Questions — Edexcel C12 (247 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 C12 2016 June Q2
2 \log _ { 2 } y = 5 - \log _ { 2 } x
\log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)
Edexcel C12 2014 January Q1
  1. Find the first 3 terms in ascending powers of \(x\) of
$$\left( 2 - \frac { x } { 2 } \right) ^ { 6 }$$ giving each term in its simplest form.
Edexcel C12 2014 January Q2
2. $$\mathrm { f } ( x ) = \frac { 8 } { x ^ { 2 } } - 4 \sqrt { x } + 3 x - 1 , \quad x > 0$$ Giving your answers in their simplest form, find
  1. \(\mathrm { f } ^ { \prime } ( x )\)
  2. \(\int \mathrm { f } ( x ) \mathrm { d } x\)
Edexcel C12 2014 January Q3
3. $$f ( x ) = 10 x ^ { 3 } + 27 x ^ { 2 } - 13 x - 12$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
    1. \(x - 2\)
    2. \(x + 3\)
  2. Hence factorise \(\mathrm { f } ( x )\) completely.
Edexcel C12 2014 January Q4
4. Answer this question without the use of a calculator and show all your working.
  1. Show that $$\frac { 4 } { 2 \sqrt { 2 } - \sqrt { 6 } } = 2 \sqrt { 2 } ( 2 + \sqrt { 3 } )$$
  2. Show that $$\sqrt { 27 } + \sqrt { 21 } \times \sqrt { 7 } - \frac { 6 } { \sqrt { 3 } } = 8 \sqrt { 3 }$$
Edexcel C12 2014 January Q5
5. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of
  1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. \(u _ { 61 }\)
  3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)
Edexcel C12 2014 January Q6
6. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a b = 25
\log _ { 4 } a - \log _ { 4 } b = 3 \end{gathered}$$ Show each step of your working, giving exact values for \(a\) and \(b\).
Edexcel C12 2014 January Q7
7. (a) Show that $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ may be expressed in the form $$12 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence, using trigonometry, find all the solutions in the interval \(0 \leqslant x \leqslant 360 ^ { \circ }\) of $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ Give each solution, in degrees, to 1 decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-15_106_97_2615_1784}
Edexcel C12 2014 January Q8
8. Find the range of values of \(k\) for which the quadratic equation $$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$ has no real roots.
Edexcel C12 2014 January Q9
9. In the first month after opening, a mobile phone shop sold 300 phones. A model for future sales assumes that the number of phones sold will increase by \(5 \%\) per month, so that \(300 \times 1.05\) will be sold in the second month, \(300 \times 1.05 ^ { 2 }\) in the third month, and so on. Using this model, calculate
  1. the number of phones sold in the 24th month,
  2. the total number of phones sold over the whole 24 months. This model predicts that, in the \(N\) th month, the number of phones sold in that month exceeds 3000 for the first time.
  3. Find the value of \(N\).
Edexcel C12 2014 January Q10
10. The curve \(C\) has equation \(y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi\)
  1. In the space below, sketch the curve \(C\).
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
  3. Solve, for \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\), $$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.
Edexcel C12 2014 January Q11
11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.
  1. Show that \(p = 9\)
  2. Find the value of the 20th term of this series.
  3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}
Edexcel C12 2014 January Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-28_549_542_212_703} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\).
\(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find
  1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
  2. the perimeter of the platform to 3 significant figures,
  3. the area of the platform to 3 significant figures.
Edexcel C12 2014 January Q13
13. The curve \(C\) has equation $$y = \frac { ( x - 3 ) ( 3 x - 25 ) } { x } , \quad x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a fully simplified form.
  2. Hence find the coordinates of the turning point on the curve \(C\).
  3. Determine whether this turning point is a minimum or maximum, justifying your answer. The point \(P\), with \(x\) coordinate \(2 \frac { 1 } { 2 }\), lies on the curve \(C\).
