1.08b Integrate x^n: where n != -1 and sums

6. Given that \(\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0\),

1. find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 5\).
2. Show that \(\mathrm { f } ( x )\) may be written in the form \(A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C\), where \(A , B\) and \(C\) are constants to be found.
3. Hence evaluate \(\int _ { 1 } ^ { 2 } f ( x ) d x\).

2.

1. Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
2. Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
   

2. Show that $$\int _ { 2 } ^ { 3 } \left( 6 \sqrt { x } - \frac { 4 } { \sqrt { x } } \right) \mathrm { d } x = k \sqrt { 3 } ,$$ where \(k\) is an integer to be found.

8. \begin{figure}[h] \end{figure} Figure 3 shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).

1. Find the coordinates of the minimum point of the curve.
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).

1. Evaluate

$$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } \mathrm {~d} x .$$

2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is given by \(k \left( 4 - \sqrt { \frac { x } { a } } \right)\), where \(k\) is a constant. Find the distance from \(A\) of the centre of mass of the rod.

3 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = a \mathrm { e } ^ { \frac { x } { a } }\) and the line \(x = a \ln 2\) (where \(a\) is a positive constant). A uniform solid of revolution, of mass \(M\), is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find, in terms of \(M\) and \(a\), the moment of inertia about the \(x\)-axis of this solid of revolution.
[0pt] [8]

2 A uniform solid circular cone has mass \(M\) and base radius \(R\).

1. Show by integration that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M R ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis and that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).) The axis of symmetry of the cone is fixed vertically and the cone is rotating about its axis at an angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A frictional couple of constant moment 0.027 Nm is applied to the cone bringing it to rest. Given that the mass of the cone is 2 kg and its base radius is 0.3 m , find
2. the constant angular deceleration of the cone,
3. the time taken for the cone to come to rest from the instant that the couple is applied.

3 The region bounded by the \(y\)-axis and the curves \(y = \sin 2 x\) and \(y = \sqrt { 2 } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) is occupied by a uniform lamina. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.

4 A uniform square lamina has mass \(m\) and sides of length \(2 a\).

1. Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-3_693_640_434_715} The uniform square lamina has centre \(C\) and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners \(A\). The lamina is initially held such that \(A C\) is vertical with \(C\) above \(A\). The lamina is slightly disturbed from rest from this initial position. When \(A C\) makes an angle \(\theta\) with the upward vertical, the force exerted by the axis on the lamina has components \(X\) parallel to \(A C\) and \(Y\) perpendicular to \(A C\) (see diagram).
2. Show that the angular speed, \(\omega\), of the lamina satisfies \(a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )\).
3. Find \(X\) and \(Y\) in terms of \(m , g\) and \(\theta\). \section*{Question 5 begins on page 4.}
   \includegraphics[max width=\textwidth, alt={}]{639c658e-0aca-4161-9e77-0f4c494b0b55-4_767_337_248_863}
   A pendulum consists of a uniform rod \(A B\) of length \(4 a\) and mass \(4 m\) and a spherical shell of radius \(a\), mass \(m\) and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the surface of the shell in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\) (see diagram).

1. Show that the moment of inertia of the pendulum about the axis of rotation is \(47 m a ^ { 2 }\). A particle of mass \(m\) is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed \(\sqrt { k g a }\), where \(k\) is a positive constant, and strikes the rod at a distance \(3 a\) from \(A\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(P\) starts to rotate.
2. Show that the initial angular speed of \(P\) is \(\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }\).
3. For the case \(k = 4\), find the angle that \(P\) has turned through when \(P\) first comes to instantaneous rest.
4. Find the least value of \(k\) such that the rod reaches the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-5_437_903_269_573} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\) is attached to \(A\). The string passes over a small smooth fixed pulley \(C\), where \(A C\) is horizontal and \(A C = a\). The other end of the string is attached to the rod at its mid-point \(D\). The rod makes an angle \(\theta\) below the horizontal (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
2. Show that \(\theta = \frac { 1 } { 6 } \pi\) is a position of stable equilibrium for the system. The system is making small oscillations about the equilibrium position.
3. By differentiating the energy equation with respect to time, show that $$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
4. Using the substitution \(\theta = \phi + \frac { 1 } { 6 } \pi\), show that the motion is approximately simple harmonic, and find the approximate period of the oscillations. \section*{END OF QUESTION PAPER}

2 The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\), and the curve \(y = k x ^ { 2 }\), where \(k\) is a positive constant, is occupied by a uniform lamina.

1. Find the exact \(x\)-coordinate of the centre of mass of the lamina.
2. Given that the \(x\) - and \(y\)-coordinates of the centre of mass of the lamina are equal, find the exact value of \(k\).

