Questions C4 (1219 questions)

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OCR MEI C4 2008 June Q6
8 marks Standard +0.3
6
  1. Find the first three non-zero terms of the binomial series expansion of \(\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }\), and state the set of values of \(x\) for which the expansion is valid.
  2. Hence find the first three non-zero terms of the series expansion of \(\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }\).
OCR MEI C4 2008 June Q7
6 marks Standard +0.3
7 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).
OCR MEI C4 2008 June Q8
18 marks Standard +0.3
8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-03_1004_1397_493_374} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)
  1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
  2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
  4. Write down a vector equation of the line RS. Calculate the coordinates of S.
OCR MEI C4 2008 June Q9
18 marks Standard +0.3
9 A skydiver drops from a helicopter. Before she opens her parachute, her speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after time \(t\) seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When \(t = 0 , v = 0\).
  1. Find \(v\) in terms of \(t\).
  2. According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Her speed \(t\) seconds after this is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
  3. Express \(\frac { 1 } { ( w - 4 ) ( w + 5 ) }\) in partial fractions.
  4. Using this result, show that \(\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }\).
  5. According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
    WEDNESDAY 21 MAY 2008
    Afternoon
    Time: Up to 1 hour
    Additional materials: Rough paper
    MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
    \includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
    This document consists of \(\mathbf { 6 }\) printed pages, \(\mathbf { 2 }\) blank pages and an insert. 1 Complete these Latin square puzzles.
  6. 213
    3
  7. \includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square
    1234
    3142
    2413
    4321
    could not be the solution to a Sudoku puzzle.
    Explain this briefly.
    3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.
    1234
    4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.
    4231
    24
    42
    2413
    5 The number of \(9 \times 9\) Sudokus is given in line 121 .
    Without doing any calculations, explain why you would expect 9! to be a factor of this number.
    6 In the table below, \(M\) represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). \(s\) is the side length of the Sudoku grid and \(b\) is the side length of its blocks.
    Block side
    length, \(b\)
    Sudoku,
    \(s \times s\)
    		\(M\)
    1\(1 \times 1\)-
    2\(4 \times 4\)12
    3\(9 \times 9\)
    4\(16 \times 16\)
    5
  8. Complete the table.
  9. Give a formula for \(M\) in terms of \(b\).
    7 A man is setting a Sudoku puzzle and starts with this solution.
    123456789
    456897312
    789312564
    231564897
    564978123
    897123645
    312645978
    645789231
    978231456
    He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
    [0pt] [2,2]
  10. 12469
    4891
    86
    2147
    647812
    8924
    16497
    64791
    982146
  11. 123456789
    456897312
    789564
    231564897
    564978123
    897645
    312645978
    645789231
    978456
  12. \(\_\_\_\_\)
  13. \(\_\_\_\_\)
Edexcel C4 2014 June Q1
7 marks Standard +0.3
A curve \(C\) has the equation $$x^3 + 2xy - x - y^3 - 20 = 0$$
  1. [(a)] Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). \hfill [5]
  2. [(b)] Find an equation of the tangent to \(C\) at the point \((3, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. \hfill [2]
Edexcel C4 2014 June Q2
5 marks Moderate -0.3
Given that the binomial expansion of \((1 + kx)^{-4}\), \(|kx| < 1\), is $$1 - 6x + Ax^2 + \ldots$$
  1. [(a)] find the value of the constant \(k\), \hfill [2]
  2. [(b)] find the value of the constant \(A\), giving your answer in its simplest form. \hfill [3]
Edexcel C4 2014 June Q3
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = \frac{10}{2x + 5\sqrt{x}}\), \(x > 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the lines with equations \(x = 1\) and \(x = 4\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac{10}{2x + 5\sqrt{x}}\)
\(x\)1234
\(y\)1.428570.903260.55556
  1. [(a)] Complete the table above by giving the missing value of \(y\) to 5 decimal places. \hfill [1]
  2. [(b)] Use the trapezium rule, with all the values of \(y\) in the completed table, to find an estimate for the area of \(R\), giving your answer to 4 decimal places. \hfill [3]
  3. [(c)] By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of \(R\). \hfill [1]
  4. [(d)] Use the substitution \(u = \sqrt{x}\), or otherwise, to find the exact value of $$\int_1^4 \frac{10}{2x + 5\sqrt{x}} dx$$ \hfill [6]
