Questions — Edexcel C2 (579 questions)

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Edexcel C2 Q7
12 marks Moderate -0.8
7. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
  1. Prove that the sum of the first \(n\) terms of this series is given by \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The second and fourth terms of the series are 3 and 1.08 respectively.
    Given that all terms in the series are positive, find
  2. the value of \(r\) and the value of \(a\),
  3. the sum to infinity of the series. \includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299} Fig. 2 shows part of the curve with equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\). The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).
    1. Show that the equation of the curve may be written as \(y = x ( x - 3 ) ^ { 2 }\), and hence write down the coordinates of \(A\).
    2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
    3. Find the area of \(R\).
Edexcel C2 Q1
6 marks Moderate -0.3
  1. (a) Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \(( 1 + 3 x ) ^ { n }\), where \(n > 2\).
Given that the coefficient of \(x ^ { 3 }\) in this expansion is ten times the coefficient of \(x ^ { 2 }\),
(b) find the value of \(n\),
(c) find the coefficient of \(x ^ { 4 }\) in the expansion.
Edexcel C2 Q2
7 marks Moderate -0.5
2. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), the remainder is 14 . When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .
  1. Find the value of \(a\) and the value of \(b\).
  2. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
    [0pt] [P3 June 2002 Question 1]
Edexcel C2 Q3
8 marks Moderate -0.8
3. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),
  1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
    [0pt] [P1 June 2002 Question 3]
Edexcel C2 Q4
10 marks Moderate -0.8
4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$
  1. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\) axis.
  2. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
  3. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\) [0pt] [P1 June 2002 Question 5]
Edexcel C2 Q5
10 marks Standard +0.3
5. (a) Given that \(3 + 2 \log _ { 2 } x = \log _ { 2 } y\), show that \(y = 8 x ^ { 2 }\).
(b) Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$ (c) Show that \(\log _ { 2 } \alpha = - 2\).
(d) Calculate \(\log _ { 2 } \beta\), giving your answer to 3 significant figures.
Edexcel C2 Q6
11 marks Standard +0.3
6. Given that \(\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0\),
  1. find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 5\).
  2. Show that \(\mathrm { f } ( x )\) may be written in the form \(A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C\), where \(A , B\) and \(C\) are constants to be found.
  3. Hence evaluate \(\int _ { 1 } ^ { 2 } f ( x ) d x\).
Edexcel C2 Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ba5cb933-dedd-4ad9-9e66-49636870b3de-3_739_1272_826_328}
\end{figure} Fig. 1 shows the cross-section \(A B C D\) of a chocolate bar, where \(A B , C D\) and \(A D\) are straight lines and \(M\) is the mid-point of \(A D\). The length \(A D\) is 28 mm , and \(B C\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B , C\) and \(D\) have coordinates (7,24), (21,24) and (28,0) respectively.
  1. Show that the length of \(B M\) is 25 mm .
  2. Show that, to 3 significant figures, \(\angle B M C = 0.568\) radians.
  3. Hence calculate, in \(\mathrm { mm } ^ { 2 }\), the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
  4. calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the volume of the bar.
Edexcel C2 Q8
13 marks Moderate -0.8
8. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be \(\pounds 54000\),
(d) find the predicted profit for the year 2011.
Edexcel C2 Q2
7 marks Moderate -0.3
2. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
(b) Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
Edexcel C2 Q3
7 marks Easy -1.2
3. Every \(\pounds 1\) of money invested in a savings scheme continuously gains interest at a rate of \(4 \%\) per year. Hence, after \(x\) years, the total value of an initial \(\pounds 1\) investment is \(\pounds y\), where $$y = 1.04 ^ { x }$$
  1. Sketch the graph of \(y = 1.04 ^ { x } , x \geq 0\).
  2. Calculate, to the nearest \(\pounds\), the total value of an initial \(\pounds 800\) investment after 10 years.
  3. Use logarithms to find the number of years it takes to double the total value of any initial investment.
Edexcel C2 Q4
7 marks Moderate -0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-2_554_561_1635_762}
\end{figure} Fig. 1 shows the sector \(A O B\) of a circle, with centre \(O\) and radius 6.5 cm , and \(\angle A O B = 0.8\) radians.
