Questions — Edexcel (9670 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 Specimen Q1
Simplify fully
  1. \(\left( 25 x ^ { 4 } \right) ^ { \frac { 1 } { 2 } }\),
  2. \(\left( 25 x ^ { 4 } \right) ^ { - \frac { 3 } { 2 } }\).
Edexcel C12 Specimen Q3
3. Answer this question without the use of a calculator and show all your working.
  1. Show that $$( 5 - \sqrt { 8 } ) ( 1 + \sqrt { 2 } ) \equiv a + b \sqrt { 2 }$$ giving the values of the integers \(a\) and \(b\).
  2. Show that $$\sqrt { 80 } + \frac { 30 } { \sqrt { 5 } } \equiv c \sqrt { 5 } , \text { where } c \text { is an integer. }$$
Edexcel C12 Specimen Q4
4. Given that \(y = 2 x ^ { 5 } + 7 + \frac { 1 } { x ^ { 3 } } , x \neq 0\), find, in their simplest form,
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\int y \mathrm {~d} x\).
Edexcel C12 Specimen Q5
5. $$y = \frac { 5 } { 3 x ^ { 2 } - 2 }$$ The table below gives values of \(y\) rounded to 3 decimal places where necessary.
\(x\)22.252.52.753
\(y\)0.50.3790.2990.2420.2
Use the trapezium rule, with all the values of \(y\) from the table above, to find an approximate value for $$\int _ { 2 } ^ { 3 } \frac { 5 } { 3 x ^ { 2 } - 2 } d x$$ © Pearson Education Limited 2013
Sample Assessment Materials
Edexcel C12 Specimen Q6
6. $$\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } + 2 x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\), the remainder is 7
  1. Show that \(a + b = 3\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ), the remainder is - 8
  2. Find the value of \(a\) and the value of \(b\).
Edexcel C12 Specimen Q7
7. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{gathered} a _ { 1 } = 2
a _ { n + 1 } = 3 a _ { n } - c \end{gathered}$$ where \(c\) is a constant.
  1. Find an expression for \(a _ { 2 }\) in terms of \(c\). Given that \(\sum _ { i = 1 } ^ { 3 } a _ { i } = 0\)
  2. find the value of \(c\).
Edexcel C12 Specimen Q8
8. The equation $$( k + 3 ) x ^ { 2 } + 6 x + k = 5 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).
  1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 24 < 0$$
  2. Hence find the set of possible values of \(k\).
Edexcel C12 Specimen Q9
9. Given that \(y = 3 x ^ { 2 }\),
  1. show that \(\log _ { 3 } y = 1 + 2 \log _ { 3 } x\)
  2. Hence, or otherwise, solve the equation $$1 + 2 \log _ { 3 } x = \log _ { 3 } ( 28 x - 9 )$$
Edexcel C12 Specimen Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-18_508_812_306_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 2 } ( 9 - 2 x ) .$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).
  1. Write down the coordinates of the point \(A\).
  2. On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( 3 x )\). On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
  3. Write down the value of \(k\).
Edexcel C12 Specimen Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-22_337_892_278_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The straight line with equation \(y = x + 4\) cuts the curve with equation \(y = - x ^ { 2 } + 2 x + 24\) at the points \(A\) and \(B\), as shown in Figure 2.
  1. Use algebra to find the coordinates of the points \(A\) and \(B\). The finite region \(R\) is bounded by the straight line and the curve and is shown shaded in Figure 2.
  2. Use calculus to find the exact area of \(R\).
Edexcel C12 Specimen Q12
12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)
  1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
  2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
    Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
  3. find the length of \(P Q\), giving your answer in its simplest surd form.
