Questions — OCR C3 (339 questions)

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OCR C3 2015 June Q8
11 marks Standard +0.3
8 The functions \(f\) and \(g\) are defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = 2 + \ln ( x + 3 ) \text { for } x \geqslant 0 \\ \mathrm {~g} ( x ) = a x ^ { 2 } \text { for all real values of } x , \text { where } a \text { is a positive constant. } \end{gathered}$$
  1. Given that \(\operatorname { gf } \left( \mathrm { e } ^ { 4 } - 3 \right) = 9\), find the value of \(a\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Given that \(\mathrm { ff } \left( \mathrm { e } ^ { N } - 3 \right) = \ln \left( 53 \mathrm { e } ^ { 2 } \right)\), find the value of \(N\).
OCR C3 2015 June Q9
13 marks Standard +0.8
9 It is given that \(\mathrm { f } ( \theta ) = \sin \left( \theta + 30 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right)\).
  1. Show that \(\mathrm { f } ( \theta ) = \cos \theta\). Hence show that $$f ( 4 \theta ) + 4 f ( 2 \theta ) \equiv 8 \cos ^ { 4 } \theta - 3 .$$
  2. Hence
    1. determine the greatest and least values of \(\frac { 1 } { \mathrm { f } ( 4 \theta ) + 4 \mathrm { f } ( 2 \theta ) + 7 }\) as \(\theta\) varies,
    2. solve the equation $$\sin \left( 12 \alpha + 30 ^ { \circ } \right) + \cos \left( 12 \alpha + 60 ^ { \circ } \right) + 4 \sin \left( 6 \alpha + 30 ^ { \circ } \right) + 4 \cos \left( 6 \alpha + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < \alpha < 60 ^ { \circ }\). \section*{END OF QUESTION PAPER}
OCR C3 2016 June Q1
5 marks Moderate -0.3
1 Find the equation of the tangent to the curve $$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$ at the point \(( - 1,3 )\), giving your answer in the form \(y = m x + c\).
OCR C3 2016 June Q2
5 marks Moderate -0.8
2 Find
  1. \(\int \left( 2 - \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\),
  2. \(\int ( 4 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
OCR C3 2016 June Q3
6 marks Moderate -0.8
3 The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).
\(t\)051025
\(m\)200160
  1. Find the values missing from the table.
  2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams.
OCR C3 2016 June Q4
8 marks Standard +0.8
4 It is given that \(A\) and \(B\) are angles such that $$\sec ^ { 2 } A - \tan A = 13 \quad \text { and } \quad \sin B \sec ^ { 2 } B = 27 \cos B \operatorname { cosec } ^ { 2 } B$$ Find the possible exact values of \(\tan ( A - B )\).
OCR C3 2016 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{6d15cb4d-f540-488b-b94e-7a494f192ba5-2_469_721_1932_662} The diagram shows the curves \(y = \mathrm { e } ^ { 2 x }\) and \(y = 8 \mathrm { e } ^ { - x }\). The shaded region is bounded by the curves and the \(y\)-axis. Without using a calculator,
  1. solve an appropriate equation to show that the curves intersect at a point for which \(x = \ln 2\),
  2. find the area of the shaded region, giving your answer in simplified form.
OCR C3 2016 June Q6
8 marks Standard +0.3
6 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \ln ( 4 x - 7 ) + 18 \quad \text { and } \quad y = a \left( x ^ { 2 } + b \right) ^ { \frac { 1 } { 2 } }$$ respectively, where \(a\) and \(b\) are positive constants. The point \(P\) lies on both curves and has \(x\)-coordinate 2 . It is given that the gradient of \(C _ { 1 }\) at \(P\) is equal to the gradient of \(C _ { 2 }\) at \(P\). Find the values of \(a\) and \(b\).
OCR C3 2016 June Q7
11 marks Standard +0.3
7
  1. By sketching the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) (where \(y\) is in radians) in a single diagram, show that the equation \(x ( 2 x + 5 ) = \cos ^ { - 1 } x\) has exactly one real root.
  2. Use the iterative formula $$x _ { n + 1 } = \frac { \cos ^ { - 1 } x _ { n } } { 2 x _ { n } + 5 } \text { with } x _ { 1 } = 0.25$$ to find the root correct to 3 significant figures. Show the result of each iteration correct to at least 4 significant figures.
  3. Two new curves are obtained by transforming each of the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) by the pair of transformations:
    reflection in the \(x\)-axis followed by reflection in the \(y\)-axis.
    State an equation of each of the new curves and determine the coordinates of their point of intersection, giving each coordinate correct to 3 significant figures.
OCR C3 2016 June Q8
10 marks Standard +0.3
8 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = | 2 x + a | + 3 a \quad \text { and } \quad \mathrm { g } ( x ) = 5 x - 4 a$$ where \(a\) is a positive constant.
