Questions C2 (1410 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 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 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 Q3
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
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
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
  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
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
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
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
  1. Evaluate
$$\int _ { - 2 } ^ { 0 } ( 3 x - 1 ) ^ { 2 } \mathrm {~d} x .$$
Edexcel C2 Q2
2. $$f ( x ) = x ^ { 3 } + k x - 20 .$$ Given that \(\mathrm { f } ( x )\) is exactly divisible by ( \(x + 1\) ),
  1. find the value of the constant \(k\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\).
Edexcel C2 Q3
3. (a) Given that $$5 \cos \theta - 2 \sin \theta = 0 ,$$ show that \(\tan \theta = 2.5\)
(b) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.
Edexcel C2 Q4
4. Solve each equation, giving your answers to an appropriate degree of accuracy.
  1. \(3 ^ { x - 2 } = 5\)
  2. \(\quad \log _ { 2 } ( 6 - y ) = 3 - \log _ { 2 } y\)
Edexcel C2 Q5
5. A geometric series has third term 36 and fourth term 27. Find
  1. the common ratio of the series,
  2. the fifth term of the series,
  3. the sum to infinity of the series.
Edexcel C2 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{857bf144-b03e-4b46-b043-1119b30f9e78-3_572_954_246_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \left( x - \log _ { 10 } x \right) ^ { 2 } , x > 0\).
  1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.
    \(x\)23456
    \(y\)2.896.36
    The shaded area is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).
  2. Use the trapezium rule with all the values in your table to estimate the area of the shaded region.
  3. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area.
Edexcel C2 Q7
7. $$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$
  1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
  2. Determine whether each stationary point is a maximum or minimum point.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions.
Edexcel C2 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{857bf144-b03e-4b46-b043-1119b30f9e78-4_533_685_242_497} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the circle \(C\) and the straight line \(l\). The centre of \(C\) lies on the \(x\)-axis and \(l\) intersects \(C\) at the points \(A ( 2,4 )\) and \(B ( 8 , - 8 )\).
  1. Find the gradient of \(l\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the centre of \(C\).
  4. Show that \(C\) has the equation \(x ^ { 2 } + y ^ { 2 } - 18 x + 16 = 0\).
Edexcel C2 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{857bf144-b03e-4b46-b043-1119b30f9e78-4_365_888_1484_479} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a design painted on the wall at a karting track. The sign consists of triangle \(A B C\) and two circular sectors of radius 2 metres and 1 metre with centres \(A\) and \(B\) respectively. Given that \(A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}\) and \(\angle A C B = 2.2\) radians,
  1. use the sine rule to find the size of \(\angle A B C\) in radians to 3 significant figures,
  2. show that \(\angle B A C = 0.588\) radians to 3 significant figures,
  3. find the area of triangle \(A B C\),
  4. find the area of the wall covered by the design.
Edexcel C2 Q1
  1. A circle has the equation \(x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0\).
    1. Find the coordinates of the centre of the circle.
    2. Find the radius of the circle.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{da9672c1-d1d8-4c2f-89ef-2055da37f720-2_369_689_552_511} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the sector \(O A B\) of a circle, centre \(O\), in which \(\angle A O B = 2.5\) radians.
    Given that the perimeter of the sector is 36 cm ,
  2. find the length \(O A\),
  3. find the area of the shaded segment.
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{da9672c1-d1d8-4c2f-89ef-2055da37f720-2_531_844_1473_447} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).
  1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
  2. find the coordinates of \(Q\) and \(R\).
Edexcel C2 Q4
4. (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
(b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
  1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
  2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\)
Edexcel C2 Q5
5. (a) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(b) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(c) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).
Edexcel C2 Q6
6. A curve has the equation $$y = x ^ { 3 } + a x ^ { 2 } - 15 x + b$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \(( - 1,12 )\),
  1. find the values of \(a\) and \(b\),
  2. find the coordinates of the other stationary point of the curve.
Edexcel C2 Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{da9672c1-d1d8-4c2f-89ef-2055da37f720-4_565_814_251_495} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , \quad x \neq 0 .$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
  3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).
Edexcel C2 Q8
8. (a) Given that \(\sin \theta = 2 - \sqrt { 2 }\), find the value of \(\cos ^ { 2 } \theta\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
(b) Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos \left( 2 x - \frac { \pi } { 6 } \right) = \frac { 1 } { 2 } .$$
Edexcel C2 Q9
  1. The second and fifth terms of a geometric series are - 48 and 6 respectively.
    1. Find the first term and the common ratio of the series.
    2. Find the sum to infinity of the series.
    3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).
Edexcel C2 Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9215e382-406c-41a3-8907-f465b134dd87-2_509_538_248_657} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the sector \(O A B\) of a circle of radius 9.2 cm and centre \(O\).
Given that the area of the sector is \(37.4 \mathrm {~cm} ^ { 2 }\), find to 3 significant figures
  1. the size of \(\angle A O B\) in radians,
  2. the perimeter of the sector.