Edexcel C4 (Core Mathematics 4)

1. Use integration by parts to find

$$\int x ^ { 2 } \sin x d x$$

1. Given that \(y = - 2\) when \(x = 1\), solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

3. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\). (8)
3. continued

4. (a) Expand \(( 1 + a x ) ^ { - 3 } , | a x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). Give each coefficient as simply as possible in terms of the constant \(a\). Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1\), is 3 ,
(b) find the two possible values of \(a\). Given also that \(a < 0\),
(c) show that the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }\) is \(\frac { 14 } { 9 }\).
4. continued

5. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 5\).

1. Find the area of the shaded region. The shaded region is rotated completely about the \(x\)-axis.
2. Find the volume of the solid formed, giving your answer in the form \(k \pi \ln 2\), where \(k\) is a simplified fraction.
   5. continued

6. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$

1. Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
2. Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
   6. continued

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).

1. Find the value of \(\theta\) at \(P\).
2. Show that the normal to the curve at \(P\) passes through the origin.
3. Find a cartesian equation for the curve.
   7. continued

8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).
   8. continued
   8. continued