Questions C34 (197 questions)

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Edexcel C34 2017 January Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-24_515_750_264_598} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The curve \(C\) shown in Figure 4 has parametric equations $$x = 1 + \sqrt { 3 } \tan \theta , \quad y = 5 \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(y\)-axis at \(A\) and has a minimum turning point at \(B\), as shown in Figure 4.
  1. Find the exact coordinates of \(A\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \sin \theta\), giving the exact value of the constant \(\lambda\).
  3. Find the coordinates of \(B\).
  4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y = k \sqrt { \left( x ^ { 2 } - 2 x + 4 \right) }$$ where \(k\) is a simplified surd to be found.
Edexcel C34 2017 January Q14
  1. \(A B C D\) is a parallelogram with \(A B\) parallel to \(D C\) and \(A D\) parallel to \(B C\). The position vectors of \(A , B , C\), and \(D\) relative to a fixed origin \(O\) are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) respectively.
Given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + 6 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$$
  1. find the position vector \(\mathbf { d }\),
  2. find the angle between the sides \(A B\) and \(B C\) of the parallelogram,
  3. find the area of the parallelogram \(A B C D\). The point \(E\) lies on the line through the points \(C\) and \(D\), so that \(D\) is the midpoint of \(C E\).
  4. Use your answer to part (c) to find the area of the trapezium \(A B C E\).
Edexcel C34 2018 January Q1
  1. A curve \(C\) has equation
$$3 ^ { x } + x y = x + y ^ { 2 } , \quad y > 1$$ The point \(P\) with coordinates \(( 4,11 )\) lies on \(C\).
Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer in the form \(a + b \ln 3\), where \(a\) and \(b\) are rational numbers.
Edexcel C34 2018 January Q2
2. $$f ( x ) = ( 125 - 5 x ) ^ { \frac { 2 } { 3 } } \quad | x | < 25$$
  1. Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) as simplified fractions.
  2. Use your expansion to find an approximate value for \(120 ^ { \frac { 2 } { 3 } }\), stating the value of \(x\) which you have used and showing your working. Give your answer to 5 decimal places.
Edexcel C34 2018 January Q3
3. $$\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 4 } + \ln ( 2 x ) , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 4 } x ^ { 2 } }$$ The equation \(\mathrm { f } ( x ) = 0\) has a root near 0.5
  2. Starting with \(x _ { 1 } = 0.5\) use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 4 } x _ { n } ^ { 2 } }$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  3. Using a suitable interval, show that 0.473 is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
Edexcel C34 2018 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-08_771_1189_212_379} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\)
The graph consists of two half lines that meet at the point \(P ( 2 , - 3 )\), the vertex of the graph.
The graph cuts the \(y\)-axis at the point \(( 0 , - 1 )\) and the \(x\)-axis at the points \(( - 1,0 )\) and \(( 5,0 )\).
Sketch, on separate diagrams, the graph of
  1. \(y = \mathrm { f } ( | x | )\),
  2. \(y = 2 \mathrm { f } ( x + 5 )\). In each case, give the coordinates of the points where the graph crosses or meets the coordinate axes. Also give the coordinates of any vertices corresponding to the point \(P\).
Edexcel C34 2018 January Q5
  1. (a) Express \(\frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } }\) in partial fractions.
Given that $$\mathrm { f } ( x ) = \frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } } , \quad x \in \mathbb { R } , \quad - \frac { 4 } { 3 } < x < \frac { 4 } { 3 }$$ (b) express \(\int \mathrm { f } ( x ) \mathrm { d } x\) in the form \(\ln ( \mathrm { g } ( x ) )\), where \(\mathrm { g } ( x )\) is a rational function.
Edexcel C34 2018 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-14_768_712_212_616} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve shown in Figure 2 has equation $$y ^ { 2 } = 3 \tan \left( \frac { x } { 2 } \right) , \quad 0 < x < \pi , \quad y > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac { \pi } { 3 }\) the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\)
The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to generate a solid of revolution.
Show that the exact value of the volume of the solid generated may be written as \(A \ln \left( \frac { 3 } { 2 } \right)\), where \(A\) is a constant to be found.
Edexcel C34 2018 January Q7
7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 13 \mathbf { i } + 15 \mathbf { j } - 8 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 7 \mathbf { i } - 6 \mathbf { j } + 14 \mathbf { k } ) + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection, \(B\).
