Questions — Edexcel C2 (579 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
Edexcel C2 Q4
7 marks Moderate -0.3
4. Solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$
Edexcel C2 Q5
9 marks Moderate -0.3
  1. The circle \(C\) has centre \(( - 1,6 )\) and radius \(2 \sqrt { 5 }\).
    1. Find an equation for \(C\).
    The line \(y = 3 x - 1\) intersects \(C\) at the points \(A\) and \(B\).
  2. Find the \(x\)-coordinates of \(A\) and \(B\).
  3. Show that \(A B = 2 \sqrt { 10 }\).
Edexcel C2 Q6
10 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e44996a-4635-46f6-bd45-7799a8c49463-3_589_894_248_397} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).
  1. Find the coordinates of the minimum point of the curve. The shaded region \(R\) is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
  2. Use the trapezium rule with three intervals of equal width to estimate the area of \(R\).
Edexcel C2 Q7
10 marks Moderate -0.3
7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes. Assuming that the times he takes to complete successive papers form a geometric sequence,
  1. find, to the nearest minute, how long he will take to complete the fifth paper,
  2. show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,
  3. find the least number of papers he must work through if he is to complete a paper in less than one hour.
Edexcel C2 Q8
10 marks Standard +0.3
8. Figure 2 Figure 2 shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm} , C D = 8 \mathrm {~cm}\), \(A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).
  1. Using the cosine rule, show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
  2. Find the size of \(\angle B C D\) in degrees.
  3. Find the area of quadrilateral \(A B C D\).
Edexcel C2 Q9
13 marks Standard +0.3
9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16 .$$
  1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
Edexcel C2 Q1
4 marks Moderate -0.8
  1. Evaluate
$$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$
Edexcel C2 Q2
5 marks Moderate -0.3
2. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
Edexcel C2 Q3
6 marks Moderate -0.8
3. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\quad \log _ { 2 } 0.3\)
Edexcel C2 Q4
8 marks Moderate -0.3
4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.
  1. Find the value of \(k\). Using this value of \(k\),
  2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375,
  3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$
Edexcel C2 Q5
9 marks Standard +0.3
  1. (a) Write down the exact value of \(\cos \frac { \pi } { 6 }\).
The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(b) Use the trapezium rule with three equally-spaced ordinates to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(c) Using your answer to part (b), find an estimate for the area of \(S\).
Edexcel C2 Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{288b99b5-1198-4463-baed-f0a4bf03e485-3_335_890_246_456} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows triangle \(A B C\) in which \(A C = 8 \mathrm {~cm}\) and \(\angle B A C = \angle B C A = 30 ^ { \circ }\).
  1. Find the area of triangle \(A B C\) in the form \(k \sqrt { 3 }\). The point \(M\) is the mid-point of \(A C\) and the points \(N\) and \(O\) lie on \(A B\) and \(B C\) such that \(M N\) and \(M O\) are arcs of circles with centres \(A\) and \(C\) respectively.
  2. Show that the area of the shaded region \(B N M O\) is \(\frac { 8 } { 3 } ( 2 \sqrt { 3 } - \pi ) \mathrm { cm } ^ { 2 }\).
Edexcel C2 Q7
10 marks Standard +0.3
7. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 8 y + k = 0 ,$$ where \(k\) is a constant. Given that the point with coordinates \(( - 6,5 )\) lies on \(C\),
  1. find the value of \(k\),
  2. find the coordinates of the centre and the radius of \(C\). A straight line which passes through the point \(A ( 2,3 )\) is a tangent to \(C\) at the point \(B\).
  3. Find the length \(A B\) in the form \(k \sqrt { 3 }\).
Edexcel C2 Q8
12 marks Standard +0.3
8. Amy plans to join a savings scheme in which she will pay in \(\pounds 500\) at the start of each year. One scheme that she is considering pays 6\% interest on the amount in the account at the end of each year. For this scheme,
  1. find the amount of interest paid into the account at the end of the second year,
  2. show that after interest is paid at the end of the eighth year, the amount in the account will be \(\pounds 5246\) to the nearest pound. Another scheme that she is considering pays \(0.5 \%\) interest on the amount in the account at the end of each month.
  3. Find, to the nearest pound, how much more or less will be in the account at the end of the eighth year under this scheme.
Edexcel C2 Q9
12 marks Standard +0.3
9. The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where \(k\) is a constant.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) the remainder is \(r\).
When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) the remainder is \(3 r\).
  1. Find the value of \(k\).
  2. Find the value of \(r\).
  3. Show that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\).
  4. Show that there is only one real solution to the equation \(\mathrm { f } ( x ) = 0\). END
Edexcel C2 Q1
5 marks Easy -1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-2_383_707_246_488} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\).
Find, to 3 significant figures,
  1. the length \(B C\),
  2. the area of triangle \(A B C\).
Edexcel C2 Q2
6 marks Moderate -0.5
2. Show that $$\int _ { 2 } ^ { 3 } \left( 6 \sqrt { x } - \frac { 4 } { \sqrt { x } } \right) \mathrm { d } x = k \sqrt { 3 } ,$$ where \(k\) is an integer to be found.
Edexcel C2 Q3
7 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-2_439_848_1560_461} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).
  1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
  2. find an estimate for the volume of the support.
Edexcel C2 Q4
7 marks Moderate -0.3
4. (a) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(b) Hence expand ( \(\left. 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.
Edexcel C2 Q5
8 marks Standard +0.3
5. (a) Given that $$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0 .$$ (b) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0 .$$
Edexcel C2 Q6
9 marks Moderate -0.3
  1. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for
    1. \(3 ^ { x + 1 }\),
    2. \(3 ^ { 2 x - 1 }\).
      (b) Hence, or otherwise, solve the equation
    $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$ giving non-exact answers to 2 decimal places.
Edexcel C2 Q7
9 marks Standard +0.3
7. The circle \(C\) has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).
  1. Find the length of the diameter of \(C\).
  2. Find an equation for \(C\).
  3. Show that the line \(y = 2 x - 3\) is a tangent to \(C\) and find the coordinates of the point of contact.
Edexcel C2 Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-4_536_883_248_486} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
  1. Find the coordinates of the minimum point of the curve.
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).
Edexcel C2 Q9
12 marks Standard +0.3
9. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
  3. find the common ratio of the series,
  4. find the sum of the first eight terms of the series.
Edexcel C2 Q1
5 marks Moderate -0.8
  1. Evaluate
$$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } \mathrm {~d} x .$$