Questions — OCR MEI C3 (366 questions)

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OCR MEI C3 2007 January Q2
2 Evaluate \(\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x\), giving your answer in an exact form.
OCR MEI C3 2007 January Q3
3 The value \(\pounds V\) of a car is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(t\) is the age of the car in years and \(A\) and \(k\) are constants. Its value when new is \(\pounds 10000\), and after 3 years its value is \(\pounds 6000\).
  1. Find the values of \(A\) and \(k\).
  2. Find the age of the car when its value is \(\pounds 2000\).
OCR MEI C3 2007 January Q4
4 Use the method of exhaustion to prove the following result.
No 1 - or 2 -digit perfect square ends in \(2,3,7\) or 8
State a generalisation of this result.
OCR MEI C3 2007 January Q5
5 The equation of a curve is \(y = \frac { x ^ { 2 } } { 2 x + 1 }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( x + 1 ) } { ( 2 x + 1 ) ^ { 2 } }\).
  2. Find the coordinates of the stationary points of the curve. You need not determine their nature.
OCR MEI C3 2007 January Q6
6 Fig. 6 shows the triangle OAP , where O is the origin and A is the point \(( 0,3 )\). The point \(\mathrm { P } ( x , 0 )\) moves on the positive \(x\)-axis. The point \(\mathrm { Q } ( 0 , y )\) moves between O and A in such a way that \(\mathrm { AQ } + \mathrm { AP } = 6\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-3_490_839_438_612} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Write down the length AQ in terms of \(y\). Hence find AP in terms of \(y\), and show that $$( y + 3 ) ^ { 2 } = x ^ { 2 } + 9 .$$
  2. Use this result to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x } { y + 3 }\).
  3. When \(x = 4\) and \(y = 2 , \frac { \mathrm {~d} x } { \mathrm {~d} t } = 2\). Calculate \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) at this time. Section B (36 marks)
OCR MEI C3 2007 January Q7
7 Fig. 7 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x \sqrt { 1 + x }\). The curve meets the \(x\)-axis at the origin and at the point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-4_491_881_476_588} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Verify that the point P has coordinates \(( - 1,0 )\). Hence state the domain of the function \(\mathrm { f } ( x )\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + 3 x } { 2 \sqrt { 1 + x } }\).
  3. Find the exact coordinates of the turning point of the curve. Hence write down the range of the function.
  4. Use the substitution \(u = 1 + x\) to show that $$\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$ Hence find the area of the region enclosed by the curve and the \(x\)-axis.
OCR MEI C3 2007 January Q8
8 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \text { for } x \geqslant 0 .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-5_707_876_440_593} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the origin and at the point \(( \ln 2,1 )\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \ln ( 1 + \sqrt { x } )\) for \(x \geqslant 0\).
  2. Show that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are inverse functions. Hence sketch the graph of \(y = \mathrm { g } ( x )\). Write down the gradient of the curve \(y = \mathrm { g } ( x )\) at the point \(( 1 , \ln 2 )\).
  3. Show that \(\int \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { x } + x + c\). Hence evaluate \(\int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } - 1 \right) ^ { 2 } \mathrm {~d} x\), giving your answer in an exact form.
  4. Using your answer to part (iii), calculate the area of the region enclosed by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis and the line \(x = 1\).
OCR MEI C3 2006 June Q1
1 Solve the equation \(| 3 x - 2 | = x\).
OCR MEI C3 2006 June Q2
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 2006 June Q3
3 Fig. 3 shows the curve defined by the equation \(y = \arcsin ( x - 1 )\), for \(0 \leqslant x \leqslant 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72de4365-c3f0-4106-8daf-3047afeab723-2_684_551_829_758} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find \(x\) in terms of \(y\), and show that \(\frac { \mathrm { d } x } { \mathrm {~d} y } = \cos y\).
  2. Hence find the exact gradient of the curve at the point where \(x = 1.5\).
OCR MEI C3 2006 June Q4
4 Fig. 4 is a diagram of a garden pond. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72de4365-c3f0-4106-8daf-3047afeab723-3_296_746_351_657} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The volume \(V \mathrm {~m} ^ { 3 }\) of water in the pond when the depth is \(h\) metres is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 3 - h )$$
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} h }\). Water is poured into the pond at the rate of \(0.02 \mathrm {~m} ^ { 3 }\) per minute.
