Questions — OCR MEI (4333 questions)

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OCR MEI M1 2013 June Q5
7 marks Standard +0.3
5 Fig. 5 shows blocks of mass 4 kg and 6 kg on a smooth horizontal table. They are connected by a light, inextensible string. As shown, a horizontal force \(F \mathrm {~N}\) acts on the 4 kg block and a horizontal force of 30 N acts on the 6 kg block. The magnitude of the acceleration of the system is \(2 \mathrm {~ms} ^ { - 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-3_106_1107_1708_479} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Find the two possible values of \(F\).
  2. Find the tension in the string in each case.
OCR MEI M1 2013 June Q6
6 marks Moderate -0.3
6 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time \(t \mathrm {~s}\), its displacement from O is \(x \mathrm {~m}\) and its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 24 - 18 t + 3 t ^ { 2 } .$$
  1. Find the times, \(T _ { 1 } \mathrm {~s}\) and \(T _ { 2 } \mathrm {~s}\) (where \(T _ { 2 } > T _ { 1 }\) ), at which the particle is stationary.
  2. Find an expression for \(x\) at time \(t \mathrm {~s}\). Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\). Section B (36 marks) \(7 \quad\) Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-4_773_1071_429_497} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
    \end{figure} Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.
OCR MEI M1 2013 June Q8
18 marks Standard +0.3
8 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_122_849_456_609} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).
  1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The resistance to motion remains 100 N .
  2. Find the speed of the sledge
    (A) 1.6 seconds after the rope breaks,
    (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_259_853_1457_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
  3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
  4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.
OCR MEI FP1 2009 January Q1
5 marks Easy -1.2
1
  1. Find the roots of the quadratic equation \(z ^ { 2 } - 6 z + 10 = 0\) in the form \(a + b \mathrm { j }\).
  2. Express these roots in modulus-argument form.
OCR MEI FP1 2009 January Q2
4 marks Moderate -0.8
2 Find the values of \(A , B\) and \(C\) in the identity \(2 x ^ { 2 } - 13 x + 25 \equiv A ( x - 3 ) ^ { 2 } - B ( x - 2 ) + C\).
OCR MEI FP1 2009 January Q3
5 marks Moderate -0.3
3 Fig. 3 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-2_465_531_806_806} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Write down the matrix \(\mathbf { P }\) representing this transformation.
  2. The parallelogram \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is transformed by the matrix \(\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\). Find the coordinates of the vertices of its image, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\), following this transformation.
  3. Describe fully the transformation represented by \(\mathbf { Q P }\).
OCR MEI FP1 2009 January Q4
3 marks Moderate -0.8
4 Write down the equation of the locus represented in the Argand diagram shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-2_474_497_1932_824} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI FP1 2009 January Q5
6 marks Standard +0.3
5 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , - 3 \alpha\) and \(\alpha + 3\). Find the values of \(\alpha , p\) and \(q\).
OCR MEI FP1 2009 January Q6
6 marks Standard +0.3
6 Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) show that $$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$
OCR MEI FP1 2009 January Q7
7 marks Standard +0.3
7 Prove by induction that \(12 + 36 + 108 + \ldots + 4 \times 3 ^ { n } = 6 \left( 3 ^ { n } - 1 \right)\) for all positive integers \(n\).
OCR MEI FP1 2009 January Q8
12 marks Standard +0.3
8 Fig. 8 shows part of the graph of \(y = \frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) }\). Two sections of the graph have been omitted. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-3_725_1025_1160_559} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  3. Copy Fig. 8 and draw in the two missing sections.
  4. Solve the inequality \(\frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) } \leqslant 0\).
OCR MEI FP1 2009 January Q9
12 marks Standard +0.3
9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 + \mathrm { j }\) and \(\beta = 2 - \mathrm { j }\).
  1. Express \(\alpha + \beta , \alpha \alpha ^ { * }\) and \(\frac { \alpha + \beta } { \alpha }\) in the form \(a + b \mathrm { j }\).
  2. Find a quadratic equation with roots \(\alpha\) and \(\alpha ^ { * }\).
  3. \(\alpha\) and \(\beta\) are roots of a quartic equation with real coefficients. Write down the two other roots and find this quartic equation in the form \(z ^ { 4 } + A z ^ { 3 } + B z ^ { 2 } + C z + D = 0\).
OCR MEI FP1 2009 January Q10
12 marks Standard +0.3
10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 3 & 4 & - 1 \\ 1 & - 1 & k \\ - 2 & 7 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r c } 11 & - 5 & - 7 \\ 1 & 11 & 5 + k \\ - 5 & 29 & 7 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } 42 & \alpha & 4 k - 8 \\ 10 - 5 k & - 16 + 29 k & - 12 + 6 k \\ 0 & 0 & \beta \end{array} \right)\).