Edexcel C4 2014 June Q4
5 marks Moderate -0.3
\includegraphics{figure_2} A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is \(h\) cm, the volume of water \(V\) cm\(^3\) is given by $$V = 4\pi h(h + 4), \quad 0 \leq h \leq 25$$ Water flows into the vase at a constant rate of \(80\pi\) cm\(^3\)s\(^{-1}\) Find the rate of change of the depth of the water, in cm s\(^{-1}\), when \(h = 6\) \hfill [5]
Edexcel C4 2014 June Q5
5 marks Moderate -0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 4\cos\left(t + \frac{\pi}{6}\right), \quad y = 2\sin t, \quad 0 \leq t < 2\pi$$
  1. [(a)] Show that $$x + y = 2\sqrt{3}\cos t$$ \hfill [3]
  2. [(b)] Show that a cartesian equation of \(C\) is $$(x + y)^2 + ay^2 = b$$ where \(a\) and \(b\) are integers to be determined. \hfill [2]
Edexcel C4 2014 June Q6
12 marks Standard +0.3
  1. [(i)] Find $$\int xe^{4x} dx$$ \hfill [3]
  2. [(ii)] Find $$\int \frac{8}{(2x - 1)^3} dx, \quad x > \frac{1}{2}$$ \hfill [2]
  3. [(iii)] Given that \(y = \frac{\pi}{6}\) at \(x = 0\), solve the differential equation $$\frac{dy}{dx} = e^x \cosec 2y \cosec y$$ \hfill [7] \end{enumerate}
Edexcel C4 2014 June Q7
15 marks Challenging +1.2
\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point \(P\) lies on \(C\) and has coordinates \((3, 2)\). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
  1. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. [(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p\pi + q\pi^2\), where \(p\) and \(q\) are rational numbers to be determined. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}
Edexcel C4 2014 June Q8
15 marks Standard +0.3
Relative to a fixed origin \(O\), the point \(A\) has position vector \(\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}\) and the point \(B\) has position vector \(\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}\) The line \(l_1\) passes through the points \(A\) and \(B\).
  1. [(a)] Find the vector \(\overrightarrow{AB}\). \hfill [2]
  2. [(b)] Hence find a vector equation for the line \(l_1\) \hfill [1]
The point \(P\) has position vector \(\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}\) Given that angle \(PBA\) is \(\theta\),
  1. [(c)] show that \(\cos\theta = \frac{1}{3}\) \hfill [3]
The line \(l_2\) passes through the point \(P\) and is parallel to the line \(l_1\)
  1. [(d)] Find a vector equation for the line \(l_2\) \hfill [2]
The points \(C\) and \(D\) both lie on the line \(l_2\) Given that \(AB = PC = DP\) and the \(x\) coordinate of \(C\) is positive,
  1. [(e)] find the coordinates of \(C\) and the coordinates of \(D\). \hfill [3]
  2. [(f)] find the exact area of the trapezium \(ABCD\), giving your answer as a simplified surd. \hfill [4] \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate}
Edexcel C4 Q1
5 marks Moderate -0.3
Use the binomial theorem to expand $$\sqrt{(4-9x)}, \quad |x| < \frac{4}{9},$$ in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying each term. [5]
Edexcel C4 Q2
7 marks Standard +0.3
A curve has equation $$x^2 + 2xy - 3y^2 + 16 = 0.$$ Find the coordinates of the points on the curve where \(\frac{dy}{dx} = 0\). [7]
Edexcel C4 Q3
8 marks Moderate -0.3
  1. Express \(\frac{5x + 3}{(2x - 3)(x + 2)}\) in partial fractions. [3]
  2. Hence find the exact value of \(\int_0^1 \frac{5x + 3}{(2x - 3)(x + 2)} dx\), giving your answer as a single logarithm. [5]
Edexcel C4 Q4
7 marks Challenging +1.2
Use the substitution \(x = \sin \theta\) to find the exact value of $$\int_0^1 \frac{1}{(1-x^2)^{3/2}} dx.$$ [7]
Edexcel C4 Q5
10 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the graph of the curve with equation $$y = xe^x, \quad x \geq 0.$$ The finite region \(R\) bounded by the lines \(x = 1\), the \(x\)-axis and the curve is shown shaded in Figure 1.
  1. Use integration to find the exact value of the area for \(R\). [5]
  2. Complete the table with the values of \(y\) corresponding to \(x = 0.4\) and \(0.8\).
    \(x\)00.20.40.60.8
    \(y = xe^x\)00.298361.99207
    [1]
  3. Use the trapezium rule with all the values in the table to find an approximate value for this area, giving your answer to 4 significant figures. [4]
Edexcel C4 Q6
12 marks Standard +0.3
A curve has parametric equations $$x = 2\cot t, \quad y = 2\sin^2 t, \quad 0 < t \leq \frac{\pi}{2}.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of the parameter \(t\). [4]
  2. Find an equation of the tangent to the curve at the point where \(t = \frac{\pi}{4}\). [4]
  3. Find a cartesian equation of the curve in the form \(y = f(x)\). State the domain on which the curve is defined. [4]
Edexcel C4 Q7
13 marks Standard +0.3
The line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$$ and the line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},$$ where \(\lambda\) and \(\mu\) are parameters. The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).
