Questions — OCR C2 (306 questions)

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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]
OCR C2 Q4
7 marks Moderate -0.3
  1. Given that \(y = \log_2 x\), find expressions in terms of \(y\) for
    1. \(\log_2 \left(\frac{x}{2}\right)\), [2]
    2. \(\log_2 (\sqrt{x})\). [2]
  2. Hence, or otherwise, solve the equation $$2 \log_2 \left(\frac{x}{2}\right) + \log_2 (\sqrt{x}) = 8.$$ [3]
OCR C2 Q5
8 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,
  1. find the length \(OA\), [2]
  2. find the perimeter and the area of the shaded segment. [6]
OCR C2 Q6
8 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}} - x\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \((a, 0)\).
  1. Show that \(a = 8\). [3]
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis. [5]
OCR C2 Q7
10 marks Moderate -0.3
  1. Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]
    1. Write down the formula for the sum of the first \(n\) positive integers. [1]
    2. Using this formula, find the sum of the integers from 100 to 200 inclusive. [3]
    3. Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3. [2]
OCR C2 Q8
12 marks Standard +0.3
The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]
Using \(x = 6\),
  1. find the common ratio of the series, [1]
  2. find the sum of the first eight terms of the series. [2]
OCR C2 Q9
13 marks Moderate -0.3
  1. Evaluate $$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$ giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [6]
  2. The gradient of a curve is given by $$\frac{dy}{dx} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that the curve passes through the points \((0, -2)\) and \((2, 18)\), show that \(k = 2\) and find an equation for the curve. [7]
OCR C2 Q1
4 marks Easy -1.2
A geometric progression has first term 75 and second term \(-15\).
  1. Find the common ratio. [2]
  2. Find the sum to infinity. [2]
OCR C2 Q2
6 marks Moderate -0.3
Find the area of the finite region enclosed by the curve \(y = 5x - x^2\) and the \(x\)-axis. [6]
OCR C2 Q3
6 marks Moderate -0.8
During one day, a biological culture is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times (1.06)^t.$$ Using this model,
  1. find the number of bacteria present at 11 a.m., [2]
  2. find, to the nearest minute, the time when the initial number of bacteria will have doubled. [4]