Questions — OCR C2 (306 questions)

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OCR C2 2015 June Q3
9 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-2_576_599_1062_733} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 8 cm . The angle \(A O B\) is 1.2 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively such that \(O C = 5.2 \mathrm {~cm}\) and \(O D = 2.6 \mathrm {~cm} . C D\) is a straight line.
  1. Find the area of the shaded region \(A C D B\).
  2. Find the perimeter of the shaded region \(A C D B\).
OCR C2 2015 June Q4
7 marks Moderate -0.3
4
  1. Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
  2. In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).
OCR C2 2015 June Q5
7 marks Moderate -0.3
5 A curve has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }\) for all positive values of \(x\). The point \(P ( 4,1 )\) lies on the curve, and the gradient of the curve at \(P\) is 5 . Find the equation of the curve.
OCR C2 2015 June Q6
10 marks Moderate -0.3
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30\).
  1. Given that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = 0\), express \(\mathrm { f } ( x )\) as the product of 3 linear factors.
  2. Use integration to find the exact value of \(\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\).
  3. Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 5 \leqslant x \leqslant 3\).
OCR C2 2015 June Q7
11 marks Standard +0.3
7 In an arithmetic progression the first term is 5 and the common difference is 3 . The \(n\)th term of the progression is denoted by \(u _ { n }\).
  1. Find the value of \(u _ { 20 }\).
  2. Show that \(\sum _ { n = 10 } ^ { 20 } u _ { n } = 517\).
  3. Find the value of \(N\) such that \(\sum _ { n = N } ^ { 2 N } u _ { n } = 2750\).
OCR C2 2015 June Q8
9 marks Moderate -0.3
8
  1. Use logarithms to solve the equation $$2 ^ { n - 3 } = 18000$$ giving your answer correct to 3 significant figures.
  2. Solve the simultaneous equations $$\log _ { 2 } x + \log _ { 2 } y = 8 , \quad \log _ { 2 } \left( \frac { x ^ { 2 } } { y } \right) = 7$$
OCR C2 2015 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-4_406_625_248_721} The diagram shows part of the curve \(y = 2 \cos \frac { 1 } { 3 } x\), where \(x\) is in radians, and the line \(y = k\).
  1. The smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\),
    1. the next smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\),
    2. the smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = - k\).
    3. The curve \(y = 2 \cos \frac { 1 } { 3 } x\) is shown in the Printed Answer Book. On the diagram, and for the same values of \(x\), sketch the curve of \(y = \sin \frac { 1 } { 3 } x\).
    4. Calculate the \(x\)-coordinates of the points of intersection of the curves in part (ii). Give your answers in radians correct to 3 significant figures. \section*{END OF QUESTION PAPER}
OCR C2 2016 June Q1
4 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_250_611_356_721} The diagram shows triangle \(A B C\), with \(A C = 8 \mathrm {~cm}\) and angle \(C A B = 30 ^ { \circ }\).
  1. Given that the area of the triangle is \(20 \mathrm {~cm} ^ { 2 }\), find the length of \(A B\).
  2. Find the length of \(B C\), giving your answer correct to 3 significant figures.
OCR C2 2016 June Q2
5 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-2_417_476_1030_790} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is \(54 ^ { \circ }\). The perimeter of the sector is 60 cm .
  1. Express \(54 ^ { \circ }\) exactly in radians, simplifying your answer.
  2. Find the value of \(r\), giving your answer correct to 3 significant figures.
OCR C2 2016 June Q3
6 marks Moderate -0.3
3
  1. Find the binomial expansion of \(( 3 + k x ) ^ { 3 }\), simplifying the terms.
  2. It is given that, in the expansion of \(( 3 + k x ) ^ { 3 }\), the coefficient of \(x ^ { 2 }\) is equal to the constant term. Find the possible values of \(k\), giving your answers in an exact form.
OCR C2 2016 June Q4
6 marks Moderate -0.8
4
  1. Express \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 )\) as a single logarithm.
  2. Hence solve the equation \(2 \log _ { 3 } x - \log _ { 3 } ( x + 4 ) = 2\).
OCR C2 2016 June Q5
8 marks Standard +0.3
5
  1. Find \(\int \left( x ^ { 2 } + 2 \right) ( 2 x - 3 ) \mathrm { d } x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 1 } ^ { a } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\), where \(a\) is a constant greater than 1 .
    2. Deduce the value of \(\int _ { 1 } ^ { \infty } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\).
OCR C2 2016 June Q6
11 marks Standard +0.8
6 An arithmetic progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 1.5\) for \(n \geqslant 1\).
  1. Given that \(u _ { k } = 140\), find the value of \(k\). A geometric progression \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { n } = 120 \times ( 0.9 ) ^ { n - 1 }\) for \(n \geqslant 1\).
