Questions — OCR (4619 questions)

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OCR Mechanics 1 2018 March Q7
7 Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle are given by $$\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - 2 a \mathbf { j } ) \mathrm { N } , \quad \mathbf { F } _ { 2 } = ( 2 b \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } \quad \text { and } \quad \mathbf { F } _ { 3 } = ( - 2 \mathbf { i } + b \mathbf { j } ) \mathrm { N } .$$ The particle is in equilibrium under the action of these three forces.
Find the value of \(a\) and the value of \(b\).
OCR Mechanics 1 2018 March Q8
8 A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The jogger then runs at a constant velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before decelerating at a constant rate of \(0.08 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) back to rest. The jogger runs a total distance of 880 m in 250 s .
  1. Sketch the velocity-time graph for the jogger's journey.
  2. Show that \(3 V ^ { 2 } - 100 V + 352 = 0\).
  3. Hence find the value of \(V\), giving a reason for your answer.
OCR Mechanics 1 2018 March Q9
9 Two particles \(A\) and \(B\) have position vectors \(\mathbf { r } _ { A }\) metres and \(\mathbf { r } _ { B }\) metres at time \(t\) seconds, where $$\mathbf { r } _ { A } = t ^ { 2 } \mathbf { i } + ( 3 t - 1 ) \mathbf { j } \quad \text { and } \quad \mathbf { r } _ { B } = \left( 1 - 2 t ^ { 2 } \right) \mathbf { i } + \left( 3 t - 2 t ^ { 2 } \right) \mathbf { j } , \quad \text { for } t \geqslant 0$$
  1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed.
  2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d ^ { 2 } = 13 t ^ { 4 } - 10 t ^ { 2 } + 2$$
  3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion.
OCR Mechanics 1 2018 March Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{467d7747-6a07-40ea-bb47-41ea3117f233-7_442_1006_251_532} A uniform \(\operatorname { rod } A B\), of weight \(W \mathrm {~N}\) and length \(2 a \mathrm {~m}\), rests with the end \(A\) on a rough horizontal table. A small object of weight \(2 W \mathrm {~N}\) is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).
  1. Find the least possible value of the coefficient of friction between the rod and the table.
  2. Given that the magnitude of the contact force at \(A\) is \(\sqrt { 39 } \mathrm {~N}\), find the value of \(W\).
OCR Mechanics 1 2018 March Q11
11 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{467d7747-6a07-40ea-bb47-41ea3117f233-7_239_1164_1299_452} A football \(P\) is kicked with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\) from a point \(A\) on horizontal ground. At the same instant a second football \(Q\) is kicked with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(2 \alpha\) from a point \(B\) on the same horizontal ground, where \(A B = 72 \mathrm {~m}\). The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point \(C\) (see diagram).
  1. Calculate the height of \(C\) above the ground.
  2. Find the direction of motion of \(P\) at the moment of impact.
  3. Suggest one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA
OCR Pure 1 2018 September Q1
1 Solve the following inequalities.
  1. \(- 5 < 3 x + 1 < 14\)
  2. \(4 x ^ { 2 } + 3 > 28\)
OCR Pure 1 2018 September Q2
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 Pure 1 2018 September Q3
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 Pure 1 2018 September Q4
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 Pure 1 2018 September Q5
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 Pure 1 2018 September Q6
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 Pure 1 2018 September Q7
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 Pure 1 2018 September Q8
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 Pure 1 2018 September Q9
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 Pure 1 2018 September Q10
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 Pure 1 2018 September Q11
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 Pure 1 2018 September Q12
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 Stats 1 2018 September Q1
1
  1. Differentiate the following with respect to \(x\).
    (a) \(\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }\)
    (b) \(\frac { \ln ( x + 2 ) } { x }\)
  2. Find \(\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x\).
OCR Stats 1 2018 September Q2
2
  1. Ben saves his pocket money as follows.
    Each week he puts money into his piggy bank (which pays no interest). In the first week he puts in 10p. In the second week he puts in 12p. In the third week he puts in 14p, and so on. How much money does Ben have in his piggy bank after 25 weeks?
  2. On January 1st Shirley invests \(\pounds 500\) in a savings account that pays compound interest at \(3 \%\) per annum. She makes no further payments into this account. The interest is added on 31st December each year.
    (a) Find the number of years after which her investment will first be worth more than \(\pounds 600\).
    (b) State an assumption that you have made in answering part (ii)(a).
OCR Stats 1 2018 September Q3
3 Use small angle approximations to estimate the solution of the equation \(\frac { \cos \frac { 1 } { 2 } \theta } { 1 + \sin \theta } = 0.825\), if \(\theta\) is small enough to neglect terms in \(\theta ^ { 3 }\) or above.
OCR Stats 1 2018 September Q4
4 Prove that the sum of the squares of any two consecutive integers is of the form \(4 k + 1\), where \(k\) is an integer.
OCR Stats 1 2018 September Q5
5 The diagram shows the graph of \(y = \sin x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\).
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-4_597_965_1909_539} Find an equation for the transformed curve when the curve \(y = \sin x ^ { \circ }\) is reflected in
  1. the \(x\)-axis,
  2. the line \(y = 0.5\).
OCR Stats 1 2018 September Q6
6
  1. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(( 3 x - 2 ) ^ { 10 }\).
  2. In the expansion of \(( 1 + 2 x ) ^ { n }\), where \(n\) is a positive integer, the coefficients of \(x ^ { 7 }\) and \(x ^ { 8 }\) are equal. Find the value of \(n\).
  3. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { \sqrt { 4 + x } }\).
OCR Stats 1 2018 September Q7
7 The diagram shows part of the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762} The area \(A\) of the region enclosed by the curve, the \(x\)-axis and the line \(x = p\) is given approximately by the sum \(S\) of the areas of \(n\) rectangles, each of width \(h\), where \(h\) is small and \(n h = p\). The first three such rectangles are shown in the diagram.
  1. Find an expression for \(S\) in terms of \(n\) and \(h\).
  2. Use the identity \(\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )\).
  3. Show how to use this result to find \(A\) in terms of \(p\).
OCR Stats 1 2018 September Q8
8 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), relative to an origin \(O\), in three dimensions. The figure \(O A P B S C T U\) is a cuboid, with vertices labelled as in the following diagram. \(M\) is the midpoint of \(A U\).
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_557_1221_2087_420}