Questions — OCR C2 (306 questions)

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OCR C2 Q8
11 marks Standard +0.3
  1. (i) Given that
$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
(ii) Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.
OCR C2 Q9
11 marks Standard +0.3
9. The second and fifth terms of a geometric series are - 48 and 6 respectively.
  1. Find the first term and the common ratio of the series.
  2. Find the sum to infinity of the series.
  3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2 ^ { 6 - n }\).
OCR C2 Q1
6 marks Easy -1.2
A sequence \(S\) has terms \(u_1, u_2, u_3, \ldots\) defined by $$u_n = 3n - 1,$$ for \(n \geqslant 1\).
  1. Write down the values of \(u_1, u_2\) and \(u_3\), and state what type of sequence \(S\) is. [3]
  2. Evaluate \(\sum_{n=1}^{100} u_n\). [3]
OCR C2 Q2
7 marks Moderate -0.3
\includegraphics{figure_2} A sector \(OAB\) of a circle of radius \(r\) cm has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is 36 cm² (see diagram).
  1. Write down two equations involving \(r\) and \(\theta\). [2]
  2. Hence show that \(r = 6\), and state the value of \(\theta\). [2]
  3. Find the area of the segment bounded by the arc \(AB\) and the chord \(AB\). [3]
OCR C2 Q3
7 marks Moderate -0.8
  1. Find \(\int (2x + 1)(x + 3) \, dx\). [4]
  2. Evaluate \(\int_0^9 \frac{1}{\sqrt{x}} \, dx\). [3]
OCR C2 Q4
8 marks Standard +0.3
\includegraphics{figure_4} In the diagram, \(ABCD\) is a quadrilateral in which \(AD\) is parallel to \(BC\). It is given that \(AB = 9\), \(BC = 6\), \(CA = 5\) and \(CD = 15\).
  1. Show that \(\cos BCA = -\frac{1}{3}\), and hence find the value of \(\sin BCA\). [4]
  2. Find the angle \(ADC\) correct to the nearest \(0.1°\). [4]
OCR C2 Q5
8 marks Moderate -0.3
The cubic polynomial \(f(x)\) is given by $$f(x) = x^3 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 1)\) is a factor of \(f(x)\) and that the remainder when \(f(x)\) is divided by \((x - 3)\) is 16.
  1. Find the values of \(a\) and \(b\). [5]
  2. Hence verify that \(f(2) = 0\), and factorise \(f(x)\) completely. [3]
OCR C2 Q6
8 marks Moderate -0.8
  1. Find the binomial expansion of \(\left(x^2 + \frac{1}{x}\right)^3\), simplifying the terms. [4]
  2. Hence find \(\int \left(x^2 + \frac{1}{x}\right)^3 dx\). [4]
OCR C2 Q7
7 marks Moderate -0.8
  1. Evaluate \(\log_3 15 + \log_3 20 - \log_3 12\). [3]
  2. Given that \(y = 3 \times 10^{2x}\), show that \(x = a \log_{10}(by)\), where the values of the constants \(a\) and \(b\) are to be found. [4]
OCR C2 Q8
9 marks Moderate -0.3
The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9. The amount pumped in 2001 was 100 000 barrels.
  1. Calculate the amount pumped in 2004. [2]
It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.
  1. Calculate in which year the amount pumped will fall below 5000 barrels. [4]
  2. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production. [3]
OCR C2 Q9
12 marks Standard +0.2
    1. Write down the exact values of \(\cos \frac{1}{6}\pi\) and \(\tan \frac{1}{6}\pi\) (where the angles are in radians). Hence verify that \(x = \frac{1}{6}\pi\) is a solution of the equation $$2 \cos x = \tan 2x.$$ [3]
    2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2x.$$ [4]
    1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.) [4]
    2. State with a reason whether this approximation is an underestimate or an overestimate. [1]
OCR C2 2007 January Q1
4 marks Moderate -0.8
In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms. [4]
OCR C2 2007 January Q2
5 marks Easy -1.2
\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle, centre \(O\) and radius 8 cm. The angle \(AOB\) is \(46°\).
