Questions 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 2017 June Q2
5 marks Easy -1.3
2 Express each of the following as a single logarithm.
  1. \(\log 3 + \log 4 - \log 2\)
  2. \(2 \log x - 3 \log y + 2 \log z\)
Pre-U Pre-U 9794/1 2017 June Q3
6 marks Moderate -0.8
3 A triangle \(A B C\) has sides \(A B , B C\) and \(C A\) of lengths \(7 \mathrm {~cm} , 6 \mathrm {~cm}\) and 8 cm respectively.
  1. Show that \(\cos A B C = \frac { 1 } { 4 }\).
  2. Find the area of triangle \(A B C\).
Pre-U Pre-U 9794/1 2017 June Q4
4 marks Moderate -0.3
4 Solve the equation \(\sin 2 x = \sqrt { 3 } \cos x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
Pre-U Pre-U 9794/1 2017 June Q5
5 marks Standard +0.3
5 Solve \(| x - \sqrt { 3 } | < | x + 2 \sqrt { 3 } |\) giving the answer in exact form.
Pre-U Pre-U 9794/1 2017 June Q6
7 marks Standard +0.3
6
  1. Expand \(( 1 + x ) ^ { \frac { 1 } { 2 } }\), for \(| x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
  2. In the expansion of \(( 2 + k x ) ( 1 + x ) ^ { \frac { 1 } { 2 } }\) the coefficient of \(x ^ { 3 }\) is 1 . Find the value of \(k\).
Pre-U Pre-U 9794/1 2017 June Q7
5 marks Standard +0.3
7
  1. Describe the transformation which maps the graph of \(y = \ln x\) onto the graph of \(y = \ln ( 1 + x )\).
  2. By sketching the curves \(y = \ln ( 1 + x )\) and \(y = 4 - x\) on a single diagram, show that the equation $$\ln ( 1 + x ) = 4 - x$$ has exactly one root.
  3. Use the Newton-Raphson method with \(x _ { 0 } = 2\) to find the root of the equation \(\ln ( 1 + x ) = 4 - x\) correct to 3 decimal places. Show the result of each iteration.
Pre-U Pre-U 9794/1 2017 June Q8
7 marks Standard +0.3
8 The curve \(C\) has equation \(y ^ { 3 } + 6 y ^ { 2 } - 2 y = 3 x ^ { 2 } + 2 x\). Show that the equation of the normal to \(C\) at the point \(( 1,1 )\) can be written in the form \(8 y + 13 x - 21 = 0\).
Pre-U Pre-U 9794/1 2017 June Q9
9 marks Standard +0.3
9 Solve the equation \(z ^ { 3 } + 6 z - 20 = 0\). Find the modulus and argument of each root and illustrate the roots on an Argand diagram.
Pre-U Pre-U 9794/1 2017 June Q10
7 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{a3cad2ad-e06b-4aa4-a3a9-a2840cd54893-3_529_527_264_810} The diagram shows the region \(R\) in the first quadrant bounded by the curves \(y = \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right)\) and \(y = \frac { 1 } { 5 } \left( 9 - x ^ { 2 } \right)\). \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Calculate the volume of the solid formed.
Pre-U Pre-U 9794/1 2017 June Q11
10 marks Standard +0.3
11 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }\) respectively, relative to the origin \(O\). The point \(P\) lies on \(O A\) extended so that \(\overrightarrow { O P } = 3 \overrightarrow { O A }\) and the point \(Q\) lies on \(O B\) extended so that \(\overrightarrow { O Q } = 2 \overrightarrow { O B }\).
  1. Find the coordinates of the point of intersection of the lines \(A Q\) and \(B P\).
  2. Find the acute angle between the lines \(A Q\) and \(B P\).
Pre-U Pre-U 9794/1 2017 June Q12
8 marks Standard +0.3
12 Boyle's Law states that when a gas is kept at a constant temperature, its pressure \(P\) pascals is inversely proportional to its volume \(V \mathrm {~m} ^ { 3 }\). When the volume of a certain gas is \(80 \mathrm {~m} ^ { 3 }\), its pressure is 5 pascals and the rate at which the volume is increasing is \(10 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the pressure is decreasing at this volume.
