Questions — Edexcel (10514 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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 C4 Q1
8 marks Standard +0.3
  1. A curve has the equation
$$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\). (8)
Edexcel C4 Q2
9 marks Standard +0.3
2.
  1. Expand \(( 4 - x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying each coefficient.
  2. State the set of values of \(x\) for which your expansion is valid.
  3. Use your expansion with \(x = 0.01\) to find the value of \(\sqrt { 399 }\), giving your answer to 9 significant figures.
Edexcel C4 Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-04_560_1059_146_406} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \ln ( 2 + \cos x ) , 0 \leq x \leq \pi\).
  1. Complete the table below for points on the curve, giving the \(y\) values to 4 decimal places.
  2. Giving your answers to 3 decimal places, find estimates for the area of the region bounded by the curve and the coordinate axes using the trapezium rule with
    1. 1 strip,
    2. 2 strips,
    3. 4 strips.
  3. Making your reasoning clear, suggest a value to 2 decimal places for the actual area of the region bounded by the curve and the coordinate axes.
    \(x\)0\(\frac { \pi } { 4 }\)\(\frac { \pi } { 2 }\)\(\frac { 3 \pi } { 4 }\)\(\pi\)
    \(y\)1.09860
    1. continued
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-06_563_983_146_379} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Edexcel C4 Q5
11 marks Standard +0.3
5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.
  1. Show that \(A , B\) and \(C\) all lie on a single straight line.
  2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
  3. Show that \(A D\) is perpendicular to \(B D\).
  4. Find the exact area of triangle \(A B D\).
    5. continued
Edexcel C4 Q6
11 marks Standard +0.3
6.
  1. Use the substitution \(x = 2 \sin u\) to evaluate $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$$
  2. Use integration by parts to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \cos x d x$$ 6. continued
Edexcel C4 Q7
15 marks Standard +0.8
7. When a plague of locusts attacks a wheat crop, the proportion of the crop destroyed after \(t\) hours is denoted by \(x\). In a model, it is assumed that the rate at which the crop is destroyed is proportional to \(x ( 1 - x )\). A plague of locusts is discovered in a wheat crop when one quarter of the crop has been destroyed. Given that the rate of destruction at this instant is such that if it remained constant, the crop would be completely destroyed in a further six hours,
  1. show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 } x ( 1 - x )\),
  2. find the percentage of the crop destroyed three hours after the plague of locusts is first discovered.
    7. continued
    7. continued
Edexcel C4 Q1
6 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{80bef9d4-b84c-4d3a-a093-67a466c6d1b9-02_615_791_146_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Find the volume of the solid formed when the shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.
□
Edexcel C4 Q2
7 marks Standard +0.3
2.
  1. Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
  2. Hence, or otherwise, show that for small \(x\), $$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$
Edexcel C4 Q3
11 marks Standard +0.3
3. $$f ( x ) = \frac { 7 + 3 x + 2 x ^ { 2 } } { ( 1 - 2 x ) ( 1 + x ) ^ { 2 } } , \quad | x | > \frac { 1 } { 2 }$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that $$\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = p - \ln q$$ where \(p\) is rational and \(q\) is an integer.
    3. continued
Edexcel C4 Q4
11 marks Standard +0.3
4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \end{aligned}$$
Edexcel C4 Q5
12 marks Moderate -0.8

& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right)
Edexcel C4 Q7
15 marks Moderate -0.5
7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + \mu \left( \begin{array} { c } - 5
Edexcel C4 Q14
Standard +0.3
14
2 \end{array} \right) , $$ and\\ where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.\\ Given that the two lines intersect,\\
  1. find the position vector of their point of intersection,
  2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
  3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\ 4. continued\\ 5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
  2. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  3. Find the coordinates of \(Q\).\\ 5. continued\\ 6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
\\
  1. Write down a differential equation connecting \(N\) and \(t\). Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
  2. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
  3. find the value of \(k\),
  4. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.\\ 6. continued\\ 7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$
  1. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
  2. find a cartesian equation for the curve.
  3. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
  4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    7. continued
    7. continued
Edexcel S1 Q1
7 marks Moderate -0.8
  1. A histogram is to be drawn to represent the following grouped continuous data:
Group\(0 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 50\)\(50 - 100\)
Frequency\(2 x\)\(3 x\)\(5 x\)\(6 x\)\(2 x\)\(x\)
The ' \(10 - 20\) ' bar has height 6 cm and width 4 cm . Calculate
  1. the height of the ' \(20 - 25\) ' bar,
  2. the total area under the histogram.
Edexcel S1 Q2
7 marks Moderate -0.8
2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find
  1. \(\mathrm { P } ( B )\),
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
  4. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
Edexcel S1 Q3
9 marks Moderate -0.8
3. The random variable \(X\) has the discrete uniform distribution over the set of consecutive integers \(\{ - 7 , - 6 , \ldots , 10 \}\).
