Questions — Edexcel (10514 questions)

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Edexcel C12 2018 January Q3
4 marks Easy -1.3
3. Simplify fully
  1. \(\left( 3 x ^ { \frac { 1 } { 2 } } \right) ^ { 4 }\)
  2. \(\frac { 2 y ^ { 7 } \times ( 4 y ) ^ { - 2 } } { 3 y }\)
Edexcel C12 2018 January Q4
7 marks Moderate -0.3
4. The equation \(( p - 2 ) x ^ { 2 } + 8 x + ( p + 4 ) = 0 , \quad\) where \(p\) is a constant has no real roots.
  1. Show that \(p\) satisfies \(p ^ { 2 } + 2 p - 24 > 0\)
  2. Hence find the set of possible values of \(p\).
Edexcel C12 2018 January Q5
11 marks Standard +0.3
5. (In this question, solutions based entirely on graphical or numerical methods are not acceptable.)
  1. Solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$5 \sin 3 \theta - 7 \cos 3 \theta = 0$$ Give each solution, in radians, to 3 significant figures.
  2. Solve, for \(0 < x < 360 ^ { \circ }\) $$9 \cos ^ { 2 } x + 5 \cos x = 3 \sin ^ { 2 } x$$ Give each solution, in degrees, to one decimal place.
Edexcel C12 2018 January Q6
9 marks Moderate -0.3
6. $$f ( x ) = a x ^ { 3 } - 8 x ^ { 2 } + b x + 6$$ 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 \(( x - 2 )\) the remainder is - 12
  1. Find the value of \(a\) and the value of \(b\).
  2. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel C12 2018 January Q7
10 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-18_675_1408_292_358} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a rectangular sheet of metal of negligible thickness, which measures 25 cm by 15 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open cuboid box, as shown in Figure 2.
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the box is given by $$V = 4 x ^ { 3 } - 80 x ^ { 2 } + 375 x$$
  2. Use calculus to find the value of \(x\), to 3 significant figures, for which the volume of the box is a maximum.
  3. Justify that this value of \(x\) gives a maximum value for \(V\).
  4. Find, to 3 significant figures, the maximum volume of the box.
    \section*{8.} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-22_670_1004_292_392} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at the point \(( 0,5 )\) and crosses the \(x\)-axis at the point \(( 6,0 )\). The curve has a minimum point at \(( 1,3 )\) and a maximum point at \(( 4,7 )\). On separate diagrams, sketch the curve with equation
Edexcel C12 2018 January Q9
10 marks Standard +0.3
  1. The first term of a geometric series is 20 and the common ratio is 0.9
    1. Find the value of the fifth term.
    2. Find the sum of the first 8 terms, giving your answer to one decimal place.
    Given that \(S _ { \infty } - S _ { N } < 0.04\), where \(S _ { N }\) is the sum of the first \(N\) terms of this series, (c) show that \(0.9 ^ { N } < 0.0002\)
  2. Hence find the smallest possible value of \(N\).
Edexcel C12 2018 January Q10
10 marks Standard +0.3
10. (i) Use the laws of logarithms to solve the equation $$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$ (ii) Using algebra, find, in terms of logarithms, the exact value of \(y\) for which $$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$
Edexcel C12 2018 January Q11
9 marks Standard +0.8
11. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$ The centre of \(C\) is at the point \(T\).
  1. Find
    1. the coordinates of the point \(T\),
    2. the radius of the circle \(C\). The point \(M\) has coordinates \(( 20,12 )\).
  2. Find the exact length of the line \(M T\). Point \(P\) lies on the circle \(C\) such that the tangent at \(P\) passes through the point \(M\).
