Multi-part: volume and tangent/normal

10 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-14_631_689_274_721} The diagram shows part of the curve with equation \(y = \frac { 4 } { ( 2 x - 1 ) ^ { 2 } }\) and parts of the lines \(x = 1\) and \(y = 1\). The curve passes through the points \(A ( 1,4 )\) and \(B , \left( \frac { 3 } { 2 } , 1 \right)\).

1. Find the exact volume generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. A triangle is formed from the tangent to the curve at \(B\), the normal to the curve at \(B\) and the \(x\)-axis. Find the area of this triangle.

7 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-3_646_841_593_651} The diagram shows part of the graph of \(y = \frac { 18 } { x }\) and the normal to the curve at \(P ( 6,3 )\). This normal meets the \(x\)-axis at \(R\). The point \(Q\) on the \(x\)-axis and the point \(S\) on the curve are such that \(P Q\) and \(S R\) are parallel to the \(y\)-axis.

1. Find the equation of the normal at \(P\) and show that \(R\) is the point ( \(4 \frac { 1 } { 2 } , 0\) ).
2. Show that the volume of the solid obtained when the shaded region \(P Q R S\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(18 \pi\).

10 The equation of a curve is \(y = \frac { 4 } { 2 x - 1 }\).

1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Given that the line \(2 y = x + c\) is a normal to the curve, find the possible values of the constant \(c\).

10 \includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-18_510_410_260_863} The diagram shows part of the curve \(y = \frac { 4 } { 5 - 3 x }\).

1. Find the equation of the normal to the curve at the point where \(x = 1\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   The shaded region is bounded by the curve, the coordinate axes and the line \(x = 1\).
2. Find, showing all necessary working, the volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-18_643_969_258_587} The diagram shows part of the curve \(y = \frac { x } { 2 } + \frac { 6 } { x }\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).

1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in terms of \(\pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 \includegraphics[max width=\textwidth, alt={}, center]{c8ac31bc-0f76-4d00-a28b-4a07758f9663-16_611_529_260_808} The diagram shows part of the curve with equation \(y = \sqrt { } \left( x ^ { 3 } + x ^ { 2 } \right)\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. \(P\) is the point on the curve with \(x\)-coordinate 3 . Find the \(y\)-coordinate of the point where the normal to the curve at \(P\) crosses the \(y\)-axis.

9 \includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-4_563_679_938_733} The diagram shows points \(A ( 0,4 )\) and \(B ( 2,1 )\) on the curve \(y = \frac { 8 } { 3 x + 2 }\). The tangent to the curve at \(B\) crosses the \(x\)-axis at \(C\). The point \(D\) has coordinates \(( 2,0 )\).

1. Find the equation of the tangent to the curve at \(B\) and hence show that the area of triangle \(B D C\) is \(\frac { 4 } { 3 }\).
2. Show that the volume of the solid formed when the shaded region \(O D B A\) is rotated completely about the \(x\)-axis is \(8 \pi\).

9 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-4_719_670_264_735} The diagram shows the curve \(y = \sqrt { } ( 3 x + 1 )\) and the points \(P ( 0,1 )\) and \(Q ( 1,2 )\) on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 2\).

1. Find the area of the shaded region.
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Tangents are drawn to the curve at the points \(P\) and \(Q\).
3. Find the acute angle, in degrees correct to 1 decimal place, between the two tangents.

10 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-4_799_1390_255_376} The diagram shows the curve \(y = \sqrt { } ( 1 + 2 x )\) meeting the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The \(y\)-coordinate of the point \(C\) on the curve is 3 .

1. Find the coordinates of \(B\) and \(C\).
2. Find the equation of the normal to the curve at \(C\).
3. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

9 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_502_663_255_740} The diagram shows part of the curve \(y = \frac { 9 } { 2 x + 3 }\), crossing the \(y\)-axis at the point \(B ( 0,3 )\). The point \(A\) on the curve has coordinates \(( 3,1 )\) and the tangent to the curve at \(A\) crosses the \(y\)-axis at \(C\).

1. Find the equation of the tangent to the curve at \(A\).
2. Determine, showing all necessary working, whether \(C\) is nearer to \(B\) or to \(O\).
3. Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-4_547_1057_255_543} The diagram shows the curve \(y = \sqrt { } \left( x ^ { 4 } + 4 x + 4 \right)\).

1. Find the equation of the tangent to the curve at the point ( 0,2 ).
2. Show that the \(x\)-coordinates of the points of intersection of the line \(y = x + 2\) and the curve are given by the equation \(( x + 2 ) ^ { 2 } = x ^ { 4 } + 4 x + 4\). Hence find these \(x\)-coordinates.
3. The region shaded in the diagram is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of revolution.

10 \includegraphics[max width=\textwidth, alt={}, center]{9cdb00a6-1e86-4185-bb73-ed3ecab981ba-4_634_937_696_603} The diagram shows part of the curve \(y = \sqrt { } \left( 9 - 2 x ^ { 2 } \right)\). The point \(P ( 2,1 )\) lies on the curve and the normal to the curve at \(P\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Show that \(B\) is the mid-point of \(A P\). The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\).
2. Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

10 \includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-16_648_823_262_660} The diagram shows part of the curve \(y = 2 ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\) and the lines \(x = \frac { 2 } { 3 }\) and \(x = 3\). The curve and the line \(x = \frac { 2 } { 3 }\) intersect at the point \(A\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the equation of the normal to the curve at \(A\), giving your answer in the form \(y = m x + c\).

9 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689} The diagram shows the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\). The shaded region between the curve, the \(x\)-axis and the line \(x = \mathrm { e }\) is denoted by \(R\).

1. Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = m x + c\).
2. Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

11 \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-18_643_969_258_587} The diagram shows part of the curve \(y = \frac { x } { 2 } + \frac { 6 } { x }\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).

1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in terms of \(\pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.