Questions — OCR (4628 questions)

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OCR C2 2011 January Q6
8 marks Moderate -0.3
6
  1. Find \(\int \frac { x ^ { 3 } + 3 x ^ { \frac { 1 } { 2 } } } { x } \mathrm {~d} x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 2 } ^ { a } 6 x ^ { - 4 } \mathrm {~d} x\), where \(a\) is a constant greater than 2 .
    2. Deduce the value of \(\int _ { 2 } ^ { \infty } 6 x ^ { - 4 } \mathrm {~d} x\).
OCR C2 2011 January Q7
8 marks Moderate -0.3
7 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(3 \tan 2 x = 1\)
  2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)
OCR C2 2011 January Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-03_420_729_1027_708} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 5 cm . Angle \(A O B\) is \(\theta\) radians. The area of triangle \(A O B\) is \(8 \mathrm {~cm} ^ { 2 }\).
  1. Given that the angle \(\theta\) is obtuse, find \(\theta\). The shaded segment in the diagram is bounded by the chord \(A B\) and the arc \(A B\).
  2. Find the area of the segment, giving your answer correct to 3 significant figures.
  3. Find the perimeter of the segment, giving your answer correct to 3 significant figures.
OCR C2 2011 January Q9
12 marks Moderate -0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-04_584_785_255_680} The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = - 4 x ^ { 3 } + 9 x ^ { 2 } + 10 x - 3\).
  1. Verify that the curve crosses the \(x\)-axis at ( 3,0 ) and hence state a factor of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  3. Hence find the other two points of intersection of the curve with the \(x\)-axis.
  4. The region enclosed by the curve and the \(x\)-axis is shaded in the diagram. Use integration to find the total area of this region.
OCR C2 2012 January Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_319_454_246_810} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 12 cm . The reflex angle \(A O B\) is 4.2 radians.
  1. Find the perimeter of the sector.
  2. Find the area of the sector.
OCR C2 2012 January Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_536_917_1016_577} The diagram shows the curve \(y = \log _ { 10 } ( 2 x + 1 )\).
  1. Use the trapezium rule with 4 strips each of width 1.5 to find an approximation to the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 4\) and \(x = 10\). Give your answer correct to 3 significant figures.
  2. Explain why this approximation is an under-estimate.
OCR C2 2012 January Q3
6 marks Moderate -0.8
3 One of the terms in the binomial expansion of \(( 4 + a x ) ^ { 6 }\) is \(160 x ^ { 3 }\).
  1. Find the value of \(a\).
  2. Using this value of \(a\), find the first two terms in the expansion of \(( 4 + a x ) ^ { 6 }\) in ascending powers of \(x\).
OCR C2 2012 January Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-3_622_513_244_776} The diagram shows two points \(A\) and \(B\) on a straight coastline, with \(A\) being 2.4 km due north of \(B\). A stationary ship is at point \(C\), on a bearing of \(040 ^ { \circ }\) and at a distance of 2 km from \(B\).
  1. Find the distance \(A C\), giving your answer correct to 3 significant figures.
  2. Find the bearing of \(C\) from \(A\).
  3. Find the shortest distance from the ship to the coastline.
OCR C2 2012 January Q5
8 marks Moderate -0.3
5 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\).
  2. Given that \(\mathrm { f } ( 2 ) = 0\), express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  3. Determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.
OCR C2 2012 January Q6
11 marks Standard +0.3
6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
  3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
  4. Find the sum to infinity of the geometric progression in part (iii).
OCR C2 2012 January Q7
11 marks Standard +0.3
7
  1. Find \(\int \left( x ^ { 2 } + 4 \right) ( x - 6 ) \mathrm { d } x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-4_449_551_349_758} The diagram shows the curve \(y = 6 x ^ { \frac { 3 } { 2 } }\) and part of the curve \(y = \frac { 8 } { x ^ { 2 } } - 2\), which intersect at the point \(( 1,6 )\). Use integration to find the area of the shaded region enclosed by the two curves and the \(x\)-axis.
OCR C2 2012 January Q8
10 marks Moderate -0.3
8
  1. Use logarithms to solve the equation \(7 ^ { w - 3 } - 4 = 180\), giving your answer correct to 3 significant figures.
  2. Solve the simultaneous equations $$\log _ { 10 } x + \log _ { 10 } y = \log _ { 10 } 3 , \quad \log _ { 10 } ( 3 x + y ) = 1$$
OCR C2 2012 January Q9
9 marks Standard +0.3
9
  1. Sketch the graph of \(y = \tan \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\) on the axes provided.
    On the same axes, sketch the graph of \(y = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\), indicating the point of intersection with the \(y\)-axis.
