Questions H240/01 (136 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
OCR H240/01 2018 March Q3
6 marks Moderate -0.3
3 The equation \(k x ^ { 2 } + ( k - 6 ) x + 2 = 0\) has two distinct real roots. Find the set of possible values of the constant \(k\), giving your answer in set notation.
OCR H240/01 2018 March Q4
9 marks Moderate -0.3
4
  1. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided in the Printed Answer Booklet.
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of intersection of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).
OCR H240/01 2018 March Q5
10 marks Moderate -0.8
5 An ice cream seller expects that the number of sales will increase by the same amount every week from May onwards. 150 ice creams are sold in Week 1 and 166 ice creams are sold in Week 2. The ice cream seller makes a profit of \(\pounds 1.25\) for each ice cream sold.
  1. Find the expected profit in Week 10.
  2. In which week will the total expected profits first exceed \(\pounds 5000\) ?
  3. Give two reasons why this model may not be appropriate.
OCR H240/01 2018 March Q6
5 marks Standard +0.5
6 Prove by contradiction that \(\sqrt { 7 }\) is irrational.
OCR H240/01 2018 March Q7
7 marks Standard +0.3
7 Two lifeboat stations, \(P\) and \(Q\), are situated on the coastline with \(Q\) being due south of \(P\). A stationary ship is at sea, at a distance of 4.8 km from \(P\) and a distance of 2.2 km from \(Q\). The ship is on a bearing of \(155 ^ { \circ }\) from \(P\).
  1. Find any possible bearings of the ship from \(Q\).
  2. Find the shortest distance from the ship to the line \(P Q\).
  3. Give a reason why the actual distance from the ship to the coastline may be different to your answer to part (ii).
OCR H240/01 2018 March Q8
9 marks Standard +0.8
8
  1. Given that \(y = \sec x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  2. In this question you must show detailed reasoning. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } ( \sec 2 x + \tan 2 x ) ^ { 2 } \mathrm {~d} x\).
OCR H240/01 2018 March Q9
11 marks Standard +0.8
9
  1. Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
  2. Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
  3. State the set of values for which the expansion in part (ii) is valid.
OCR H240/01 2018 March Q10
10 marks Challenging +1.2
10 In this question you must show detailed reasoning.
Show that the curve with equation \(x ^ { 2 } - 4 x y + 8 y ^ { 3 } - 4 = 0\) has exactly one stationary point.
OCR H240/01 2018 March Q11
12 marks Moderate -0.3
11 The height, in metres, of the sea at a coastal town during a day may be modelled by the function $$\mathrm { f } ( t ) = 1.7 + 0.8 \sin ( 30 t ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
  1. (a) Find the maximum height of the sea as given by this model.
    (b) Find the time of day at which this maximum height first occurs.
  2. Determine the time when, according to this model, the height of the sea will first be 1.2 m . The height, in metres, at a different coastal town during a day may be modelled by the function $$\mathrm { g } ( t ) = a + b \sin ( c t + d ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
  3. It is given that at this different coastal town the maximum height of the sea is 3.1 m , and this height occurs at 0500 and 1700. The minimum height of the sea is 0.7 m , and this height occurs at 1100 and 2300 . Find the values of the constants \(a , b , c\) and \(d\).
  4. It is instead given that the maximum height of the sea actually occurs at 0500 and 1709 . State, with a reason, how this will affect the value of \(c\) found in part (iii). \includegraphics[max width=\textwidth, alt={}, center]{74a37bca-0b28-4c48-bd21-a9304f31b8f8-6_563_568_322_751} The diagram shows the curve \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\) for \(x \geqslant 0\).
  5. Use the substitution \(u ^ { 2 } = x + 1\) to find \(\int \mathrm { e } ^ { \sqrt { x + 1 } } \mathrm {~d} x\).
  6. Make \(x\) the subject of the equation \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\).
  7. Hence show that \(\int _ { \mathrm { e } } ^ { \mathrm { e } ^ { 4 } } \left( ( \ln y ) ^ { 2 } - 1 \right) \mathrm { d } y = 9 \mathrm { e } ^ { 4 }\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA
OCR H240/01 2018 September Q1
5 marks Easy -1.8
1 Solve the following inequalities.
  1. \(- 5 < 3 x + 1 < 14\)
  2. \(4 x ^ { 2 } + 3 > 28\)
OCR H240/01 2018 September Q2
6 marks Easy -1.2
2 Vector \(\mathbf { v } = a \mathbf { i } + 0.6 \mathbf { j }\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf { v }\) is \(45 ^ { \circ }\), state the value of \(a\).
  2. Given instead that \(\mathbf { v }\) is parallel to \(8 \mathbf { i } + 3 \mathbf { j }\), find the value of \(a\).
  3. Given instead that \(\mathbf { v }\) is a unit vector, find the possible values of \(a\).
OCR H240/01 2018 September Q3
6 marks Moderate -0.3
3
  1. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\). \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-4_423_682_1110_694} On the diagram in your Printed Answer Booklet, give the coordinates of the points of intersection of the graphs with the coordinate axes.
  2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).
