x\)-coordinates of \(P\) and \(Q\) are \(a\) and \(b\) respectively.

1. Show that \(x = a\) and \(x = b\) are roots of the equation \(x ^ { 6 } - 9 x ^ { 3 } + 8 = 0\). Solve this equation and hence state the value of \(a\) and the value of \(b\).
2. Find the area of the shaded region between the two curves.
3. The tangents to the two curves at \(x = c\) (where \(a < c < b\) ) are parallel to each other. Find the value of \(c\).

11 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-4_995_905_260_621} The diagram shows parts of the curves \(y = ( 4 x + 1 ) ^ { \frac { 1 } { 2 } }\) and \(y = \frac { 1 } { 2 } x ^ { 2 } + 1\) intersecting at points \(P ( 0,1 )\) and \(Q ( 2,3 )\). The angle between the tangents to the two curves at \(Q\) is \(\alpha\).

1. Find \(\alpha\), giving your answer in degrees correct to 3 significant figures.
2. Find by integration the area of the shaded region.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = x ^ { 3 } - 6 x + 9 & x \geqslant 0 \\ C _ { 2 } : y = - 2 x ^ { 2 } + 7 x - 1 & x \geqslant 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\) as shown in Figure 1 .
The point \(A\) has coordinates (1,4). Using algebra and showing all steps of your working,

1. find the coordinates of the point \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Use algebraic integration to find the exact area of \(R\).

8. Figure 2 Figure 2 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { l l } C _ { 1 } : y = 10 x - x ^ { 2 } - 8 & x > 0 \\ C _ { 2 } : y = x ^ { 3 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\).

1. Verify that the point \(A\) has coordinates (1, 1)
2. Use algebra to find the coordinates of the point \(B\) The finite region \(R\) is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
3. Use calculus to find the exact area of \(R\) \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-23_936_759_118_582} \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-26_2674_1948_107_118}

4 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_547_511_1813_817} The diagram shows a sketch of parts of the curves \(y = \frac { 16 } { x ^ { 2 } }\) and \(y = 17 - x ^ { 2 }\).

1. Verify that these curves intersect at the points \(( 1,16 )\) and \(( 4,1 )\).
2. Calculate the exact area of the shaded region between the curves.

9. \includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-3_538_872_1790_447} The diagram shows the curves \(y = 2 x ^ { 2 } - 6 x - 3\) and \(y = 9 + 3 x - x ^ { 2 }\).

1. Find the coordinates of the points where the two curves intersect.
2. Find the area of the shaded region bounded by the two curves.

5 \includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_646_839_255_653} The diagram shows parts of the curves \(y = x ^ { 2 } + 1\) and \(y = 11 - \frac { 9 } { x ^ { 2 } }\), which intersect at \(( 1,2 )\) and \(( 3,10 )\). Use integration to find the exact area of the shaded region enclosed between the two curves.

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1. Find \(\int \left( x ^ { 2 } + 4 \right) ( x - 6 ) \mathrm { d } x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-4_449_551_349_758} The diagram shows the curve \(y = 6 x ^ { \frac { 3 } { 2 } }\) and part of the curve \(y = \frac { 8 } { x ^ { 2 } } - 2\), which intersect at the point \(( 1,6 )\). Use integration to find the area of the shaded region enclosed by the two curves and the \(x\)-axis.

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 2 x ^ { 3 } + 10 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 42 x - 15 x ^ { 2 } - 7 \quad x > 0$$

1. Verify that the curves intersect at \(x = \frac { 1 } { 2 }\) The curves intersect again at the point \(P\)
2. Using algebra and showing all stages of working, find the exact \(x\) coordinate of \(P\)

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

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = x ^ { 2 } + 3 \quad x > 0$$ and part of the curve \(C _ { 2 }\) with equation $$y = 13 - \frac { 9 } { x ^ { 2 } } \quad x > 0$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\) as shown in Figure 1 .

1. Use algebra to find the \(x\) coordinate of \(P\) and the \(x\) coordinate of \(Q\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Use algebraic integration to find the exact area of \(R\).

Two curves are defined by \(y = x^k\) and \(y = x^{\frac{1}{k}}\), for \(x \geqslant 0\), where \(k > 0\).

1. Prove that, except for one value of \(k\), the curves intersect in exactly two points. [4]

The two curves enclose a finite region \(R\).

1. Find the area, \(A\), of \(R\), giving your answer in the form \(A = f(k)\) and distinguishing clearly between the cases \(k < 1\) and \(k > 1\). [4]
2. Determine the set of values of \(k\) for which \(A \leqslant 0.5\). [3]
3. The function \(f\) is given by \(f : x \mapsto A\) with \(k > 1\). Prove that \(f\) is one-one and determine its inverse. [4]