Questions — OCR C2 (306 questions)

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OCR C2 Q4
6 marks Standard +0.3
4. (i) Solve the inequality $$x ^ { 2 } - 13 x + 30 < 0$$ (ii) Hence find the set of values of \(y\) such that $$2 ^ { 2 y } - 13 \left( 2 ^ { y } \right) + 30 < 0 .$$
OCR C2 Q5
7 marks Moderate -0.3
5.
\includegraphics[max width=\textwidth, alt={}]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-2_515_771_246_438}
The diagram shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates \(( - 4,0 )\),
  1. show that \(k = - 7\),
  2. find the coordinates of \(B\) and \(C\).
OCR C2 Q6
8 marks Moderate -0.8
6. Given that $$\mathrm { f } ^ { \prime } ( x ) = 5 + \frac { 4 } { x ^ { 2 } } , \quad x \neq 0$$
  1. find an expression for \(\mathrm { f } ( x )\). Given also that $$\mathrm { f } ( 2 ) = 2 \mathrm { f } ( 1 ) ,$$
  2. find \(\mathrm { f } ( 4 )\).
OCR C2 Q7
10 marks Standard +0.3
7. \includegraphics[max width=\textwidth, alt={}, center]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-3_376_892_221_427} The diagram shows a design painted on the wall at a karting track. The sign consists of triangle \(A B C\) and two circular sectors of radius 2 metres and 1 metre with centres \(A\) and \(B\) respectively. Given that \(A B = 7 \mathrm {~m} , A C = 3 \mathrm {~m}\) and \(\angle A C B = 2.2\) radians,
  1. find the size of \(\angle A B C\) in radians to 3 significant figures,
  2. show that \(\angle B A C = 0.588\) radians to 3 significant figures,
  3. find the area of triangle \(A B C\),
  4. find the area of the wall covered by the design.
OCR C2 Q8
12 marks Moderate -0.3
8. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).
  1. Use the trapezium rule with three intervals, each of width 2 , to estimate to 3 significant figures the area of \(R\).
  2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
  3. Find the percentage error in the estimate made in part (a).
OCR C2 Q9
12 marks Standard +0.3
9. The first two terms of a geometric progression are 2 and \(x\) respectively, where \(x \neq 2\).
  1. Find an expression for the third term in terms of \(x\). The first and third terms of arithmetic progression are 2 and \(x\) respectively.
  2. Find an expression for the 11th term in terms of \(x\). Given that the third term of the geometric progression and the 11th term of the arithmetic progression have the same value,
  3. find the value of \(x\),
  4. find the sum of the first 50 terms of the arithmetic progression.
OCR C2 Q1
4 marks Moderate -0.3
  1. Solve the equation
$$\log _ { 5 } ( 4 x + 3 ) - \log _ { 5 } ( x - 1 ) = 2$$
OCR C2 Q2
5 marks Moderate -0.8
  1. Find the coefficient of \(x ^ { 2 }\) in the expansion of
$$( 1 + x ) ( 1 - x ) ^ { 6 }$$
OCR C2 Q3
7 marks Moderate -0.8
  1. (i) Evaluate
$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.
OCR C2 Q4
7 marks Moderate -0.3
4. The diagram shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,
  1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
  2. show that \(Q R = 2 \sqrt { 6 }\),
  3. find \(\angle P Q R\) in degrees to 1 decimal place.
OCR C2 Q5
7 marks Moderate -0.3
5. (i) Find $$\int \left( 8 x - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x$$ The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 8 x - \frac { 2 } { x ^ { 3 } } , \quad x \neq 0$$ and the curve passes through the point \(( 1,1 )\).
(ii) Show that the equation of the curve can be written in the form $$y = \left( a x + \frac { b } { x } \right) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found.
OCR C2 Q6
9 marks Moderate -0.3
6. Given that $$f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + p x - 6$$ and that \(x = - 3\) is a solution to the equation \(\mathrm { f } ( x ) = 0\),
  1. find the value of the constant \(p\),
  2. show that when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) there is a remainder of 50 ,
  3. find the other solutions to the equation \(\mathrm { f } ( x ) = 0\), giving your answers to 2 decimal places.
OCR C2 Q7
10 marks Standard +0.3
7. The second and third terms of a geometric series are \(\log _ { 3 } 4\) and \(\log _ { 3 } 16\) respectively.
  1. Find the common ratio of the series.
  2. Show that the first term of the series is \(\log _ { 3 } 2\).
  3. Find, to 3 significant figures, the sum of the first six terms of the series.
OCR C2 Q8
11 marks Standard +0.3
8. (i) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\tan 2 x = 3$$ (ii) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y$$
OCR C2 Q9
12 marks Standard +0.3
9.
\includegraphics[max width=\textwidth, alt={}]{0744b3cf-2941-45cb-b6df-2aaf44588e5c-3_592_771_683_541}
The diagram shows the curve \(C\) with equation \(y = 3 x - 4 \sqrt { x } + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,
  1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2 x - 2\). The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the \(y\)-axis.
  2. Find the area of the shaded region.
OCR C2 Q1
6 marks Standard +0.8
  1. (i) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
    (ii) Hence state how many solutions exist to the equation
$$\sin 2 x = \tan \frac { x } { 2 } ,$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.
OCR C2 Q2
6 marks Standard +0.8
2. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593} The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).
OCR C2 Q3
7 marks Standard +0.3
3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = \left( u _ { n } \right) ^ { 2 } - 1 , \quad n \geq 1 .$$ Given that \(u _ { 1 } = k\), where \(k\) is a constant,
  1. find expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given also that \(u _ { 2 } + u _ { 3 } = 11\),
  2. find the possible values of \(k\).
OCR C2 Q4
7 marks Moderate -0.8
4. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-2_465_844_246_516} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).
  1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
  2. find an estimate for the volume of the support.
OCR C2 Q5
7 marks Moderate -0.3
5. (i) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (ii) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.
OCR C2 Q6
9 marks Moderate -0.8
6. (i) Evaluate $$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$ (ii) Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).
OCR C2 Q7
9 marks Moderate -0.3
7. \includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-3_499_721_248_552} The diagram shows part of the curve \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , x \neq 0\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
  3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).
OCR C2 Q8
10 marks Moderate -0.3
8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence. Given that sales total \(\pounds 8100\) during the first six months, use the model to
  1. find the value of \(x\),
  2. find the expected value of sales in the eighth month,
  3. show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
  4. Explain why this model cannot be valid over a long period of time.
OCR C2 Q9
11 marks Standard +0.3
9. \(f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + x + 2\).
  1. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\).
  4. Find, in terms of \(\pi\), the values of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) for which $$2 \sin ^ { 3 } \theta - 5 \sin ^ { 2 } \theta + \sin \theta + 2 = 0$$
OCR C2 2009 January Q1
6 marks Easy -1.2
1 Find
  1. \(\int \left( x ^ { 3 } + 8 x - 5 \right) \mathrm { d } x\),
  2. \(\int 12 \sqrt { x } \mathrm {~d} x\).