  2. Find the sum of the first 16 terms of this geometric progression, giving your answer correct to 3 significant figures.
  3. Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { N } u _ { n } > \sum _ { n = 1 } ^ { \infty } w _ { n }\).
OCR C2 2016 June Q7
12 marks Standard +0.3
7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
  1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
  2. Hence find the three roots of the equation \(\mathrm { f } ( x ) = 0\). \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-3_540_718_1466_660} The diagram shows the curve \(C\) with equation \(y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9\).
  3. Show that the \(x\)-coordinates of the stationary points on \(C\) are given by \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0\).
  4. Use integration to find the exact area of the region enclosed by \(C\) and the \(x\)-axis.
OCR C2 2016 June Q8
12 marks Moderate -0.8
8
  1. The curve \(y = 3 ^ { x }\) can be transformed to the curve \(y = 3 ^ { x - 2 }\) by a translation. Give details of the translation.
  2. Alternatively, the curve \(y = 3 ^ { x }\) can be transformed to the curve \(y = 3 ^ { x - 2 }\) by a stretch. Give details of the stretch.
  3. Sketch the curve \(y = 3 ^ { x - 2 }\), stating the coordinates of any points of intersection with the axes.
  4. The point \(P\) on the curve \(y = 3 ^ { x - 2 }\) has \(y\)-coordinate equal to 180 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
  5. Use the trapezium rule, with 2 strips each of width 1.5, to find an estimate for \(\int _ { 1 } ^ { 4 } 3 ^ { x - 2 } \mathrm {~d} x\). Give your answer correct to 3 significant figures.
OCR C2 2016 June Q9
8 marks Moderate -0.3
9 A curve has equation \(y = \sin ( a x )\), where \(a\) is a positive constant and \(x\) is in radians.
  1. State the period of \(y = \sin ( a x )\), giving your answer in an exact form in terms of \(a\).
  2. Given that \(x = \frac { 1 } { 5 } \pi\) and \(x = \frac { 2 } { 5 } \pi\) are the two smallest positive solutions of \(\sin ( a x ) = k\), where \(k\) is a positive constant, find the values of \(a\) and \(k\).
  3. Given instead that \(\sin ( a x ) = \sqrt { 3 } \cos ( a x )\), find the two smallest positive solutions for \(x\), giving your answers in an exact form in terms of \(a\). \section*{END OF QUESTION PAPER}
OCR C2 Q5
8 marks Moderate -0.3
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
OCR C2 Q7
10 marks Standard +0.3
  1. Show that the common difference is 5 .
  2. Find the 12th term. Another arithmetic sequence has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two sequences are equal,
  3. find the value of \(n\).
OCR C2 Q1
5 marks Moderate -0.8
  1. Giving your answers in terms of \(\pi\), solve the equation
$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\).
OCR C2 Q2
6 marks Moderate -0.8
2. Given that \(p = \log _ { 2 } 3\) and \(q = \log _ { 2 } 5\), find expressions in terms of \(p\) and \(q\) for
  1. \(\quad \log _ { 2 } 45\),
  2. \(\log _ { 2 } 0.3\)
OCR C2 Q3
6 marks Moderate -0.3
3. For the binomial expansion in ascending powers of \(x\) of \(\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }\), where \(n\) is an integer and \(n \geq 2\),
  1. find and simplify the first three terms,
  2. find the value of \(n\) for which the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\).
OCR C2 Q4
8 marks Standard +0.3
4. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations \(y = 7 - 2 x - 3 x ^ { 2 }\) and \(y = \frac { 2 } { x }\).
The two curves intersect at the points \(P , Q\) and \(R\).
  1. Show that the \(x\)-coordinates of \(P , Q\) and \(R\) satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that \(P\) has coordinates \(( - 2 , - 1 )\),
  2. find the coordinates of \(Q\) and \(R\).
OCR C2 Q5
8 marks Moderate -0.8
5. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$
  1. Find \(\mathrm { f } ( x )\).
  2. Show that the area of the finite region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is \(4 \frac { 1 } { 2 }\).
OCR C2 Q6
8 marks Standard +0.3
6. \includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle \(A B C\) in which \(A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(\angle A B C = 1.7\) radians.
  1. Find the size of \(\angle A C B\) in radians. The point \(D\) lies on \(A C\) such that \(B D\) is an arc of a circle, centre \(C\).
  2. Find the perimeter of the shaded region bounded by the arc \(B D\) and the straight lines \(A B\) and \(A D\).
OCR C2 Q7
9 marks Moderate -0.3
7. (a) Given that \(y = 3 ^ { x }\), find expressions in terms of \(y\) for
  1. \(3 ^ { x + 1 }\),
  2. \(3 ^ { 2 x - 1 }\).
    (b) Hence, or otherwise, solve the equation $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$