Questions — Pre-U Pre-U 9794/1 (194 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
Pre-U Pre-U 9794/1 2019 Specimen Q4
1 marks Moderate -0.8
4
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2019 Specimen Q7
4 marks Moderate -0.3
7
  1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x - 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
  2. Hence show that \(\equiv \frac { 5 x - 1 } { \overline { 2 } } \frac { 8 x - 1 ) ( x + 1 ) } { ( 2 x - \ln 24 \text {. } }\)
Pre-U Pre-U 9794/1 2019 Specimen Q9
2 marks Easy -1.2
9 The complex number 3-4i is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(2 z + z ^ { * }\),
  2. \(\frac { 5 } { z }\).
  3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.
Pre-U Pre-U 9794/1 2019 Specimen Q10
4 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \neq } { 4 } \right) + \frac { \sin \left( \theta + \frac { \neq } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \neq } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2019 Specimen Q11
7 marks Standard +0.8
11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).
  1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
  2. Find the sum to infinity of the geometric progression in terms of \(a\).
Pre-U Pre-U 9794/1 2019 Specimen Q12
2 marks Standard +0.8
12
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
  2. Find
    1. \(\quad \int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
    2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2020 Specimen Q6
6 marks Moderate -0.5
6 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - 2 x y + y ^ { 2 } = 9 .$$
Pre-U Pre-U 9794/1 2020 Specimen Q7
4 marks Moderate -0.3
7
  1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
  2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).
Pre-U Pre-U 9794/1 2020 Specimen Q9
2 marks Easy -1.2
9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(2 z + z ^ { * }\),
  2. \(\frac { 5 } { z }\).
  3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.
Pre-U Pre-U 9794/1 2020 Specimen Q10
4 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2020 Specimen Q12
2 marks Standard +0.8
12
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
  2. Find
    1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
    2. \(\int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 Specimen Q1
3 marks Easy -1.8
1 Find the set of all real values of \(x\) which satisfy the equation $$| 2 x + 5 | < 7$$
Pre-U Pre-U 9794/1 Specimen Q2
4 marks Moderate -0.3
2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 x + 1 } { x - 3 }\) for all real \(x , x \neq 3\). Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
Pre-U Pre-U 9794/1 Specimen Q3
5 marks Moderate -0.8
3 Show that the equation of the tangent to the curve \(y = \ln \left( x ^ { 2 } + 3 \right)\) at the point \(( 1 , \ln 4 )\) is $$2 y - x = \ln ( 16 ) - 1$$
Pre-U Pre-U 9794/1 Specimen Q4
4 marks Standard +0.8
4 The diagram shows triangle \(A B C\), in which \(A B = 1\) unit , \(A C = k\) units and \(B C = 2\) units .
  1. Express \(\cos C\) in terms of \(k\).
  2. Given that \(\cos C < \frac { 7 } { 8 }\), show that \(2 k ^ { 2 } - 7 k + 6 < 0\) and find the set of possible values of \(k\).
Pre-U Pre-U 9794/1 Specimen Q5
4 marks Moderate -0.3
5
  1. Show that the equation \(4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x\) can be expressed in the form $$3 \sin ^ { 2 } x - 4 \sin x + 1 = 0$$
  2. Hence find all values of \(x\) for which \(0 < x < \pi\) that satisfy the equation $$4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x$$
Pre-U Pre-U 9794/1 Specimen Q6
6 marks Moderate -0.3
6 The equation \(x ^ { 3 } - x - 1 = 0\) has exactly one real root in the interval \(0 \leq x \leq 3\).
  1. Denoting this root by \(\alpha\), find the integer \(n\) such that \(n < \alpha < n + 1\).
  2. Taking \(n\) as a first approximation, use the Newton-Raphson method to find \(\alpha\), correct to 2 decimal places. You must show the result of each iteration correct to an appropriate degree of accuracy.
Pre-U Pre-U 9794/1 Specimen Q7
8 marks Standard +0.3
7 Express \(\frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in the form \(\frac { A } { x + 2 } + \frac { B x + C } { x ^ { 2 } + 1 }\) where the numerical values of \(A , B\) and \(C\) are to be found. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x = \ln 3 - \frac { 5 } { 2 } \ln 2\).