Edexcel C12 Specimen Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-28_374_410_278_776} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius \(x\) metres with two equal rectangles attached to it along its radii. Each rectangle has length equal to \(x\) metres and width equal to \(y\) metres. Given that the area of the flowerbed is \(4 \mathrm {~m} ^ { 2 }\),
  1. show that $$y = \frac { 16 - \pi x ^ { 2 } } { 8 x }$$
  2. Hence show that the perimeter \(P\) metres of the flowerbed is given by the equation $$P = \frac { 8 } { x } + 2 x$$
  3. Use calculus to find the minimum value of \(P\).
Edexcel C12 Specimen Q14
  1. In this question you must show all stages of your working. (Solutions based entirely on graphical or numerical methods are not acceptable.)
    1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place,
    $$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$
  2. Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians.
    \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-35_108_95_2572_1804}
Edexcel C12 Specimen Q15
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1528bec3-7a7a-42c5-bac2-756ff3493818-36_394_608_287_676} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The triangle \(X Y Z\) in Figure 4 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 4 is a major arc of the circle with centre \(X\) and radius 4 cm .
  1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
  2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 4. Calculate
  3. the area of this shaded region,
  4. the perimeter \(Z W Y Z\) of this shaded region.
    \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-39_90_54_2576_1868}
Edexcel C12 Specimen Q16
16. Maria trains for a triathlon, which involves swimming, cycling and running. On the first day of training she swims 1.5 km and then she swims 1.5 km on each of the following days.
  1. Find the total distance that Maria swims in the first 17 days of training. Maria also runs 1.5 km on the first day of training and on each of the following days she runs 0.25 km further than on the previous day. So she runs 1.75 km on the second day and 2 km on the third day and so on.
  2. Find how far Maria runs on the 17th day of training. Maria also cycles 1.5 km on the first day of training and on each of the following days she cycles \(5 \%\) further than on the previous day.
  3. Find the total distance that Maria cycles in the first 17 days of training.
  4. Find the total distance Maria travels by swimming, running and cycling in the first 17 days of training. Maria needs to cycle 40 km in the triathlon.
  5. On which day of training does Maria first cycle more than 40 km ?
Edexcel C1 2005 January Q1
  1. Write down the value of \(16 ^ { \frac { 1 } { 2 } }\).
  2. Find the value of \(16 ^ { - \frac { 3 } { 2 } }\).
Edexcel C1 2005 January Q3
3. Given that the equation \(k x ^ { 2 } + 12 x + k = 0\), where \(k\) is a positive constant, has equal roots, find the value of \(k\).
Edexcel C1 2005 January Q4
4. Solve the simultaneous equations $$\begin{gathered} x + y = 2
x ^ { 2 } + 2 y = 12 \end{gathered}$$
Edexcel C1 2005 January Q5
5. The \(r\) th term of an arithmetic series is ( \(2 r - 5\) ).
  1. Write down the first three terms of this series.
  2. State the value of the common difference.
  3. Show that \(\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )\).
Edexcel C1 2005 January Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-10_515_714_292_609}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\). In separate diagrams sketch the curve with equation
  1. \(y = - \mathrm { f } ( x )\),
  2. \(y = \mathrm { f } ( 2 x )\). On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.
Edexcel C1 2005 January Q7
7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .
  1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
  2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
  3. Find the value of \(k\).
Edexcel C1 2005 January Q8
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{bace07ee-1eb8-43d6-8229-152d1f74ab59-14_687_1196_280_388}
\end{figure} The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
  2. Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
  3. Find the exact \(x\)-coordinate of \(E\).
Edexcel C1 2005 January Q9
9. The gradient of the curve \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 } .$$ The point \(P ( 1,4 )\) lies on \(C\).
  1. Find an equation of the normal to \(C\) at \(P\).
  2. Find an equation for the curve \(C\) in the form \(y = \mathrm { f } ( x )\).
  3. Using \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }\), show that there is no point on \(C\) at which the tangent is parallel to the line \(y = 1 - 2 x\).
Edexcel C1 2005 January Q10
10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$
  1. express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geqslant 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
  2. In the space provided on page 19, sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
  3. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.
Edexcel C1 2006 January Q1
  1. Factorise completely
$$x ^ { 3 } - 4 x ^ { 2 } + 3 x .$$