  3. Find the exact area of triangle \(M T P\), giving your answer as a simplified surd.
Edexcel C12 2018 January Q12
9 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(x + 3 y - 11 = 0\)
The point \(A\) and the point \(B\) lie on \(l _ { 1 }\) Given that \(A\) has coordinates ( \(- 1 , p\) ) and \(B\) has coordinates ( \(q , 2\) ), where \(p\) and \(q\) are integers,
  1. find the value of \(p\) and the value of \(q\),
  2. find the length of \(A B\), giving your answer as a simplified surd. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\).
  3. Find an equation for \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
Edexcel C12 2018 January Q13
7 marks Moderate -0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-42_840_1010_287_571} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the position of two stationary boats, \(A\) and \(B\), and a port \(P\) which are assumed to be in the same horizontal plane. Boat \(A\) is 8.7 km on a bearing of \(314 ^ { \circ }\) from port \(P\).
Boat \(B\) is 3.5 km on a bearing of \(052 ^ { \circ }\) from port \(P\).
  1. Show that angle \(A P B\) is \(98 ^ { \circ }\)
  2. Find the distance of boat \(B\) from boat \(A\), giving your answer to one decimal place.
  3. Find the bearing of boat \(B\) from boat \(A\), giving your answer to the nearest degree.
Edexcel C12 2018 January Q14
13 marks Standard +0.3
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-46_812_1091_292_429} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the line \(l\) with equation \(y = 8 - x\) and part of the curve \(C\) with equation \(y = 14 + 3 x - 2 x ^ { 2 }\) The line \(l\) and the curve \(C\) intersect at the point \(A\) and the point \(B\) as shown.
  1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the coordinate axes, the line \(l\), and the curve \(C\).
  2. Use algebraic integration to calculate the exact area of \(R\).
Edexcel C12 2018 January Q15
10 marks Standard +0.3
15. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.
  1. Show that \(n k ( n - 1 ) = 252\)
  2. Find the value of \(k\) and the value of \(n\).
  3. Using the values of \(k\) and \(n\), find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 1 + k x ) ^ { n }\)
Edexcel C12 2019 January Q1
3 marks Easy -1.3
  1. A line \(l\) passes through the points \(A ( 5 , - 2 )\) and \(B ( 1,10 )\).
Find the equation of \(l\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants.
(3)
VI4V SIHI NI JIIIM ION OCVI4V SIHI NI JALYM ION OCVJYV SIHI NI JLIYM ION OO
VIIIV SIHI NI JIIIM ION OCVIAV SIHI NI JALM IONOOVJYV SIHL NI GLIYM LON OO
Edexcel C12 2019 January Q2
4 marks Easy -1.2
2. Given \(y = 2 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.
  1. \(2 ^ { 2 x }\)
  2. \(2 ^ { x + 3 }\)
  3. \(\frac { 1 } { 4 ^ { 2 x - 3 } }\)
Edexcel C12 2019 January Q3
4 marks Moderate -0.8
3. A curve has equation $$y = \sqrt { 2 } x ^ { 2 } - 6 \sqrt { x } + 4 \sqrt { 2 } , \quad x > 0$$ Find the gradient of the curve at the point \(P ( 2,2 \sqrt { 2 } )\).
Write your answer in the form \(a \sqrt { 2 }\), where \(a\) is a constant.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \(L\)
Edexcel C12 2019 January Q4
6 marks Moderate -0.8
4. A sequence is defined by $$\begin{aligned} u _ { 1 } & = k , \text { where } k \text { is a constant } \\ u _ { n + 1 } & = 4 u _ { n } - 3 , n \geqslant 1 \end{aligned}$$
  1. Find \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\), simplifying your answers as appropriate. Given \(\sum _ { n = 1 } ^ { 3 } u _ { n } = 18\)
  2. find \(k\).
Edexcel C12 2019 January Q5
7 marks Moderate -0.8
  1. (a) Use the binomial theorem to find the first 4 terms, in ascending powers of \(x\), of the expansion of
$$\left( 1 - \frac { x } { 2 } \right) ^ { 8 }$$ Give each term in its simplest form.
