Questions C4 (1219 questions)

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Edexcel C4 Q6
11 marks Moderate -0.3
6. Liquid is poured into a container at a constant rate of \(30 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds liquid is leaking from the container at a rate of \(\frac { 2 } { 15 } V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\), where \(V \mathrm {~cm} ^ { 3 }\) is the volume of liquid in the container at that time.
  1. Show that $$- 15 \frac { \mathrm {~d} V } { \mathrm {~d} t } = 2 V - 450$$ Given that \(V = 1000\) when \(t = 0\),
  2. find the solution of the differential equation, in the form \(V = \mathrm { f } ( t )\).
  3. Find the limiting value of \(V\) as \(t \rightarrow \infty\).
Edexcel C4 Q7
11 marks Standard +0.3
7. The curve \(C\) has equation \(y = \frac { x } { 4 + x ^ { 2 } }\).
  1. Use calculus to find the coordinates of the turning points of \(C\). Using the result \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }\), or otherwise,
  2. determine the nature of each of the turning points.
  3. Sketch the curve \(C\).
Edexcel C4 Q8
13 marks Standard +0.3
8. (i) Given that \(\cos ( x + 30 ) ^ { \circ } = 3 \cos ( x - 30 ) ^ { \circ }\), prove that tan \(x ^ { \circ } = - \frac { \sqrt { 3 } } { 2 }\).
(ii) (a) Prove that \(\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta\).
(b) Verify that \(\theta = 180 ^ { \circ }\) is a solution of the equation \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
(c) Using the result in part (a), or otherwise, find the other two solutions, \(0 < \theta < 360 ^ { \circ }\), of the equation using \(\sin 2 \theta = 2 - 2 \cos 2 \theta\).
Edexcel C4 Q9
14 marks Standard +0.8
9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
  2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
  3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END
Edexcel C4 Q1
6 marks Moderate -0.8
  1. The following is a table of values for \(y = \sqrt { } ( 1 + \sin x )\), where \(x\) is in radians.
\(x\)00.511.52
\(y\)11.216\(p\)1.413\(q\)
  1. Find the value of \(p\) and the value of \(q\).
    (2)
  2. Use the trapezium rule and all the values of \(y\) in the completed table to obtain an estimate of \(I\), where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$ (4)
Edexcel C4 Q2
7 marks Moderate -0.3
2. (a) Use integration by parts to find $$\int x \cos 2 x d x$$ (b) Prove that the answer to part (a) may be expressed as $$\frac { 1 } { 2 } \sin x ( 2 x \cos x - \sin x ) + C ,$$ where \(C\) is an arbitrary constant.
Edexcel C4 Q3
8 marks Standard +0.3
3. (a) Expand \(( 1 + 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
(b) Hence, or otherwise, find the first three terms in the expansion of \(\frac { x + 4 } { ( 1 + 3 x ) ^ { 2 } }\) as a series in ascending powers of \(x\).
Edexcel C4 Q4
12 marks Standard +0.3
4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).
  1. Find the vector \(\overrightarrow { A B }\).
  2. Calculate the cosine of \(\angle O A B\).
  3. Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
  4. Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
  5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).
Edexcel C4 Q5
11 marks Challenging +1.2
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-3_668_1172_1231_354}
\end{figure} Figure 1 shows a graph of \(y = x \sqrt { } \sin x , 0 < x < \pi\). The maximum point on the curve is \(A\).
  1. Show that the \(x\)-coordinate of the point \(A\) satisfies the equation \(2 \tan x + x = 0\). The finite region enclosed by the curve and the \(x\)-axis is shaded as shown in Fig. 1.
    A solid body \(S\) is generated by rotating this region through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the exact value of the volume of \(S\).
    (7)
Edexcel C4 Q6
12 marks Moderate -0.8
6. A radioactive isotope decays in such a way that the rate of change of the number \(N\) of radioactive atoms present after \(t\) days, is proportional to \(N\).
  1. Write down a differential equation relating \(N\) and \(t\).
  2. Show that the general solution may be written as \(N = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants. Initially the number of radioactive atoms present is \(7 \times 10 ^ { 18 }\) and 8 days later the number present is \(3 \times 10 ^ { 17 }\).
  3. Find the value of \(k\).
  4. Find the number of radioactive atoms present after a further 8 days.
Edexcel C4 Q7
8 marks Standard +0.3
7. Given that $$\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) } \equiv \frac { A } { 1 - 2 x } + \frac { B } { 2 + x }$$
  1. find the values of the constants \(A\) and \(B\).
