Questions — OCR (4619 questions)

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OCR C2 Q3
3. The sides of a triangle have lengths of \(7 \mathrm {~cm} , 8 \mathrm {~cm}\) and 10 cm . Find the area of the triangle correct to 3 significant figures.
OCR C2 Q4
4. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which $$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$ giving non-exact answers to 1 decimal place.
OCR C2 Q5
5. (i) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(ii) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(iii) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).
OCR C2 Q6
6. Evaluate
  1. \(\quad \int _ { 1 } ^ { 4 } \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\),
  2. \(\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 4 } } \mathrm {~d} x\).
OCR C2 Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-2_364_666_1338_568} The diagram shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.
  1. Find the radius of the fourth circle to be drawn.
  2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
  3. Find the total area of the design in square centimetres when ten circles have been drawn.
OCR C2 Q8
8. Given that for small values of \(x\) $$( 1 + a x ) ^ { n } \approx 1 - 24 x + 270 x ^ { 2 }$$ where \(n\) is an integer and \(n > 1\),
  1. show that \(n = 16\) and find the value of \(a\),
  2. use your value of \(a\) and a suitable value of \(x\) to estimate the value of (0.9985) \({ } ^ { 16 }\), giving your answer to 5 decimal places.
OCR C2 Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-3_559_732_824_388} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$f ^ { \prime } ( x ) = 4 - 6 x - 3 x ^ { 2 }$$
  1. find an expression for \(y\) in terms of \(x\),
  2. show that \(A\) has coordinates ( \(- 4,0\) ) and find the coordinates of \(B\),
  3. find the total area of the two regions bounded by the curve and the \(x\)-axis.
OCR M1 2007 January Q1
1 A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Given that the force exerted on the trailer by the tow-bar is 700 N , find the resistance to motion of the trailer.
  2. Given also that the driving force of the car is 2100 N , find the resistance to motion of the car.
OCR M1 2007 January Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_583_785_676_680} Three horizontal forces of magnitudes \(15 \mathrm {~N} , 11 \mathrm {~N}\) and 13 N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8\) and \(\cos \beta = 0.6\).
  1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero.
  2. Find the magnitude of the resultant of the three forces.
  3. State the direction of the resultant of the three forces.
    \includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_348_711_1804_717} A block \(B\) of mass 0.4 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).
  4. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table.
  5. A horizontal force of magnitude \(X \mathrm {~N}\), acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\).
OCR M1 2007 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-3_216_1146_269_502} Three uniform spheres \(L , M\) and \(N\) have masses \(0.8 \mathrm {~kg} , 0.6 \mathrm {~kg}\) and 0.7 kg respectively. The spheres are moving in a straight line on a smooth horizontal table, with \(M\) between \(L\) and \(N\). The sphere \(L\) is moving towards \(M\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the spheres \(M\) and \(N\) are moving towards \(L\) with speeds \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).
  1. \(L\) collides with \(M\). As a result of this collision the direction of motion of \(M\) is reversed, and its speed remains \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(L\) after the collision.
  2. \(M\) then collides with \(N\).
    (a) Find the total momentum of \(M\) and \(N\) in the direction of \(M\) 's motion before this collision takes place, and deduce that the direction of motion of \(N\) is reversed as a result of this collision.
    (b) Given that \(M\) is at rest immediately after this collision, find the speed of \(N\) immediately after this collision.
OCR M1 2007 January Q5
5 A particle starts from rest at a point \(A\) at time \(t = 0\), where \(t\) is in seconds. The particle moves in a straight line. For \(0 \leqslant t \leqslant 4\) the acceleration is \(1.8 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and for \(4 \leqslant t \leqslant 7\) the particle has constant acceleration \(7.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leqslant t \leqslant 4\).
  2. Show that the displacement of the particle from \(A\) is 19.2 m when \(t = 4\).
  3. Find the displacement of the particle from \(A\) when \(t = 7\).
OCR M1 2007 January Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-4_556_1373_269_386} The diagram shows the ( \(t , v\) ) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s , coming to rest 8 m above ground level.
  1. Find the greatest speed reached by the hoist during this stage. The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at \(0.02 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 40 s of the third stage, reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  2. the value of \(V\),
  3. the length of time during the third stage for which the hoist is moving at constant speed,
  4. the deceleration of the hoist in the final part of the third stage.