  1. Find the coordinates of \(B\). [4]
  2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. [4]
The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\). The point \(D\) is such that \(ABCD\) is a parallelogram.
  1. Show that \(|\overrightarrow{AB}| = |\overrightarrow{BC}|\). [3]
  2. Find the position vector of the point \(D\). [2]
Edexcel C4 Q8
13 marks Standard +0.3
Liquid is pouring into a container at a constant rate of \(20\text{ cm}^3\text{s}^{-1}\) and is leaking out at a rate proportional to the volume of the liquid already in the container.
  1. Explain why, at time \(t\) seconds, the volume, \(V\text{ cm}^3\), of liquid in the container satisfies the differential equation $$\frac{dV}{dt} = 20 - kV,$$ where \(k\) is a positive constant. [2]
The container is initially empty.
  1. By solving the differential equation, show that $$V = A + Be^{-kt},$$ giving the values of \(A\) and \(B\) in terms of \(k\). [6]
Given also that \(\frac{dV}{dt} = 10\) when \(t = 5\),
  1. find the volume of liquid in the container at 10 s after the start. [5]
Edexcel C4 2013 June Q1
8 marks Moderate -0.3
  1. Find the binomial expansion of $$\sqrt{(9 + 8x)}, \quad |x| < \frac{9}{8}$$ in ascending powers of \(x\), up to and including the term in \(x^2\). Give each coefficient as a simplified fraction. [5]
  2. Use your expansion to estimate the value of \(\sqrt{11}\), giving your answer as a single fraction. [3]
Edexcel C4 2013 June Q2
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation \(y = xe^{-\frac{1}{2}x}\), \(x > 0\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, and the line \(x = 4\). The table shows corresponding values of \(x\) and \(y\) for \(y = xe^{-\frac{1}{2}x}\).
\(x\)01234
\(y\)0\(e^{-\frac{1}{2}}\)\(3e^{-\frac{3}{2}}\)\(4e^{-2}\)
  1. Complete the table with the value of \(y\) corresponding to \(x = 2\) [1]
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places. [4]
    1. Find \(\int xe^{-\frac{1}{2}x} \, dx\).
    2. Hence find the exact area of \(R\), giving your answer in the form \(a + be^{-2}\), where \(a\) and \(b\) are integers. [6]
Edexcel C4 2013 June Q3
7 marks Moderate -0.3
A curve \(C\) has parametric equations $$x = 2t + 5, \quad y = 3 + \frac{4}{t}, \quad t \neq 0$$
  1. Find the value of \(\frac{dy}{dx}\) at the point on \(C\) with coordinates \((9, 5)\). [4]
  2. Find a cartesian equation of the curve in the form $$y = \frac{ax + b}{cx + d}$$ where \(a\), \(b\), \(c\) and \(d\) are integers. [3]
Edexcel C4 2013 June Q4
10 marks Moderate -0.3
With respect to a fixed origin \(O\), the line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$ where \(\mu\) is a scalar parameter. The point \(A\) is on \(l_1\) where \(\mu = 2\).
  1. Write down the coordinates of \(A\). [1] The acute angle between \(OA\) and \(l_1\) is \(\theta\), where \(O\) is the origin.
  2. Find the value of \(\cos \theta\). [3] The point \(B\) is such that \(\overrightarrow{OB} = 3\overrightarrow{OA}\). The line \(l_2\) passes through the point \(B\) and is parallel to the line \(l_1\).
  3. Find a vector equation of \(l_2\). [2]
  4. Find the length of \(OB\), giving your answer as a simplified surd. [1] The point \(X\) lies on \(l_2\). Given that the vector \(\overrightarrow{OX}\) is perpendicular to \(l_2\),
  5. find the length of \(OX\), giving your answer to 3 significant figures. [3]
Edexcel C4 2013 June Q5
9 marks Standard +0.3
The curve \(C\) has the equation $$\sin(\pi y) - y - x^2 y = -5, \quad x > 0$$
  1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5] The point \(P\) with coordinates \((2, 1)\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
  2. Find the exact value of the \(x\)-coordinate of \(A\). [4]