Questions — OCR (4907 questions)

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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]
OCR C2 Q4
7 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve with equation \(y = (x - \log_{10} x)^2\), \(x > 0\).
  1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.
    \(x\)23456
    \(y\)2.896.36
    [2]
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).
  1. Use the trapezium rule with all the values in your table to estimate the area of the shaded region. [3]
  2. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area. [2]
OCR C2 Q5
8 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos 3x = \frac{\sqrt{3}}{2}.$$ [5]
OCR C2 Q6
8 marks Standard +0.3
\includegraphics{figure_6} The diagram shows triangle \(ABC\) in which \(AC = 8\) cm and \(\angle BAC = \angle BCA = 30°\).
  1. Find the area of triangle \(ABC\) in the form \(k\sqrt{3}\). [4]
The point \(M\) is the mid-point of \(AC\) and the points \(N\) and \(O\) lie on \(AB\) and \(BC\) such that \(MN\) and \(MO\) are arcs of circles with centres \(A\) and \(C\) respectively.
  1. Show that the area of the shaded region \(BNMO\) is \(\frac{8}{3}(2\sqrt{3} - \pi)\) cm\(^2\). [4]
OCR C2 Q7
9 marks Moderate -0.8
  1. Expand \((2 + x)^4\) in ascending powers of \(x\), simplifying each coefficient. [4]
  2. Find the integers \(A\), \(B\) and \(C\) such that $$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
  3. Find the real values of \(x\) for which $$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]
OCR C2 Q8
12 marks Moderate -0.3
  1. The gradient of a curve is given by $$\frac{dy}{dx} = 3 - \frac{2}{x^2}, \quad x \neq 0.$$ Find an equation for the curve given that it passes through the point \((2, 6)\). [6]
  2. Show that $$\int_2^3 (6\sqrt{x} - \frac{4}{\sqrt{x}}) \, dx = k\sqrt{3},$$ where \(k\) is an integer to be found. [6]
OCR C2 Q9
12 marks Standard +0.3
The polynomial f(x) is given by $$\text{f}(x) = x^3 + kx^2 - 7x - 15,$$ where \(k\) is a constant. When f(x) is divided by \((x + 1)\) the remainder is \(r\). When f(x) is divided by \((x - 3)\) the remainder is \(3r\).
  1. Find the value of \(k\). [5]
  2. Find the value of \(r\). [1]
  3. Show that \((x - 5)\) is a factor of f(x). [2]
  4. Show that there is only one real solution to the equation f(x) = 0. [4]
OCR C3 Q1
4 marks Moderate -0.8
The function f is defined for all real values of \(x\) by $$f(x) = 10 - (x + 3)^2.$$
  1. State the range of f. [1]
  2. Find the value of ff(-1). [3]
OCR C3 Q2
4 marks Moderate -0.3
Find the exact solutions of the equation \(|6x - 1| = |x - 1|\). [4]
OCR C3 Q3
6 marks Moderate -0.3
The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180e^{-0.017t}.$$
  1. Find the value of \(t\) for which the mass is 25 grams. [3]
  2. Find the rate at which the mass is decreasing when \(t = 55\). [3]
OCR C3 Q4
8 marks Standard +0.2
  1. \includegraphics{figure_4a} The diagram shows the curve \(y = \frac{2}{\sqrt{x}}\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1\), \(x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed. [4]
  2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int_1^5 \sqrt{(x^2 + 1)} \, dx,$$ giving your answer correct to 3 decimal places. [4]
OCR C3 Q5
8 marks Standard +0.3
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac{5}{2}\), giving all solutions for which \(0° < \theta < 360°\). [5]
OCR C3 Q6
7 marks Moderate -0.3
  1. Find the exact value of the \(x\)-coordinate of the stationary point of the curve \(y = x \ln x\). [4]
  2. The equation of a curve is \(y = \frac{4x + c}{4x - c}\), where \(c\) is a non-zero constant. Show by differentiation that this curve has no stationary points. [3]
OCR C3 Q7
9 marks Standard +0.3
  1. Write down the formula for \(\cos 2x\) in terms of \(\cos x\). [1]
  2. Prove the identity \(\frac{4 \cos 2x}{1 + \cos 2x} = 4 - 2 \sec^2 x\). [3]
  3. Solve, for \(0 < x < 2\pi\), the equation \(\frac{4 \cos 2x}{1 + \cos 2x} = 3 \tan x - 7\). [5]
OCR C3 Q8
16 marks Standard +0.3
\includegraphics{figure_8} The diagram shows part of each of the curves \(y = e^{\frac{1}{3}x}\) and \(y = \sqrt[3]{(3x + 8)}\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.
  1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3. [3]
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac{2}{3} \ln(3x + 8)\). [2]
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places. [3]
  4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\). [5]
OCR C3 Q9
13 marks Challenging +1.2
\includegraphics{figure_9} The function f is defined by \(f(x) = \sqrt{(mx + 7)} - 4\), where \(x \geq -\frac{7}{m}\) and \(m\) is a positive constant. The diagram shows the curve \(y = f(x)\).
  1. A sequence of transformations maps the curve \(y = \sqrt{x}\) to the curve \(y = f(x)\). Give details of these transformations. [4]
  2. Explain how you can tell that f is a one-one function and find an expression for \(f^{-1}(x)\). [4]
  3. It is given that the curves \(y = f(x)\) and \(y = f^{-1}(x)\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5]
OCR C3 Q1
4 marks Moderate -0.5
Show that \(\int_2^8 \frac{3}{x} \, dx = \ln 64\). [4]