Questions — Edexcel C12 (247 questions)

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Edexcel C12 2019 June Q12
12. (a) Show that $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ may be expressed in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ where \(a , b\) and \(c\) are constants to be found.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ giving your answers in radians to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
\includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}
Edexcel C12 2019 June Q13
13. Given that \(p = \log _ { a } 9\) and \(q = \log _ { a } 10\), where \(a\) is a constant, find in terms of \(p\) and \(q\),
  1. \(\log _ { a } 900\)
  2. \(\log _ { a } 0.3\)
    VIIIV SIHI NI III IM ION OCVIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM ION OO
Edexcel C12 2019 June Q14
14. The 5 th term of an arithmetic series is \(4 k\), where \(k\) is a constant. The sum of the first 8 terms of this series is \(20 k + 16\)
    1. Find, in terms of \(k\), an expression for the common difference of the series.
    2. Show that the first term of the series is \(16 - 8 k\) Given that the 9th term of the series is 24, find
  2. the value of \(k\),
  3. the sum of the first 20 terms.
    \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-40_2257_54_314_1977}
Edexcel C12 2019 June Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-44_537_679_258_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram not drawn to scale Figure 3 shows the plan view of a garden. The shape of this garden consists of a rectangle joined to a semicircle. The rectangle has length \(x\) metres and width \(y\) metres.
The area of the garden is \(100 \mathrm {~m} ^ { 2 }\).
  1. Show that the perimeter, \(P\) metres, of the garden is given by $$P = \frac { 1 } { 4 } x ( 4 + \pi ) + \frac { 200 } { x } \quad x > 0$$
  2. Use calculus to find the exact value of \(x\) for which the perimeter of the garden is a minimum.
  3. Justify that the value of \(x\) found in part (b) gives a minimum value for \(P\).
  4. Find the minimum perimeter of the garden, giving your answer in metres to one decimal place.
Edexcel C12 2019 June Q16
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-48_855_780_267_580} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(y = 2 x ^ { 2 } - 11 x + 12\). The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the points \(B\) and \(C\).
  1. Find the coordinates of the points \(A , B\) and \(C\). The point \(D\) lies on the curve such that the line \(A D\) is parallel to the \(x\)-axis. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line \(A C\) and the line \(A D\).
  2. Use algebraic integration to find the exact area of \(R\).
Edexcel C12 2016 October Q1
1. $$f ( x ) = 3 x ^ { 2 } + x - \frac { 4 } { \sqrt { x } } + 6 x ^ { - 3 } , \quad x > 0$$ Find \(\int \mathrm { f } ( x ) \mathrm { d } x\), simplifying each term.
Edexcel C12 2016 October Q2
2. Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which
  1. \(7 ^ { 2 x } = 14\)
  2. \(\log _ { 5 } ( 3 x + 1 ) = - 2\)
Edexcel C12 2016 October Q3
3. Answer this question without the use of a calculator and show your method clearly.
  1. Show that $$\sqrt { 45 } - \frac { 20 } { \sqrt { 5 } } + \sqrt { 6 } \sqrt { 30 } = 5 \sqrt { 5 }$$
  2. Show that $$\frac { 17 \sqrt { 2 } } { \sqrt { 2 } + 6 } = 3 \sqrt { 2 } - 1$$
Edexcel C12 2016 October Q4
4. $$f ( x ) = 6 x ^ { 3 } - 7 x ^ { 2 } - 43 x + 30$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
    1. \(2 x + 1\)
    2. \(x - 3\)
  2. Hence factorise \(\mathrm { f } ( x )\) completely.
Edexcel C12 2016 October Q5
5. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$ where \(a\) is a positive constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 3 }\),
(b) find the exact value of \(a\) in its simplest form.
Edexcel C12 2016 October Q6
6. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 36
u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1 \end{aligned}$$
  1. Find the exact simplified values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. Write down the common ratio of the sequence.
  3. Find, giving your answer to 4 significant figures, the value of \(u _ { 11 }\)
  4. Find the exact value of \(\sum _ { i = 1 } ^ { 6 } u _ { i }\)
  5. Find the value of \(\sum _ { i = 1 } ^ { \infty } u _ { i }\)
Edexcel C12 2016 October Q7
  1. (a) Sketch the graph of \(y = 3 ^ { x - 2 } , x \in \mathbb { R }\)
Give the exact values for the coordinates of the point where your graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = 3 ^ { x - 2 }\)
The values of \(y\) are rounded to 3 decimal places where necessary.
\(x\)0.511.522.53
\(y\)0.1920.3330.57711.7323
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { 0.5 } ^ { 3 } 3 ^ { x - 2 } \mathrm {~d} x$$ Give your answer to 2 decimal places.
