Questions Further Pure Core (114 questions)

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OCR MEI Further Pure Core Specimen Q12
12 In this question you must show detailed reasoning.
  1. Given that \(y = \arctan x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\). Fig. 12 shows the curve \(y = \frac { 1 } { 1 + x ^ { 2 } }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-5_444_1435_1371_332} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure}
  2. Find, in exact form, the mean value of the function \(\mathrm { f } ( x ) = \frac { 1 } { 1 + x ^ { 2 } }\) for \(- 1 \leq x \leq 1\).
  3. The region bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = - 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, in exact form, the volume of the solid of revolution generated.
OCR MEI Further Pure Core Specimen Q13
13 Matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c c } k & 1 & - 5
2 & 3 & - 3
- 1 & 2 & 2 \end{array} \right)\), where \(k\) is a constant.
  1. Show that \(\operatorname { det } \mathbf { M } = 12 ( k - 3 )\).
  2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$\begin{aligned} 4 x ^ { 2 } + y ^ { 2 } - 5 z ^ { 2 } & = 6
    2 x ^ { 2 } + 3 y ^ { 2 } - 3 z ^ { 2 } & = 6
    - x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } & = - 6 \end{aligned}$$
  3. (A) Verify that the point ( \(2,0,1\) ) lies on each of the following three planes. $$\begin{aligned} 3 x + y - 5 z & = 1
    2 x + 3 y - 3 z & = 1
    - x + 2 y + 2 z & = 0 \end{aligned}$$ (B) Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer.
  4. Find the values of \(k\) for which the transformation represented by \(\mathbf { M }\) has a volume scale factor of 6 .
OCR MEI Further Pure Core Specimen Q16
16 A small object is attached to a spring and performs oscillations in a vertical line. The displacement of the object at time \(t\) seconds is denoted by \(x \mathrm {~cm}\). Preliminary observations suggest that the object performs simple harmonic motion (SHM) with a period of 2 seconds about the point at which \(x = 0\).
  1. (A) Write down a differential equation to model this motion.
    (B) Give the general solution of the differential equation in part (i) (A). Subsequent observations indicate that the object's motion would be better modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( k ^ { 2 } + 9 \right) x = 0$$ where \(k\) is a positive constant.
  2. (A) Obtain the general solution of (*).
    (B) State two ways in which the motion given by this model differs from that in part (i). The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.
  3. Find the value of \(k\). At the start of the object's motion, \(x = 0\) and the velocity is \(12 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction.
  4. Find an equation for \(x\) as a function of \(t\).
  5. Without doing any further calculations, explain why, according to this model, the greatest distance of the object from its starting point in the subsequent motion will be slightly less than 4 cm . \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series.
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OCR MEI Further Pure Core 2020 November Q10
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
OCR MEI Further Pure Core Specimen Q7
  1. Use the Maclaurin series for \(\ln ( 1 + x )\) up to the term in \(x ^ { 3 }\) to obtain an approximation to \(\ln 1.5\).
  2. (A) Find the error in the approximation in part (i).
    (B) Explain why the Maclaurin series in part (i), with \(x = 2\), should not be used to find an approximation to \(\ln 3\).
  3. Find a cubic approximation to \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
  4. (A) Use the approximation in part (iii) to find approximations to
    • ln 1.5 and
    • \(\quad \ln 3\).
      (B) Comment on your answers to part (iv) (A).
OCR MEI Further Pure Core Specimen Q14
  1. Starting with the result $$\mathrm { e } ^ { \mathrm { i } \theta } = \cos \theta + \mathrm { i } \sin \theta$$ show that
    (A) \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta\)
    (B) \(\cos \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right)\).
  2. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity
  3. Using the result in part (i) (B), obtain the values of the constants \(P , Q , R\) and \(S\) in the identity
  4. Show that \(\cos \frac { \pi } { 12 } = \left( \frac { 26 + 15 \sqrt { 3 } } { 64 } \right) ^ { \frac { 1 } { 6 } }\).
  5. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity $$\cos 6 \theta \equiv a \cos ^ { 6 } \theta + b \cos ^ { 4 } \theta + c \cos ^ { 2 } \theta + d$$ $$\cos ^ { 6 } \theta \equiv P \cos 6 \theta + Q \cos 4 \theta + R \cos 2 \theta + S$$
OCR MEI Further Pure Core 2019 June Q4
4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01a574f1-f6f6-40f5-baa5-535c36269731-2_501_670_1329_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
OCR MEI Further Pure Core 2019 June Q8
8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
  1. Evaluate, in exact form, the roots of the equation.
  2. Find \(k\).
OCR MEI Further Pure Core 2019 June Q10
10 In this question you must show detailed reasoning.
  1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
  2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
  3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).
OCR MEI Further Pure Core 2019 June Q15
15 In this question you must show detailed reasoning. Show that \(\int _ { \frac { 3 } { 4 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 4 x + 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { 3 + \sqrt { 5 } } { 2 } \right)\).
OCR MEI Further Pure Core 2023 June Q15
15 In this question you must show detailed reasoning. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 1 + 2 x - x ^ { 2 } } } d x\), giving your answer in terms of \(\pi\).
OCR MEI Further Pure Core 2024 June Q16
16 In this question you must show detailed reasoning. Show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } } \mathrm { dx } = \ln \left( \frac { \mathrm { a } + \mathrm { b } \sqrt { 3 } } { \mathrm { c } } \right)\), where \(a , b\) and \(c\) are integers to be determined.
OCR MEI Further Pure Core 2020 November Q11
11 In this question you must show detailed reasoning. In Fig. 11, the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }\) and FA are \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-5_807_872_443_239} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
OCR MEI Further Pure Core Specimen Q15
15 In this question you must show detailed reasoning. Show that $$\int _ { 0 } ^ { \frac { 2 } { 3 } } \operatorname { arsinh } 2 x \mathrm {~d} x = \frac { 2 } { 3 } \ln 3 - \frac { 1 } { 3 }$$