Pre-U Pre-U 9794/1 Specimen Q8
9 marks Standard +0.3
8
  1. Show that the lines $$\mathbf { r } = - 3 \mathbf { i } + \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + \mathbf { 6 } \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } + \mu ( - 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ intersect and write down the coordinates of their point of intersection.
  2. Find in degrees the obtuse angle between the two lines.
Pre-U Pre-U 9794/1 Specimen Q9
5 marks Moderate -0.3
9
  1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
  2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.
Pre-U Pre-U 9794/1 Specimen Q10
7 marks Standard +0.8
10
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin 3 x - 3 x \cos 3 x ) = 9 x \sin 3 x\). The curve shown in the figure below is part of the graph of the function \(y = x \sin 3 x\). \includegraphics[max width=\textwidth, alt={}, center]{3e4281d1-dbad-46a2-bbb7-97706bda2dfa-3_508_1136_1939_466}
  2. Show that \(\int _ { 0 } ^ { \frac { 2 \pi } { 3 } } | x \sin 3 x | \mathrm { d } x = \frac { 4 \pi } { 9 }\).
Pre-U Pre-U 9794/1 Specimen Q11
11 marks Challenging +1.8
11 A sequence of terms \(x _ { n }\) generated by a recurrence relation is said to be strictly increasing if, for each \(x _ { n } , x _ { n + 1 } > x _ { n }\).
  1. Let a recurrence relation be defined by $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 } \quad \text { and } \quad x _ { 0 } = \frac { 1 } { 2 } \quad \text { for } n \geq 0$$ Calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) correct to 3 significant figures where appropriate.
  2. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 }$$ show that the sequence is strictly increasing when \(x _ { n } > 2\) or \(x _ { n } < 1\).
  3. If \(- 1 < x _ { 0 } < 1\), then the sequence \(x _ { n } ( n \geq 0 )\) converges to a limit. Explain briefly why this limit is 1 .
  4. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + k } { m } \text { with } m > 0$$ prove that \(x _ { n }\) is a strictly increasing sequence for all \(x _ { n }\) if \(m ^ { 2 } < 4 k\).
Pre-U Pre-U 9794/1 Specimen Q12
6 marks Moderate -0.8
12 A set of data is shown in the table below.
\(x\)012345678
frequency3104320001
  1. Calculate the mean and standard deviation of the data. The value 8 may be regarded as an outlier.
  2. Explain how you would treat this outlier if the data represents
    1. the difference of the scores obtained when throwing a pair of ordinary dice,
    2. the number of thunderstorms per year in Cambridgeshire over a 23-year period.
    3. Without doing any further calculations state what effect, if any, removing the outlier would have on the mean and standard deviation.
Pre-U Pre-U 9794/1 Specimen Q13
9 marks Moderate -0.3
13 A seed company investigated how well African Marigold seeds germinated when the seeds were past their sell-by date. The table shows the average number of seeds which germinated per packet, \(y\), and the number of months past their sell-by date, \(t\).
\(t\)1020304050
\(y\)24.524.021.718.612.4
The summary data for the investigation were as follows. $$\Sigma t = 150 \quad \Sigma t ^ { 2 } = 5500 \quad \Sigma y = 101.2 \quad \Sigma y ^ { 2 } = 2146.86 \quad \Sigma t y = 2740$$
  1. Calculate the equation of the regression line of \(y\) on \(t\).
  2. Use your regression line to calculate \(y\) when \(t = 10\). Compare your answer with the value of \(y\) when \(t = 10\) in the table and comment on the result.
  3. Use your regression line to calculate \(y\) when \(t = 100\). Comment on the validity of this result.
  4. Suggest with reasons whether the regression line provides a good model for predicting the germination of seeds past their sell-by date.
Pre-U Pre-U 9794/1 Specimen Q14
14 marks Moderate -0.3
14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.
  1. Find the probability that the maximum pressure for a randomly chosen patient is more than 160.
  2. If the maximum pressure is found to be \(t\) or more, the patient must be referred to a consultant. If \(5 \%\) of the patients are referred to a consultant, find the value of \(t\).
  3. Find the percentage of patients whose maximum pressure is between 130 and 160 . The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .
  4. Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.
  5. If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.