  2. Show that the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) can be expressed in the form $$3 \sin ^ { 2 } \left( \frac { 1 } { 2 } x \right) + \sin \left( \frac { 1 } { 2 } x \right) - 3 = 0$$ Hence solve the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).
OCR C2 2013 January Q1
4 marks Moderate -0.8
1 The diagram shows triangle \(A B C\), with \(A C = 14 \mathrm {~cm} , B C = 10 \mathrm {~cm}\) and angle \(A B C = 63 ^ { \circ }\).
  1. Find angle \(C A B\).
  2. Find the length of \(A B\).
OCR C2 2013 January Q2
6 marks Moderate -0.5
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 7 \text { and } u _ { n + 1 } = u _ { n } + 4 \text { for } n \geqslant 1 .$$
  1. Show that \(u _ { 17 } = 71\).
  2. Show that \(\sum _ { n = 1 } ^ { 35 } u _ { n } = \sum _ { n = 36 } ^ { 50 } u _ { n }\).
OCR C2 2013 January Q3
7 marks Moderate -0.8
3 A curve has an equation which satisfies \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k x ( 2 x - 1 )\) for all values of \(x\). The point \(P ( 2,7 )\) lies on the curve and the gradient of the curve at \(P\) is 9 .
  1. Find the value of the constant \(k\).
  2. Find the equation of the curve.
OCR C2 2013 January Q4
7 marks Moderate -0.3
4
  1. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
  2. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).
OCR C2 2013 January Q5
7 marks Standard +0.3
5
  1. Show that the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\) can be expressed in the form $$6 \cos ^ { 2 } x - \cos x - 2 = 0 .$$
  2. Hence solve the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2013 January Q6
11 marks Moderate -0.3
6
  1. The first three terms of an arithmetic progression are \(2 x , x + 4\) and \(2 x - 7\) respectively. Find the value of \(x\).
  2. The first three terms of another sequence are also \(2 x , x + 4\) and \(2 x - 7\) respectively.
    (a) Verify that when \(x = 8\) the terms form a geometric progression and find the sum to infinity in this case.
    (b) Find the other possible value of \(x\) that also gives a geometric progression.
OCR C2 2013 January Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-3_412_707_751_680} The diagram shows two circles of radius 7 cm with centres \(A\) and \(B\). The distance \(A B\) is 12 cm and the point \(C\) lies on both circles. The region common to both circles is shaded.
  1. Show that angle \(C A B\) is 0.5411 radians, correct to 4 significant figures.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
OCR C2 2013 January Q8
9 marks Moderate -0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-4_489_697_274_667} The diagram shows the curves \(y = \log _ { 2 } x\) and \(y = \log _ { 2 } ( x - 3 )\).
  1. Describe the geometrical transformation that transforms the curve \(y = \log _ { 2 } x\) to the curve \(y = \log _ { 2 } ( x - 3 )\).
  2. The curve \(y = \log _ { 2 } x\) passes through the point ( \(a , 3\) ). State the value of \(a\).
  3. The curve \(y = \log _ { 2 } ( x - 3 )\) passes through the point ( \(b , 1.8\) ). Find the value of \(b\), giving your answer correct to 3 significant figures.
  4. The point \(P\) lies on \(y = \log _ { 2 } x\) and has an \(x\)-coordinate of \(c\). The point \(Q\) lies on \(y = \log _ { 2 } ( x - 3 )\) and also has an \(x\)-coordinate of \(c\). Given that the distance \(P Q\) is 4 units find the exact value of \(c\).
OCR C2 2013 January Q9
12 marks Standard +0.3
9 The positive constant \(a\) is such that \(\int _ { a } ^ { 2 a } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 4 } { x ^ { 2 } } \mathrm {~d} x = 0\).
  1. Show that \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\).
  2. Show that \(a = 1\) is a root of \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\), and hence find the other possible value of \(a\), giving your answer in simplified surd form.
OCR C2 2009 June Q1
5 marks Moderate -0.8
1 The lengths of the three sides of a triangle are \(6.4 \mathrm {~cm} , 7.0 \mathrm {~cm}\) and 11.3 cm .
  1. Find the largest angle in the triangle.
  2. Find the area of the triangle.
OCR C2 2009 June Q2
6 marks Moderate -0.8
2 The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term of the progression is 44 .
  1. Find the first term and the common difference.
  2. Find the sum of the first 50 terms.
OCR C2 2009 June Q3
5 marks Moderate -0.8
3 Use logarithms to solve the equation \(7 ^ { x } = 2 ^ { x + 1 }\), giving the value of \(x\) correct to 3 significant figures.