4

1. Write down the moment of inertia of a uniform circular disc of mass \(m\) and radius \(2 a\) about a diameter. A uniform solid cylinder has mass \(M\), radius \(2 r\) and height \(h\).
2. Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is $$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$ and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-3_919_897_260_591} A smooth circular wire hoop, with centre \(O\) and radius \(r\), is fixed in a vertical plane. The highest point on the wire is \(H\). A small bead \(B\) of mass \(m\) is free to move along the wire. A light inextensible string of length \(a\), where \(a > 2 r\), has one end attached to the bead. The other end of the string passes over a small smooth pulley at \(H\) and carries at its end a particle \(P\) of mass \(\lambda m\), where \(\lambda\) is a positive constant. The part of the string \(H P\) is vertical and the part of the string \(B H\) makes an angle \(\theta\) radians with the downward vertical where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\) (see diagram). You may assume that \(P\) remains above the lowest point of the wire.

3 \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-2_439_444_1318_822} A uniform rod \(A B\) has mass \(m\) and length \(4 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) is attached to \(B\). The other end of the string is attached to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(3 a\) above \(A\). The string is always vertical and the rod makes an angle \(\theta\) radians with the horizontal, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, find an expression for the total potential energy \(V\) of the system, and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \cos \theta ( 4 \lambda ( 1 + 2 \sin \theta ) - 1 ) .$$ Determine the positions of equilibrium and the nature of their stability in the cases
2. \(\lambda > \frac { 1 } { 12 }\),
3. \(\lambda < \frac { 1 } { 12 }\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-3_392_689_269_671} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \ln x\). The region \(R\), shaded in the diagram, is bounded by the curve, the \(x\)-axis and the line \(x = 4\). A uniform solid of revolution is formed by rotating \(R\) completely about the \(y\)-axis to form a solid of volume \(V\).

1. Show that \(V = \frac { 1 } { 4 } \pi ( 64 \ln 2 - 15 )\).
2. Find the exact \(y\)-coordinate of the centre of mass of the solid. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_385_741_269_646} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows part of the line \(y = \frac { a } { h } x\), where \(a\) and \(h\) are constants. The shaded region bounded by the line, the \(x\)-axis and the line \(x = h\) is rotated about the \(x\)-axis to form a uniform solid cone of base radius \(a\), height \(h\) and volume \(\frac { 1 } { 3 } \pi a ^ { 2 } h\). The mass of the cone is \(M\).

1. Show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 3 } { 20 } M \left( a ^ { 2 } + 4 h ^ { 2 } \right)\). (You may assume the standard formula \(\frac { 1 } { 4 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about a diameter.) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_501_556_1238_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A uniform solid cone has mass 3 kg , base radius 0.4 m and height 1.2 m . The cone can rotate about a fixed vertical axis passing through its centre of mass with the axis of the cone moving in a horizontal plane. The cone is rotating about this vertical axis at an angular speed of \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A stationary particle of mass \(m \mathrm {~kg}\) becomes attached to the vertex of the cone (see Fig. 2). The particle being attached to the cone causes the angular speed to change instantaneously from \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(7.8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the value of \(m\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-5_534_501_255_767} A triangular frame \(A B C\) consists of three uniform rods \(A B , B C\) and \(C A\), rigidly joined at \(A , B\) and \(C\). Each rod has mass \(m\) and length \(2 a\). The frame is free to rotate in a vertical plane about a fixed horizontal axis passing through \(A\). The frame is initially held such that the axis of symmetry through \(A\) is vertical and \(B C\) is below the level of \(A\). The frame starts to rotate with an initial angular speed of \(\omega\) and at time \(t\) the angle between the axis of symmetry through \(A\) and the vertical is \(\theta\) (see diagram).

1. Show that the moment of inertia of the frame about the axis through \(A\) is \(6 m a ^ { 2 }\).
2. Show that the angular speed \(\dot { \theta }\) of the frame when it has turned through an angle \(\theta\) satisfies $$a \dot { \theta } ^ { 2 } = a \omega ^ { 2 } - k g \sqrt { 3 } ( 1 - \cos \theta ) ,$$ stating the exact value of the constant \(k\).
   Hence find, in terms of \(a\) and \(g\), the set of values of \(\omega ^ { 2 }\) for which the frame makes complete revolutions. At an instant when \(\theta = \frac { 1 } { 6 } \pi\), the force acting on the frame at \(A\) has magnitude \(F\).
3. Given that \(\omega ^ { 2 } = \frac { 2 g } { a \sqrt { 3 } }\), find \(F\) in terms of \(m\) and \(g\). \section*{END OF QUESTION PAPER}

8 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:

1. \(\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x\);
2. \(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).

