Questions — OCR C3 (285 questions)

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OCR C3 2006 June Q7
7
  1. Find the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 } { ( 4 x - 1 ) ^ { 2 } } \mathrm {~d} x\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-3_563_753_1681_735} The diagram shows part of the curve \(y = \frac { 1 } { x }\). The point \(P\) has coordinates \(\left( a , \frac { 1 } { a } \right)\) and the point \(Q\) has coordinates \(\left( 2 a , \frac { 1 } { 2 a } \right)\), where \(a\) is a positive constant. The point \(R\) is such that \(P R\) is parallel to the \(x\)-axis and \(Q R\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(P R\) and \(Q R\). Show that the area of this shaded region is \(\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\).
OCR C3 2006 June Q8
8
  1. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\).
  3. Solve, for \(0 ^ { \circ } < x < 360 ^ { \circ }\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1 ^ { \circ }\).
OCR C3 2006 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{ebfdf170-99c6-4785-b9d7-201c3425b4c9-4_556_720_676_715} The diagram shows the curve with equation \(y = 2 \ln ( x - 1 )\). The point \(P\) has coordinates ( \(0 , p\) ). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0 , y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(\boldsymbol { y }\)-axis to form a solid.
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the solid is given by $$V = \pi \left( \mathrm { e } ^ { p } + 4 \mathrm { e } ^ { \frac { 1 } { 2 } p } + p - 5 \right) .$$
  2. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \mathrm {~cm} \mathrm {~min} ^ { - 1 }\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures.
OCR C3 2007 June Q1
1 Differentiate each of the following with respect to \(x\).
  1. \(x ^ { 3 } ( x + 1 ) ^ { 5 }\)
  2. \(\sqrt { 3 x ^ { 4 } + 1 }\)
OCR C3 2007 June Q2
2 Solve the inequality \(| 4 x - 3 | < | 2 x + 1 |\).
OCR C3 2007 June Q3
3 The function \(f\) is defined for all non-negative values of \(x\) by $$f ( x ) = 3 + \sqrt { x }$$
  1. Evaluate ff(169).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( \mathrm { x } )\) in terms of x .
  3. On a single diagram sketch the graphs of \(y = f ( x )\) and \(y = f ^ { - 1 } ( x )\), indicating how the two graphs are related.
OCR C3 2007 June Q4
4 The integral I is defined by $$I = \int _ { 0 } ^ { 13 } ( 2 x + 1 ) ^ { \frac { 1 } { 3 } } d x$$
  1. Use integration to find the exact value of I .
  2. Use Simpson's rule with two strips to find an approximate value for I. Give your answer correct to 3 significant figures.
OCR C3 2007 June Q5
5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula $$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$
  1. Find the time taken for the substance to halve its mass.
  2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year.
OCR C3 2007 June Q6
6
  1. Given that \(\int _ { 0 } ^ { \mathrm { a } } \left( 6 \mathrm { e } ^ { 2 \mathrm { x } } + \mathrm { x } \right) \mathrm { dx } = 42\), show that \(\mathrm { a } = \frac { 1 } { 2 } \ln \left( 15 - \frac { 1 } { 6 } \mathrm { a } ^ { 2 } \right)\).
  2. Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 decimal places. Use a starting value of 1 and show the result of each iteration.
OCR C3 2007 June Q7
7
  1. Sketch the graph of \(y = \sec x\) for \(0 \leqslant x \leqslant 2 \pi\).
  2. Solve the equation \(\sec x = 3\) for \(0 \leqslant x \leqslant 2 \pi\), giving the roots correct to 3 significant figures.
  3. Solve the equation \(\sec \theta = 5 \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving the roots correct to 3 significant figures.
OCR C3 2007 June Q8
8
  1. Given that \(\mathrm { y } = \frac { 4 \ln \mathrm { x } - 3 } { 4 \ln \mathrm { x } + 3 }\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 24 } { \mathrm { x } ( 4 \ln \mathrm { x } + 3 ) ^ { 2 } }\).
  2. Find the exact value of the gradient of the curve \(y = \frac { 4 \ln x - 3 } { 4 \ln x + 3 }\) at the point where it crosses the \(x\)-axis.

  3. \includegraphics[max width=\textwidth, alt={}, center]{133c38fb-307f-4f20-86cb-1bd57cc4f870-3_524_830_941_699} The diagram shows part of the curve with equation $$\mathrm { y } = \frac { 2 } { \mathrm { x } ^ { \frac { 1 } { 2 } } ( 4 \ln \mathrm { x } + 3 ) }$$ The region shaded in the diagram is bounded by the curve and the lines \(x = 1 , x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the x -axis.
OCR C3 2007 June Q9
9
  1. Prove the identity $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta - 3 } { 1 - 3 \tan ^ { 2 } \theta }$$
  2. Solve, for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = 4 \sec ^ { 2 } \theta - 3 ,$$ giving your answers correct to the nearest \(0.1 ^ { \circ }\).
  3. Show that, for all values of the constant k , the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = \mathrm { K } ^ { 2 }$$ has two roots in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
OCR C3 2008 June Q1
1 Find the exact solutions of the equation \(| 4 x - 5 | = | 3 x - 5 |\).
OCR C3 2008 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-2_529_855_397_646} The diagram shows the graph of \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 3 ) = 0\) and \(\mathrm { f } ( 0 ) = 2\). Sketch, on separate diagrams, the following graphs, indicating in each case the coordinates of the points where the graph crosses the axes:
  1. \(y = \mathrm { f } ^ { - 1 } ( x )\),
  2. \(y = - 2 \mathrm { f } ( x )\).
OCR C3 2008 June Q3
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
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
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
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
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
3 marks
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
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
1 Solve the inequality \(| 2 x + 1 | > | x - 1 |\).
OCR C3 Specimen Q2
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
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
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.