  1. Show that \(\alpha = 0\) and \(\beta = 28 + 7 k\).
  2. Find \(\mathbf { A B }\) when \(k = 2\).
  3. For the case when \(k = 2\) write down the matrix \(\mathbf { A } ^ { - 1 }\).
  4. Use the result from part (iii) to solve the following simultaneous equations. $$\begin{aligned} 3 x + 4 y - z & = 1 \\ x - y + 2 z & = - 9 \\ - 2 x + 7 y - 3 z & = 26 \end{aligned}$$ \footnotetext{OCR
    RECOGNISING ACHIEVEMENT
OCR MEI FP1 2010 January Q1
5 marks Moderate -0.8
1 Two complex numbers are given by \(\alpha = - 3 + \mathrm { j }\) and \(\beta = 5 - 2 \mathrm { j }\).
Find \(\alpha \beta\) and \(\frac { \alpha } { \beta }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.
OCR MEI FP1 2010 January Q2
7 marks Moderate -0.8
2 You are given that \(\mathbf { A } = \left( \begin{array} { r } 4 \\ - 2 \\ 4 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 5 & 1 \\ 2 & - 3 \end{array} \right) , \mathbf { C } = \left( \begin{array} { l l l } 5 & 1 & 8 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { r r } - 2 & 0 \\ 4 & 1 \end{array} \right)\).
  1. Calculate, where they exist, \(\mathbf { A B } , \mathbf { C A } , \mathbf { B } + \mathbf { D }\) and \(\mathbf { A C }\) and indicate any that do not exist.
  2. Matrices \(\mathbf { B }\) and \(\mathbf { D }\) represent transformations B and D respectively. Find the single matrix that represents transformation B followed by transformation D.
OCR MEI FP1 2010 January Q3
6 marks Standard +0.3
3 The roots of the cubic equation \(4 x ^ { 3 } - 12 x ^ { 2 } + k x - 3 = 0\) may be written \(a - d , a\) and \(a + d\). Find the roots and the value of \(k\).
OCR MEI FP1 2010 January Q4
6 marks Moderate -0.3
4 You are given that if \(\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1 \\ - 6 & 1 & 1 \\ 5 & 2 & 5 \end{array} \right)\) then \(\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1 \\ - 35 & - 15 & 10 \\ 17 & 8 & - 4 \end{array} \right)\).
Find the value of \(k\). Hence solve the following simultaneous equations. $$\begin{aligned} 4 x + z & = 9 \\ - 6 x + y + z & = 32 \\ 5 x + 2 y + 5 z & = 81 \end{aligned}$$
OCR MEI FP1 2010 January Q5
6 marks Moderate -0.8
5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r - 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } - 19 \right)\).
OCR MEI FP1 2010 January Q6
6 marks Standard +0.3
6 Prove by induction that \(1 \times 2 + 2 \times 3 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }\) for all positive integers \(n\). Section B (36 marks)
OCR MEI FP1 2010 January Q7
12 marks Standard +0.3
7 A curve has equation \(y = \frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) }\).
  1. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your answers.
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0\).
OCR MEI FP1 2010 January Q8
12 marks Standard +0.3
8
  1. Fig. 8 shows an Argand diagram. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{df275813-15de-496f-9742-427a9e03f431-3_892_899_1048_664} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
    1. Write down the equation of the locus represented by the circumference of circle B.
    2. Write down the two inequalities that define the shaded region between, but not including, circles A and B.
    1. Draw an Argand diagram to show the region where $$\frac { \pi } { 4 } < \arg ( z - ( 2 + \mathrm { j } ) ) < \frac { 3 \pi } { 4 }$$
    2. Determine whether the point \(43 + 47 \mathrm { j }\) lies within this region.
OCR MEI FP1 2010 January Q9
12 marks Standard +0.3
9
  1. Verify that \(\frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 2 } { r } - \frac { 3 } { r + 1 } + \frac { 1 } { r + 2 }\).
  2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 3 } { 2 } - \frac { 2 } { n + 1 } + \frac { 1 } { n + 2 } .$$
  3. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\) converges as \(n\) tends to infinity.
  4. Find \(\sum _ { r = 50 } ^ { 100 } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\), giving your answer to 3 significant figures.
OCR MEI FP1 2011 January Q1
5 marks Moderate -0.8
1 Find the values of \(P , Q , R\) and \(S\) in the identity \(3 x ^ { 3 } + 18 x ^ { 2 } + P x + 31 \equiv Q ( x + R ) ^ { 3 } + S\).
OCR MEI FP1 2011 January Q2
7 marks Moderate -0.3
2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 4 & 0 \\ - 1 & 3 \end{array} \right)\).
  1. The transformation associated with \(\mathbf { M }\) is applied to a figure of area 3 square units. Find the area of the transformed figure.
  2. Find \(\mathbf { M } ^ { - 1 }\) and \(\operatorname { det } \mathbf { M } ^ { - 1 }\).
  3. Explain the significance of \(\operatorname { det } \mathbf { M } \times \operatorname { det } \mathbf { M } ^ { - 1 }\) in terms of transformations.
OCR MEI FP1 2011 January Q3
7 marks Standard +0.8
3 The roots of the cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).