  2. Find the value of \(\frac { \mathrm { d } h } { \mathrm {~d} t }\) when \(h = 0.4\).
OCR MEI C3 2006 June Q5
5 Positive integers \(a , b\) and \(c\) are said to form a Pythagorean triple if \(a ^ { 2 } + b ^ { 2 } = c ^ { 2 }\).
  1. Given that \(t\) is an integer greater than 1 , show that \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\) form a Pythagorean triple.
  2. The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form \(2 t , t ^ { 2 } - 1\) and \(t ^ { 2 } + 1\).
OCR MEI C3 2006 June Q6
6 The mass \(M \mathrm {~kg}\) of a radioactive material is modelled by the equation $$M = M _ { 0 } \mathrm { e } ^ { - k t } ,$$ where \(M _ { 0 }\) is the initial mass, \(t\) is the time in years, and \(k\) is a constant which measures the rate of radioactive decay.
  1. Sketch the graph of \(M\) against \(t\).
  2. For Carbon \(14 , k = 0.000121\). Verify that after 5730 years the mass \(M\) has reduced to approximately half the initial mass. The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the initial mass.
  3. Show that, in general, the half-life \(T\) is given by \(T = \frac { \ln 2 } { k }\).
  4. Hence find the half-life of Plutonium 239, given that for this material \(k = 2.88 \times 10 ^ { - 5 }\).
OCR MEI C3 2007 June Q1
1
  1. Differentiate \(\sqrt { 1 + 2 x }\).
  2. Show that the derivative of \(\ln \left( 1 - \mathrm { e } ^ { - x } \right)\) is \(\frac { 1 } { \mathrm { e } ^ { x } - 1 }\).
OCR MEI C3 2007 June Q2
2 Given that \(\mathrm { f } ( x ) = 1 - x\) and \(\mathrm { g } ( x ) = | x |\), write down the composite function \(\mathrm { gf } ( x )\).
On separate diagrams, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { gf } ( x )\).
OCR MEI C3 2007 June Q3
3 A curve has equation \(2 y ^ { 2 } + y = 9 x ^ { 2 } + 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point \(\mathrm { A } ( 1,2 )\).
  2. Find the coordinates of the points on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
OCR MEI C3 2007 June Q4
4 A cup of water is cooling. Its initial temperature is \(100 ^ { \circ } \mathrm { C }\). After 3 minutes, its temperature is \(80 ^ { \circ } \mathrm { C }\).
  1. Given that \(T = 25 + a \mathrm { e } ^ { - k t }\), where \(T\) is the temperature in \({ } ^ { \circ } \mathrm { C } , t\) is the time in minutes and \(a\) and \(k\) are constants, find the values of \(a\) and \(k\).
  2. What is the temperature of the water
    (A) after 5 minutes,
    (B) in the long term?
OCR MEI C3 2007 June Q5
5 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.
OCR MEI C3 2007 June Q6
6 Fig. 6 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \arctan x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-3_378_725_367_669} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Find the range of the function \(\mathrm { f } ( x )\), giving your answer in terms of \(\pi\).
  2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\). Find the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the origin.
  3. Hence write down the gradient of \(y = \frac { 1 } { 2 } \arctan x\) at the origin.
OCR MEI C3 2007 June Q7
7 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]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-3_654_1034_1505_497} \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 2007 June Q8
8 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]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-4_421_965_349_550} \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 2008 June Q1
1 Solve the inequality \(| 2 x - 1 | \leqslant 3\).
OCR MEI C3 2008 June Q2
2 Find \(\int x \mathrm { e } ^ { 3 x } \mathrm {~d} x\).
OCR MEI C3 2008 June Q3
3
  1. State the algebraic condition for the function \(\mathrm { f } ( x )\) to be an even function.
    What geometrical property does the graph of an even function have?
  2. State whether the following functions are odd, even or neither.
    (A) \(\mathrm { f } ( x ) = x ^ { 2 } - 3\)
    (B) \(\mathrm { g } ( x ) = \sin x + \cos x\)
    (C) \(\mathrm { h } ( x ) = \frac { 1 } { x + x ^ { 3 } }\)
OCR MEI C3 2008 June Q4
4 Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 6\).