Questions C4 (1219 questions)

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Edexcel C4 2018 June Q7
15 marks Standard +0.8
7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),
  1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
  2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
  3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
  4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
  5. find the coordinates of the point \(Q\).
Edexcel C4 2018 June Q8
9 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-28_680_1266_118_482} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Find \(\int x \cos 4 x d x\) Figure 3 shows part of the curve with equation \(y = \sqrt { x } \sin 2 x , \quad x \geqslant 0\) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 4 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact value of the volume of this solid of revolution, giving your answer in its simplest form.
    (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-32_2630_1828_121_121}
Edexcel C4 Specimen Q1
5 marks Moderate -0.3
Use the binomial theorem to expand \(( 4 - 3 x ) ^ { - \frac { 1 } { 2 } }\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
Edexcel C4 Specimen Q3
8 marks Standard +0.3
3. Use the substitution \(x = \tan \theta\) to show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \frac { \pi } { 8 } + \frac { 1 } { 4 }$$ (8)
Edexcel C4 Specimen Q4
9 marks Moderate -0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-3_458_1552_415_223}
\end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$
  1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
  2. Find an equation of the normal to the curve at \(P\).
  3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).
Edexcel C4 Specimen Q5
11 marks Standard +0.3
5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \\ & \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find
  1. the coordinates of the point of intersection,
  2. the value of the constant \(a\),
  3. the acute angle between the two lines.
Edexcel C4 Specimen Q6
11 marks Standard +0.3
6. Given that $$\frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \equiv \frac { A } { ( 1 - x ) ^ { 2 } } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 2 + 3 x ) }$$
  1. find the values of \(A , B\) and \(C\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \mathrm { d } x\), giving your answer in the form \(k + \ln a\), where \(k\) is an integer and \(a\) is a simplified fraction.
Edexcel C4 Specimen Q7
12 marks Challenging +1.2
7. (a) Given that \(u = \frac { x } { 2 } - \frac { 1 } { 8 } \sin 4 x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sin ^ { 2 } 2 x\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-5_697_1239_587_367}
\end{figure} Figure 2 shows the finite region bounded by the curve \(y = x ^ { \frac { 1 } { 2 } } \sin 2 x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis. This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Using the result in part (a), or otherwise, find the exact value of the volume generated.
(8)
Edexcel C4 Specimen Q8
13 marks Standard +0.3
8. A circular stain grows in such a way that the rate of increase of its radius is inversely proportional to the square of the radius. Given that the area of the stain at time \(t\) seconds is \(A \mathrm {~cm} ^ { 2 }\),
  1. show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } \propto \frac { 1 } { \sqrt { A } }\).
    (6) Another stain, which is growing more quickly, has area \(S \mathrm {~cm} ^ { 2 }\) at time \(t\) seconds. It is given that $$\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 2 \mathrm { e } ^ { 2 t } } { \sqrt { S } }$$ Given that, for this second stain, \(S = 9\) at time \(t = 0\),
  2. solve the differential equation to find the time at which \(S = 16\). Give your answer to 2 significant figures. \section*{END}
Edexcel C4 2009 January Q2
9 marks Moderate -0.3
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-03_410_552_205_694} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve \(y = \frac { 3 } { \sqrt { } ( 1 + 4 x ) }\). The region \(R\) is bounded by the curve, the \(x\)-axis, and the lines \(x = 0\) and \(x = 2\), as shown shaded in Figure 1.
  1. Use integration to find the area of \(R\). The region \(R\) is rotated \(360 ^ { \circ }\) about the \(x\)-axis.
  2. Use integration to find the exact value of the volume of the solid formed.
Edexcel C4 2013 June Q2
9 marks Moderate -0.3
  1. Use the binomial expansion to show that $$\left. \sqrt { ( } \frac { 1 + x } { 1 - x } \right) \approx 1 + x + \frac { 1 } { 2 } x ^ { 2 } , \quad | x | < 1$$
  2. Substitute \(x = \frac { 1 } { 26 }\) into $$\sqrt { \left( \frac { 1 + x } { 1 - x } \right) = 1 + x + \frac { 1 } { 2 } x ^ { 2 } }$$ to obtain an approximation to \(\sqrt { } 3\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
Edexcel C4 Specimen Q2
6 marks Standard +0.3
The curve \(C\) has equation $$13 x ^ { 2 } + 13 y ^ { 2 } - 10 x y = 52$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) and \(y\), simplifying your answer.
