Questions — OCR MEI (4333 questions)

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OCR MEI C1 2016 June Q7
5 marks Easy -1.3
7
  1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
  2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.
OCR MEI C1 2016 June Q8
5 marks Moderate -0.8
8 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + a x + c\) and that \(\mathrm { f } ( 2 ) = 11\). The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 8 . Find the values of \(a\) and \(c\).
OCR MEI C1 2016 June Q9
13 marks Moderate -0.3
9 Fig. 9 shows the curves \(y = \frac { 1 } { x + 2 }\) and \(y = x ^ { 2 } + 7 x + 7\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-3_1255_1470_434_299} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Use Fig. 9 to estimate graphically the roots of the equation \(\frac { 1 } { x + 2 } = x ^ { 2 } + 7 x + 7\).
  2. Show that the equation in part (i) may be simplified to \(x ^ { 3 } + 9 x ^ { 2 } + 21 x + 13 = 0\). Find algebraically the exact roots of this equation.
  3. The curve \(y = x ^ { 2 } + 7 x + 7\) is translated by \(\binom { 3 } { 0 }\).
    (A) Show graphically that the translated curve intersects the curve \(y = \frac { 1 } { x + 2 }\) at only one point. Estimate the coordinates of this point.
    (B) Find the equation of the translated curve, simplifying your answer.
OCR MEI C1 2016 June Q10
11 marks Moderate -0.3
10 Fig. 10 shows a sketch of the points \(\mathrm { A } ( 2,7 ) , \mathrm { B } ( 0,3 )\) and \(\mathrm { C } ( 8 , - 1 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2ebf7ad2-638f-4378-b98d-aadd0de4c766-4_579_748_301_657} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Prove that angle ABC is \(90 ^ { \circ }\).
  2. Find the equation of the circle which has AC as a diameter.
  3. Find the equation of the tangent to this circle at A . Give your answer in the form \(a y = b x + c\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2016 June Q11
12 marks Standard +0.3
11
  1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
  2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
  3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}
OCR MEI C2 2009 January Q1
4 marks Easy -1.3
1 Find \(\int \left( 20 x ^ { 4 } + 6 x ^ { - \frac { 3 } { 2 } } \right) \mathrm { d } x\).
[0pt] [4]
OCR MEI C2 2009 January Q2
4 marks Easy -1.2
2 Fig. 2 shows the coordinates at certain points on a curve. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-2_645_1146_589_497} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Use the trapezium rule with 6 strips to calculate an estimate of the area of the region bounded by this curve and the axes.
OCR MEI C2 2009 January Q3
2 marks Easy -1.8
3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 2009 January Q4
3 marks Moderate -0.8
4 Solve the equation \(\sin 2 x = - 0.5\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR MEI C2 2009 January Q6
5 marks Moderate -0.8
6 An arithmetic progression has first term 7 and third term 12.
  1. Find the 20th term of this progression.
  2. Find the sum of the 21st to the 50th terms inclusive of this progression.
OCR MEI C2 2009 January Q7
5 marks Moderate -0.8
7 Differentiate \(4 x ^ { 2 } + \frac { 1 } { x }\) and hence find the \(x\)-coordinate of the stationary point of the curve \(y = 4 x ^ { 2 } + \frac { 1 } { x }\).
OCR MEI C2 2009 January Q8
5 marks Moderate -0.8
8 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } \end{aligned}$$
  1. Find the third term of this sequence and state what type of sequence it is.
  2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.
OCR MEI C2 2009 January Q9
4 marks Easy -1.8
9
  1. State the value of \(\log _ { a } a\).
  2. Express each of the following in terms of \(\log _ { a } x\).
    (A) \(\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }\) (B) \(\log _ { a } \frac { 1 } { x }\) Section B (36 marks)
OCR MEI C2 2009 January Q10
13 marks Standard +0.3
10 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-4_609_908_1338_621} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
  2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
  3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10.