  2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(- 5 \mathbf { i } - 3 \mathbf { j } + 16 \mathbf { k }\)
  3. Show that \(A\) lies on \(l _ { 1 }\) The point \(C\) lies on the line \(l _ { 1 }\) where \(\overrightarrow { A B } = \overrightarrow { B C }\)
  4. Find the position vector of \(C\).
    \section*{"}
Edexcel C34 2018 January Q8
  1. Given that
$$y = 8 \tan ( 2 x ) , \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \frac { A } { B + y ^ { 2 } }$$ where \(A\) and \(B\) are integers to be found.
Edexcel C34 2018 January Q9
  1. (a) Show that
$$\frac { \cot ^ { 2 } x } { 1 + \cot ^ { 2 } x } \equiv \cos ^ { 2 } x$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\frac { \cot ^ { 2 } x } { 1 + \cot ^ { 2 } x } = 8 \cos 2 x + 2 \cos x$$ Give each solution in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2018 January Q10
  1. It is given that
$$\begin{gathered} \mathrm { f } ( x ) = \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R }
\mathrm {~g} ( x ) = \frac { x } { x - 3 } \quad x > 3 \end{gathered}$$
  1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any points where the graph crosses the axes.
  2. Find the range of g
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 }\)
  4. Using algebra, find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = 3\)
Edexcel C34 2018 January Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-32_858_743_118_603} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The curve \(C\) shown in Figure 3 has parametric equations $$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leqslant t \leqslant 2 \pi$$ The curve \(C\) meets the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 3 .
  1. Write down the coordinates of \(A\) and \(B\).
  2. Find the values of \(t\) at which the curve passes through the origin.
  3. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), and hence find the gradient of the curve when \(t = \frac { \pi } { 6 }\)
  4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y ^ { 2 } = a x ^ { 2 } \left( b - x ^ { 2 } \right)$$ where \(a\) and \(b\) are integers to be determined.
Edexcel C34 2018 January Q12
  1. (a) Express \(2 \sin x - 4 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 significant figures. In a town in Norway, a student records the number of hours of daylight every day for a year. He models the number of hours of daylight, \(H\), by the continuous function given by the formula $$H = 12 + 4 \sin \left( \frac { 2 \pi t } { 365 } \right) - 8 \cos \left( \frac { 2 \pi t } { 365 } \right) , \quad 0 \leqslant t \leqslant 365$$ where \(t\) is the number of days since he began recording.
(b) Using your answer to part (a), or otherwise, find the maximum and minimum number of hours of daylight given by this formula. Give your answers to 3 significant figures.
(c) Use the formula to find the values of \(t\) when \(H = 17\), giving your answers to the nearest integer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
VIIIV SIHI NI JIIHM 10 N OCVIIV 5141 NI 3114 M I ON OCVI4V SIHIL NI JIIYM ION OC
Edexcel C34 2018 January Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-40_495_634_207_657} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 1 } { 2 x } \ln 2 x , \quad x > \frac { 1 } { 2 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = \mathrm { e }\) and \(x = 5 \mathrm { e }\). The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { 1 } { 2 x } \ln 2 x\). The values for \(y\) are given to 4 significant figures.
\(x\)e2 e3 e4 e5 e
\(y\)0.31140.21950.17120.14160.1215
  1. Use the trapezium rule with all the \(y\) values in the table to find an approximate value for the area of \(R\), giving your answer to 3 significant figures.
  2. Using the substitution \(u = \ln 2 x\), or otherwise, find \(\int \frac { 1 } { 2 x } \ln 2 x \mathrm {~d} x\)
  3. Use your answer to part (b) to find the true area of \(R\), giving your answer to 3 significant figures.
  4. Using calculus, find an equation for the tangent to the curve at the point where \(x = \frac { \mathrm { e } ^ { 2 } } { 2 }\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are exact multiples of powers of e.
Edexcel C34 2018 January Q14
14. The volume of a spherical balloon of radius \(r \mathrm {~cm}\) is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)
  1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) The volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 9000 \pi } { ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$
  2. Using the chain rule, or otherwise, show that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$ where \(k\) and \(n\) are constants to be found. Initially, the radius of the balloon is 3 cm .
