Questions — OCR C4 (317 questions)

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OCR C4 Q2
5 marks Standard +0.3
A curve has the equation $$x^2 + 2xy^2 + y = 4.$$ Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
OCR C4 Q3
5 marks Standard +0.3
Evaluate $$\int_0^{\frac{\pi}{4}} \sin 2x \cos x \, dx.$$ [5]
OCR C4 Q4
7 marks Standard +0.3
A curve has parametric equations $$x = \cos 2t, \quad y = \cosec t, \quad 0 < t < \frac{\pi}{2}.$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac{1}{2}\).
  1. Find the value of the parameter \(t\) at \(P\). [2]
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 2x + 1.$$ [5]
OCR C4 Q5
8 marks Standard +0.8
  1. Express \(\frac{2 + 20x}{1 + 2x - 8x^2}\) as a sum of partial fractions. [3]
  2. Hence find the series expansion of \(\frac{2 + 20x}{1 + 2x - 8x^2}\), \(|x| < \frac{1}{4}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
OCR C4 Q6
8 marks Standard +0.3
Use the substitution \(x = 2 \tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]
OCR C4 Q7
9 marks Standard +0.3
A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.
  1. Find the position vector of \(C\). [5]
Given that 1 unit on each coordinate axis represents 200 metres,
  1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]
OCR C4 Q8
12 marks Challenging +1.2
  1. Find \(\int \tan^2 x \, dx\). [3]
  2. Show that $$\int \tan x \, dx = \ln |\sec x| + c,$$ where \(c\) is an arbitrary constant. [4]
\includegraphics{figure_8} The diagram shows part of the curve with equation \(y = x^{\frac{1}{2}} \tan x\). The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{3}\) is rotated through \(360°\) about the \(x\)-axis.
  1. Show that the volume of the solid formed is \(\frac{1}{18}\pi^2(6\sqrt{3} - \pi) - \pi \ln 2\). [5]
OCR C4 Q9
14 marks Standard +0.3
An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.
  1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]
Given that after one week there are 360 insects in the colony,
  1. find the value of \(k\) to 3 significant figures. [2]
Given also that after two and three weeks there are 440 and 600 insects respectively,
  1. comment on suitability of the modelling assumption. [2]
An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25 \cos 0.5t).$$
  1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [3]
  2. Compare the suitability of the two models. [2]
OCR C4 Q1
6 marks Moderate -0.3
Express \(3\cos\theta + 4\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Hence find the range of the function \(f(\theta)\), where $$f(\theta) = 7 + 3\cos\theta + 4\sin\theta \quad \text{for } 0 \leqslant \theta \leqslant 2\pi.$$ Write down the greatest possible value of \(\frac{1}{7 + 3\cos\theta + 4\sin\theta}\). [6]
OCR C4 Q2
7 marks Moderate -0.3
Express \(3\sin x + 2\cos x\) in the form \(R\sin(x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = f(x)\), where $$f(x) = 3\sin x + 2\cos x, \quad 0 \leqslant x \leqslant \pi.$$ [7]
OCR C4 Q3
3 marks Moderate -0.8
Show that \(\frac{\sin 2\theta}{1 + \cos 2\theta} = \tan\theta\). [3]
OCR C4 Q4
7 marks Moderate -0.3
The angle \(\theta\) satisfies the equation \(\sin(\theta + 45°) = \cos\theta\).
  1. Using the exact values of \(\sin 45°\) and \(\cos 45°\), show that \(\tan\theta = \sqrt{2} - 1\). [5]
  2. Find the values of \(\theta\) for \(0° < \theta < 360°\). [2]
OCR C4 Q5
6 marks Standard +0.3
Solve the equation \(2\sin 2\theta + \cos 2\theta = 1\), for \(0° \leqslant \theta < 360°\). [6]
OCR C4 Q6
7 marks Standard +0.3
Express \(6\cos 2\theta + \sin\theta\) in terms of \(\sin\theta\). Hence solve the equation \(6\cos 2\theta + \sin\theta = 0\), for \(0° \leqslant \theta \leqslant 360°\). [7]
OCR C4 Q7
8 marks Standard +0.3
  1. Show that \(\cos(\alpha + \beta) = \frac{1 - \tan\alpha\tan\beta}{\sec\alpha\sec\beta}\). [3]
  2. Hence show that \(\cos 2\alpha = \frac{1 - \tan^2\alpha}{1 + \tan^2\alpha}\). [2]
  3. Hence or otherwise solve the equation \(\frac{1 - \tan^2\theta}{1 + \tan^2\theta} = \frac{1}{2}\) for \(0° \leqslant \theta \leqslant 180°\). [3]
OCR C4 Q8
16 marks Standard +0.3
In Fig. 6, OAB is a thin bent rod, with OA = \(a\) metres, AB = \(b\) metres and angle OAB = 120°. The bent rod lies in a vertical plane. OA makes an angle \(\theta\) above the horizontal. The vertical height BD of B above O is \(h\) metres. The horizontal through A meets BD at C and the vertical through A meets OD at E. \includegraphics{figure_6}
  1. Find angle BAC in terms of \(\theta\). Hence show that $$h = a\sin\theta + b\sin(\theta - 60°).$$ [3]
  2. Hence show that \(h = (a + \frac{1}{2}b)\sin\theta - \frac{\sqrt{3}}{2}b\cos\theta\). [3]
The rod now rotates about O, so that \(\theta\) varies. You may assume that the formulae for \(h\) in parts (i) and (ii) remain valid.
  1. Show that OB is horizontal when \(\tan\theta = \frac{\sqrt{3}b}{2a + b}\). [3]
In the case when \(a = 1\) and \(b = 2\), \(h = 2\sin\theta - \sqrt{3}\cos\theta\).
  1. Express \(2\sin\theta - \sqrt{3}\cos\theta\) in the form \(R\sin(\theta - \alpha)\). Hence, for this case, write down the maximum value of \(h\) and the corresponding value of \(\theta\). [7]
OCR C4 Q9
8 marks Standard +0.3
  1. Express \(\cos\theta + \sqrt{3}\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(\alpha\) is acute, expressing \(\alpha\) in terms of \(\pi\). [4]
  2. Write down the derivative of \(\tan\theta\). Hence show that \(\int_0^{\frac{\pi}{3}} \frac{1}{(\cos\theta + \sqrt{3}\sin\theta)^2} \, d\theta = \frac{\sqrt{3}}{4}\). [4]