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OCR MEI C3 Q1
18 marks Standard +0.3
1 Fig. 8 shows a sketch of part of the curve \(y = x \sin 2 x\), where \(x\) is in radians.
The curve crosses the \(x\)-axis at the point P . The tangent to the curve at P crosses the \(y\)-axis at Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35646966-3747-4f1d-bf94-60e9e3130afe-1_706_920_489_606} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that the \(x\)-coordinates of the turning points of the curve satisfy the equation \(\tan 2 x + 2 x = 0\).
  2. Find, in terms of \(\pi\), the \(x\)-coordinate of the point P . Show that the tangent PQ has equation \(2 \pi x + 2 y = \pi ^ { 2 }\).
    Find the exact coordinates of Q.
  3. Show that the exact value of the area shaded in Fig. 8 is \(\frac { 1 } { 8 } \pi \left( \pi ^ { 2 } - 2 \right)\).
OCR MEI C3 Q2
18 marks Standard +0.8
2
  1. Use the substitution \(u = 1 + x\) to show that $$\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { 1 + x } \mathrm {~d} x = \int _ { a } ^ { b } \left( u ^ { 2 } - 3 u + 3 - \frac { 1 } { u } \right) \mathrm { d } u$$ where \(a\) and \(b\) are to be found.
    Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 3 } } { 1 + x } \mathrm {~d} x\), giving your answer in exact form. Fig. 8 shows the curve \(y = x ^ { 2 } \ln ( 1 + x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{35646966-3747-4f1d-bf94-60e9e3130afe-2_829_806_944_706} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Verify that the origin is a stationary point of the curve.
  3. Using integration by parts, and the result of part (i), find the exact area enclosed by the curve \(y = x ^ { 2 } \ln ( 1 + x )\), the \(x\)-axis and the line \(x = 1\).
OCR MEI C3 Q3
8 marks Standard +0.3
3
  1. Differentiate \(\frac { \ln x } { x ^ { 2 } }\), simplifying your answer.
  2. Using integration by parts, show that \(\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = - \frac { 1 } { x } ( 1 + \ln x ) + c\).
OCR MEI C3 Q4
8 marks Moderate -0.3
4 Evaluate the following integrals, giving your answers in exact form. \begin{displayquote}
  1. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x ^ { 2 } + 1 } \mathrm {~d} x\)
  2. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x + 1 } \mathrm {~d} x\) \end{displayquote}
OCR MEI C3 Q2
17 marks Standard +0.3
2 Fig. 8 shows the curve \(y = 3 \ln x + x - x ^ { 2 }\).
The curve crosses the \(x\)-axis at P and Q , and has a turning point at R . The \(x\)-coordinate of Q is approximately 2.05 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72893fd5-bc8e-433b-8358-f7979b2da636-2_717_830_606_693} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Verify that the coordinates of P are \(( 1,0 )\).
  2. Find the coordinates of R , giving the \(y\)-coordinate correct to 3 significant figures. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that R is a maximum point.
  3. Find \(\int \ln x \mathrm {~d} x\). Hence calculate the area of the region enclosed by the curve and the \(x\)-axis between P and Q , giving your answer to 2 significant figures.
OCR MEI C3 Q3
19 marks Standard +0.3
3 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { 2 x } } { 1 + \mathrm { e } ^ { 2 x } }\). The curve crosses the \(y\)-axis at P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72893fd5-bc8e-433b-8358-f7979b2da636-3_594_1230_514_494} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find the coordinates of P .
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying your answer. Hence calculate the gradient of the curve at P .
  3. Show that the area of the region enclosed by \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is \(\frac { 1 } { 2 } \ln \left( \frac { 1 + \mathrm { e } ^ { 2 } } { 2 } \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \frac { 1 } { 2 } \left( \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } \right)\).
  4. Prove algebraically that \(\mathrm { g } ( x )\) is an odd function. Interpret this result graphically.
  5. (A) Show that \(\mathrm { g } ( x ) + \frac { 1 } { 2 } = \mathrm { f } ( x )\).