(6)
OCR C4 2006 January Q1
3 marks Easy -1.2
1 Simplify \(\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }\).
OCR C4 2006 January Q2
5 marks Standard +0.3
2 Given that \(\sin y = x y + x ^ { 2 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 2006 January Q3
6 marks Moderate -0.3
3
  1. Find the quotient and the remainder when \(3 x ^ { 3 } - 2 x ^ { 2 } + x + 7\) is divided by \(x ^ { 2 } - 2 x + 5\).
  2. Hence, or otherwise, determine the values of the constants \(a\) and \(b\) such that, when \(3 x ^ { 3 } - 2 x ^ { 2 } + a x + b\) is divided by \(x ^ { 2 } - 2 x + 5\), there is no remainder.
OCR C4 2006 January Q4
7 marks Standard +0.3
4
  1. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
  2. Hence find \(\int x \tan ^ { 2 } x \mathrm {~d} x\).
OCR C4 2006 January Q5
8 marks Moderate -0.3
5 A curve is given parametrically by the equations \(x = t ^ { 2 } , y = 2 t\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Show that the equation of the tangent to the curve at \(\left( p ^ { 2 } , 2 p \right)\) is $$p y = x + p ^ { 2 } .$$
  3. Find the coordinates of the point where the tangent at \(( 9,6 )\) meets the tangent at \(( 25 , - 10 )\).
OCR C4 2006 January Q6
9 marks Standard +0.8
6
  1. Show that the substitution \(x = \sin ^ { 2 } \theta\) transforms \(\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\) to \(\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta\).
  2. Hence find \(\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\).
OCR C4 2006 January Q7
10 marks Standard +0.3
7 The expression \(\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }\) is denoted by \(\mathrm { f } ( x )\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }\), where \(A , B\) and \(C\) are constants.
  2. Given that \(| x | < 1\), find the first 3 terms in the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\).
OCR C4 2006 January Q8
11 marks Moderate -0.3
8
  1. Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - x } { y - 3 }$$ giving the particular solution that satisfies the condition \(y = 4\) when \(x = 5\).
  2. Show that this particular solution can be expressed in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where the values of the constants \(a , b\) and \(k\) are to be stated.
  3. Hence sketch the graph of the particular solution, indicating clearly its main features.
OCR C4 2006 January Q9
13 marks Standard +0.3
9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4 \\ 2 \\ - 6 \end{array} \right) + t \left( \begin{array} { r } - 8 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2 \\ a \\ - 2 \end{array} \right) + s \left( \begin{array} { r } - 9 \\ 2 \\ - 5 \end{array} \right) ,$$ where \(a\) is a constant.
  1. Calculate the acute angle between the lines.
  2. Given that these two lines intersect, find \(a\) and the point of intersection.
OCR C4 2008 January Q1
4 marks Moderate -0.5
1 Find the angle between the vectors \(\mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) and \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
OCR C4 2008 January Q2
5 marks Moderate -0.8
2
  1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
  2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).
OCR C4 2008 January Q3
5 marks Standard +0.3
3 When \(x ^ { 4 } - 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x + a\) is divided by \(x ^ { 2 } + 2 x - 1\), the quotient is \(x ^ { 2 } + b x + 2\) and the remainder is \(c x + 7\). Find the values of the constants \(a , b\) and \(c\).
OCR C4 2008 January Q4
6 marks Standard +0.3
4 Find the equation of the normal to the curve $$x ^ { 3 } + 4 x ^ { 2 } y + y ^ { 3 } = 6$$ at the point \(( 1,1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.