Questions C3 (1200 questions)

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OCR MEI C3 Q9
9 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = 2 x - 1$$ for all real values of \(x\).
  1. State the ranges of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\). Explain why \(\mathrm { f } ( x )\) has no inverse.
  2. Find an expression for the inverse function \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\). Sketch the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\) on the same axes.
  3. Find expressions for \(\operatorname { gf } ( x )\) and \(\operatorname { fg } ( x )\).
  4. Solve the equation \(\operatorname { gf } ( x ) = \mathrm { fg } ( x )\). Sketch the graphs of \(y = \operatorname { gf } ( x )\) and \(y = \operatorname { fg } ( x )\) on the same axes to illustrate your answer.
  5. Show that the equation \(\mathrm { f } ( x + a ) = \mathrm { g } ^ { 2 } ( x )\) has no solution if \(a > \frac { 1 } { 4 }\).
OCR C3 Q1
  1. Show that
$$\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x = \ln 3$$
OCR C3 Q2
  1. Find the set of values of \(x\) such that
$$| 3 x + 1 | \leq | x - 2 |$$
OCR C3 Q3
  1. Find all values of \(\theta\) in the interval \(- 180 < \theta < 180\) for which
$$\tan ^ { 2 } \theta ^ { \circ } + \sec \theta ^ { \circ } = 1$$
OCR C3 Q4
  1. Solve each equation, giving your answers in exact form.
    1. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
    2. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
    3. (i) Prove, by counter-example, that the statement
      "cosec \(\theta - \sin \theta > 0\) for all values of \(\theta\) in the interval \(0 < \theta < \pi\) " is false.
    4. Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that
    $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.
OCR C3 Q6
6. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).
  1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
  2. Find the \(x\)-coordinates of the stationary points of \(C\).
OCR C3 Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{6cdbc2bc-8863-4003-a218-44552d75d137-2_556_777_246_468} The diagram shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).
  1. Showing the coordinates of any stationary points, sketch the curve with equation \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
  2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ }$$ where \(R > 0\) and \(0 < \alpha < 90\),
  3. state the value of \(A\),
  4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
OCR C3 Q8
8. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$
  1. State the range of f .
  2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3$$
  4. Evaluate fg(-3).
  5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x )$$
OCR C3 Q9
  1. A curve has the equation \(y = ( 2 x + 3 ) \mathrm { e } ^ { - x }\).
    1. Find the exact coordinates of the stationary point of the curve.
    The curve crosses the \(y\)-axis at the point \(P\).
  2. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) meets the curve again at \(Q\).
  3. Show that the \(x\)-coordinate of \(Q\) lies between - 2 and - 1 .
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 3 - 3 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { x _ { n } } - 2 }$$ with \(x _ { 0 } = - 1\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 2 decimal places.
  5. Show that your value for \(x _ { 4 }\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places.
OCR C3 Q1
  1. (i) Differentiate \(x ^ { 3 } \ln x\) with respect to \(x\).
    (ii) Given that
$$x = \frac { y + 1 } { 3 - 2 y }$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).
OCR C3 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-1_638_677_749_443} The diagram shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.
  1. Write down the coordinates of \(A\) and \(B\). The two curves intersect at the point \(C\).
  2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).
OCR C3 Q3
3. The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } ,
& \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } . \end{aligned}$$
  1. Evaluate \(\mathrm { gf } ( 1 )\).
  2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
  3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$
OCR C3 Q4
  1. (i) Use the identity for \(\cos ( A + B )\) to prove that
$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for \(\cos x \neq 0\), $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\) for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$
OCR C3 Q5
  1. (i) Show that the equation
$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for \(x\) in the interval \(0 \leq x \leq \pi\).
OCR C3 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
  1. Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
  2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).
OCR C3 Q7
7. (i) Sketch on the same diagram the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$
OCR C3 Q8
  1. A curve has the equation \(y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , x \neq 0\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      [0pt]
    2. Show that the curve has a stationary point in the interval [1.3,1.4].
    The point \(A\) on the curve has \(x\)-coordinate 2 .
  2. Show that the tangent to the curve at \(A\) passes through the origin. The tangent to the curve at \(A\) intersects the curve again at the point \(B\).
    The \(x\)-coordinate of \(B\) is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with \(x _ { 0 } = - 1\).
  3. Find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 7 significant figures and hence state the \(x\)-coordinate of \(B\) to 5 significant figures.
OCR MEI C3 Q1
1 Find \(\int \sqrt [ 3 ] { 2 x - 1 } \mathrm {~d} x\).
OCR MEI C3 Q2
2 Fig. 8 shows the line \(y = 1\) and the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x }\). The curve touches the \(x\)-axis at \(\mathrm { P } ( 2,0 )\) and has another turning point at the point Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ea594c5-52ba-4467-a098-cb66004b5a38-1_959_1469_748_317} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 4 } { x ^ { 2 } }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\). Hence find the coordinates of Q and, using \(\mathrm { f } ^ { \prime \prime } ( x )\), verify that it is a maximum point.
