Edexcel C4 (Core Mathematics 4) 2013 June

1. (a) Find the binomial expansion of

$$\sqrt { } ( 9 + 8 x ) , \quad | x | < \frac { 9 } { 8 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate the value of \(\sqrt { } ( 11 )\), giving your answer as a single fraction.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = x \mathrm { e } ^ { - \frac { 1 } { 2 } x } , x \geqslant 0\).
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = x e ^ { - \frac { 1 } { 2 } x }\).

1. Complete the table with the value of \(y\) corresponding to \(x = 2\)
2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
3. 1. Find \(\int x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x\).
   2. Hence find the exact area of \(R\), giving your answer in the form \(a + b \mathrm { e } ^ { - 2 }\), where \(a\) and \(b\) are integers.

1. A curve \(C\) has parametric equations

$$x = 2 t + 5 , \quad y = 3 + \frac { 4 } { t } , \quad t \neq 0$$

1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates \(( 9,5 )\).
2. Find a cartesian equation of the curve in the form $$y = \frac { a x + b } { c x + d }$$ where \(a\), \(b\), \(c\) and \(d\) are integers.

4. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 9

5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).
   The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\).
3. Find a vector equation of \(l _ { 2 }\).
4. Find the length of \(O B\), giving your answer as a simplified surd. The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
5. find the length of \(O X\), giving your answer to 3 significant figures.
   5. The curve \(C\) has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).
   The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
7. Find the exact value of the \(x\)-coordinate of \(A\).

6. (i) (a) Express \(\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }\) in partial fractions.
(b) Given that \(x > \frac { 1 } { 2 }\), find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$

8
5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).
   The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\).
3. Find a vector equation of \(l _ { 2 }\).
4. Find the length of \(O B\), giving your answer as a simplified surd. The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
5. find the length of \(O X\), giving your answer to 3 significant figures.
   5. The curve \(C\) has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).
   The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
7. Find the exact value of the \(x\)-coordinate of \(A\).
   6. (i) (a) Express \(\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }\) in partial fractions.
8. Given that \(x > \frac { 1 } { 2 }\), find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$ 7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-11_703_1164_373_492} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = 1 + 2 \cos 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(x\)-axis at \(( \sqrt { } 3,0 )\). The finite shaded region \(S\) shown in Figure 2 is bounded by \(C\), the line \(x = 1\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
9. Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( 16 \cos ^ { 2 } \theta - 8 + \sec ^ { 2 } \theta \right) d \theta$$ where \(k\) is a constant.
10. Hence, use integration to find the exact value for this volume.
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-13_869_545_312_811} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm . At time \(t\) minutes, the depth of liquid in the tank is \(h\) centimetres. The liquid leaks from a hole \(P\) at the bottom of the tank. The liquid leaks from the tank at a rate of \(32 \pi \sqrt { } h \mathrm {~cm} ^ { 3 } \mathrm {~min} ^ { - 1 }\).
11. Show that at time \(t\) minutes, the height \(h \mathrm {~cm}\) of liquid in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - 0.02 \sqrt { } h$$
12. Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm .
    
    \includegraphics[max width=\textwidth, alt={}]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-14_2639_1834_214_217}