Questions — OCR (4907 questions)

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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]
OCR C2 Specimen Q5
8 marks Moderate -0.3
  1. Show that the equation \(15\cos^2\theta = 13 + \sin\theta\) may be written as a quadratic equation in \(\sin\theta\). [2]
  2. Hence solve the equation, giving all values of \(\theta\) such that \(0 \leq \theta \leq 360\). [6]
OCR C2 Specimen Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows triangle \(ABC\), in which \(AB = 3\) cm, \(AC = 5\) cm and angle \(ABC = 2.1\) radians. Calculate
  1. angle \(ACB\), giving your answer in radians, [2]
  2. the area of the triangle. [3]
An arc of a circle with centre \(A\) and radius 3 cm is drawn, cutting \(AC\) at the point \(D\).
  1. Calculate the perimeter and the area of the sector \(ABD\). [4]
OCR C2 Specimen Q7
9 marks Moderate -0.8
\includegraphics{figure_7} The diagram shows the curves \(y = -3x^2 - 9x + 30\) and \(y = x^2 + 3x - 10\).
  1. Verify that the curves intersect at the points \(A(-5, 0)\) and \(B(2, 0)\). [2]
  2. Show that the area of the shaded region between the curves is given by \(\int_{-5}^{2} (-4x^2 - 12x + 40) dx\). [2]
  3. Hence or otherwise show that the area of the shaded region between the curves is \(228\frac{2}{3}\). [5]
OCR C2 Specimen Q8
10 marks Moderate -0.3
\includegraphics{figure_8} The diagram shows the curve \(y = 1.25^x\).
  1. A point on the curve has y-coordinate 2. Calculate its x-coordinate. [3]
  2. Use the trapezium rule with 4 intervals to estimate the area of the shaded region, bounded by the curve, the axes, and the line \(x = 4\). [4]
  3. State, with a reason, whether the estimate found in part (ii) is an overestimate or an underestimate. [2]
  4. Explain briefly how the trapezium rule could be used to find a more accurate estimate of the area of the shaded region. [1]
OCR C2 Specimen Q9
11 marks Standard +0.3
The cubic polynomial \(x^3 + ax^2 + bx - 6\) is denoted by f\((x)\).
  1. The remainder when f\((x)\) is divided by \((x - 2)\) is equal to the remainder when f\((x)\) is divided by \((x + 2)\). Show that \(b = -4\). [3]
  2. Given also that \((x - 1)\) is a factor of f\((x)\), find the value of \(a\). [2]
  3. With these values of \(a\) and \(b\), express f\((x)\) as a product of a linear factor and a quadratic factor. [3]
  4. Hence determine the number of real roots of the equation f\((x) = 0\), explaining your reasoning. [3]
OCR C2 Q1
4 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the sector \(OAB\) of a circle of radius 9.2 cm and centre \(O\). Given that the area of the sector is 37.4 cm\(^2\), find to 3 significant figures
  1. the size of \(\angle AOB\) in radians, [2]
  2. the perimeter of the sector. [2]
OCR C2 Q2
6 marks Moderate -0.3
$$f(x) = x^3 + kx - 20.$$ Given that f(x) is exactly divisible by \((x + 1)\),
  1. find the value of the constant \(k\), [2]
  2. solve the equation \(f(x) = 0\). [4]
OCR C2 Q3
7 marks Moderate -0.3
Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{4}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]
OCR C2 Q4
8 marks Moderate -0.3
A geometric progression has third term 36 and fourth term 27. Find
  1. the common ratio, [2]
  2. the fifth term, [2]
  3. the sum to infinity. [4]
OCR C2 Q5
8 marks Standard +0.3
  1. Solve the equation $$\log_2 (6 - x) = 3 - \log_2 x.$$ [4]
  2. Find the smallest integer \(n\) such that $$3^{n-2} > 8^{250}.$$ [4]
OCR C2 Q6
8 marks Moderate -0.8
$$f(x) = \cos 2x, \quad 0 \leq x \leq \pi.$$
  1. Sketch the curve \(y = f(x)\). [2]
  2. Write down the coordinates of any points where the curve \(y = f(x)\) meets the coordinate axes. [3]
  3. Solve the equation \(f(x) = 0.5\), giving your answers in terms of \(\pi\). [3]
OCR C2 Q7
9 marks Moderate -0.8
  1. Find $$\int \left( x + 5 + \frac{3}{\sqrt{x}} \right) dx.$$ [4]
  2. Evaluate $$\int_{-2}^{0} (3x - 1)^2 dx.$$ [5]
OCR C2 Q8
11 marks Moderate -0.3
  1. An arithmetic series has a common difference of 7. Given that the sum of the first 20 terms of the series is 530, find
    1. the first term of the series, [3]
    2. the smallest positive term of the series. [2]
  2. The terms of a sequence are given by $$u_n = (n + k)^2, \quad n \geq 1,$$ where \(k\) is a positive constant. Given that \(u_2 = 2u_1\),
    1. find the value of \(k\), [4]
    2. show that \(u_3 = 11 + 6\sqrt{2}\). [2]
OCR C2 Q9
11 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows the curve \(y = 2x^2 + 6x + 7\) and the straight line \(y = 2x + 13\).
  1. Find the coordinates of the points where the curve and line intersect. [4]
  2. Show that the area of the shaded region bounded by the curve and line is given by $$\int_{-3}^{1} (6 - 4x - 2x^2) dx.$$ [2]
  3. Hence find the area of the shaded region. [5]
OCR C2 Q2
4 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows the curve with equation \(y = 2^x\). Use the trapezium rule with four intervals, each of width 1, to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = -2\) and \(x = 2\). [4]
OCR C2 Q3
6 marks Moderate -0.8
  1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
  2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]