  1. Express \(46°\) in radians, correct to 3 significant figures. [2]
  2. Find the length of the arc \(AB\). [1]
  3. Find the area of the sector \(OAB\). [2]
OCR C2 2007 January Q3
5 marks Easy -1.2
  1. Find \(\int (4x - 5) dx\). [2]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = 4x - 5\). The curve passes through the point \((3, 7)\). Find the equation of the curve. [3]
OCR C2 2007 January Q4
6 marks Moderate -0.8
In a triangle \(ABC\), \(AB = 5\sqrt{2}\) cm, \(BC = 8\) cm and angle \(B = 60°\).
  1. Find the exact area of the triangle, giving your answer as simply as possible. [3]
  2. Find the length of \(AC\), correct to 3 significant figures. [3]
OCR C2 2007 January Q5
8 marks Moderate -0.8
    1. Express \(\log_3(4x + 7) - \log_3 x\) as a single logarithm. [1]
    2. Hence solve the equation \(\log_3(4x + 7) - \log_3 x = 2\). [3]
  2. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int_3^9 \log_{10} x \, dx,$$ giving your answer correct to 3 significant figures. [4]
OCR C2 2007 January Q6
7 marks Moderate -0.8
  1. Find and simplify the first four terms in the expansion of \((1 + 4x)^7\) in ascending powers of \(x\). [4]
  2. In the expansion of $$(3 + ax)(1 + 4x)^7,$$ the coefficient of \(x^2\) is 1001. Find the value of \(a\). [3]
OCR C2 2007 January Q7
8 marks Moderate -0.8
    1. Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0° \leq x \leq 360°\), indicating the coordinates of any points where the curve meets the axes. [2]
    2. Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0°\) and \(360°\). [3]
  2. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(-180°\) and \(180°\). [3]
OCR C2 2007 January Q8
9 marks Moderate -0.3
The polynomial f(x) is defined by \(f(x) = x^3 - 9x^2 + 7x + 33\).
  1. Find the remainder when f(x) is divided by \((x + 2)\). [2]
  2. Show that \((x - 3)\) is a factor of f(x). [1]
  3. Solve the equation f(x) = 0, giving each root in an exact form as simply as possible. [6]
OCR C2 2007 January Q9
10 marks Standard +0.3
On its first trip between Maltby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses 2% more coal than the previous trip.
  1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures. [2]
  2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02^N < 1.52.$$ [4]
  3. Hence, by using logarithms, find the greatest number of trips possible. [4]
OCR C2 2007 January Q10
10 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the graph of \(y = 1 - 3x^{-\frac{1}{2}}\).
  1. Verify that the curve intersects the \(x\)-axis at \((9, 0)\). [1]
  2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\)). Given that the area of the shaded region is 4 square units, find the value of \(a\). [9]
OCR C2 Specimen Q1
5 marks Easy -1.2
Expand \((1-2x)^4\) in ascending powers of \(x\), simplifying the coefficients. [5]
OCR C2 Specimen Q2
6 marks Easy -1.2
  1. Find \(\int \frac{1}{x^2} dx\). [3]
  2. The gradient of a curve is given by \(\frac{dy}{dx} = \frac{1}{x^2}\). Find the equation of the curve, given that it passes through the point \((1, 3)\). [3]
OCR C2 Specimen Q3
7 marks Moderate -0.8
  1. Express each of the following in terms of \(\log_2 x\):
    1. \(\log_2(x^2)\), [1]
    2. \(\log_2(8x^2)\). [3]
  2. Given that \(y^2 = 27\), find the value of \(\log_3 y\). [3]
OCR C2 Specimen Q4
7 marks Moderate -0.8
Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher's forecasts for
  1. the number of copies that will be sold in the 20th week after publication, [3]
  2. the total number of copies sold during the first 20 weeks after publication, [2]
  3. the total number of copies that will ever be sold. [2]