Questions — OCR MEI (4333 questions)

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OCR MEI C4 Q6
6 marks Standard +0.2
6 The function \(\mathrm { f } ( \theta ) = 3 \sin \theta + 4 \cos \theta\) is to be expressed in the form \(r \sin ( \theta + \alpha )\) where \(r > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  1. Find the values of \(r\) and \(\alpha\).
  2. Write down the maximum and minimum value of \(\mathrm { f } ( \theta )\).
  3. Solve the equation \(\mathrm { f } ( \theta ) = 1\) for \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).
OCR MEI C4 Q7
6 marks Moderate -0.3
7
  1. Show that \(\frac { 1 } { \sqrt { 25 - x } } = \frac { 1 } { 5 } \left( 1 - \frac { x } { 25 } \right) ^ { - \frac { 1 } { 2 } }\).
  2. Hence expand \(\frac { 1 } { \sqrt { 25 - x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
  3. Write down the range of values of \(x\) for which the expansion is valid.
OCR MEI C4 Q8
19 marks Standard +0.3
8 The new price of a particular make of car is \(\pounds 10000\). When its age is \(t\) years, the list price is \(\pounds V\). When \(t = 5 , V = 5000\). Aloke, Ben and Charlie all run outlets for used cars. Each of them has a different model for the depreciation.
  1. Aloke claims that the rate of depreciation is constant. Write this claim as a differential equation.
    Solve the differential equation and hence find the value of a car that is 7 years old according to this model.
    Explain why this model breaks down for large \(t\).
  2. Ben believes that the rate of depreciation is inversely proportional to the square root of the age of the car. Express this claim as a differential equation and hence find the value of a car that is 7 years old according to this model.
    Does this model ever break down?
  3. Charlie believes that a better model is given by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = k V$$ Solve this differential equation and find the value of the car after 7 years according to this model.
    Does this model ever break down?
  4. Further investigation reveals that the average value of this particular type of car when 8 years old is \(\pounds 3000\). Find the value of \(V\) when \(t = 8\) for the three models above. Which of the three models best predicts the value of \(V\) at this time?
OCR MEI C4 Q9
17 marks Standard +0.3
9 Beside a major route into a county town the authorities decide to build a large pyramid. Fig. 9.1 shows this pyramid, ABCDE O is the centre point of the horizontal base BCDE . A coordinate system is defined with O as the origin. The \(x\)-axis and \(y\)-axis are horizontal and the \(z\)-axis is vertical, as shown in Fig. 9.1 The vertices of the pyramid are $$A ( 0,0,6 ) , B ( - 4 , - 4,0 ) , C ( 4 , - 4,0 ) , D ( 4,4,0 ) \text { and } E ( - 4,4,0 ) .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78993065-a6cd-4b77-b21f-c9ccc82fb37a-4_668_878_493_623} \captionsetup{labelformat=empty} \caption{Fig.9.1}
\end{figure} The pyramid is supported by a vertical pole OA and there are also support rods from O to points on the triangular faces \(\mathrm { ABC } , \mathrm { ACD } , \mathrm { ADE }\) and AEB . One of the rods, ON , is shown in fig.9.2 which shows one quarter of the pyramid. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{78993065-a6cd-4b77-b21f-c9ccc82fb37a-4_428_675_1521_831} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
\end{figure} M is the mid-point of the line BC .
  1. Write down the coordinates of M.
  2. Write down the vector \(\overrightarrow { \mathrm { AM } }\) and hence the coordinates of the point N which divides \(\overrightarrow { \mathrm { AM } }\) so that the ratio \(\mathrm { AN } : \mathrm { NM } = 2 : 1\).
  3. Show that ON is perpendicular to both AM and BC .
  4. Hence write down the equation of the plane ABC in its simplest form.
  5. Find the angle between the face ABC and the ground.
OCR MEI C4 Q1
4 marks Moderate -0.5
1 Solve the equation for values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). $$\cot 2 \theta = 5$$
OCR MEI C4 Q2
4 marks Moderate -0.3
2 Find where the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)\) meets the plane \(2 x + 3 y - 4 z - 5 = 0\).
OCR MEI C4 Q3
6 marks Moderate -0.3
3 The graph shows part of the curve \(y ^ { 2 } = ( x - 1 )\). \includegraphics[max width=\textwidth, alt={}, center]{73112db3-7b05-48db-9fff-fdbac7dbd564-2_428_860_973_616} Find the volume when the area between this curve, the \(x\)-axis and the line \(x = 5\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
OCR MEI C4 Q4
7 marks Moderate -0.3
4 You are given that \(\mathbf { a } = 2 \mathbf { i } + 6 \mathbf { j } + 9 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\).
  1. Write down a unit vector parallel to a.
  2. Find the value of \(\lambda\) such that \(\mathbf { a } + \lambda \mathbf { b }\) is parallel to \(\mathbf { k }\).
  3. Calculate the size of the angle between \(\mathbf { a }\) and \(\mathbf { b }\).
OCR MEI C4 Q5
5 marks Moderate -0.3
5
  1. Simplify \(\frac { x ^ { 3 } - x ^ { 2 } - 3 x - 9 } { x - 3 }\).
  2. Hence or otherwise solve the equation \(x ^ { 3 } - x ^ { 2 } - 3 x - 9 = 6 ( x - 3 )\).
OCR MEI C4 Q6
6 marks Moderate -0.8
6 Prove that
  1. \(\frac { \sin 2 \theta } { 2 \tan \theta } + \sin ^ { 2 } \theta = 1\),
  2. \(\quad \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 45 ^ { \circ } \right)\).