OCR H240/01 2018 September Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-5_487_789_251_639} The diagram shows the triangle \(A O B\), in which angle \(A O B = 0.8\) radians, \(O A = 7 \mathrm {~cm}\) and \(O B = 10 \mathrm {~cm}\). \(C D\) is the arc of a circle with centre \(O\) and radius \(O C\). The area of the triangle \(A O B\) is twice the area of the sector COD
  1. Find the length \(O C\).
  2. Find the perimeter of the region \(A B C D\).
OCR H240/01 2018 September Q5
6 marks Moderate -0.3
5 A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\ & 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x = 2 \\ & x = 8 \end{aligned}$$
  1. Identify the two mistakes that the student has made.
  2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.
OCR H240/01 2018 September Q6
7 marks Standard +0.3
6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).
OCR H240/01 2018 September Q7
11 marks Moderate -0.3
7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
  2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
OCR H240/01 2018 September Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-6_533_524_246_772} The diagram shows a container which consists of a cylinder with a solid base and a hemispherical top. The radius of the cylinder is \(r \mathrm {~cm}\) and the height is \(h \mathrm {~cm}\). The container is to be made of thin plastic. The volume of the container is \(45 \pi \mathrm {~cm} ^ { 3 }\).
  1. Show that the surface area of the container, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { 5 } { 3 } \pi r ^ { 2 } + \frac { 90 \pi } { r } .$$ [The volume of a sphere is \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) and the surface area of a sphere is \(S = 4 \pi r ^ { 2 }\).]
  2. Use calculus to find the minimum surface area of the container, justifying that it is a minimum.
  3. Suggest a reason why the manufacturer would wish to minimise the surface area.
OCR H240/01 2018 September Q9
8 marks Standard +0.3
9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.
  1. Find a model to represent the annual sales, defining any variables used.
  2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
  3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.
OCR H240/01 2018 September Q10
13 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-7_579_764_255_651} The diagram shows the graph of \(\mathrm { f } ( x ) = \ln ( 3 x + 1 ) - x\), which has a stationary point at \(x = \alpha\). A student wishes to find the non-zero root \(\beta\) of the equation \(\ln ( 3 x + 1 ) - x = 0\) using the Newton-Raphson method.
  1. (a) Determine the value of \(\alpha\).
    (b) Explain why the Newton-Raphson method will fail if \(\alpha\) is used as the initial value.
  2. Show that the Newton-Raphson iterative formula for finding \(\beta\) can be written as $$x _ { n + 1 } = \frac { 3 x _ { n } - \left( 3 x _ { n } + 1 \right) \ln \left( 3 x _ { n } + 1 \right) } { 2 - 3 x _ { n } } .$$
  3. Apply the iterative formula in part (ii) with initial value \(x _ { 1 } = 1\) to find the value of \(\beta\) correct to 5 significant figures. You should show the result of each iteration.
  4. Use a change of sign method to verify that the value of \(\beta\) found in part (iii) is correct to 5 significant figures.
OCR H240/01 2018 September Q11
12 marks Challenging +1.2
11 In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g _ { n }\) and the \(n\)th term of an arithmetic progression is denoted by \(a _ { n }\). It is given that \(g _ { 1 } = a _ { 1 } = 1 + \sqrt { 5 } , g _ { 3 } = a _ { 2 }\) and \(g _ { 4 } + a _ { 3 } = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2 \sqrt { 5 }\).
OCR H240/01 2018 September Q12
10 marks Challenging +1.2
12 The gradient function of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } \sin 2 x } { 2 \cos ^ { 2 } 4 y - 1 }\).
  1. Find an equation for the curve in the form \(\mathrm { f } ( y ) = g ( x )\). The curve passes through the point \(\left( \frac { 1 } { 4 } \pi , \frac { 1 } { 12 } \pi \right)\).
  2. Find the smallest positive value of \(y\) for which \(x = 0\). \section*{END OF QUESTION PAPER}
OCR H240/01 2018 December Q1
4 marks Easy -1.2
1 In this question you must show detailed reasoning. Andrea is comparing the prices charged by two different taxi firms.
Firm A charges \(\pounds 20\) for a 5 mile journey and \(\pounds 30\) for a 10 mile journey, and there is a linear relationship between the price and the length of the journey.
Firm B charges a pick-up fee of \(\pounds 3\) and then \(\pounds 2.40\) for each mile travelled.
Find the length of journey for which both firms would charge the same amount.
OCR H240/01 2018 December Q2
6 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-4_661_579_831_246} The diagram shows a patio. The perimeter of the patio has to be less than 44 m .
The area of the patio has to be at least \(45 \mathrm {~m} ^ { 2 }\).
  1. Write down, in terms of \(x\), an inequality satisfied by
    1. the perimeter of the patio,
    2. the area of the patio.
  2. Hence determine the set of possible values of \(x\).
OCR H240/01 2018 December Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning.
  1. Given that \(\sin \alpha = \frac { 2 } { 3 }\), find the exact values of \(\cos \alpha\).
  2. Given that \(2 \tan ^ { 2 } \beta - 7 \sec \beta + 5 = 0\), find the exact value of \(\sec \beta\).
OCR H240/01 2018 December Q4
5 marks Standard +0.8
4 In this question you must show detailed reasoning. Solve the simultaneous equations \(\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 3\) \(\mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { 2 y } = 33\). Give your answer in an exact form.