Questions C3 (1200 questions)

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OCR MEI C3 2006 January Q2
2 A population is \(P\) million at time \(t\) years. \(P\) is modelled by the equation $$P = 5 + a \mathrm { e } ^ { - b t }$$ where \(a\) and \(b\) are constants.
The population is initially 8 million, and declines to 6 million after 1 year.
  1. Use this information to calculate the values of \(a\) and \(b\), giving \(b\) correct to 3 significant figures.
  2. What is the long-term population predicted by the model?
OCR MEI C3 2006 January Q3
3
  1. Express \(2 \ln x + \ln 3\) as a single logarithm.
  2. Hence, given that \(x\) satisfies the equation $$2 \ln x + \ln 3 = \ln ( 5 x + 2 )$$ show that \(x\) is a root of the quadratic equation \(3 x ^ { 2 } - 5 x - 2 = 0\).
  3. Solve this quadratic equation, explaining why only one root is a valid solution of $$2 \ln x + \ln 3 = \ln ( 5 x + 2 ) .$$
OCR MEI C3 2006 January Q4
4 Fig. 4 shows a cone. The angle between the axis and the slant edge is \(30 ^ { \circ }\). Water is poured into the cone at a constant rate of \(2 \mathrm {~cm} ^ { 3 }\) per second. At time \(t\) seconds, the radius of the water surface is \(r \mathrm {~cm}\) and the volume of water in the cone is \(V \mathrm {~cm} ^ { 3 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a4c3f3b-a298-4b13-b97e-b52f8d9d527b-3_369_401_431_831} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Write down the value of \(\frac { \mathrm { d } V } { \mathrm {~d} t }\).
  2. Show that \(V = \frac { \sqrt { 3 } } { 3 } \pi r ^ { 3 }\), and find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\).
    [0pt] [You may assume that the volume of a cone of height \(h\) and radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  3. Use the results of parts (i) and (ii) to find the value of \(\frac { \mathrm { d } r } { \mathrm {~d} t }\) when \(r = 2\).
OCR MEI C3 2006 January Q5
5 A curve is defined implicitly by the equation $$y ^ { 3 } = 2 x y + x ^ { 2 }$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( x + y ) } { 3 y ^ { 2 } - 2 x }\).
  2. Hence write down \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(x\) and \(y\).
OCR MEI C3 2006 January Q6
6 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 1 + 2 \sin x\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(\mathrm { f } ^ { - 1 } ( x ) = \arcsin \left( \frac { x - 1 } { 2 } \right)\) and state the domain of this function. Fig. 6 shows a sketch of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6a4c3f3b-a298-4b13-b97e-b52f8d9d527b-4_506_561_705_751} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
  2. Write down the coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C .
OCR MEI C3 2006 January Q7
7 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]{6a4c3f3b-a298-4b13-b97e-b52f8d9d527b-5_531_1262_671_536} \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 . \section*{[Question 8 is printed overleaf.]}
OCR MEI C3 2007 January Q1
1 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-2_490_844_493_607} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Write down the coordinates of P .
  2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).
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.