  2. Hence, or otherwise, find the series expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of \(\frac { 10 ( 2 - 3 x ) } { ( 1 - 2 x ) ( 2 + x ) }\), for \(| x | < \frac { 1 } { 2 }\).
Edexcel C4 Q8
15 marks Standard +0.8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-5_609_1210_248_374}
\end{figure} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta , \quad y = 4 \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where \(\theta = \alpha , \theta = - \alpha , \theta = \pi - \alpha , \theta = - \pi + \alpha\).
  1. Find an equation of the tangent to the ellipse at ( \(5 \cos \alpha , 4 \sin \alpha\) ), and show that it can be written in the form $$5 y \sin \alpha + 4 x \cos \alpha = 20 .$$
  2. Find by integration the area enclosed by the ellipse.
  3. Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac { 80 } { \sin 2 \alpha } - 20 \pi$$
  4. Given that \(0 < \alpha < \frac { \pi } { 4 }\), find the value of \(\alpha\) for which the areas of two types of wood are equal.
Edexcel C4 Q1
6 marks Standard +0.3
  1. A curve has the equation
$$x ^ { 2 } ( 2 + y ) - y ^ { 2 } = 0 .$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
Edexcel C4 Q2
8 marks Moderate -0.3
2. $$f ( x ) = \frac { 3 } { \sqrt { 1 - x } } , | x | < 1$$
  1. Show that \(\mathrm { f } \left( \frac { 1 } { 10 } \right) = \sqrt { 10 }\).
  2. Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  3. Use your expansion to find an approximate value for \(\sqrt { 10 }\), giving your answer to 8 significant figures.
  4. Find, to 1 significant figure, the percentage error in your answer to part (c).
Edexcel C4 Q3
11 marks Standard +0.3
3. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),
  1. find the values of \(p\) and \(q\),
  2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
  3. Find the coordinates of \(C\).
  4. Find the ratio \(A C : C B\) 3. continued
Edexcel C4 Q4
12 marks Challenging +1.2
4. During a chemical reaction, a compound is being made from two other substances. At time \(t\) hours after the start of the reaction, \(x \mathrm {~g}\) of the compound has been produced. Assuming that \(x = 0\) initially, and that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 2 ( x - 6 ) ( x - 3 )$$
  1. show that it takes approximately 7 minutes to produce 2 g of the compound.
  2. Explain why it is not possible to produce 3 g of the compound.
    4. continued
Edexcel C4 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-08_617_917_146_475} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 2\).
  1. Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region. The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Find, in terms of \(\pi\) and e, the exact volume of the solid formed.
    5. continued
Edexcel C4 Q6
12 marks Standard +0.3
6. (a) Find $$\int 2 \sin 3 x \sin 2 x d x$$ (b) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$ 6. continued
Edexcel C4 Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e877dc80-4cfc-4c8b-9640-9b186cd7ab13-12_556_860_246_452} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with parametric equations $$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).
  1. Find the value of the parameter \(t\) at \(P\).
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$ The shaded region is bounded by the curve, the coordinate axes and the line \(x = \frac { 1 } { 2 }\).
  3. Show that the area of the shaded region is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$ where \(k\) is a positive integer to be found.
  4. Hence find the exact area of the shaded region.
    7. continued
    7. continued
Edexcel C4 Q1
6 marks Standard +0.8
  1. Use integration by parts to find
$$\int x ^ { 2 } \sin x d x$$
Edexcel C4 Q2
7 marks Moderate -0.3
  1. Given that \(y = - 2\) when \(x = 1\), solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 Q3
8 marks Standard +0.3
3. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\). (8)
3. continued
Edexcel C4 Q4
9 marks Standard +0.3
4. (a) Expand \(( 1 + a x ) ^ { - 3 } , | a x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). Give each coefficient as simply as possible in terms of the constant \(a\). Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } } , | a x | < 1\), is 3 ,
(b) find the two possible values of \(a\). Given also that \(a < 0\),
(c) show that the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 6 - x } { ( 1 + a x ) ^ { 3 } }\) is \(\frac { 14 } { 9 }\).
4. continued
Edexcel C4 Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b71c9832-e502-4a25-85fb-a49c03ea9209-08_663_899_146_495} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 5\).
  1. Find the area of the shaded region. The shaded region is rotated completely about the \(x\)-axis.
  2. Find the volume of the solid formed, giving your answer in the form \(k \pi \ln 2\), where \(k\) is a simplified fraction.
    5. continued
Edexcel C4 Q6
11 marks Standard +0.3
6. $$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$
  1. Find the values of the constants \(A , B\) and \(C\) such that $$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
  2. Find the value of $$\int _ { - 1 } ^ { 0 } f ( x ) d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.
    6. continued