(b) Use the answer to part (a) to find an approximate value to \(0.9 ^ { 8 }\) Write your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers.
Edexcel C12 2019 January Q6
7 marks Easy -1.2
6. (a) Sketch the graph of \(y = 1 + \cos x , \quad 0 \leqslant x \leqslant 2 \pi\) Show on your sketch the coordinates of the points where your graph meets the coordinate axes.
(b) Use the trapezium rule, with 6 strips of equal width, to find an approximate value for $$\int _ { 0 } ^ { 2 \pi } ( 1 + \cos x ) d x$$
Edexcel C12 2019 January Q7
5 marks Moderate -0.3
7. The equation \(2 x ^ { 2 } + 5 p x + p = 0\), where \(p\) is a constant, has no real roots. Find the set of possible values for \(p\).
Edexcel C12 2019 January Q8
5 marks Standard +0.3
8. Given \(k > 3\) and $$\int _ { 3 } ^ { k } \left( 2 x + \frac { 6 } { x ^ { 2 } } \right) \mathrm { d } x = 10 k$$ show that \(k ^ { 3 } - 10 k ^ { 2 } - 7 k - 6 = 0\)
Edexcel C12 2019 January Q9
8 marks Moderate -0.3
9. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 9 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\). The point \(P ( - 2,7 )\) lies on \(C\).
  3. Find an equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C12 2019 January Q10
11 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-20_761_1475_331_239} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the design for a shop sign \(A B C D A\). The sign consists of a triangle \(A O D\) joined to a sector of a circle \(D O B C D\) with radius 1.8 m and centre \(O\). The points \(A , B\) and \(O\) lie on a straight line.
Given that \(A D = 3.9 \mathrm {~m}\) and angle \(B O D\) is 0.84 radians,
  1. calculate the size of angle \(D A O\), giving your answer in radians to 3 decimal places.
  2. Show that, to one decimal place, the length of \(A O\) is 4.9 m .
  3. Find, in \(\mathrm { m } ^ { 2 }\), the area of the shop sign, giving your answer to one decimal place.
  4. Find, in m , the perimeter of the shop sign, giving your answer to one decimal place.
Edexcel C12 2019 January Q11
8 marks Moderate -0.3
11. (i) Given that \(x\) is a positive real number, solve the equation $$\log _ { x } 324 = 4$$ writing your answer as a simplified surd.
(ii) Given that $$\log _ { a } ( 5 y - 4 ) - \log _ { a } ( 2 y ) = 3 \quad y > 0.8,0 < a < 1$$ express \(y\) in terms of \(a\).
Edexcel C12 2019 January Q12
9 marks Moderate -0.5
12. Karen is going to raise money for a charity. She aims to cycle a total distance of 1000 km over a number of days.
On day one she cycles 25 km .
She increases the distance that she cycles each day by \(10 \%\) of the distance cycled on the previous day, until she reaches the total distance of 1000 km . She reaches the total distance of 1000 km on day \(N\), where \(N\) is a positive integer.
  1. Find the value of \(N\). On day one, 50 people donated money to the charity. Each day, 20 more people donated to the charity than did so on the previous day, so that 70 people donated money on day two, 90 people donated money on day three, and so on.
  2. Find the number of people who donated to the charity on day fifteen. Each day, the donation given by each person was \(\pounds 5\)
  3. Find the total amount of money donated by the end of day fifteen.
Edexcel C12 2019 January Q13
10 marks Moderate -0.3
13. \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + c x + 12\), where \(c\) is a constant Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
  1. show that \(c = - 14\)
  2. Write \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$ where \(\mathrm { Q } ( x )\) is a quadratic function.
  3. Use the answer to part (b) to prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real solution. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-32_595_915_1034_518} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). On separate diagrams sketch the curve with equation
    1. \(y = \mathrm { f } ( 3 x )\)
    2. \(y = - \mathrm { f } ( \mathrm { x } )\) On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.