Questions — OCR (4907 questions)

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OCR C2 Q8
10 marks Moderate -0.3
8. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,
  1. find an expression for \(b\) in terms of \(a\). Given also that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
OCR C2 Q9
12 marks Moderate -0.3
9. \includegraphics[max width=\textwidth, alt={}, center]{33f9663f-26bb-445e-af6e-ca5ca927f7dd-3_638_757_1064_493} The diagram shows the curve with equation \(y = 5 + x - x ^ { 2 }\) and the normal to the curve at the point \(P ( 1,5 )\).
  1. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
  2. Find the coordinates of the point \(Q\), where the normal to the curve at \(P\) intersects the curve again.
  3. Show that the area of the shaded region bounded by the curve and the straight line \(P Q\) is \(\frac { 4 } { 3 }\).
OCR M1 2005 January Q1
6 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-2_200_537_269_804} A box of weight 100 N rests in equilibrium on a plane inclined at an angle \(\alpha\) to the horizontal. It is given that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). A force of magnitude \(P \mathrm {~N}\) acts on the box parallel to the plane in the upwards direction (see diagram). The coefficient of friction between the box and the plane is 0.25 .
  1. Find the magnitude of the normal force acting on the box.
  2. Given that the equilibrium is limiting, show that the magnitude of the frictional force acting on the box is 24 N .
  3. Find the value of \(P\) for which the box is on the point of slipping
    1. down the plane,
    2. up the plane.
OCR M1 2005 January Q2
8 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-2_221_1153_1340_497} Three small uniform spheres \(A , B\) and \(C\) have masses \(0.4 \mathrm {~kg} , 1.2 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres move in the same straight line on a smooth horizontal table, with \(B\) between \(A\) and \(C\). Sphere \(A\) is moving towards \(B\) with speed \(6 \mathrm {~ms} ^ { - 1 } , B\) is moving towards \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(C\) is moving towards \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Spheres \(A\) and \(B\) collide. After this collision \(B\) moves with speed \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(C\).
  1. Find the speed with which \(A\) moves after the collision and state the direction of motion of \(A\).
  2. Spheres \(B\) and \(C\) now collide and move away from each other with speeds \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find the value of \(m\).
OCR M1 2005 January Q3
9 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-3_638_839_269_653} Three coplanar forces of magnitudes \(5 \mathrm {~N} , 8 \mathrm {~N}\) and 8 N act at the origin \(O\) of rectangular coordinate axes. The directions of the forces are as shown in the diagram.
  1. Find the component of the resultant of the three forces in
    1. the \(x\)-direction,
    2. the \(y\)-direction.
    3. Find the magnitude and direction of the resultant.
OCR M1 2005 January Q4
9 marks Moderate -0.8
4 A particle moves in a straight line. Its velocity \(t \mathrm {~s}\) after leaving a fixed point on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t + 0.1 t ^ { 2 }\). Find
  1. an expression for the acceleration of the particle at time \(t\),
  2. the distance travelled by the particle from time \(t = 0\) until the instant when its acceleration is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
OCR M1 2005 January Q5
10 marks Moderate -0.3
5 Two particles \(A\) and \(B\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(A\) and \(B\) are \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. Write down expressions for the heights above the ground of \(A\) and \(B\) at time \(t\) seconds after projection.
  2. Hence find a simplified expression for the difference in the heights of \(A\) and \(B\) at time \(t\) seconds after projection.
  3. Find the difference in the heights of \(A\) and \(B\) when \(A\) is at its maximum height. At the instant when \(B\) is 3.5 m above \(A\), find
  4. whether \(A\) is moving upwards or downwards,
  5. the height of \(A\) above the ground.
OCR M1 2005 January Q6
13 marks Moderate -0.3
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_664_969_264_589} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A cyclist \(P\) travels along a straight road starting from rest at \(A\) and accelerating at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(B 20 \mathrm {~s}\) after leaving \(A\). Fig. 1 shows the ( \(t , v\) ) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\). Find
  1. the time for which \(P\) is accelerating,
  2. the distance \(A B\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_607_937_1420_605} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Another cyclist \(Q\) travels along the same straight road in the opposite direction. She starts at rest from \(B\) at the same instant that \(P\) leaves \(A\). Cyclist \(Q\) accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(8 \mathrm {~ms} ^ { - 1 }\) and continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(A 20 \mathrm {~s}\) after leaving \(B\). Fig. 2 shows the \(( t , x )\) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\), where \(x\) is the displacement of \(Q\) from \(A\) towards \(B\).
  3. Sketch a copy of Fig. 1 and add to your copy a sketch of the ( \(t , v\) ) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\).
  4. Sketch a copy of Fig. 2 and add to your copy a sketch of the \(( t , x )\) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\).
  5. Find the value \(t\) at the instant that \(P\) and \(Q\) pass each other. \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-5_447_739_269_703} The upper edge of a smooth plane inclined at \(70 ^ { \circ }\) to the horizontal is joined to an edge of a rough horizontal table. Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth pulley which is fixed at the top of the smooth inclined plane. Particle \(A\) is held in contact with the rough horizontal table and particle \(B\) is in contact with the smooth inclined plane with the string taut (see diagram). The coefficient of friction between \(A\) and the horizontal table is 0.4 . Particle \(A\) is released from rest and the system starts to move.
  6. Find the acceleration of \(A\) and the tension in the string. The string breaks when the speed of the particles is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  7. Assuming \(A\) does not reach the pulley, find the distance travelled by \(A\) after the string breaks.
  8. Assuming \(B\) does not reach the ground before \(A\) stops, find the distance travelled by \(B\) from the time the string breaks to the time that \(A\) stops.
OCR C2 Q1
4 marks Moderate -0.8
  1. \(f ( x ) = 3 x ^ { 3 } - 2 x ^ { 2 } + k x + 9\).
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\) there is a remainder of - 35 ,
  1. find the value of the constant \(k\),
  2. find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 3 x - 2 )\).
OCR C2 Q2
4 marks Easy -1.2
2. \includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-1_554_848_685_461} The diagram shows the curve with equation \(y = 4 x + \frac { 1 } { x } , x > 0\).
Use the trapezium rule with three intervals, each of width 1 , to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
OCR C2 Q3
5 marks Moderate -0.3
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
8 marks Standard +0.3
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
9 marks Moderate -0.3
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
10 marks Standard +0.3
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
10 marks Moderate -0.3
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
10 marks Standard +0.3
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
12 marks Moderate -0.3
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 C2 Q1
5 marks Moderate -0.8
  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
6 marks Standard +0.3
2. Given that $$y = 2 x ^ { \frac { 3 } { 2 } } - 1 ,$$ find $$\int y ^ { 2 } \mathrm {~d} x .$$
OCR C2 Q3
6 marks Moderate -0.8
  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
6 marks Standard +0.3
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
7 marks Moderate -0.3
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
8 marks Moderate -0.8
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
10 marks Standard +0.3
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
12 marks Moderate -0.3
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).