Questions (30179 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 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 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
Sort by: Default | Easiest first | Hardest first
Edexcel C12 2016 October Q5
7 marks Moderate -0.3
5. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$ where \(a\) is a positive constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 3 }\),
(b) find the exact value of \(a\) in its simplest form.
Edexcel C12 2016 October Q6
9 marks Easy -1.2
6. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 36 \\ u _ { n + 1 } & = \frac { 2 } { 3 } u _ { n } , \quad n \geqslant 1 \end{aligned}$$
  1. Find the exact simplified values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
  2. Write down the common ratio of the sequence.
  3. Find, giving your answer to 4 significant figures, the value of \(u _ { 11 }\)
  4. Find the exact value of \(\sum _ { i = 1 } ^ { 6 } u _ { i }\)
  5. Find the value of \(\sum _ { i = 1 } ^ { \infty } u _ { i }\)
Edexcel C12 2016 October Q7
6 marks Easy -1.2
  1. (a) Sketch the graph of \(y = 3 ^ { x - 2 } , x \in \mathbb { R }\)
Give the exact values for the coordinates of the point where your graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = 3 ^ { x - 2 }\) The values of \(y\) are rounded to 3 decimal places where necessary.
\(x\)0.511.522.53
\(y\)0.1920.3330.57711.7323
(b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { 0.5 } ^ { 3 } 3 ^ { x - 2 } \mathrm {~d} x$$ Give your answer to 2 decimal places.
Edexcel C12 2016 October Q8
8 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-20_545_1048_212_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The compound shape \(A B C D A\), shown in Figure 1, consists of a triangle \(A B D\) joined along its edge \(B D\) to a sector \(D B C\) of a circle with centre \(B\) and radius 6 cm . The points \(A , B\) and \(C\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B C = 6 \mathrm {~cm}\). Angle \(D A B = 1.1\) radians.
  1. Show that angle \(A B D = 1.20\) radians to 3 significant figures.
  2. Find the area of the compound shape, giving your answer to 3 significant figures.
Edexcel C12 2016 October Q9
8 marks Moderate -0.8
  1. In a large theatre there are 20 rows of seats.
The number of seats in the first row is \(a\), where \(a\) is a constant. In the second row the number of seats is \(( a + d )\), where \(d\) is a constant. In the third row the number of seats is \(( a + 2 d )\), and on each subsequent row there are \(d\) more seats than on the previous row. The number of seats in each row forms an arithmetic sequence. The total number of seats in the first 10 rows is 395
  1. Use this information to show that \(10 a + 45 d = 395\) The total number of seats in the first 18 rows is 927
  2. Use this information to write down a second simplified equation relating \(a\) and \(d\).
  3. Solve these equations to find the value of \(a\) and the value of \(d\).
  4. Find the number of seats in the 20th row of the theatre.
Edexcel C12 2016 October Q10
8 marks Standard +0.3
10. (a) Given that $$8 \tan x = - 3 \cos x$$ show that $$3 \sin ^ { 2 } x - 8 \sin x - 3 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$8 \tan 2 \theta = - 3 \cos 2 \theta$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-29_124_37_2615_1882}
Edexcel C12 2016 October Q11
8 marks Standard +0.3
11. The equation \(5 x ^ { 2 } + 6 = k \left( 13 x ^ { 2 } - 12 x \right)\), where \(k\) is a constant, has two distinct real roots.
  1. Show that \(k\) satisfies the inequality $$6 k ^ { 2 } + 13 k - 5 > 0$$
  2. Find the set of possible values for \(k\).
Edexcel C12 2016 October Q12
11 marks Moderate -0.3
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-32_748_883_274_477} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$ The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).
  1. Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
  2. Use calculus to find the coordinates of the points \(C\) and \(D\). The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
  3. Write down the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-35_29_37_182_1914}
Edexcel C12 2016 October Q13
13 marks Standard +0.3
13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).