3 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:

1. \(\int _ { 9 } ^ { \infty } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
2. \(\int _ { 9 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).

1 In this question you must show detailed reasoning. Show that \(\int _ { 4 } ^ { 9 } ( 2 x + \sqrt { x } ) \mathrm { d } x = \frac { 233 } { 3 }\).

5

1. Find \(\int \left( x ^ { 3 } - 6 x \right) \mathrm { d } x\).
2. 1. Find \(\int \left( \frac { 4 } { x ^ { 2 } } - 1 \right) \mathrm { d } x\).
   2. The diagram shows part of the curve \(y = \frac { 4 } { x ^ { 2 } } - 1\). \includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-04_707_1283_708_415} The curve crosses the \(x\)-axis at \(( 2,0 )\).
      The shaded region is bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = 5\). Calculate the area of the shaded region.

20. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$$ and that \(C\) passes through the point \(P ( 2,6 )\),

1. find \(y\) in terms of \(x\).
2. Verify that \(C\) passes through the point ( 4,0 ).
3. Find an equation of the tangent to \(C\) at \(P\). The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).
4. Calculate the exact \(x\)-coordinate of \(Q\).
   21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find \(\int y \mathrm {~d} x\).
   

22.

1. Given that \(3 ^ { x } = 9 ^ { y - 1 }\), show that \(x = 2 y - 2\).
2. Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2 \\ x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
   1. The straight line \(l _ { 1 }\) with equation \(y = \frac { 3 } { 2 } x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \(( 5 , - 3 )\).
   The straight line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(Q\).

1. Calculate the coordinates of the mid-point of \(P Q\).
2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integer constants. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\).
3. Calculate the exact coordinates of \(R\).
   24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$

1. Use integration to find \(y\) in terms of \(x\).
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\).
   25. Find the set of values for \(x\) for which

1. \(6 x - 7 < 2 x + 3\),
2. \(2 x ^ { 2 } - 11 x + 5 < 0\),
3. both \(6 x - 7 < 2 x + 3\) and \(2 x ^ { 2 } - 11 x + 5 < 0\).
   [0pt] [P1 June 2003 Question 2]
   26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \(( 280 + x )\) phones will be sold in the second month, \(( 280 + 2 x )\) in the third month, and so on. Using this model with \(x = 5\), calculate

1. 1. the number of phones sold in the 36th month,
   2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
      Using the same model,
2. find the least value of \(x\) required to achieve this target.
   [0pt] [P1 June 2003 Question 3]
   27. The points \(A\) and \(B\) have coordinates \(( 4,6 )\) and \(( 12,2 )\) respectively. The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a\), b and \(c\) are integers. The straight line \(l _ { 2 }\) passes through the origin and has gradient - 4 .
2. Write down an equation for \(l _ { 2 }\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intercept at the point \(C\).
3. Find the exact coordinates of the mid-point of \(A C\).
   28. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
2. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   [0pt] [P1 June 2003 Question 8*]
   29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$

1. Write down the first two terms of the series.
2. Find the common difference of the series. Given that \(n = 50\),
3. find the sum of the series.
   

30.

1. Solve the equation \(4 x ^ { 2 } + 12 x = 0\). $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where \(c\) is a constant.
2. Given that \(\mathrm { f } ( x ) = 0\) has equal roots, find the value of \(c\) and hence solve \(\mathrm { f } ( x ) = 0\).
   31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0 \\ & x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
   1. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.
   Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-15_668_748_358_699}
   The points \(A\) and \(B\) have coordinates \(( 2 , - 3 )\) and \(( 8,5 )\) respectively, and \(A B\) is a chord of a circle with centre \(C\), as shown in Fig. 1.

1. Find the gradient of \(A B\). The point \(M\) is the mid-point of \(A B\).
2. Find an equation for the line through \(C\) and \(M\). Given that the \(x\)-coordinate of \(C\) is 4 ,
3. find the \(y\)-coordinate of \(C\),
4. show that the radius of the circle is \(\frac { 5 \sqrt { } 17 } { 4 }\).
   34. The first three terms of an arithmetic series are \(p , 5 p - 8\), and \(3 p + 8\) respectively.

1. Show that \(p = 4\).
2. Find the value of the 40th term of this series.
3. Prove that the sum of the first \(n\) terms of the series is a perfect square.
   35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$

1. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has no real solutions. Given that \(k = 4\),
2. express \(\mathrm { f } ( x )\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found,
   36. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$

1. Show that, when \(x = 8\), the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(9 \sqrt { } 2\). The curve \(C\) passes through the point \(( 4,30 )\).
2. Using integration, find \(\mathrm { f } ( x )\).
   37. \section*{Figure 2}
   \includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-17_687_1074_351_539}
   Figure 2 shows the curve with equation \(y ^ { 2 } = 4 ( x - 2 )\) and the line with equation \(2 x - 3 y = 12\).
   The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\).
2. Find, using algebra, the coordinates of \(P\) and \(Q\).
3. Show that \(\angle P A Q\) is a right angle.
   38. A sequence is defined by the recurrence relation $$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ where \(a\) is a constant.