    (B) Describe the transformation which maps the curve \(y = \mathrm { g } ( x )\) onto the curve \(y = \mathrm { f } ( x )\).
    (C) What can you conclude about the symmetry of the curve \(y = \mathrm { f } ( x )\) ?
OCR MEI C3 Q1
5 marks Standard +0.3
1 Evaluate \(\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x\), giving your answer in an exact form.
OCR MEI C3 Q2
16 marks Standard +0.3
2 Fig. 7 shows the curve \(y = \frac { x ^ { 2 } } { 1 + 2 x ^ { 3 } }\). It is undefined at \(x = a\); the line \(x = a\) is a vertical asymptote. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00c12cc4-f7ee-4219-8d34-a1854284f65d-1_647_1027_832_534} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Calculate the value of \(a\), giving your answer correct to 3 significant figures.
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x - 2 x ^ { 4 } } { \left( 1 + 2 x ^ { 3 } \right) ^ { 2 } }\). Hence determine the coordinates of the turning points of the curve.
  3. Show that the area of the region between the curve and the \(x\)-axis from \(x = 0\) to \(x = 1\) is \(\frac { 1 } { 6 } \ln 3\).
OCR MEI C3 Q3
20 marks Standard +0.3
3 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00c12cc4-f7ee-4219-8d34-a1854284f65d-2_425_974_478_591} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the exact coordinates of P .
  2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
  3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
  5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
  6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.
OCR MEI C3 Q1
18 marks Standard +0.3
1 Fig. 7 shows the curve $$y = 2 x - x \ln x , \text { where } x > 0 .$$ The curve crosses the \(x\)-axis at A , and has a turning point at B . The point C on the curve has \(x\)-coordinate 1 . Lines CD and BE are drawn parallel to the \(y\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-1_529_1259_657_602} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the \(x\)-coordinate of A , giving your answer in terms of e .
  2. Find the exact coordinates of B .
  3. Show that the tangents at A and C are perpendicular to each other.
  4. Using integration by parts, show that $$\int x \ln x \mathrm {~d} x = \frac { 1 } { 2 } x ^ { 2 } \ln x - \frac { 1 } { 4 } x ^ { 2 } + c$$ Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines CD and BE .
OCR MEI C3 Q2
6 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } x \sin 2 x \mathrm {~d} x = \frac { 3 \sqrt { 3 } \pi } { 24 }\).
OCR MEI C3 Q3
17 marks Standard +0.3
3 Fig. 8 shows part of the curve \(y = x \sin 3 x\). It crosses the \(x\)-axis at P . The point on the curve with \(x\)-coordinate \(\frac { 1 } { 6 } \pi\) is Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-2_420_780_549_655} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find the \(x\)-coordinate of P .
  2. Show that Q lies on the line \(y = x\).
  3. Differentiate \(x \sin 3 x\). Hence prove that the line \(y = x\) touches the curve at Q .
  4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac { 1 } { 72 } \left( \pi ^ { 2 } - 8 \right)\).
  1. Differentiate \(x \cos 2 x\) with respect to \(x\).
  2. Integrate \(x \cos 2 x\) with respect to \(x\).
Edexcel S1 2004 January Q1
13 marks Moderate -0.8
  1. An office has the heating switched on at 7.00 a.m. each morning. On a particular day, the temperature of the office, \(t { } ^ { \circ } \mathrm { C }\), was recorded \(m\) minutes after 7.00 a.m. The results are shown in the table below.
\(m\)01020304050
\(t\)6.08.911.813.515.316.1
  1. Calculate the exact values of \(S _ { m t }\) and \(S _ { m m }\).
  2. Calculate the equation of the regression line of \(t\) on \(m\) in the form \(t = a + b m\).
  3. Use your equation to estimate the value of \(t\) at 7.35 a.m.
  4. State, giving a reason, whether or not you would use the regression equation in (b) to estimate the temperature
    1. at 9.00 a.m. that day,
    2. at 7.15 a.m. one month later.
