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. 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
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
  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
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 \(x ^ { 2 }\).
Edexcel C2 Q4
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
  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
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
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
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
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
  1. Find the coefficient of \(x ^ { 2 }\) in the expansion of
$$( 1 + x ) ( 1 - x ) ^ { 6 }$$
Edexcel C2 Q2
  1. A geometric series has common ratio \(\frac { 1 } { 3 }\).
Given that the sum of the first four terms of the series is 200,
  1. find the first term of the series,
  2. find the sum to infinity of the series.
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ccac020a-c378-45db-80f4-c63b5c213e1d-2_513_775_945_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates ( \(- 4,0\) ),
  1. show that \(k = - 7\),
  2. find the coordinates of \(B\) and \(C\).
Edexcel C2 Q4
4. (a) (i) Sketch the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
(ii) Write down the coordinates of the turning points of the curve in this interval.
(b) Find all values of \(x\) in the interval \(- 180 \leq x \leq 180\) for which $$\sin ( x - 30 ) ^ { \circ } = 0.35$$ giving your answers to 1 decimal place.
Edexcel C2 Q5
5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0 .$$
Edexcel C2 Q6
  1. \(\quad \mathrm { f } ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0\).
    1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
    2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
    3. Determine whether the turning point is a maximum or minimum point.
    4. The points \(P , Q\) and \(R\) have coordinates \(( - 5,2 ) , ( - 3,8 )\) and \(( 9,4 )\) respectively.
    5. Show that \(\angle P Q R = 90 ^ { \circ }\).
    Given that \(P , Q\) and \(R\) all lie on circle \(C\),
  2. find the coordinates of the centre of \(C\),
  3. show that the equation of \(C\) can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.
Edexcel C2 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ccac020a-c378-45db-80f4-c63b5c213e1d-4_549_517_246_605} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(120 ^ { \circ }\) at the centre of the circle.
  1. Find the exact length of the major arc \(P Q\).
  2. Show that the perimeter of the shaded minor segment is given by \(k ( 2 \pi + 3 \sqrt { 3 } ) \mathrm { cm }\), where \(k\) is an integer to be found.
  3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle.
Edexcel C2 Q9
9. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).
  1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\).
  2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
  3. Find the percentage error in the estimate made in part (a).
Edexcel C2 Q1
  1. Expand \(( 3 - 2 x ) ^ { 4 }\) in ascending powers of \(x\) and simplify each coefficient.
Figure 1 Figure 1 shows triangle \(P Q R\) in which \(P Q = x , P R = 7 - x , Q R = x + 1\) and \(\angle P Q R = 60 ^ { \circ }\). Using the cosine rule, find the value of \(x\).
Edexcel C2 Q3
3. Find the coordinates of the stationary point of the curve with equation $$y = x + \frac { 4 } { x ^ { 2 } } .$$
Edexcel C2 Q4
  1. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which
$$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$
Edexcel C2 Q5
  1. (a) Sketch the curve \(y = 5 ^ { x - 1 }\), showing the coordinates of any points of intersection with the coordinate axes.
    (b) Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5 ^ { x - 1 }\) intersects
    1. the straight line \(y = 10\),
    2. the curve \(y = 2 ^ { x }\).
    $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1 .$$
Edexcel C2 Q7
7. (a) Prove that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is given by $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ (b) Evaluate \(\quad \sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)\).
Edexcel C2 Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f824c38-ae19-4889-a2e8-05a3707e9b27-3_496_716_1407_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = 5 + x - x ^ { 2 }\) and the normal to the curve at the point \(P ( 1,5 )\).
  1. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
  3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).
Edexcel C2 Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f824c38-ae19-4889-a2e8-05a3707e9b27-4_757_855_246_482} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$
  1. Find the coordinates of the centre and the radius of \(C\).
    \(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).
  2. Find the coordinates of \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(\theta\) at the centre of \(C\).
  3. Using the cosine rule, show that \(\cos \theta = \frac { 7 } { 25 }\).
  4. Find the area of the shaded minor segment bounded by \(C\) and the chord \(P Q\). END