OCR MEI C2 2009 January Q11
11 marks Standard +0.3
11
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_469_878_274_671} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} Fig. 11.1 shows the surface ABCD of a TV presenter's desk. AB and CD are arcs of circles with centre O and sector angle 2.5 radians. \(\mathrm { OC } = 60 \mathrm {~cm}\) and \(\mathrm { OB } = 140 \mathrm {~cm}\).
    (A) Calculate the length of the arc CD.
    (B) Calculate the area of the surface ABCD of the desk.
  2. The TV presenter is at point P , shown in Fig. 11.2. A TV camera can move along the track EF , which is of length 3.5 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{838d6b15-69a9-4e67-bc36-5bf60254a767-5_378_877_1334_675} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} When the camera is at E , the TV presenter is 1.6 m away. When the camera is at F , the TV presenter is 2.8 m away.
    (A) Calculate, in degrees, the size of angle EFP.
    (B) Calculate the shortest possible distance between the camera and the TV presenter.
OCR MEI C2 2010 January Q1
3 marks Easy -1.2
1 Find \(\int \left( x - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x\).
OCR MEI C2 2010 January Q2
3 marks Easy -1.2
2 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 55th term of this sequence, showing your method.
  2. Find the sum of the first 55 terms of the sequence.
OCR MEI C2 2010 January Q3
3 marks Moderate -0.8
3 You are given that \(\sin \theta = \frac { \sqrt { 2 } } { 3 }\) and that \(\theta\) is an acute angle. Find the exact value of \(\tan \theta\).
OCR MEI C2 2010 January Q4
3 marks Moderate -0.8
4 A sector of a circle has area \(8.45 \mathrm {~cm} ^ { 2 }\) and sector angle 0.4 radians. Calculate the radius of the sector.
OCR MEI C2 2010 January Q5
4 marks Moderate -0.8
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-2_547_991_1340_577} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Fig. 5 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { P } , \mathrm { Q }\) and R .
  1. \(y = \mathrm { f } ( 2 x )\)
  2. \(y = \frac { 1 } { 4 } \mathrm { f } ( x )\)
OCR MEI C2 2010 January Q6
5 marks Easy -1.3
6
  1. Find the 51 st term of the sequence given by $$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = u _ { n } + 4 \end{aligned}$$
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-3_531_969_744_589} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 shows triangle ABC , with \(\mathrm { AB } = 8.4 \mathrm {~cm}\). D is a point on AC such that angle \(\mathrm { ADB } = 79 ^ { \circ }\), \(\mathrm { BD } = 5.6 \mathrm {~cm}\) and \(\mathrm { CD } = 7.8 \mathrm {~cm}\). Calculate
  3. angle BAD ,
  4. the length BC .
OCR MEI C2 2010 January Q8
5 marks Moderate -0.8
8 Find the equation of the tangent to the curve \(y = 6 \sqrt { x }\) at the point where \(x = 16\).
OCR MEI C2 2010 January Q9
5 marks Easy -1.2
9
  1. Sketch the graph of \(y = 3 ^ { x }\).
  2. Use logarithms to solve \(3 ^ { 2 x + 1 } = 10\), giving your answer correct to 2 decimal places.
OCR MEI C2 2010 January Q10
11 marks Moderate -0.8
10
  1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
  2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
  3. Sketch the curve.
OCR MEI C2 2010 January Q11
12 marks Moderate -0.8
11 Fig. 11 shows the cross-section of a school hall, with measurements of the height in metres taken at 1.5 m intervals from O . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{053009a4-e88f-4711-ad97-cebb1740744b-4_579_1381_861_383} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Use the trapezium rule with 8 strips to calculate an estimate of the area of the cross-section.
  2. Use 8 rectangles to calculate a lower bound for the area of the cross-section. The curve of the roof may be modelled by \(y = - 0.013 x ^ { 3 } + 0.16 x ^ { 2 } - 0.082 x + 2.4\), where \(x\) metres is the horizontal distance from O across the hall, and \(y\) metres is the height.
  3. Use integration to find the area of the cross-section according to this model.
  4. Comment on the accuracy of this model for the height of the hall when \(x = 7.5\).