Edexcel S1 2004 January Q2
7 marks Easy -1.2
2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
  1. Write down 3 properties of the distribution of \(X\). Given that \(\mu = 27\) and \(\sigma = 10\)
  2. find \(\mathrm { P } ( 26 < X < 28 )\).
Edexcel S1 2004 January Q3
10 marks Easy -1.3
3. A discrete random variable \(X\) has the probability function shown in the table below.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 12 }\)
Find
  1. \(\mathrm { P } ( 1 < X \leq 3 )\),
  2. \(\mathrm { F } ( 2.6 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\mathrm { E } ( 2 X - 3 )\),
  5. \(\operatorname { Var } ( X )\)
Edexcel S1 2004 January Q4
11 marks Moderate -0.8
4. \(\quad\) The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }\) and \(\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }\).
  1. Find
    1. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
    2. \(\mathrm { P } ( A \cap B )\),
    3. \(\mathrm { P } ( A \cup B )\),
    4. \(\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)\).
  2. State, with a reason, whether or \(\operatorname { not } A\) and \(B\) are
    1. mutually exclusive,
    2. independent.
Edexcel S1 2004 January Q5
18 marks Moderate -0.3
5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.
SalesNumber of days
\(1 - 200\)166
\(201 - 400\)100
\(401 - 700\)59
\(701 - 1000\)30
\(1001 - 1500\)5
  1. Draw a histogram to represent these data.
  2. Use interpolation to estimate the median and inter-quartile range of daily sales.
  3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
  4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
    (2)
Edexcel S1 2004 January Q6
16 marks Moderate -0.3
6. One of the objectives of a computer game is to collect keys. There are three stages to the game. The probability of collecting a key at the first stage is \(\frac { 2 } { 3 }\), at the second stage is \(\frac { 1 } { 2 }\), and at the third stage is \(\frac { 1 } { 4 }\).
  1. Draw a tree diagram to represent the 3 stages of the game.
  2. Find the probability of collecting all 3 keys.
  3. Find the probability of collecting exactly one key in a game.
  4. Calculate the probability that keys are not collected on at least 2 successive stages in a game.
OCR C4 2006 January Q1
3 marks Easy -1.2
1 Simplify \(\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }\).
OCR C4 2006 January Q2
5 marks Standard +0.3
2 Given that \(\sin y = x y + x ^ { 2 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 2006 January Q3
6 marks Moderate -0.3
3
  1. Find the quotient and the remainder when \(3 x ^ { 3 } - 2 x ^ { 2 } + x + 7\) is divided by \(x ^ { 2 } - 2 x + 5\).
  2. Hence, or otherwise, determine the values of the constants \(a\) and \(b\) such that, when \(3 x ^ { 3 } - 2 x ^ { 2 } + a x + b\) is divided by \(x ^ { 2 } - 2 x + 5\), there is no remainder.
OCR C4 2006 January Q4
7 marks Standard +0.3
4
  1. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
  2. Hence find \(\int x \tan ^ { 2 } x \mathrm {~d} x\).
OCR C4 2006 January Q5
8 marks Moderate -0.3
5 A curve is given parametrically by the equations \(x = t ^ { 2 } , y = 2 t\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Show that the equation of the tangent to the curve at \(\left( p ^ { 2 } , 2 p \right)\) is $$p y = x + p ^ { 2 } .$$
  3. Find the coordinates of the point where the tangent at \(( 9,6 )\) meets the tangent at \(( 25 , - 10 )\).
OCR C4 2006 January Q6
9 marks Standard +0.8
6
  1. Show that the substitution \(x = \sin ^ { 2 } \theta\) transforms \(\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\) to \(\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find \(\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\).
OCR C4 2006 January Q7
10 marks Standard +0.3
7 The expression \(\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
  2. Given that \(| x | < 1\), find the first 3 terms in the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\).