  1. State the range of f and the range of g .
  2. State why f has no inverse, and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
  3. Solve for \(x\) the equation \(\operatorname { gf } ( x ) = 31 a\).
  4. Show that \(\sin 2 \theta ( \tan \theta + \cot \theta ) \equiv 2\).
  5. Hence
    1. find the exact value of \(\tan \frac { 1 } { 12 } \pi + \tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 12 } \pi + \cot \frac { 1 } { 8 } \pi\),
    2. solve the equation \(\sin 4 \theta ( \tan \theta + \cot \theta ) = 1\) for \(0 < \theta < \frac { 1 } { 2 } \pi\),
    3. express \(( 1 - \cos 2 \theta ) ^ { 2 } \left( \tan \frac { 1 } { 2 } \theta + \cot \frac { 1 } { 2 } \theta \right) ^ { 3 }\) in terms of \(\sin \theta\).
OCR C3 Q1
4 marks Moderate -0.8
The function f is defined for all real values of \(x\) by $$f(x) = 10 - (x + 3)^2.$$
  1. State the range of f. [1]
  2. Find the value of ff(-1). [3]
OCR C3 Q2
4 marks Moderate -0.3
Find the exact solutions of the equation \(|6x - 1| = |x - 1|\). [4]
OCR C3 Q3
6 marks Moderate -0.3
The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180e^{-0.017t}.$$
  1. Find the value of \(t\) for which the mass is 25 grams. [3]
  2. Find the rate at which the mass is decreasing when \(t = 55\). [3]
OCR C3 Q4
8 marks Standard +0.2
  1. \includegraphics{figure_4a} The diagram shows the curve \(y = \frac{2}{\sqrt{x}}\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1\), \(x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed. [4]
  2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int_1^5 \sqrt{(x^2 + 1)} \, dx,$$ giving your answer correct to 3 decimal places. [4]
OCR C3 Q5
8 marks Standard +0.3
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac{5}{2}\), giving all solutions for which \(0° < \theta < 360°\). [5]
OCR C3 Q6
7 marks Moderate -0.3
  1. Find the exact value of the \(x\)-coordinate of the stationary point of the curve \(y = x \ln x\). [4]
  2. The equation of a curve is \(y = \frac{4x + c}{4x - c}\), where \(c\) is a non-zero constant. Show by differentiation that this curve has no stationary points. [3]
OCR C3 Q7
9 marks Standard +0.3
  1. Write down the formula for \(\cos 2x\) in terms of \(\cos x\). [1]
  2. Prove the identity \(\frac{4 \cos 2x}{1 + \cos 2x} = 4 - 2 \sec^2 x\). [3]
  3. Solve, for \(0 < x < 2\pi\), the equation \(\frac{4 \cos 2x}{1 + \cos 2x} = 3 \tan x - 7\). [5]
OCR C3 Q8
16 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of each of the curves \(y = e^{\frac{1}{3}x}\) and \(y = \sqrt[3]{(3x + 8)}\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.
  1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3. [3]
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac{2}{3} \ln(3x + 8)\). [2]
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places. [3]
  4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). [5]
OCR C3 Q9
13 marks Challenging +1.2
\includegraphics{figure_9} The function f is defined by \(f(x) = \sqrt{(mx + 7)} - 4\), where \(x \geq -\frac{7}{m}\) and \(m\) is a positive constant. The diagram shows the curve \(y = f(x)\).
  1. A sequence of transformations maps the curve \(y = \sqrt{x}\) to the curve \(y = f(x)\). Give details of these transformations. [4]
  2. Explain how you can tell that f is a one-one function and find an expression for \(f^{-1}(x)\). [4]
  3. It is given that the curves \(y = f(x)\) and \(y = f^{-1}(x)\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5]
OCR C3 Q1
4 marks Moderate -0.5
Show that \(\int_2^8 \frac{3}{x} \, dx = \ln 64\). [4]
OCR C3 Q2
5 marks Standard +0.3
Solve, for \(0° < \theta < 360°\), the equation \(\sec^2 \theta = 4 \tan \theta - 2\). [5]
OCR C3 Q3
6 marks Moderate -0.3
  1. Differentiate \(x^2(x + 1)^6\) with respect to \(x\). [3]
  2. Find the gradient of the curve \(y = \frac{x^2 + 3}{x^2 - 3}\) at the point where \(x = 1\). [3]
OCR C3 Q4
5 marks Moderate -0.3
\includegraphics{figure_4} The function f is defined by \(f(x) = 2 - \sqrt{x}\) for \(x \geq 0\). The graph of \(y = f(x)\) is shown above.
  1. State the range of f. [1]
  2. Find the value of ff(4). [2]
  3. Given that the equation \(|f(x)| = k\) has two distinct roots, determine the possible values of the constant \(k\). [2]
OCR C3 Q5
8 marks Standard +0.8
\includegraphics{figure_5} The diagram shows the curves \(y = (1 - 2x)^5\) and \(y = e^{2x-1} - 1\). The curves meet at the point \((\frac{1}{2}, 0)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve. [8]
OCR C3 Q6
9 marks Moderate -0.3
  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 \(t = 20\). [3]
  2. The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80e^{-0.02t}.$$
    1. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures. [3]
    2. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures. [3]