  1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
  2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
  3. Find the coordinates of the point \(Q\).
Edexcel C12 2016 October Q14
11 marks Challenging +1.2
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-40_456_689_269_623} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = - x ^ { 2 } + 6 x - 8\). The normal to \(C\) at the point \(P ( 5 , - 3 )\) is the line \(l\), which is also shown in Figure 3.
  1. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. The finite region \(R\), shown shaded in Figure 3, is bounded below by the line \(l\) and the curve \(C\), and is bounded above by the \(x\)-axis.
  2. Find the exact value of the area of \(R\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C12 2016 October Q15
13 marks Standard +0.3
15. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-44_647_917_260_484} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a solid wooden block. The block is a right prism with length \(h \mathrm {~cm}\). The cross-section of the block is a semi-circle with radius \(r \mathrm {~cm}\). The total surface area of the block, including the curved surface, the two semi-circular ends and the rectangular base, is \(200 \mathrm {~cm} ^ { 2 }\)
  1. Show that the volume \(V \mathrm {~cm} ^ { 3 }\) of the block is given by $$V = \frac { \pi r \left( 200 - \pi r ^ { 2 } \right) } { 4 + 2 \pi }$$
  2. Use calculus to find the maximum value of \(V\). Give your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
  3. Justify, by further differentiation, that the value of \(V\) that you have found is a maximum.
Edexcel C12 2017 October Q1
4 marks Easy -1.2
  1. The line \(l _ { 1 }\) has equation
$$8 x + 2 y - 15 = 0$$
  1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 3 } { 4 } , 16 \right)\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel C12 2017 October Q2
4 marks Easy -1.3
2. The point \(P ( 2,3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\). State the coordinates of the image of \(P\) under the transformation represented by the curve with equation
  1. \(y = \mathrm { f } ( x + 2 )\)
  2. \(y = - \mathrm { f } ( x )\)
  3. \(2 y = f ( x )\)
  4. \(y = \mathrm { f } ( x ) - 4\) State the coordinates of the image of \(P\) under the transformation represented by the curve
    with equation (a) \(y = \mathrm { f } ( x + 2 )\)
Edexcel C12 2017 October Q3
6 marks Moderate -0.8
3. (a) Express \(\frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } }\) in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants.
(b) Hence find $$\int \frac { x ^ { 3 } + 4 } { 2 x ^ { 2 } } d x$$ simplifying your answer.
Edexcel C12 2017 October Q4
6 marks Moderate -0.5
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-08_287_689_255_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a triangle \(A B C\) with \(A B = 3 x \mathrm {~cm} , A C = x \mathrm {~cm}\) and angle \(C A B = 60 ^ { \circ }\) Given that the area of triangle \(A B C = 24 \sqrt { 3 }\)
  1. show that \(x = 4 \sqrt { 2 }\)
  2. Hence find the exact length of \(B C\), giving your answer as a simplified surd.
Edexcel C12 2017 October Q5
8 marks Moderate -0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-10_678_1076_248_434} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 27 \sqrt { x } - 2 x ^ { 2 } , \quad x \in \mathbb { R } , x > 0$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The curve has a maximum turning point \(P\), as shown in Figure 2.
  2. Use the answer to part (a) to find the exact coordinates of \(P\).
Edexcel C12 2017 October Q6
7 marks Moderate -0.8
  1. Each year Lin pays into a savings scheme. In year 1 she pays in \(\pounds 600\). Her payments then increase by \(\pounds 80\) a year, so that she pays \(\pounds 680\) into the savings scheme in year \(2 , \pounds 760\) in year 3 and so on. In year \(N\), Lin pays \(\pounds 1000\) into the savings scheme.
    1. Find the value of \(N\).
    2. Find the total amount that Lin pays into the savings scheme from year 1 to year 15 inclusive.
    Saima starts paying into a different savings scheme at the same time as Lin starts paying into her savings scheme. In year 1 she pays in \(\pounds A\). Her payments increase by \(\pounds A\) each year so that she pays \(\pounds 2 A\) in year \(2 , \pounds 3 A\) in year 3 and so on. Given that Saima and Lin have each paid, in total, the same amount of money into their savings schemes after 15 years,
  2. find the value of \(A\).
