Questions — OCR (4907 questions)

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OCR C3 Q3
8 marks Moderate -0.3
3. (a) Given that \(y = \ln x\),
  1. find an expression for \(\ln \frac { x ^ { 2 } } { \mathrm { e } }\) in terms of \(y\),
  2. show that \(\log _ { 2 } x = \frac { y } { \ln 2 }\).
    (b) Hence, or otherwise, solve the equation $$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$ giving your answer to 2 decimal places.
OCR C3 Q4
9 marks Standard +0.8
4. \includegraphics[max width=\textwidth, alt={}, center]{c0b79c3c-9537-4c71-903b-01434dfb5d26-1_492_803_1562_452} The diagram shows the curves \(y = ( x - 1 ) ^ { 2 }\) and \(y = 2 - \frac { 2 } { x } , x > 0\).
  1. Verify that the two curves meet at the points where \(x = 1\) and where \(x = 2\). The shaded region bounded by the two curves is rotated completely about the \(x\)-axis.
  2. Find the exact volume of the solid formed.
OCR C3 Q5
10 marks Moderate -0.3
5. \(\mathrm { f } ( x ) = 5 + \mathrm { e } ^ { 2 x - 3 } , x \in \mathbb { R }\).
  1. State the range of f .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 7\).
  4. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(y = 7\).
OCR C3 Q6
10 marks Standard +0.3
6.
  1. Express \(\sqrt { 3 } \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
  2. State the maximum value of \(\sqrt { 3 } \sin \theta + \cos \theta\) and the smallest positive value of \(\theta\) for which this maximum value occurs.
  3. Solve the equation $$\sqrt { 3 } \sin \theta + \cos \theta + \sqrt { 3 } = 0$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\), giving your answers in terms of \(\pi\).
OCR C3 Q7
10 marks Standard +0.3
7. $$\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$
  1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
  3. Use Simpson's rule with six strips to find an approximate value for $$\int _ { 0 } ^ { 6 } f ( x ) d x$$
OCR C3 Q8
14 marks Standard +0.3
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$
  1. State the range of f .
  2. Evaluate fg(-2).
  3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
  4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
  5. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
  6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures.
OCR C3 Q1
4 marks Moderate -0.5
  1. Use Simpson's rule with four strips to estimate the value of the integral
$$\int _ { 0 } ^ { 3 } \mathrm { e } ^ { \cos x } \mathrm {~d} x$$
OCR C3 Q2
7 marks Standard +0.8
  1. Giving your answers to 1 decimal place, solve the equation
$$5 \tan ^ { 2 } 2 \theta - 13 \sec 2 \theta = 1 ,$$ for \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).
OCR C3 Q3
7 marks Standard +0.3
3. \includegraphics[max width=\textwidth, alt={}, center]{14ec6709-e1cb-42d7-af99-91365e50e4fc-1_535_810_877_406} The diagram shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at \(( - 3,2 )\) and a minimum point at \(( 2 , - 4 )\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 3 \mathrm { f } ( 2 x )\).
  2. Write down the values of the constants \(a\) and \(b\) such that the curve with equation \(y = a + \mathrm { f } ( x + b )\) has a minimum point at the origin \(O\).
OCR C3 Q4
7 marks Standard +0.8
4. Find the values of \(x\) in the interval \(- 180 < x < 180\) for which $$\tan ( x + 45 ) ^ { \circ } - \tan x ^ { \circ } = 4 ,$$ giving your answers to 1 decimal place.
OCR C3 Q5
8 marks Moderate -0.3
5. The finite region \(R\) is bounded by the curve with equation \(y = \sqrt [ 3 ] { 3 x - 1 }\), the \(x\)-axis and the lines \(x = \frac { 2 } { 3 }\) and \(x = 3\).
  1. Find the area of \(R\).
  2. Find, in terms of \(\pi\), the volume of the solid formed when \(R\) is rotated through four right angles about the \(x\)-axis.
OCR C3 Q6
9 marks Standard +0.3
6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 1 - a x , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \rightarrow x ^ { 2 } + 2 a x + 2 , \quad x \in \mathbb { R } \end{aligned}$$ where \(a\) is a constant.
Find, in terms of \(a\),
  1. an expression for \(\mathrm { f } ^ { - 1 } ( x )\),
  2. the range of g . Given that \(g f ( 3 ) = 7\),
  3. find the two possible values of \(a\).
OCR C3 Q7
9 marks Standard +0.8
7. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).
