OCR MEI Further Pure Core (Further Pure Core) 2019 June

1 Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.

2 The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).

3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1
2 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } k & 1
2 & 0 \end{array} \right)\), where \(k\) is a constant.

1. Verify the result \(( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }\) in this case.
2. Investigate whether \(\mathbf { A }\) and \(\mathbf { B }\) are commutative under matrix multiplication.

4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h] \end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.

5 Using the Maclaurin series for \(\cos 2 x\), show that, for small values of \(x\), \(\sin ^ { 2 } x \approx a x ^ { 2 } + b x ^ { 4 } + c x ^ { 6 }\),
where the values of \(a , b\) and \(c\) are to be given in exact form.

6 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).

7 A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 c ^ { 2 } x y\), where \(c\) is a positive constant.

1. Show that the polar equation of the curve is \(r ^ { 2 } = c ^ { 2 } \sin 2 \theta\).
2. Sketch the curves \(r = c \sqrt { \sin 2 \theta }\) and \(r = - c \sqrt { \sin 2 \theta }\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
3. Find the area of the region enclosed by one of the loops in part (b). Section B (110 marks)
   Answer all the questions.

8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).

1. Evaluate, in exact form, the roots of the equation.
2. Find \(k\).

9 Prove by induction that \(5 ^ { n } + 2 \times 11 ^ { n }\) is divisible by 3 for all positive integers \(n\).

10 In this question you must show detailed reasoning.

1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).

11

1. Specify fully the transformations represented by the following matrices.
   • \(\mathbf { M } _ { 1 } = \left( \begin{array} { r r } \frac { 3 } { 5 } & - \frac { 4 } { 5 }
     \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\)
   • \(\mathbf { M } _ { 2 } = \left( \begin{array} { r r } 1 & 0
     0 & - 1 \end{array} \right)\)
   • Find the equation of the mirror line of the reflection R represented by the matrix \(\mathbf { M } _ { 3 } = \mathbf { M } _ { 1 } \mathbf { M } _ { 2 }\).
   • It is claimed that the reflection represented by the matrix \(\mathbf { M } _ { 4 } = \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }\) has the same mirror line as R . Explain whether or not this claim is correct.

12 Three intersecting lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have equations
\(L _ { 1 } : \frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 1 } , \quad L _ { 2 } : \frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { - 4 } \quad\) and \(\quad L _ { 3 } : \frac { x - 1 } { 1 } = \frac { y - 2 } { 1 } = \frac { z + 4 } { 5 }\).
Find the area of the triangle enclosed by these lines.

13

1. Using the logarithmic form of \(\operatorname { arcosh } x\), prove that the derivative of \(\operatorname { arcosh } x\) is \(\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
2. Hence find \(\int _ { 1 } ^ { 2 } \operatorname { arcosh } x \mathrm {~d} x\), giving your answer in exact logarithmic form.
3. Ali tries to evaluate \(\int _ { 0 } ^ { 1 } \operatorname { arcosh } x \mathrm {~d} x\) using his calculator, and gets an 'error'. Explain why.

14 Three planes have equations $$\begin{aligned} - x + a y & = 2
2 x + 3 y + z & = - 3
x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.

1. In the case where the planes do not intersect at a unique point,
   1. find \(b\) in terms of \(a\),
   2. find the value of \(c\) for which the planes form a sheaf.
2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).

15 In this question you must show detailed reasoning. Show that \(\int _ { \frac { 3 } { 4 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 4 x + 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { 3 + \sqrt { 5 } } { 2 } \right)\).

16

1. Show that \(\left( 2 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 2 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by
   \(C = \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \cos n \theta\),
   \(S = \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \sin n \theta\).
2. Show that \(C = \frac { 2 ^ { n } ( 2 \cos \theta - 1 ) - 2 \cos ( n + 1 ) \theta + \cos n \theta } { 2 ^ { n } ( 5 - 4 \cos \theta ) }\).

17 A cyclist accelerates from rest for 5 seconds then brakes for 5 seconds, coming to rest at the end of the 10 seconds. The total mass of the cycle and rider is \(m \mathrm {~kg}\), and at time \(t\) seconds, for \(0 \leqslant t \leqslant 10\), the cyclist's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance to motion, modelled by a force of magnitude 0.1 mvN , acts on the cyclist during the whole 10 seconds.

1. Explain why modelling the resistance to motion in this way is likely to be more realistic than assuming this force is constant. During the braking phase of the motion, for \(5 \leqslant t \leqslant 10\), the brakes apply an additional constant resistance force of magnitude \(2 m \mathrm {~N}\) and the cyclist does not provide any driving force.
2. Show that, for \(5 \leqslant t \leqslant 10 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = - 2\).
3. 1. Solve the differential equation in part (b).
   2. Hence find the velocity of the cyclist when \(t = 5\). During the acceleration phase ( \(0 \leqslant t \leqslant 5\) ), the cyclist applies a driving force of magnitude directly proportional to \(t\).
4. Show that, for \(0 \leqslant t \leqslant 5 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = \lambda t\), where \(\lambda\) is a positive constant.
5. 1. Show by integration that, for \(0 \leqslant t \leqslant 5 , v = 10 \lambda \left( t - 10 + 10 \mathrm { e } ^ { - 0.1 t } \right)\).
   2. Hence find \(\lambda\).
6. Find the total distance, to the nearest metre, travelled by the cyclist during the motion.