  4. Find the equation of the normal at \(P\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-35_90_72_2631_1873}
Edexcel C12 2014 January Q14
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-36_652_791_223_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
  2. Use integration to calculate an exact value for the area of \(R\).
Edexcel C12 2014 January Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-40_883_824_212_568} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Diagram NOT drawn to scale The points \(X\) and \(Y\) have coordinates \(( 0,3 )\) and \(( 6,11 )\) respectively. \(X Y\) is a chord of a circle \(C\) with centre \(Z\), as shown in Figure 3.
  1. Find the gradient of \(X Y\). The point \(M\) is the midpoint of \(X Y\).
  2. Find an equation for the line which passes through \(Z\) and \(M\). Given that the \(y\) coordinate of \(Z\) is 10 ,
  3. find the \(x\) coordinate of \(Z\),
  4. find the equation of the circle \(C\), giving your answer in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$$ where \(a\), \(b\) and \(c\) are constants.
Edexcel C12 2015 January Q1
Simplify the following expressions fully.
  1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
  2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)
Edexcel C12 2015 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-05_645_933_258_463} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points ( \(2.5,0\) ) and ( 0,9 ), has a stationary point at ( 1,11 ), and has an asymptote \(y = 3\) On separate diagrams, sketch the curve with equation
  1. \(y = 3 \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } ( - x )\) On each diagram show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote.
Edexcel C12 2015 January Q4
  1. (a) Find the first 4 terms in ascending powers of \(x\) of the binomial expansion of
$$\left( 2 + \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.
(b) Use your expansion to find an estimated value for \(2.025 ^ { 10 }\), stating the value of \(x\) which you have used and showing your working.
Edexcel C12 2015 January Q5
5. (a) Prove that the sum of the first \(n\) terms of an arithmetic series is given by the formula $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ where \(a\) is the first term of the series and \(d\) is the common difference between the terms.
(b) Find the sum of the integers which are divisible by 7 and lie between 1 and 500
Edexcel C12 2015 January Q6
6. Given that $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$
  1. show that $$4 x ^ { 2 } - 16 x - 9 = 0$$
  2. Hence solve the equation $$2 \log _ { 4 } ( 2 x + 3 ) = 1 + \log _ { 4 } x + \log _ { 4 } ( 2 x - 1 ) , \quad x > \frac { 1 } { 2 }$$
Edexcel C12 2015 January Q7
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 18 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\). The circle \(C\) meets the line with equation \(x = - 3\) at two points.
  3. Find the exact values for the \(y\) coordinates of these two points, giving your answers as fully simplified surds.
Edexcel C12 2015 January Q8
  1. A sequence is defined by
$$\begin{aligned} u _ { 1 } & = k
u _ { n + 1 } & = 3 u _ { n } - 12 , \quad n \geqslant 1 \end{aligned}$$ where \(k\) is a constant.
  1. Write down fully simplified expressions for \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\) in terms of \(k\). Given that \(u _ { 4 } = 15\)
  2. find the value of \(k\),
  3. find \(\sum _ { i = 1 } ^ { 4 } u _ { i }\), giving an exact numerical answer.
Edexcel C12 2015 January Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-13_460_698_269_625} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} In Figure 3, the points \(A\) and \(B\) are the centres of the circles \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The circle \(C _ { 1 }\) has radius 10 cm and the circle \(C _ { 2 }\) has radius 5 cm . The circles intersect at the points \(X\) and \(Y\), as shown in the figure. Given that the distance between the centres of the circles is 12 cm ,
  1. calculate the size of the acute angle \(X A B\), giving your answer in radians to 3 significant figures,
  2. find the area of the major sector of circle \(C _ { 1 }\), shown shaded in Figure 3,
  3. find the area of the kite \(A Y B X\).
Edexcel C12 2015 January Q10
10. $$f ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x - 5$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) there is no remainder.
When \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is - 15
  1. Find the value of \(a\) and the value of \(b\).
  2. Factorise \(\mathrm { f } ( x )\) completely.