  2. Verify that the line \(y = 1\) meets the curve \(y = \mathrm { f } ( x )\) at the points with \(x\)-coordinates 1 and 4 . Hence find the exact area of the shaded region enclosed by the line and the curve. The curve \(y = \mathrm { f } ( x )\) is now transformed by a translation with vector \(\binom { - 1 } { - 1 }\). The resulting curve has equation \(y = \mathrm { g } ( x )\).
  3. Show that \(\mathrm { g } ( x ) = \frac { x ^ { 2 } - 3 x } { x + 1 }\).
  4. Without further calculation, write down the value of \(\int _ { 0 } ^ { 3 } \mathrm {~g} ( x ) \mathrm { d } x\), justifying your answer.
OCR MEI C3 Q3
3 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( 1 - \sin 3 x ) \mathrm { d } x\), giving your answer in exact form.
OCR MEI C3 Q4
4 Fig. 9 shows the curve \(y = x \mathrm { e } ^ { - 2 x }\) together with the straight line \(y = m x\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P . The dashed line is the tangent at P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ea594c5-52ba-4467-a098-cb66004b5a38-2_431_977_728_602} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Show that the \(x\)-coordinate of P is \(- \frac { 1 } { 2 } \ln m\).
  2. Find, in terms of \(m\), the gradient of the tangent to the curve at P . You are given that OP and this tangent are equally inclined to the \(x\)-axis.
  3. Show that \(m = \mathrm { e } ^ { - 2 }\), and find the exact coordinates of P .
  4. Find the exact area of the shaded region between the line OP and the curve.
OCR MEI C3 Q5
5 Using a suitable substitution or otherwise, show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 x } { 3 + \cos 2 x } \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
OCR MEI C3 Q1
1 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }\), with its turning point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75eebbfb-7bfa-4382-a6d7-1c5a7f3f419a-1_722_817_450_642} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the coordinates of the intercepts of \(y = \mathrm { f } ( x )\) with the \(x\) - and \(y\)-axes.
  2. Find the exact coordinates of the turning point P .
  3. Show that the exact area of the region enclosed by the curve and the \(x\) - and \(y\)-axes is \(\frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 } - 3 \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
  4. Express \(\mathrm { g } ( x )\) in terms of \(x\). Sketch the curve \(y = \mathrm { g } ( x )\) on the copy of Fig. 8, indicating the coordinates of its intercepts with the \(x\) - and \(y\)-axes and of its turning point.
  5. Write down the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\) and the \(x\) - and \(y\)-axes.
OCR MEI C3 Q2
2 Fig. 9 shows the curve with equation \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\). It has an asymptote \(x = a\) and turning point P . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75eebbfb-7bfa-4382-a6d7-1c5a7f3f419a-2_754_870_478_609} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the value of \(a\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 3 x ^ { 2 } } { 3 y ^ { 2 } ( 2 x - 1 ) ^ { 2 } }\). Hence find the coordinates of the turning point P , giving the \(y\)-coordinate to 3 significant figures.
  3. Show that the substitution \(u = 2 x - 1\) transforms \(\int \frac { x } { \sqrt [ 3 ] { 2 x - 1 } } \mathrm {~d} x\) to \(\frac { 1 } { 4 } \int \left( u ^ { \frac { 2 } { 3 } } + u ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve \(y ^ { 3 } = \frac { x ^ { 3 } } { 2 x - 1 }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4.5\).
OCR MEI C3 Q3
3 Fig. 9 shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The function \(y = \mathrm { f } ( x )\) is given by $$f ( x ) = \ln \left( \frac { 2 x } { 1 + x } \right) , x > 0$$ The curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at P , and the line \(x = 2\) at Q . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75eebbfb-7bfa-4382-a6d7-1c5a7f3f419a-3_559_644_622_745} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Verify that the \(x\)-coordinate of P is 1 . Find the exact \(y\)-coordinate of Q .
  2. Find the gradient of the curve at P. [Hint: use \(\ln \frac { a } { b } = \ln a - \ln b\).] The function \(\mathrm { g } ( x )\) is given by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } } { 2 - \mathrm { e } ^ { x } } , \quad x < \ln 2 .$$ The curve \(y = \mathrm { g } ( x )\) crosses the \(y\)-axis at the point R .
  3. Show that \(\mathrm { g } ( x )\) is the inverse function of \(\mathrm { f } ( x )\). Write down the gradient of \(y = \mathrm { g } ( x )\) at R .
  4. Show, using the substitution \(u = 2 - \mathrm { e } ^ { x }\) or otherwise, that \(\int _ { 0 } ^ { \ln \frac { 4 } { 3 } } \mathrm {~g} ( x ) \mathrm { d } x = \ln \frac { 3 } { 2 }\). Using this result, show that the exact area of the shaded region shown in Fig. 9 is \(\ln \frac { 32 } { 27 }\). [Hint: consider its reflection in \(y = x\).]