Edexcel C12 2016 October Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-20_545_1048_212_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The compound shape \(A B C D A\), shown in Figure 1, consists of a triangle \(A B D\) joined along its edge \(B D\) to a sector \(D B C\) of a circle with centre \(B\) and radius 6 cm . The points \(A , B\) and \(C\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B C = 6 \mathrm {~cm}\). Angle \(D A B = 1.1\) radians.
  1. Show that angle \(A B D = 1.20\) radians to 3 significant figures.
  2. Find the area of the compound shape, giving your answer to 3 significant figures.
Edexcel C12 2016 October Q9
  1. In a large theatre there are 20 rows of seats.
The number of seats in the first row is \(a\), where \(a\) is a constant. In the second row the number of seats is \(( a + d )\), where \(d\) is a constant. In the third row the number of seats is \(( a + 2 d )\), and on each subsequent row there are \(d\) more seats than on the previous row. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first 10 rows is 395
  1. Use this information to show that \(10 a + 45 d = 395\) The total number of seats in the first 18 rows is 927
  2. Use this information to write down a second simplified equation relating \(a\) and \(d\).
  3. Solve these equations to find the value of \(a\) and the value of \(d\).
  4. Find the number of seats in the 20th row of the theatre.
Edexcel C12 2016 October Q10
10. (a) Given that $$8 \tan x = - 3 \cos x$$ show that $$3 \sin ^ { 2 } x - 8 \sin x - 3 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$8 \tan 2 \theta = - 3 \cos 2 \theta$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-29_124_37_2615_1882}
Edexcel C12 2016 October Q11
11. The equation \(5 x ^ { 2 } + 6 = k \left( 13 x ^ { 2 } - 12 x \right)\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } + 13 k - 5 > 0$$
  2. Find the set of possible values for \(k\).
Edexcel C12 2016 October Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-32_748_883_274_477} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$ The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).
  1. Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
  2. Use calculus to find the coordinates of the points \(C\) and \(D\). The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
  3. Write down the value of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-35_29_37_182_1914}
Edexcel C12 2016 October Q13
13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).
  1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
  2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
  3. Find the coordinates of the point \(Q\).
Edexcel C12 2016 October Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-40_456_689_269_623} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = - x ^ { 2 } + 6 x - 8\). The normal to \(C\) at the point \(P ( 5 , - 3 )\) is the line \(l\), which is also shown in Figure 3.
  1. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The finite region \(R\), shown shaded in Figure 3, is bounded below by the line \(l\) and the curve \(C\), and is bounded above by the \(x\)-axis.
  2. Find the exact value of the area of \(R\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2016 October Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-44_647_917_260_484} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a solid wooden block. The block is a right prism with length \(h \mathrm {~cm}\). The cross-section of the block is a semi-circle with radius \(r \mathrm {~cm}\). The total surface area of the block, including the curved surface, the two semi-circular ends and the rectangular base, is \(200 \mathrm {~cm} ^ { 2 }\)
  1. Show that the volume \(V \mathrm {~cm} ^ { 3 }\) of the block is given by $$V = \frac { \pi r \left( 200 - \pi r ^ { 2 } \right) } { 4 + 2 \pi }$$
  2. Use calculus to find the maximum value of \(V\). Give your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
  3. Justify, by further differentiation, that the value of \(V\) that you have found is a maximum.
Edexcel C12 2017 October Q1
  1. The line \(l _ { 1 }\) has equation
$$8 x + 2 y - 15 = 0$$
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 3 } { 4 } , 16 \right)\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2017 October Q2
2. The point \(P ( 2,3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\). State the coordinates of the image of \(P\) under the transformation represented by the curve with equation
  1. \(y = \mathrm { f } ( x + 2 )\)
  2. \(y = - \mathrm { f } ( x )\)
  3. \(2 y = f ( x )\)
  4. \(y = \mathrm { f } ( x ) - 4\)
    State the coordinates of the image of \(P\) under the transformation represented by the curve
    with equation (a) \(y = \mathrm { f } ( x + 2 )\)
Edexcel C12 2017 October Q3
3. (a) Express \(\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }\) in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants.
(b) Hence find $$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$ simplifying your answer.
Edexcel C12 2017 October Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-08_287_689_255_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C = 24 \sqrt { 3 }\)
  1. show that \(x = 4 \sqrt { 2 }\)
  2. Hence find the exact length of \(B C\), giving your answer as a simplified surd.
Edexcel C12 2017 October Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-10_678_1076_248_434} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 27 \sqrt { x } - 2 x ^ { 2 } , \quad x \in \mathbb { R } , x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The curve has a maximum turning point \(P\), as shown in Figure 2.
  2. Use the answer to part (a) to find the exact coordinates of \(P\).