OCR M1 2007 January Q7
7 A particle \(P\) of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is 0.6 .
  1. Show that the magnitude of the frictional force acting on \(P\) is 2.25 N , correct to 3 significant figures.
  2. Find the acceleration of \(P\) when it is moving
    (a) up the plane,
    (b) down the plane.
  3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (a) Find the length of time before \(P\) reaches its highest point.
    (b) Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
OCR C2 Q1
  1. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$u _ { n } = 2 ^ { n } + k n ,$$ where \(k\) is a constant.
Given that \(u _ { 1 } = u _ { 3 }\),
  1. find the value of \(k\),
  2. find the value of \(u _ { 5 }\).
OCR C2 Q2
2. Given that $$y = 2 x ^ { \frac { 3 } { 2 } } - 1 ,$$ find $$\int y ^ { 2 } \mathrm {~d} x .$$
OCR C2 Q3
  1. (i) Sketch the curve \(y = \sin x ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
    (ii) Sketch on the same diagram the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
    (iii) Use your diagram to solve the equation
$$\sin x ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ for \(x\) in the interval \(- 180 \leq x \leq 180\).
OCR C2 Q4
4. (i) Solve the inequality $$x ^ { 2 } - 13 x + 30 < 0$$ (ii) Hence find the set of values of \(y\) such that $$2 ^ { 2 y } - 13 \left( 2 ^ { y } \right) + 30 < 0 .$$
OCR C2 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-2_515_771_246_438}
The diagram shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates \(( - 4,0 )\),
  1. show that \(k = - 7\),
  2. find the coordinates of \(B\) and \(C\).
OCR C2 Q6
6. Given that $$\mathrm { f } ^ { \prime } ( x ) = 5 + \frac { 4 } { x ^ { 2 } } , \quad x \neq 0$$
  1. find an expression for \(\mathrm { f } ( x )\). Given also that $$\mathrm { f } ( 2 ) = 2 \mathrm { f } ( 1 ) ,$$
  2. find \(\mathrm { f } ( 4 )\).
OCR C2 Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-3_376_892_221_427} The diagram shows a design painted on the wall at a karting track. The sign consists of triangle \(A B C\) and two circular sectors of radius 2 metres and 1 metre with centres \(A\) and \(B\) respectively. Given that \(A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}\) and \(\angle A C B = 2.2\) radians,
  1. find the size of \(\angle A B C\) in radians to 3 significant figures,
  2. show that \(\angle B A C = 0.588\) radians to 3 significant figures,
  3. find the area of triangle \(A B C\),
  4. find the area of the wall covered by the design.
OCR C2 Q8
8. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).
  1. Use the trapezium rule with three intervals, each of width 2 , to estimate to 3 significant figures the area of \(R\).
  2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
  3. Find the percentage error in the estimate made in part (a).
OCR C2 Q9
9. The first two terms of a geometric progression are 2 and \(x\) respectively, where \(x \neq 2\).
  1. Find an expression for the third term in terms of \(x\). The first and third terms of arithmetic progression are 2 and \(x\) respectively.
  2. Find an expression for the 11th term in terms of \(x\). Given that the third term of the geometric progression and the 11th term of the arithmetic progression have the same value,
  3. find the value of \(x\),
  4. find the sum of the first 50 terms of the arithmetic progression.
OCR M1 2008 January Q1
1 A man of mass 70 kg stands on the floor of a lift which is moving with an upward acceleration of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the magnitude of the force exerted by the floor on the man.
OCR M1 2008 January Q2
2 An ice skater of mass 40 kg is moving in a straight line with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when she collides with a skater of mass 60 kg moving in the opposite direction along the same straight line with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision the skaters move together with a common speed in the same straight line. Calculate their common speed, and state their direction of motion.
OCR M1 2008 January Q3
3 Two horizontal forces \(\mathbf { X }\) and \(\mathbf { Y }\) act at a point \(O\) and are at right angles to each other. \(\mathbf { X }\) has magnitude 12 N and acts along a bearing of \(090 ^ { \circ } . \mathbf { Y }\) has magnitude 15 N and acts along a bearing of \(000 ^ { \circ }\).
  1. Calculate the magnitude and bearing of the resultant of \(\mathbf { X }\) and \(\mathbf { Y }\).
  2. A third force \(\mathbf { E }\) is now applied at \(O\). The three forces \(\mathbf { X } , \mathbf { Y }\) and \(\mathbf { E }\) are in equilibrium. State the magnitude of \(\mathbf { E }\), and give the bearing along which it acts.