Questions C3 (1301 questions)

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OCR C3 2008 June Q3
6 marks Standard +0.3
3 Find, in the form \(y = m x + c\), the equation of the tangent to the curve $$y = x ^ { 2 } \ln x$$ at the point with \(x\)-coordinate e.
OCR C3 2008 June Q4
9 marks Standard +0.3
4 The gradient of the curve \(y = \left( 2 x ^ { 2 } + 9 \right) ^ { \frac { 5 } { 2 } }\) at the point \(P\) is 100 .
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = 10 \left( 2 x ^ { 2 } + 9 \right) ^ { - \frac { 3 } { 2 } }\).
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.3 and 0.4 .
  3. Use an iterative formula, based on the equation in part (i), to find the \(x\)-coordinate of \(P\) correct to 4 decimal places. You should show the result of each iteration.
OCR C3 2008 June Q5
9 marks Moderate -0.3
5
  1. Express \(\tan 2 \alpha\) in terms of \(\tan \alpha\) and hence solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation $$\tan 2 \alpha \tan \alpha = 8 .$$
  2. Given that \(\beta\) is the acute angle such that \(\sin \beta = \frac { 6 } { 7 }\), find the exact value of
    1. \(\operatorname { cosec } \beta\),
    2. \(\cot ^ { 2 } \beta\).
OCR C3 2008 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-3_586_798_267_676} The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
OCR C3 2008 June Q7
9 marks Moderate -0.3
7 It is claimed that the number of plants of a certain species in a particular locality is doubling every 9 years. The number of plants now is 42 . The number of plants is treated as a continuous variable and is denoted by \(N\). The number of years from now is denoted by \(t\).
  1. Two equivalent expressions giving \(N\) in terms of \(t\) are $$N = A \times 2 ^ { k t } \quad \text { and } \quad N = A \mathrm { e } ^ { m t } .$$ Determine the value of each of the constants \(A , k\) and \(m\).
  2. Find the value of \(t\) for which \(N = 100\), giving your answer correct to 3 significant figures.
  3. Find the rate at which the number of plants will be increasing at a time 35 years from now.
OCR C3 2008 June Q8
10 marks Standard +0.3
8 The expression \(\mathrm { T } ( \theta )\) is defined for \(\theta\) in degrees by $$\mathrm { T } ( \theta ) = 3 \cos \left( \theta - 60 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) .$$
  1. Express \(\mathrm { T } ( \theta )\) in the form \(A \sin \theta + B \cos \theta\), giving the exact values of the constants \(A\) and \(B\). [3]
  2. Hence express \(\mathrm { T } ( \theta )\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  3. Find the smallest positive value of \(\theta\) such that \(\mathrm { T } ( \theta ) + 1 = 0\).
OCR C3 2008 June Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-4_534_935_264_605} The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \frac { 15 x } { x ^ { 2 } + 5 }$$ The diagram shows the curve with equation \(y = \mathrm { f } ( x )\).
  1. Find the range of f .
  2. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm { g } ( x ) = \frac { 15 x } { x ^ { 2 } + 5 }$$ Given that g is a one-one function, state the least possible value of \(k\).
  3. Show that there is no point on the curve \(y = \mathrm { g } ( x )\) at which the gradient is - 1 .
OCR C3 Specimen Q1
5 marks Standard +0.3
1 Solve the inequality \(| 2 x + 1 | > | x - 1 |\).
OCR C3 Specimen Q2
6 marks Moderate -0.3
2
  1. Prove the identity $$\sin \left( x + 30 ^ { \circ } \right) + ( \sqrt { } 3 ) \cos \left( x + 30 ^ { \circ } \right) \equiv 2 \cos x$$ where \(x\) is measured in degrees.
  2. Hence express \(\cos 15 ^ { \circ }\) in surd form.
OCR C3 Specimen Q3
7 marks Standard +0.3
3 The sequence defined by the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 17 - 5 x _ { n } \right)$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Use the iterative formula to find \(\alpha\) correct to 2 decimal places. You should show the result of each iteration.
