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 2018 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-02_575_812_214_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 } , \quad x \geqslant - 2$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line with equation \(x = 10\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \frac { ( x + 2 ) ^ { \frac { 3 } { 2 } } } { 4 }\)
  1. Complete the table, giving values of \(y\) corresponding to \(x = 2\) and \(x = 6\)
    \(x\)- 22610
    \(y\)0\(6 \sqrt { } 3\)
  2. Use the trapezium rule, with all the values of \(y\) from the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places.
Edexcel C2 2018 June Q2
  1. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 2 + k x ) ^ { 7 }$$ where \(k\) is a non-zero constant. Give each term in its simplest form. Given that the coefficient of \(x ^ { 3 }\) in this expansion is 1890
(b) find the value of \(k\).
Edexcel C2 2018 June Q3
3. $$f ( x ) = 24 x ^ { 3 } + A x ^ { 2 } - 3 x + B$$ where \(A\) and \(B\) are constants.
When \(\mathrm { f } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 30
  1. Show that \(A + 4 B = 114\) Given also that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
  2. find another equation in \(A\) and \(B\).
  3. Find the value of \(A\) and the value of \(B\).
  4. Hence find a quadratic factor of \(\mathrm { f } ( x )\).
Edexcel C2 2018 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-10_310_716_214_621} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Not to scale Figure 2 shows a flag \(X Y W Z X\). The flag consists of a triangle \(X Y Z\) joined to a sector \(Z Y W\) of a circle with radius 5 cm and centre \(Y\). The angle of the sector, angle \(Z Y W\), is 0.7 radians. The points \(X , Y\) and \(W\) lie on a straight line with \(X Y = 7 \mathrm {~cm}\) and \(Y W = 5 \mathrm {~cm}\). Find
  1. the area of the sector \(Z Y W\) in \(\mathrm { cm } ^ { 2 }\),
  2. the area of the flag, in \(\mathrm { cm } ^ { 2 }\), to 2 decimal places,
  3. the length of the perimeter, \(X Y W Z X\), of the flag, in cm to 2 decimal places.
Edexcel C2 2018 June Q5
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 2 x + 14 y = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the exact value of the radius of \(C\),
  3. the \(y\) coordinates of the points where the circle \(C\) crosses the \(y\)-axis.
  4. Find an equation of the tangent to \(C\) at the point ( 2,0 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C2 2018 June Q6
  1. A geometric series with common ratio \(r = - 0.9\) has sum to infinity 10000 For this series,
    1. find the first term,
    2. find the fifth term,
    3. find the sum of the first twelve terms, giving this answer to the nearest integer.
Edexcel C2 2018 June Q7
7. (i) Find the value of \(y\) for which $$1.01 ^ { y - 1 } = 500$$ Give your answer to 2 decimal places.
(ii) Given that $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$
  1. show that $$9 x ^ { 2 } + 18 x - 7 = 0$$
  2. Hence solve the equation $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$ DO NOTI WRITE IN THIS AREA
Edexcel C2 2018 June Q8
8 In this question solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos \left( x + 70 ^ { \circ } \right) = 3$$ giving your answers in degrees to one decimal place.
  2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$6 \cos ^ { 2 } \theta - 5 = 6 \sin ^ { 2 } \theta + \sin \theta$$ giving your answers in radians to 3 significant figures.
Edexcel C2 2018 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-28_751_876_214_539} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 7 x ^ { 2 } ( 5 - 2 \sqrt { x } ) , \quad x \geqslant 0$$ The curve has a turning point at the point \(A\), where \(x > 0\), as shown in Figure 3.
  1. Using calculus, find the coordinates of the point \(A\). The curve crosses the \(x\)-axis at the point \(B\), as shown in Figure 3.
  2. Use algebra to find the \(x\) coordinate of the point \(B\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.
  3. Use integration to find the area of the region \(R\), giving your answer to 2 decimal places.
    END
Edexcel C2 Q1
1. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + p x + 6$$ where \(p\) is a constant.
Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find
  1. the value of \(p\),
  2. the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).
Edexcel C2 Q2
2. (a) Find \(\quad \int \left( 3 + 4 x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) \mathrm { d } x\).
(b) Hence evaluate \(\quad \int _ { 1 } ^ { 2 } \left( 3 + 4 x ^ { 3 } - \frac { 2 } { x ^ { 2 } } \right) \mathrm { d } x\).
\includegraphics[max width=\textwidth, alt={}]{9e4e1626-238b-4afd-b81c-68c5ab1624c2-04_2573_1927_146_52}
Edexcel C2 Q4
  1. Solve
$$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2 , \quad x > 2 .$$
Edexcel C2 Q5
5. The second and fifth terms of a geometric series are 9 and 1.125 respectively. For this series find
  1. the value of the common ratio,
  2. the first term,
  3. the sum to infinity.
    5.
    continued
Edexcel C2 Q6
  1. The circle \(C\), with centre \(A\), has equation
$$x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 12 = 0$$
  1. Find the coordinates of \(A\).
  2. Show that the radius of \(C\) is 5 . The points \(P , Q\) and \(R\) lie on \(C\). The length of \(P Q\) is 10 and the length of \(P R\) is 3 .
  3. Find the length of \(Q R\), giving your answer to 1 decimal place.
    \includegraphics[max width=\textwidth, alt={}]{9e4e1626-238b-4afd-b81c-68c5ab1624c2-09_2540_1718_150_93}
Edexcel C2 Q7
  1. The first four terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + k x ) ^ { n }\) are
$$1 + A x + B x ^ { 2 } + B x ^ { 3 } + \ldots$$ where \(k\) is a positive constant and \(A\), \(B\) and \(n\) are positive integers.