  1. Calculate, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(A O B\).
  2. Show that the length of the chord \(A B\) is 5.06 cm , to 3 significant figures. The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
  3. Calculate, in cm , the perimeter of \(R\).
Edexcel C2 Q5
8 marks Moderate -0.3
5. (a) Write down the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 1 + a x ) ^ { n } , n > 2 .$$ Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 ,
(b) calculate the value of \(n\) and the value of \(a\),
(c) find the coefficient of \(x ^ { 3 }\).
[0pt] [P2 November 2003 Question 3]
Edexcel C2 Q6
12 marks Standard +0.3
6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.
  1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
  2. Find the value of \(r\) for which \(A\) is a minimum.
  3. Prove that this value of \(r\) gives a minimum value of \(A\).
  4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.
Edexcel C2 Q7
17 marks Moderate -0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-3_771_972_1322_557}
\end{figure} Fig. 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x\).
The curve crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).
  1. Factorise \(\mathrm { f } ( x )\) completely.
  2. Write down the \(x\)-coordinates of the points \(A\) and \(B\).
  3. Find the gradient of \(C\) at \(A\). The region \(R\) is bounded by \(C\) and the line \(O A\), and the region \(S\) is bounded by \(C\) and the line \(A B\).
  4. Use integration to find the area of the combined regions \(R\) and \(S\), shown shaded in Fig.2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
    \end{figure} Fig. 3 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0\). The point \(P\), on \(C\), has \(x\)-coordinate 1 and the point \(Q\) is the minimum turning point of \(C\).
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find the coordinates of \(Q\).
    3. Show that \(P Q\) is parallel to the \(x\)-axis.
    4. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).
Edexcel C2 Q1
5 marks Moderate -0.8
  1. \(\quad \mathrm { f } ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).
Given that when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) there is a remainder of - 35 ,
  1. find the value of the constant \(k\),
  2. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(3 x - 2\) ).
Edexcel C2 Q2
5 marks Easy -1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-2_613_911_692_376} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).
Edexcel C2 Q3
6 marks Moderate -0.8
3. Giving your answers in terms of \(\pi\), solve the equation $$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).
Edexcel C2 Q4
7 marks Moderate -0.3
4. (a) Expand \(( 1 + 3 x ) ^ { 8 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). You should simplify each coefficient in your expansion.
(b) Use your series, together with a suitable value of \(x\) which you should state, to estimate the value of (1.003) \({ } ^ { 8 }\), giving your answer to 8 significant figures.
Edexcel C2 Q5
7 marks Moderate -0.3
5. (a) Given that \(t = \log _ { 3 } x\), find expressions in terms of \(t\) for
  1. \(\log _ { 3 } x ^ { 2 }\),
  2. \(\log _ { 9 } x\).
    (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4 .$$
Edexcel C2 Q6
10 marks Moderate -0.8
  1. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).
    1. Find an equation for \(C\).
    2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
    3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-3_664_1016_1276_376} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the curve \(y = 2 x ^ { 2 } + 6 x + 7\) and the straight line \(y = 2 x + 13\).
Edexcel C2 Q8
11 marks Standard +0.3
8. A geometric series has first term \(a\) and common ratio \(r\) where \(r > 1\). The sum of the first \(n\) terms of the series is denoted by \(S _ { n }\). Given that \(S _ { 4 } = 10 \times S _ { 2 }\),
  1. find the value of \(r\). Given also that \(S _ { 3 } = 26\),
  2. find the value of \(a\),
  3. show that \(S _ { 6 } = 728\).
Edexcel C2 Q9
13 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c7c8cf84-06ac-4059-b8f0-d68b6d1d8dcc-4_661_915_932_431} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a design consisting of two rectangles measuring \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\) joined to a circular sector of radius \(x \mathrm {~cm}\) and angle 0.5 radians. Given that the area of the design is \(50 \mathrm {~cm} ^ { 2 }\),
  1. show that the perimeter, \(P\) cm, of the design is given by $$P = 2 x + \frac { 100 } { x }$$
  2. Find the value of \(x\) for which \(P\) is a minimum.
  3. Show that \(P\) is a minimum for this value of \(x\).
  4. Find the minimum value of \(P\) in the form \(k \sqrt { 2 }\).
Edexcel C2 Q2
6 marks Moderate -0.5
  1. Given that
$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
Edexcel C2 Q3
6 marks Moderate -0.3
3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),
  1. find and simplify the first three terms,
  2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).