  2. Find a cubic equation of the form $$x ^ { 3 } + c x + d = 0$$ which has \(\alpha\) as a root.
  3. Does this cubic equation have any other real roots? Justify your answer.
OCR C3 Specimen Q4
8 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-2_419_707_1576_660} The diagram shows the curve $$y = \frac { 1 } { \sqrt { } ( 4 x + 1 ) }$$ The region \(R\) (shaded in the diagram) is enclosed by the curve, the axes and the line \(x = 2\).
  1. Show that the exact area of \(R\) is 1 .
  2. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed.
OCR C3 Specimen Q5
8 marks Moderate -0.3
5 At time \(t\) minutes after an oven is switched on, its temperature \(\theta ^ { \circ } \mathrm { C }\) is given by $$\theta = 200 - 180 \mathrm { e } ^ { - 0.1 t }$$
  1. State the value which the oven's temperature approaches after a long time.
  2. Find the time taken for the oven's temperature to reach \(150 ^ { \circ } \mathrm { C }\).
  3. Find the rate at which the temperature is increasing at the instant when the temperature reaches \(150 ^ { \circ } \mathrm { C }\).
OCR C3 Specimen Q6
8 marks Standard +0.3
6 The function f is defined by $$\mathrm { f } : x \mapsto 1 + \sqrt { } x \quad \text { for } x \geqslant 0$$
  1. State the domain and range of the inverse function \(\mathrm { f } ^ { - 1 }\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. By considering the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), show that the solution to the equation $$\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )$$ is \(x = \frac { 1 } { 2 } ( 3 + \sqrt { } 5 )\).
OCR C3 Specimen Q7
9 marks Standard +0.3
7
  1. Write down the formula for \(\tan 2 x\) in terms of \(\tan x\).
  2. By letting \(\tan x = t\), show that the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ becomes $$3 t ^ { 4 } - 8 t ^ { 2 } - 3 = 0$$
  3. Hence find all the solutions of the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ which lie in the interval \(0 \leqslant x \leqslant 2 \pi\).
OCR C3 Specimen Q8
10 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_476_608_287_756} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. The point \(P\) on the curve is the point at which the gradient takes its maximum value. Show that the tangent at \(P\) passes through the point \(( 0 , - 1 )\).
OCR C3 Specimen Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{b6b6e55a-a5ba-466c-ac9f-b5ef5bca7a3c-4_424_707_1260_724} The diagram shows the curve \(y = \tan ^ { - 1 } x\) and its asymptotes \(y = \pm a\).
  1. State the exact value of \(a\).
  2. Find the value of \(x\) for which \(\tan ^ { - 1 } x = \frac { 1 } { 2 } a\). The equation of another curve is \(y = 2 \tan ^ { - 1 } ( x - 1 )\).
  3. Sketch this curve on a copy of the diagram, and state the equations of its asymptotes in terms of \(a\).
  4. Verify by calculation that the value of \(x\) at the point of intersection of the two curves is 1.54 , correct to 2 decimal places. Another curve (which you are not asked to sketch) has equation \(y = \left( \tan ^ { - 1 } x \right) ^ { 2 }\).
  5. Use Simpson's rule, with 4 strips, to find an approximate value for \(\int _ { 0 } ^ { 1 } \left( \tan ^ { - 1 } x \right) ^ { 2 } \mathrm {~d} x\).
OCR MEI C3 2006 January Q1
4 marks Moderate -0.8
1 Given that \(y = ( 1 + 6 x ) ^ { \frac { 1 } { 3 } }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { y ^ { 2 } }\).
OCR MEI C3 2006 January Q2
6 marks Moderate -0.3
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
7 marks Moderate -0.3
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
7 marks Standard +0.3
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 marks Moderate -0.3
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
7 marks Moderate -0.8
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
18 marks Standard +0.3
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
5 marks Easy -1.2
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
5 marks Standard +0.3
2 Evaluate \(\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x\), giving your answer in an exact form.