Calculate (a) the expectation and variance of \(X\),
(b) \(\mathrm { P } ( X > 7 )\),
(c) the value of \(n\) for which \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).
Edexcel S1 Q4
9 marks Moderate -0.8
4. The marks, \(x\) out of 100 , scored by 30 candidates in an examination were as follows:
5192021232531373941
42444751565760616265
677071737577818298100
Given that \(\sum x = 1600\) and \(\sum x ^ { 2 } = 102400\),
  1. find the median, the mean and the standard deviation of these marks. The marks were scaled to give modified scores, \(y\), using the formula \(y = \frac { 4 x } { 5 } + 20\).
  2. Find the median, the mean and the standard deviation of the modified scores. \section*{STATISTICS 1 (A) TEST PAPER 1 Page 2}
Edexcel S1 Q5
12 marks Moderate -0.8
  1. The table shows the numbers of cars and vans in a company's fleet having registrations with the prefix letters shown.
Registration letter\(K\)\(L\)\(M\)\(N\)\(P\)\(R\)\(S\)\(T\)\(V\)
Number of cars \(( x )\)67911151412107
Number of vans \(( y )\)810141313151498
  1. Plot a scatter graph of this data, with the number of cars on the horizontal axis and the number of vans on the vertical axis.
  2. If there were \(4 J\)-registered cars, estimate the number of \(J\)-registered vans. Given that \(\sum x ^ { 2 } = 1001 , \sum y ^ { 2 } = 1264\) and \(\sum x y = 1106\),
  3. calculate the product-moment correlation coefficient between \(x\) and \(y\). Give a brief interpretation of your answer.
Edexcel S1 Q6
14 marks Moderate -0.3
6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables:
\(x\)012
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 10 }\)
\(y\)01
\(\mathrm { P } ( Y = y )\)\(\frac { 5 } { 8 }\)\(\frac { 3 } { 8 }\)
The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).
  1. Tabulate the probability distribution for \(Z\).
  2. Calculate \(\mathrm { E } ( Z )\).
  3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
  4. Calculate Var (3Z-4).
Edexcel S1 Q7
17 marks Standard +0.3
7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.
  1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
  2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
  3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
  4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.
Edexcel S1 Q1
8 marks Moderate -0.8
  1. Twelve observations are made of a random variable \(X\). This set of observations has mean 13 and variance \(10 \cdot 2\).
Another twelve observations of \(X\) are such that \(\sum x = 164\) and \(\sum x ^ { 2 } = 2372\).
Find the mean and the variance for all twenty-four observations.
Edexcel S1 Q2
11 marks Moderate -0.3
2. Given that \(\mathrm { P } ( A ) = \frac { 3 } { 5 } , \mathrm { P } ( B ) = \frac { 5 } { 8 } , \mathrm { P } ( A \cap B ) = \frac { 7 } { 20 } , \mathrm { P } ( A \cup C ) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid A ) = \frac { 1 } { 3 }\),
  1. determine whether \(A\) and \(B\) are independent events.
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } \left( ( A \cap C ) ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( A \mid C )\).
Edexcel S1 Q3
13 marks Moderate -0.3
3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .
Length \(( \mathrm { cm } )\)\(5 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 40\)\(40 - 60\)\(60 - 90\)
Frequency8162018201412
  1. Calculate estimates of the three quartiles of the distribution.
  2. On graph paper, draw a box and whisker plot of the data.
  3. Hence describe the skewness of the distribution.
  4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?
Edexcel S1 Q4
13 marks Challenging +1.2
4. A botanist believes that the lengths of the branches on trees of a certain species can be modelled by a normal distribution.
When he measures the lengths of 500 branches, he finds 55 which are less than 30 cm long and 200 which are more than 90 cm long.
  1. Find the mean and the standard deviation of the lengths.
  2. In a sample of 1000 branches, how many would he expect to find with lengths greater than 1 metre? \section*{STATISTICS 1 (A) TEST PAPER 7 Page 2}
Edexcel S1 Q5
13 marks Moderate -0.3
  1. Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled \(1,2,3\) and 4 . Both spinners are biased and the probability distributions for the scores \(X\) and \(Y\) obtained when they are spun are respectively:
\(x\)123
\(\mathrm { P } ( X = x )\)\(0 \cdot 2\)\(0 \cdot 4\)\(p\)
\(Y\)1234
\(\mathrm { P } ( Y = y )\)0.20.5\(q\)\(q\)
  1. Find the values of \(p\) and \(q\).
  2. Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .
  3. State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.
  4. Calculate \(\mathrm { E } ( X + Y )\).