  1. By considering the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\), show that \(3 = ( n - 2 ) k\). Given that \(A = 4\),
  2. find the value of \(n\) and the value of \(k\).
    7. continuedLeave blank
Edexcel C2 Q8
  1. (a) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(\cos \left( x - 20 ^ { \circ } \right) = - 0.437\), giving your answers to the nearest degree.
    (b) Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which
$$3 \tan \theta = 2 \cos \theta$$
\includegraphics[max width=\textwidth, alt={}]{9e4e1626-238b-4afd-b81c-68c5ab1624c2-13_2536_1737_150_98}
Edexcel C2 Q9
  1. A pencil holder is in the shape of an open circular cylinder of radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The surface area of the cylinder (including the base) is \(250 \mathrm {~cm} ^ { 2 }\).
    1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by \(V = 125 r - \frac { \pi r ^ { 3 } } { 2 }\).
    2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value.
    3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\).
    4. Calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the maximum volume of the pencil holder.
    9. continuedLeave blank
Edexcel C2 Q10
10. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{9e4e1626-238b-4afd-b81c-68c5ab1624c2-16_525_928_312_621}
\end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = 9 - 2 x - \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The point \(A ( 1,5 )\) lies on \(C\) and the curve crosses the \(x\)-axis at \(B ( b , 0 )\), where \(b\) is a constant and \(b > 0\).
  1. Verify that \(b = 4\). The tangent to \(C\) at the point \(A\) cuts the \(x\)-axis at the point \(D\), as shown in Fig. 2 .
  2. Show that an equation of the tangent to \(C\) at \(A\) is \(y + x = 6\).
  3. Find the coordinates of the point \(D\). The shaded region \(R\), shown in Fig. 2, is bounded by \(C\), the line \(A D\) and the \(x\)-axis.
  4. Use integration to find the area of \(R\).
    1. continued
    2. continued
Edexcel C2 Specimen Q1
Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + 3 x ) ^ { 6 }\).
(4)
Edexcel C2 Specimen Q4
4. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(3 \sin ^ { 2 } x = 1 + \cos x\), giving your answers to the nearest degree.
Edexcel C2 Specimen Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-1_351_663_1529_693}
\end{figure} The shaded area in Fig. 1 shows a badge \(A B C\), where \(A B\) and \(A C\) are straight lines, with \(A B = A C = 8 \mathrm {~mm}\). The curve \(B C\) is an arc of a circle, centre \(O\), where \(O B = O C =\) 8 mm and \(O\) is in the same plane as \(A B C\). The angle \(B A C\) is 0.9 radians.
  1. Find the perimeter of the badge.
  2. Find the area of the badge.
Edexcel C2 Specimen Q6
6. At the beginning of the year 2000 a company bought a new machine for \(\pounds 15000\). Each year the value of the machine decreases by \(20 \%\) of its value at the start of the year.
  1. Show that at the start of the year 2002, the value of the machine was \(\pounds 9600\). When the value of the machine falls below \(\pounds 500\), the company will replace it.
  2. Find the year in which the machine will be replaced. To plan for a replacement machine, the company pays \(\pounds 1000\) at the start of each year into a savings account. The account pays interest at a fixed rate of \(5 \%\) per annum. The first payment was made when the machine was first bought and the last payment will be made at the start of the year in which the machine is replaced.
  3. Using your answer to part (b), find how much the savings account will be worth immediately after the payment at the start of the year in which the machine is replaced.
Edexcel C2 Specimen Q7
7. (a) Use the factor theorem to show that \(( x + 1 )\) is a factor of \(x ^ { 3 } - x ^ { 2 } - 10 x - 8\).
(b) Find all the solutions of the equation \(x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0\).
(c) Prove that the value of \(x\) that satisfies $$2 \log _ { 2 } x + \log _ { 2 } ( x - 1 ) = 1 + \log _ { 2 } ( 5 x + 4 )$$ is a solution of the equation $$x ^ { 3 } - x ^ { 2 } - 10 x - 8 = 0$$ (d) State, with a reason, the value of \(x\) that satisfies equation (I).
Edexcel C2 Specimen Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-2_616_712_1658_713}
\end{figure} The line with equation \(y = x + 5\) cuts the curve with equation \(y = x ^ { 2 } - 3 x + 8\) at the points \(A\) and \(B\), as shown in Fig. 2.
  1. Find the coordinates of the points \(A\) and \(B\).
  2. Find the area of the shaded region between the curve and the line, as shown in Fig. 2.
Edexcel C2 Specimen Q9
9. Figure 3 $$( x + 1 ) ^ { 2 }$$ Figure 3 shows a triangle \(P Q R\). The size of angle \(Q P R\) is \(30 ^ { \circ }\), the length of \(P Q\) is \(( x + 1 )\) and the length of \(P R\) is \(( 4 - x ) ^ { 2 }\), where \(X \in \Re\).
  1. Show that the area \(A\) of the triangle is given by \(A = \frac { 1 } { 4 } \left( x ^ { 3 } - 7 x ^ { 2 } + 8 x + 16 \right)\)
  2. Use calculus to prove that the area of \(\triangle P Q R\) is a maximum when \(x = \frac { 2 } { 3 }\). Explain clearly how you know that this value of \(x\) gives the maximum area.
  3. Find the maximum area of \(\triangle P Q R\).
  4. Find the length of \(Q R\) when the area of \(\triangle P Q R\) is a maximum. END