Pre-U Pre-U 9794/1 2018 June Q1
4 marks Moderate -0.3
1 Solve \(5 x + 3 < | 3 x - 1 |\).
Pre-U Pre-U 9794/1 2018 June Q2
7 marks Moderate -0.3
2 It is given that \(\mathrm { f } ( x ) = 4 + 3 \sqrt { x }\), where \(x \geqslant 0\).
  1. State the range of f .
  2. State the value of \(\mathrm { ff } ( 16 )\).
  3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  4. On the same axes, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) and state how the graphs are related.
Pre-U Pre-U 9794/1 2018 June Q3
4 marks Easy -1.2
3 Given that \(z = 1\) is the real root of the equation \(z ^ { 3 } - 1 = 0\), find the two complex roots.
Pre-U Pre-U 9794/1 2018 June Q4
5 marks Moderate -0.3
4
  1. Sketch the graph of \(y = \sec \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Solve \(\sec \theta = \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2018 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{7895dcbc-2ae0-498f-8770-7b738feed7c9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).
  1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
  2. Calculate the exact area of the region bounded by the curve and the tangent.
Pre-U Pre-U 9794/1 2018 June Q6
8 marks Standard +0.3
6 Two straight lines have equations $$\mathbf { r } = - 3 \mathbf { i } + 11 \mathbf { j } - 9 \mathbf { k } + \lambda ( 4 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } )$$ and $$\mathbf { r } = 21 \mathbf { i } + 2 \mathbf { j } + 15 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$$
  1. Show that the lines intersect and find the coordinates of their point of intersection.
  2. Find the acute angle between the two lines.
Pre-U Pre-U 9794/1 2018 June Q7
8 marks Standard +0.3
7 Find the coordinates of the two stationary points of the curve $$9 x ^ { 2 } + 4 y ^ { 2 } - 6 x - 4 y = 34$$ showing that one is a maximum and one is a minimum.
Pre-U Pre-U 9794/1 2018 June Q8
7 marks Standard +0.3
8
  1. Using the quotient rule, show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tan 3 \theta ) = 3 + 3 \tan ^ { 2 } 3 \theta\) for \(- \frac { 1 } { 6 } \pi < \theta < \frac { 1 } { 6 } \pi\).
  2. Hence find the value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 9 } \pi } \tan ^ { 2 } 3 \theta \mathrm {~d} \theta\), giving your answer in the simplest exact form.
Pre-U Pre-U 9794/1 2018 June Q9
12 marks Standard +0.8
9
  1. Find the coordinates of the stationary point of the curve with equation $$y = \ln x - k x , \text { where } k > 0 \text { and } x > 0$$ and determine its nature.
  2. Hence show that the equation \(\ln x - k x = 0\) has real roots if \(0 < k \leqslant \frac { 1 } { \mathrm { e } }\).
  3. In the particular case that \(k = \frac { 1 } { 3 }\), the equation \(\ln x - k x = 0\) has two roots, one of which is near \(x = 5\). Use the Newton-Raphson process to find, correct to 3 significant figures, the root of the equation \(\ln x - \frac { 1 } { 3 } x = 0\) which is near \(x = 5\).
  4. Show that the equation \(\ln x - k x = 0\) has one real root if \(k \leqslant 0\).
  5. Explain why the equation \(\ln x - k x = 0\) has two distinct real roots if \(0 < k < \frac { 1 } { \mathrm { e } }\).
Pre-U Pre-U 9794/1 2018 June Q10
12 marks Standard +0.8
10
  1. Using partial fractions, find the general solution of the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y - y ^ { 3 } \text { for } 0 < y < 1$$ giving your solution in the form \(y = \mathrm { f } ( x )\).
  2. Determine \(\lim _ { x \rightarrow - \infty } \mathrm { f } ( x )\) and \(\lim _ { x \rightarrow + \infty } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2018 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{69214874-18a7-495d-892d-2a0a7019cbe9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).
  1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
  2. Calculate the exact area of the region bounded by the curve and the tangent.
Pre-U Pre-U 9794/1 2019 Specimen Q1
2 marks Easy -1.8
1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).
  1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
  2. Find the coordinates of the midpoint of the chord \(A B\).
Pre-U Pre-U 9794/1 2019 Specimen Q2
2 marks Moderate -0.8
2 The equation of a curve is \(y = x ^ { 3 } - 2 x ^ { 2 } - 4 x + 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the stationary points on the curve.
Pre-U Pre-U 9794/1 2019 Specimen Q3
1 marks Easy -1.3
3 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).
  1. Give a reason why f has no inverse function.
  2. Write down an expression for \(\operatorname { gf } ( x )\).
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
  4. Explain the relationship between the graph of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).