Questions — OCR C2 (296 questions)

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OCR C2 2013 January Q3
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
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
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
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
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
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
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
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
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
3 Use logarithms to solve the equation \(7 ^ { x } = 2 ^ { x + 1 }\), giving the value of \(x\) correct to 3 significant figures.
OCR C2 2009 June Q4
4
  1. Find the binomial expansion of \(\left( x ^ { 2 } - 5 \right) ^ { 3 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 2 } - 5 \right) ^ { 3 } \mathrm {~d} x\).
OCR C2 2009 June Q5
5 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(\sin 2 x = 0.5\)
  2. \(2 \sin ^ { 2 } x = 2 - \sqrt { 3 } \cos x\)
OCR C2 2009 June Q6
6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a\), where \(a\) is a constant. The curve passes through the points \(( - 1,2 )\) and \(( 2,17 )\). Find the equation of the curve.
OCR C2 2009 June Q7
7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
  2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
  4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.
OCR C2 2009 June Q8
8 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_378_467_269_840} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Fig. 1 shows a sector \(A O B\) of a circle, centre \(O\) and radius \(O A\). The angle \(A O B\) is 1.2 radians and the area of the sector is \(60 \mathrm {~cm} ^ { 2 }\).
  1. Find the perimeter of the sector. A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector \(A O B\) from Fig. 1, and the area of each successive sector is \(\frac { 3 } { 5 }\) of the area of the previous one. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. (a) Find the area of the fifth sector in the pattern.
    (b) Find the total area of the first ten sectors in the pattern.
    (c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.
OCR C2 2009 June Q9
9
  1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
  2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
  3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).
OCR C2 2010 June Q1
1 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x - 14\), where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. Using this value of \(a\), find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ).
OCR C2 2010 June Q2
2
  1. Use the trapezium rule, with 3 strips each of width 3 , to estimate the area of the region bounded by the curve \(y = \sqrt [ 3 ] { 7 + x }\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 10\). Give your answer correct to 3 significant figures.
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate of the area.
OCR C2 2010 June Q3
3
  1. Find and simplify the first four terms in the binomial expansion of \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 + 4 x + 2 x ^ { 2 } \right) \left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }\).
OCR C2 2010 June Q4
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 n + 1\).
  1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Evaluate \(\sum _ { n = 1 } ^ { 40 } u _ { n }\). Another sequence \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { 1 } = 2\) and \(w _ { n + 1 } = 5 w _ { n } + 1\).
  3. Find the value of \(p\) such that \(u _ { p } = w _ { 3 }\).
OCR C2 2010 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{570435e0-5685-4c5b-9ed8-f2bc22bdfb24-02_396_1070_1768_536} The diagram shows two congruent triangles, \(B C D\) and \(B A E\), where \(A B C\) is a straight line. In triangle \(B C D , B D = 8 \mathrm {~cm} , C D = 11 \mathrm {~cm}\) and angle \(C B D = 65 ^ { \circ }\). The points \(E\) and \(D\) are joined by an arc of a circle with centre \(B\) and radius 8 cm .
  1. Find angle \(B C D\).
  2. (a) Show that angle \(E B D\) is 0.873 radians, correct to 3 significant figures.
    (b) Hence find the area of the shaded segment bounded by the chord \(E D\) and the arc \(E D\), giving your answer correct to 3 significant figures.
OCR C2 2010 June Q6
6
  1. Use integration to find the exact area of the region enclosed by the curve \(y = x ^ { 2 } + 4 x\), the \(x\)-axis and the lines \(x = 3\) and \(x = 5\).
  2. Find \(\int ( 2 - 6 \sqrt { y } ) \mathrm { d } y\).
  3. Evaluate \(\int _ { 1 } ^ { \infty } \frac { 8 } { x ^ { 3 } } \mathrm {~d} x\).
OCR C2 2010 June Q7
7
  1. Show that \(\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv \tan ^ { 2 } x - 1\).
  2. Hence solve the equation $$\frac { \sin ^ { 2 } x - \cos ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5 - \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR C2 2010 June Q8
8
  1. Use logarithms to solve the equation \(5 ^ { 3 w - 1 } = 4 ^ { 250 }\), giving the value of \(w\) correct to 3 significant figures.
  2. Given that \(\log _ { x } ( 5 y + 1 ) - \log _ { x } 3 = 4\), express \(y\) in terms of \(x\).
OCR C2 2010 June Q9
9 A geometric progression has first term \(a\) and common ratio \(r\), and the terms are all different. The first, second and fourth terms of the geometric progression form the first three terms of an arithmetic progression.
  1. Show that \(r ^ { 3 } - 2 r + 1 = 0\).
  2. Given that the geometric progression converges, find the exact value of \(r\).
  3. Given also that the sum to infinity of this geometric progression is \(3 + \sqrt { 5 }\), find the value of the integer \(a\).