1. Given that \(a = 20\) and \(u _ { 1 } = 3\), find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\), giving your answers to 2 decimal places.
2. Given instead that \(u _ { 1 } = u _ { 2 } = 3\),
   1. calculate the value of \(a\),
   2. write down the value of \(u _ { 5 }\).
      [0pt] [P2 January 2004 Question 2]
      39. The points \(A\) and \(B\) have coordinates \(( 1,2 )\) and \(( 5,8 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find, in the form \(y = m x + c\), an equation for the straight line through \(A\) and \(B\).
   40. Giving your answers in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers, find

1. \(( 3 - \sqrt { } 8 ) ^ { 2 }\),
2. \(\frac { 1 } { 4 - \sqrt { 8 } }\).
   41. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,

1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
2. form a quadratic inequality in \(x\).
3. by solving your inequalities, find the set of possible values of \(x\).
   42. The curve \(C\) has equation \(y = x ^ { 2 } - 4\) and the straight line \(l\) has equation \(y + 3 x = 0\).

1. In the space below, sketch \(C\) and \(l\) on the same axes.
2. Write down the coordinates of the points at which \(C\) meets the coordinate axes.
3. Using algebra, find the coordinates of the points at which \(l\) intersects \(C\).
   43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$

1. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
2. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).

5 The diagram shows an open-topped water tank with a horizontal rectangular base and four vertical faces. The base has width \(x\) metres and length \(2 x\) metres, and the height of the tank is \(h\) metres. \includegraphics[max width=\textwidth, alt={}, center]{33da89e2-f74f-4d5a-8bbd-ceaa728b6c34-4_403_410_477_792} The combined internal surface area of the base and four vertical faces is \(54 \mathrm {~m} ^ { 2 }\).

1. 1. Show that \(x ^ { 2 } + 3 x h = 27\).
   2. Hence express \(h\) in terms of \(x\).
   3. Hence show that the volume of water, \(V \mathrm {~m} ^ { 3 }\), that the tank can hold when full is given by $$V = 18 x - \frac { 2 x ^ { 3 } } { 3 }$$
2. 1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
   2. Verify that \(V\) has a stationary value when \(x = 3\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) and hence determine whether \(V\) has a maximum value or a minimum value when \(x = 3\).
   (2 marks)

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
   1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
   1. State the coordinates of the point \(A\).
   2. Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
   4. Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
   5. Hence find an equation of the normal to the curve at the point \(B\).

6

1. The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5\).
   1. Use the Factor Theorem to show that \(x - 1\) is a factor of \(\mathrm { f } ( x )\).
   2. Express \(\mathrm { f } ( x )\) in the form \(( x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
   3. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root and state its value.
2. The curve with equation \(y = x ^ { 3 } + 4 x - 5\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529} The curve cuts the \(x\)-axis at the point \(A ( 1,0 )\) and the point \(B ( 2,11 )\) lies on the curve.
   1. Find \(\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x\).
   2. Hence find the area of the shaded region bounded by the curve and the line \(A B\).

5 The curve with equation \(y = 16 - x ^ { 4 }\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-3_435_663_824_685} The points \(A ( - 2,0 ) , B ( 2,0 )\) and \(C ( 1,15 )\) lie on the curve.

1. Find an equation of the straight line \(A C\).
2. 1. Find \(\int _ { - 2 } ^ { 1 } \left( 16 - x ^ { 4 } \right) \mathrm { d } x\).
   2. Hence calculate the area of the shaded region bounded by the curve and the line \(A C\).

4

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\).
   1. Find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
   2. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   3. Express \(\mathrm { p } ( x ) = x ^ { 3 } - x + 6\) in the form \(( x + 2 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
   4. The equation \(\mathrm { p } ( x ) = 0\) has one root equal to - 2 . Show that the equation has no other real roots.
2. The curve with equation \(y = x ^ { 3 } - x + 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{5f1ff5fa-b6e8-4c4f-aef7-63eb947b299f-3_529_702_945_667} The curve cuts the \(x\)-axis at the point \(A ( - 2,0 )\) and the \(y\)-axis at the point \(B\).
   1. State the \(y\)-coordinate of the point \(B\).
   2. Find \(\int _ { - 2 } ^ { 0 } \left( x ^ { 3 } - x + 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - x + 6\) and the line \(A B\).