Questions — OCR C2 (296 questions)

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OCR C2 2013 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-4_524_822_274_609} The diagram shows the curves \(y = a ^ { x }\) and \(y = 4 b ^ { x }\).
  1. (a) State the coordinates of the point of intersection of \(y = a ^ { x }\) with the \(y\)-axis.
    (b) State the coordinates of the point of intersection of \(y = 4 b ^ { x }\) with the \(y\)-axis.
    (c) State a possible value for \(a\) and a possible value for \(b\).
  2. It is now given that \(a b = 2\). Show that the \(x\)-coordinate of the point of intersection of \(y = a ^ { x }\) and \(y = 4 b ^ { x }\) can be written as $$x = \frac { 2 } { 2 \log _ { 2 } a - 1 } .$$
OCR C2 2013 June Q9
9 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ).
  2. Show that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and hence factorise \(\mathrm { f } ( x )\) completely.
  3. Solve the equation $$4 \cos ^ { 3 } \theta - 7 \cos \theta - 3 = 0$$ for \(0 \leqslant \theta \leqslant 2 \pi\). Give each solution for \(\theta\) in an exact form.
OCR C2 2014 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_426_1244_280_413} The diagram shows triangle \(A B C\), with \(A B = 8 \mathrm {~cm}\), angle \(B A C = 65 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The point \(D\) is on \(A C\) such that \(A D = 10 \mathrm {~cm}\).
  1. Find the area of triangle \(A B D\).
  2. Find the length of \(B D\).
  3. Find the length of \(B C\).
OCR C2 2014 June Q2
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 3 n - 1\), for \(n \geqslant 1\).
  1. Find the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).
OCR C2 2014 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-2_350_597_1695_735} The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 12 cm . The angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians.
  1. Find the exact length of the \(\operatorname { arc } A B\).
  2. Find the exact area of the shaded segment enclosed by the arc \(A B\) and the chord \(A B\).
OCR C2 2014 June Q4
4
  1. Show that the equation $$\sin x - \cos x = \frac { 6 \cos x } { \tan x }$$ can be expressed in the form $$\tan ^ { 2 } x - \tan x - 6 = 0 .$$
  2. Hence solve the equation \(\sin x - \cos x = \frac { 6 \cos x } { \tan x }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR C2 2014 June Q5
5 Solve the equation \(2 ^ { 4 x - 1 } = 3 ^ { 5 - 2 x }\), giving your answer in the form \(x = \frac { \log _ { 10 } a } { \log _ { 10 } b }\).
OCR C2 2014 June Q6
6
  1. Find the binomial expansion of \(\left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 } \mathrm {~d} x\).
OCR C2 2014 June Q7
7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 12 - 22 x + 9 x ^ { 2 } - x ^ { 3 }\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Show that ( \(3 - x\) ) is a factor of \(\mathrm { f } ( x )\).
  3. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  4. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in simplified surd form where appropriate.
OCR C2 2014 June Q8
8
  1. The first term of a geometric progression is 50 and the common ratio is 0.8 . Use logarithms to find the smallest value of \(k\) such that the value of the \(k\) th term is less than 0.15 .
  2. In a different geometric progression, the second term is - 3 and the sum to infinity is 4 . Show that there is only one possible value of the common ratio and hence find the first term. \section*{Question 9 begins on page 4.}
OCR C2 2014 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{9e95415c-00f5-4b52-a443-0b946602b3b4-4_387_624_287_717} The diagram shows part of the curve \(y = - 3 + 2 \sqrt { x + 4 }\). The point \(P ( 5,3 )\) lies on the curve. Region \(A\) is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 5\). Region \(B\) is bounded by the curve, the \(y\)-axis and the line \(y = 3\).
  1. Use the trapezium rule, with 2 strips each of width 2.5 , to find an approximate value for the area of region \(A\), giving your answer correct to 3 significant figures.
  2. Use your answer to part (i) to deduce an approximate value for the area of region \(B\).
  3. By first writing the equation of the curve in the form \(x = \mathrm { f } ( y )\), use integration to show that the exact area of region \(B\) is \(\frac { 14 } { 3 }\). \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {N } }\)}
OCR C2 2015 June Q1
1 A geometric progression has first term 3 and second term - 6 .
  1. State the value of the common ratio.
  2. Find the value of the eleventh term.
  3. Find the sum of the first twenty terms.
OCR C2 2015 June Q2
2
  1. Use the trapezium rule, with 4 strips each of width 1.5, to estimate the value of $$\int _ { 4 } ^ { 10 } \sqrt { 2 x - 1 } \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate.
OCR C2 2015 June Q3
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
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
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
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
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
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
\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 \(\alpha\),
    (a) the next smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\),
    (b) the smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = - k\).
  2. 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 \(y = \sin \frac { 1 } { 3 } x\).
  3. 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
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
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
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
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
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 \(\int _ { 1 } ^ { \infty } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\).