OCR MEI C4 Q7
4 marks Moderate -0.8
7 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { y }\) given that when \(x = 1 , y = 2\).
OCR MEI C4 Q8
19 marks Standard +0.3
8 Scientists predict the velocity ( \(v\) kilometres per minute) for the new "outer explorer" spacecraft over the first minute of its entry to the atmosphere of the planet Titan to be modelled by the equation: $$v = \frac { 5000 } { ( 1 + t ) ( 2 + t ) ^ { 2 } } , 0 \leq t \leq 1 \text { where } t \text { represents time in minutes. }$$
  1. Use a binomial expansion to expand \(( 1 + t ) ^ { - 1 }\) up to and including the term in \(t ^ { 2 }\).
  2. Use a binomial expansion to expand \(( 2 + t ) ^ { - 2 }\) up to and including the term in \(t ^ { 2 }\).
  3. Hence, or otherwise, show that \(v \approx 1250 \left( 1 - 2 t + \frac { 11 t ^ { 2 } } { 4 } \right)\).
  4. The displacement of the spacecraft can be found by calculating the area under the velocity time graph. Use the approximation found in part (iii) to estimate the displacement of the spacecraft over the first half minute.
  5. Write \(\frac { 1 } { ( 1 + t ) ( 2 + t ) ^ { 2 } }\) in partial fractions.
  6. The displacement of the spacecraft in the first \(T\) minutes is given by \(\int _ { 0 } ^ { T } v \mathrm {~d} t\) Calculate the exact value of the displacement of the spacecraft over the first half minute given by the model.
  7. On further investigation the scientists believe the original model may be valid for up to three minutes. Explain why the approximation in (iii) will be no longer be valid for this time interval.
OCR MEI C4 Q9
17 marks Standard +0.3
9 Two astronomers wish to model the path of motion of a particle in two dimensions.
Experimental results show that the position of the particle can be found using the parametric equations $$x = 2 \cos \theta - \sin \theta + 2 \quad y = \cos \theta + 2 \sin \theta - 1 \quad \left( 0 \leq \theta \leq 360 ^ { \circ } \right)$$ One astronomer uses trigonometry.
  1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants to be determined. Show also that, for the same values of \(R\) and \(\alpha\), $$\cos \theta + 2 \sin \theta = R \sin ( \theta + \alpha )$$
  2. Hence, or otherwise, show that the path of particle may be written in the form $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 5$$ Describe the path of the particle. The second astronomer sets up a first order differential equation with the condition that \(x = 4\) when \(y = 0\).
  3. Verify that the point with parameter \(\theta = 0\) has coordinates \(( 4,0 )\).
  4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Deduce that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x - 2 } { y + 1 }$$
  5. Solve this differential equation, using the condition that \(y = 0\) when \(x = 4\). Hence show that the two solutions give the same cartesian equation for the path of particle.
OCR MEI FP1 2005 January Q1
3 marks Moderate -0.8
1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & 3 \\ - 2 & 1 \end{array} \right)\).
Find the inverse of \(\mathbf { M }\).
The transformation associated with \(\mathbf { M }\) is applied to a figure of area 2 square units. What is the area of the transformed figure?
OCR MEI FP1 2005 January Q2
6 marks Moderate -0.5
2
  1. Show that \(\frac { 1 } { r + 1 } - \frac { 1 } { r + 2 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).
  2. Hence use the method of differences to find the sum of the series $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
OCR MEI FP1 2005 January Q3
7 marks Standard +0.3
3
  1. Solve the equation \(\frac { 1 } { x + 2 } = 3 x + 4\).
  2. Solve the inequality \(\frac { 1 } { x + 2 } \leqslant 3 x + 4\).
OCR MEI FP1 2005 January Q4
6 marks Moderate -0.3
4 Find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 )\), giving your answer in a factorised form.
OCR MEI FP1 2005 January Q5
6 marks Standard +0.3
5 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), simplifying your answer as far as you can.
OCR MEI FP1 2005 January Q6
8 marks Standard +0.8
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\).
OCR MEI FP1 2005 January Q7
14 marks Standard +0.8
7 A curve has equation \(y = \frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) }\).
  1. Write down the values of \(x\) for which \(y = 0\).
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or from below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
  5. Solve the inequality \(\frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) } \leqslant 2\).
OCR MEI FP1 2005 January Q8
12 marks Standard +0.3
8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).
  1. Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
  2. Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
  3. Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
  4. On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).
OCR MEI FP1 2005 January Q9
10 marks Standard +0.3
9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
  2. Explain how your answer to part (i) relates to this information.
  3. By investigating the invariant points of the reflection, find the equation of the mirror line.
  4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
  5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
  6. The composite transformation described in part ( \(\mathbf { v }\) ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
OCR MEI FP1 2006 January Q1
7 marks Easy -1.8
1 You are given that \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & - 3 \\ 1 & 4 \end{array} \right) , \mathbf { C } = \left( \begin{array} { r r } 1 & - 1 \\ 0 & 2 \\ 0 & 1 \end{array} \right)\).
  1. Calculate, where possible, \(2 \mathbf { B } , \mathbf { A } + \mathbf { C } , \mathbf { C A }\) and \(\mathbf { A } - \mathbf { B }\).
  2. Show that matrix multiplication is not commutative.
OCR MEI FP1 2006 January Q2
5 marks Easy -1.2
2
  1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
  2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).
OCR MEI FP1 2006 January Q3
6 marks Moderate -0.8
3 Find \(\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r - 1 )\), expressing your answer in a fully factorised form.