Questions — OCR (4619 questions)

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OCR C2 Q9
9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$
  1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
  3. Sketch the curve \(y = \mathrm { f } ( x )\).
  4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
OCR C2 Q1
  1. Evaluate
$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$
OCR C2 Q2
2. The diagram shows triangle \(A B C\) in which \(A B = 12.6 \mathrm {~cm} , \angle A B C = 107 ^ { \circ }\) and \(\angle A C B = 31 ^ { \circ }\). Find
  1. the length \(B C\),
  2. the area of triangle \(A B C\).
OCR C2 Q3
3. The curve with equation \(y = \mathrm { f } ( x )\) passes through the point (8, 7). Given that $$f ^ { \prime } ( x ) = 4 x ^ { \frac { 1 } { 3 } } - 5$$ find \(\mathrm { f } ( x )\).
OCR C2 Q4
4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\). Give your answers to 1 decimal place.
OCR C2 Q5
5. (i) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (ii) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0$$
OCR C2 Q6
\begin{enumerate} \setcounter{enumi}{5} \item (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
(b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.
  1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
  2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\) \item The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.
OCR C2 Q8
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
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 C2 Q1
  1. Expand \(( 3 - 2 x ) ^ { 4 }\) in ascending powers of \(x\) and simplify each coefficient.
\includegraphics[max width=\textwidth, alt={}]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-1_558_830_395_461}
The diagram shows the curve with equation \(y = 2 ^ { x }\).
Use the trapezium rule with four intervals, each of width 1 , to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 2\) and \(x = 2\).
OCR C2 Q3
3. (i) Given that $$5 \cos \theta - 2 \sin \theta = 0$$ show that \(\tan \theta = 2.5\)
(ii) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.
OCR C2 Q4
4. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
  2. \(\quad \log _ { 2 } ( \sqrt { x } )\).
    (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
OCR C2 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-2_439_807_242_534}
The diagram shows the sector \(O A B\) of a circle, centre \(O\), in which \(\angle A O B = 2.5\) radians. Given that the perimeter of the sector is 36 cm ,
  1. find the length \(O A\),
  2. find the perimeter and the area of the shaded segment.
OCR C2 Q6
6.
\includegraphics[max width=\textwidth, alt={}, center]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-2_503_839_1208_557} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).
  1. Show that \(a = 8\).
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.
OCR C2 Q7
7. (a) Evaluate $$\sum _ { r = 10 } ^ { 30 } ( 7 + 2 r )$$ (b) (i) Write down the formula for the sum of the first \(n\) positive integers.
(ii) Using this formula, find the sum of the integers from 100 to 200 inclusive.
(iii) Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3 .
OCR C2 Q8
8. The first three terms of a geometric series are \(( x - 2 ) , ( x + 6 )\) and \(x ^ { 2 }\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x ^ { 3 } - 3 x ^ { 2 } - 12 x - 36 = 0$$
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. Using \(x = 6\),
  3. find the common ratio of the series,
  4. find the sum of the first eight terms of the series.
OCR C2 Q9
9. (i) Evaluate $$\int _ { 1 } ^ { 3 } ( 3 - \sqrt { x } ) ^ { 2 } d x$$ giving your answer in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
(ii) The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 4 x + k$$ where \(k\) is a constant. Given that the curve passes through the points \(( 0 , - 2 )\) and \(( 2,18 )\), show that \(k = 2\) and find an equation for the curve.
OCR M1 2005 January Q1
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
    (a) down the plane,
    (b) up the plane.
OCR M1 2005 January Q2
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
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
    (a) the \(x\)-direction,
    (b) the \(y\)-direction.
  2. Find the magnitude and direction of the resultant.
OCR M1 2005 January Q4
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
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
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
  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
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 \(x = 4\).