  1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
  2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
  3. Find the \(x\)-coordinate of \(R\) in exact form.
OCR C3 Q8
10 marks Standard +0.3
8.
  1. Solve the equation $$\pi - 3 \cos ^ { - 1 } \theta = 0$$
  2. Sketch on the same diagram the curves \(y = \cos ^ { - 1 } ( x - 1 ) , 0 \leq x \leq 2\) and \(y = \sqrt { x + 2 } , x \geq - 2\). Given that \(\alpha\) is the root of the equation $$\cos ^ { - 1 } ( x - 1 ) = \sqrt { x + 2 }$$
  3. show that \(0 < \alpha < 1\),
  4. use the iterative formula $$x _ { n + 1 } = 1 + \cos \sqrt { x _ { n } + 2 }$$ with \(x _ { 0 } = 1\) to find \(\alpha\) correct to 3 decimal places.
    You should show the result of each iteration.
OCR C3 Q9
11 marks Moderate -0.8
9. The number of bacteria present in a culture at time \(t\) hours is modelled by the continuous variable \(N\) and the relationship $$N = 2000 \mathrm { e } ^ { k t }$$ where \(k\) is a constant.
Given that when \(t = 3 , N = 18000\), find
  1. the value of \(k\) to 3 significant figures,
  2. how long it takes for the number of bacteria present to double, giving your answer to the nearest minute,
  3. the rate at which the number of bacteria is increasing when \(t = 3\).
OCR C3 2006 January Q1
4 marks Easy -1.2
1 Show that \(\int _ { 2 } ^ { 8 } \frac { 3 } { x } \mathrm {~d} x = \ln 64\).
OCR C3 2006 January Q2
5 marks Standard +0.3
2 Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(\sec ^ { 2 } \theta = 4 \tan \theta - 2\).
OCR C3 2006 January Q3
6 marks Moderate -0.3
3
  1. Differentiate \(x ^ { 2 } ( x + 1 ) ^ { 6 }\) with respect to \(x\).
  2. Find the gradient of the curve \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 3 }\) at the point where \(x = 1\).
OCR C3 2006 January Q4
5 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_529_737_900_701} The function f is defined by \(\mathrm { f } ( x ) = 2 - \sqrt { x }\) for \(x \geqslant 0\). The graph of \(y = \mathrm { f } ( x )\) is shown above.
  1. State the range of f.
  2. Find the value of \(\mathrm { ff } ( 4 )\).
  3. Given that the equation \(| \mathrm { f } ( x ) | = k\) has two distinct roots, determine the possible values of the constant \(k\).
OCR C3 2006 January Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_486_746_1978_696} The diagram shows the curves \(y = ( 1 - 2 x ) ^ { 5 }\) and \(y = \mathrm { e } ^ { 2 x - 1 } - 1\). The curves meet at the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve.
OCR C3 2006 January Q6
9 marks Moderate -0.3
6
  1. \(t\)01020
    \(X\)275440
    The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\).
  2. The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80 \mathrm { e } ^ { - 0.02 t }$$
    1. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures.
    2. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures.
OCR C3 2006 January Q7
11 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-3_465_748_1133_717} The diagram shows the curve with equation \(y = \cos ^ { - 1 } x\).
  1. Sketch the curve with equation \(y = 3 \cos ^ { - 1 } ( x - 1 )\), showing the coordinates of the points where the curve meets the axes.
  2. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) has exactly one root.
  3. Show by calculation that the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) lies between 1.8 and 1.9 .
  4. The sequence defined by $$x _ { 1 } = 2 , \quad x _ { n + 1 } = 1 + \cos \left( \frac { 1 } { 3 } x _ { n } \right)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\).
OCR C3 2006 January Q8
11 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-4_787_742_276_719} The diagram shows part of the curve \(y = \ln \left( 5 - x ^ { 2 } \right)\) which meets the \(x\)-axis at the point \(P\) with coordinates \(( 2,0 )\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(P Q\) and \(x = 0\).
  1. Find the equation of the tangent to the curve at \(P\).
  2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures.
  3. Deduce an approximation to the area of the region \(B\).
OCR C3 2007 January Q1
5 marks Moderate -0.3
1 Find the equation of the tangent to the curve \(y = \frac { 2 x + 1 } { 3 x - 1 }\) at the point \(\left( 1 , \frac { 3 } { 2 } \right)\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
OCR C3 2007 January Q2
5 marks Moderate -0.8
2 It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac { 12 } { 13 }\). Find the exact value of
  1. \(\cot \theta\),
  2. \(\cos 2 \theta\).