  3. Using the values of \(k\) and \(n\) found in part (b), solve the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$ to obtain a formula for \(r\) in terms of \(t\).
  4. Hence find the radius of the balloon when \(t = 175\), giving your answer to 3 significant figures.
    (1)
  5. Find the rate of increase of the radius of the balloon when \(t = 175\). Give your answer to 3 significant figures.
    END
Edexcel C34 2019 January Q1
  1. (a) Express \(7 \sin 2 \theta - 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
    (b) Hence solve, for \(0 \leqslant \theta < 90 ^ { \circ }\), the equation
$$7 \sin 2 \theta - 2 \cos 2 \theta = 4$$ giving your answers in degrees to one decimal place.
(c) Express \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\) in the form \(a \sin 2 \theta + b \cos 2 \theta + c\), where \(a\), \(b\) and \(c\) are constants to be found.
(d) Use your answers to part (a) and part (c) to deduce the exact maximum value of \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\)
Edexcel C34 2019 January Q2
2. Given that $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \equiv A + \frac { B } { x + 1 } + \frac { C } { x - 3 }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence, or otherwise, find the series expansion of $$\frac { 3 x ^ { 2 } + 4 x - 7 } { ( x + 1 ) ( x - 3 ) } \quad | x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\)
    Give each coefficient as a simplified fraction.
Edexcel C34 2019 January Q3
3. The function f is defined by $$f : x \mapsto 2 x ^ { 2 } + 3 k x + k ^ { 2 } \quad x \in \mathbb { R } , - 4 k \leqslant x \leqslant 0$$ where \(k\) is a positive constant.
  1. Find, in terms of \(k\), the range of f . The function g is defined by $$\mathrm { g } : x \mapsto 2 k - 3 x \quad x \in \mathbb { R }$$ Given that \(\operatorname { gf } ( - 2 ) = - 12\)
  2. find the possible values of \(k\).
Edexcel C34 2019 January Q4
  1. The curve \(C\) has equation
$$81 y ^ { 3 } + 64 x ^ { 2 } y + 256 x = 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Hence find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Edexcel C34 2019 January Q5
5. The angle \(x\) and the angle \(y\) are such that $$\tan x = m \text { and } 4 \tan y = 8 m + 5$$ where \(m\) is a constant.
Given that \(16 \sec ^ { 2 } x + 16 \sec ^ { 2 } y = 537\)
  1. find the two possible values of \(m\). Given that the angle \(x\) and the angle \(y\) are acute, find the exact value of
  2. \(\sin x\)
  3. \(\cot y\)
Edexcel C34 2019 January Q6
6. Relative to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates ( \(2,1,9 ) , ( 5,2,7 )\) and \(( 4 , - 3,3 )\) respectively. The line \(l\) passes through the points \(A\) and \(B\).
  1. Find a vector equation for the line \(l\).
  2. Find, in degrees, the acute angle between the line \(I\) and the line \(A C\). The point \(D\) lies on the line \(l\) such that angle \(A C D\) is \(90 ^ { \circ }\)
  3. Find the coordinates of \(D\).
  4. Find the exact area of triangle \(A D C\), giving your answer as a fully simplified surd.
Edexcel C34 2019 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-26_615_867_292_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { x + 7 } { \sqrt { 2 x - 3 } } \quad x > \frac { 3 } { 2 }$$ The region \(R\), shown shaded in Figure 1, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 6\)
  1. Use the trapezium rule with 4 strips of equal width to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
  2. Using the substitution \(u = 2 x - 3\), or otherwise, use calculus to find the exact area of \(R\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are constants to be found.
Edexcel C34 2019 January Q8
8. A curve has parametric equations $$x = t ^ { 2 } - t , \quad y = \frac { 4 t } { 1 - t } \quad t \neq 1$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer as a simplified fraction.
  2. Find an equation for the tangent to the curve at the point \(P\) where \(t = - 1\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The tangent to the curve at \(P\) cuts the curve at the point \(Q\).
  3. Use algebra to find the coordinates of \(Q\).
Edexcel C34 2019 January Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-34_1331_1589_264_182} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} (c) Find the exact value for the volume of this solid, giving your answer as a single, simplified fraction. \section*{Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\).
The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\)
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x + \sin 2 x\). The region \(R\), shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.} \(\_\_\_\_\) simplified fraction.