Edexcel C12 2017 October Q7
9 marks Standard +0.3
7. $$g ( x ) = 2 x ^ { 3 } + a x ^ { 2 } - 18 x - 8$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\),
  1. show that \(a = - 3\)
  2. Hence, using algebra, fully factorise \(\mathrm { g } ( x )\). Using your answer to part (b),
  3. solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$2 \sin ^ { 3 } \theta - 3 \sin ^ { 2 } \theta - 18 \sin \theta = 8$$ giving each answer, in radians, as a multiple of \(\pi\).
Edexcel C12 2017 October Q8
6 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-18_387_397_255_794} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circumference of this circle. The minor arc \(A B\) subtends an angle \(\theta\) radians at \(O\), as shown in Figure 3.
Given the length of minor \(\operatorname { arc } A B\) is 6 cm and the area of minor sector \(O A B\) is \(20 \mathrm {~cm} ^ { 2 }\),
  1. write down two different equations in \(r\) and \(\theta\).
  2. Hence find the value of \(r\) and the value of \(\theta\).
Edexcel C12 2017 October Q9
10 marks Moderate -0.8
  1. (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of
$$y = a ^ { x } , \quad x \in \mathbb { R }$$ On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2) The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\)
\(x\)- 4- 2024
\(y\)0.06250.251416
(b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for $$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$ (c) Use the answer to part (b) to find an approximate value for
  1. \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
  2. \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
    \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-23_86_47_2617_1886}
Edexcel C12 2017 October Q10
9 marks Moderate -0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-24_863_929_255_511} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Diagram NOT drawn to scale The points \(A ( 7 , - 3 ) , B ( 7,20 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 4. The point \(D ( 10,5 )\) is the midpoint of \(A C\).
  1. Find the value of \(p\) and the value of \(q\). The line \(l\) passes through \(D\) and is perpendicular to \(A C\).
  2. Find an equation for \(l\), in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. Given that the line \(l\) intersects \(A B\) at \(E\),
  3. find the exact coordinates of \(E\).
Edexcel C12 2017 October Q11
7 marks Standard +0.3
11. \(\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }\), where \(a\) is a positive constant
  1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of \(\mathrm { f } ( x )\) the coefficient of \(x\) is zero,
  2. find the exact value of \(a\).
Edexcel C12 2017 October Q12
11 marks Standard +0.3
12. (i) Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(ii) (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.
Edexcel C12 2017 October Q13
9 marks Moderate -0.8
  1. The circle \(C\) has equation
$$( x - 3 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 30$$ Write down
    1. the coordinates of the centre of \(C\),
    2. the exact value of the radius of \(C\). Given that the point \(P\) with coordinates \(( 6 , k )\), where \(k\) is a constant, lies inside circle \(C\), (b) show that $$k ^ { 2 } + 8 k - 5 < 0$$
  2. Hence find the exact set of values of \(k\) for which \(P\) lies inside \(C\). \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-34_2256_52_315_1978}
Edexcel C12 2017 October Q14
10 marks Moderate -0.3
  1. A new mineral has been discovered and is going to be mined over a number of years.
A model predicts that the mass of the mineral mined each year will decrease by \(15 \%\) per year, so that the mass of the mineral mined each year forms a geometric sequence. Given that the mass of the mineral mined during year 1 is 8000 tonnes,
  1. show that, according to the model, the mass of the mineral mined during year 6 will be approximately 3550 tonnes. According to the model, there is a limit to the total mass of the mineral that can be mined.
  2. With reference to the geometric series, state why this limit exists.
  3. Calculate the value of this limit. It is decided that a total mass of 40000 tonnes of the mineral is required. This is going to be mined from year 1 to year \(N\) inclusive.
  4. Assuming the model, find the value of \(N\).