Questions — Edexcel C2 (476 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel C2 Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a911dc0b-351c-415c-a6df-2af652d5a59b-3_755_1024_934_322}
\end{figure} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac { x ^ { 2 } } { 25 } , x \geq 0\).
The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\).
  2. Find an equation of the tangent to \(C\) at \(A\). The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
  3. For points \(( x , y )\) on \(C\), express \(x\) in terms of \(y\).
  4. Use integration to find the area of \(R\).
Edexcel C2 Q9
9. (i) Solve, for \(0 ^ { \circ } < x < 180 ^ { \circ }\), the equation \(\sin \left( 2 x + 50 ^ { \circ } \right) = 0.6\), giving your answers to 1 d. p.
(ii) In the triangle \(A B C , A C = 18 \mathrm {~cm} , \angle A B C = 60 ^ { \circ }\) and \(\sin A = \frac { 1 } { 3 }\).
  1. Use the sine rule to show that \(B C = 4 \sqrt { } 3\).
  2. Find the exact value of \(\cos A\). L
Edexcel C2 Q1
  1. The point \(A\) has coordinates \(( 2,5 )\) and the point \(B\) has coordinates \(( - 2,8 )\). Find, in cartesian form, an equation of the circle with diameter \(A B\).
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{8c2ee205-d02f-413d-abf7-e259c0734353-2_613_844_479_518}
\end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.
  1. Write down the value of \(a\) and the value of \(b\).
  2. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
  3. Find, to 3 significant figures, the length of \(P T\).
Edexcel C2 Q3
3. \(f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9\), where \(p\) is a constant.
  1. Given that \(\mathrm { f } ( n )\) has a remainder of 3 when it is divided by ( \(n + 2\) ), prove that \(p = 6\).
  2. Show that \(\mathrm { f } ( n )\) can be written in the form \(( n + 2 ) ( n + q ) ( n + r ) + 3\), where \(q\) and \(r\) are integers to be found.
  3. Hence show that \(\mathrm { f } ( n )\) is divisible by 3 for all positive integer values of \(n\).
Edexcel C2 Q4
4. (a) Sketch, for \(0 \leq x \leq 360 ^ { \circ }\), the graph of \(y = \sin \left( x + 30 ^ { \circ } \right)\).
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }\).
Edexcel C2 Q5
5. The expansion of \(( 2 - p x ) ^ { 6 }\) in ascending powers of \(x\), as far as the term in \(x ^ { 2 }\), is $$64 + A x + 135 x ^ { 2 }$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\).
Edexcel C2 Q6
6. Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which $$2 \cos ^ { 2 } \theta - \cos \theta - 1 = \sin ^ { 2 } \theta$$ Give your answers to 1 decimal place where appropriate.
(8)
Edexcel C2 Q7
7. \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{8c2ee205-d02f-413d-abf7-e259c0734353-3_558_642_605_651}
Fig. 2 shows the sector \(O A B\) of a circle of radius \(r \mathrm {~cm}\). The area of the sector is \(15 \mathrm {~cm} ^ { 2 }\) and \(\angle A O B = 1.5\) radians.
  1. Prove that \(r = 2 \sqrt { 5 }\).
  2. Find, in cm , the perimeter of the sector \(O A B\). The segment \(R\), shaded in Fig 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
  3. Calculate, to 3 decimal places, the area of \(R\).
Edexcel C2 Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{8c2ee205-d02f-413d-abf7-e259c0734353-4_965_1125_301_278}
\end{figure} Fig. 3 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x ^ { 2 } - 2 x + 3\).
The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
  1. Calculate the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(R\) is bounded by the line and the curve.
  2. Calculate the area of \(R\).
    [0pt] [P1 June 2003 Question 7]
Edexcel C2 Q9
9. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. find the coordinates of each of the stationary points,
  3. determine the nature of each stationary point. The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
  4. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel C2 Q2
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
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
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
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
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
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 \(C\).
  5. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  6. Find the coordinates of \(Q\).
  7. Show that \(P Q\) is parallel to the \(x\)-axis.
  8. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).
Edexcel C2 Q1
\begin{enumerate} \item (a) Show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x - 30\).
(b) Factorise \(\mathrm { f } ( x )\) completely. \item For the binomial expansion, in descending powers of \(x\), of \(\left( x ^ { 3 } - \frac { 1 } { 2 x } \right) ^ { 12 }\),
Edexcel C2 Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{12675be8-6167-495b-a167-43b705b5ea5f-3_524_1310_808_292}
\end{figure} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 }\).
The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A ( p , 0 )\).
  1. Show that \(p = 6\).
  2. Find an equation of the tangent to \(C\) at \(A\). The curve \(C\) has a maximum at the point \(P\).
  3. Find the \(x\)-coordinate of \(P\). The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.
  4. Find the area of \(R\).
Edexcel C2 Q8
8. 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 \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The first and second terms of a geometric series \(G\) are 10 and 9 respectively.
  2. Find, to 3 significant figures, the sum of the first twenty terms of \(G\).
  3. Find the sum to infinity of \(G\). Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.
  4. Find the exact value of the common ratio of this series.
Edexcel C2 Q1
  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
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
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
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
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
  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 \(y = 2 x + 13\).