OCR C4 (Core Mathematics 4)

1. Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
2. Find the quotient and remainder when \(\left( x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 \right)\) is divided by \(\left( x ^ { 2 } + x - 6 \right)\).
3. Differentiate each of the following with respect to \(x\) and simplify your answers.
   1. \(\cot x ^ { 2 }\)
   2. \(\frac { \sin x } { 3 + 2 \cos x }\)
   3. (i) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
   4. Hence, or otherwise, show that for small \(x\),
   $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$

5. The diagram shows the curve with parametric equations $$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$ where \(a\) is a positive constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
2. Show that the area of triangle \(O A B\) is \(a ^ { 2 }\).

6. Relative to a fixed origin, two lines have the equations $$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet.
2. Find, in degrees to 1 decimal place, the acute angle between the lines.

7. At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$ where \(k\) is a positive constant,

1. Find an expression for \(y\) in terms of \(k\) and \(t\). Given that two hours after being filled the depth of water in the tank is 1.6 metres,
2. find the value of \(k\) to 4 significant figures. Given also that the hole in the tank is \(h \mathrm {~cm}\) above the base of the tank,
3. show that \(h = 79\) to 2 significant figures.

8. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
2. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
3. Find the coordinates of \(Q\).

9. (i) Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ into the integral $$\int \frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) } \mathrm { d } u$$ (ii) Express \(\frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) }\) in partial fractions.
(iii) Hence, evaluate $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers.