1.07s Parametric and implicit differentiation

761 questions

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SPS SPS FM Pure 2025 September Q4
13 marks Standard +0.8
The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [2]
The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).
  1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]
The line \(l\) intersects the curve \(C\) again at the point \(Q\).
  1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]
OCR H240/03 2018 March Q4
11 marks Standard +0.3
A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t^2, \quad y = t^3.$$
  1. Show that the equation of the tangent at the point with parameter \(t\) is $$2y = 3tx - t^3.$$ [4]
  1. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A\left(\frac{19}{2}, -\frac{15}{8}\right)\) and it meets the \(x\)-axis at the point \(B\). Find the area of triangle \(OAB\), where \(O\) is the origin. [7]
OCR H240/01 2017 Specimen Q10
8 marks Standard +0.3
A curve has equation \(x = (y + 5)\ln(2y - 7)\).
  1. Find \(\frac{dx}{dy}\) in terms of y. [3]
  2. Find the gradient of the curve where it crosses the y-axis. [5]
OCR H240/01 2017 Specimen Q12
11 marks Standard +0.8
The parametric equations of a curve are given by \(x = 2\cos\theta\) and \(y = 3\sin\theta\) for \(0 \leq \theta < 2\pi\).
  1. Find \(\frac{dy}{dx}\) in terms of \(\theta\). [2]
The tangents to the curve at the points P and Q pass through the point (2, 6).
  1. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2\sin\theta + \cos\theta = 1\). [4]
  2. Find the values of \(\theta\) at the points P and Q. [5]
OCR H240/01 2017 Specimen Q13
9 marks Challenging +1.8
In this question you must show detailed reasoning. Find the exact values of the x-coordinates of the stationary points of the curve \(x^3 + y^3 = 3xy + 35\). [9]
Pre-U Pre-U 9794/1 2010 June Q5
7 marks Standard +0.3
The parametric equations of a curve are \(x = \frac{1}{1 + t^2}\) and \(y = \frac{t}{1 + t^2}\), \(t \in \mathbb{R}\).
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [5]
  2. Hence find the coordinates of the stationary points of the curve. [2]
Pre-U Pre-U 9794/1 2010 June Q9
9 marks Standard +0.3
A curve has equation \(x^2 - xy + y^2 = 1\).
  1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
  2. Find the coordinates of the points on the curve in the second and fourth quadrants where the tangent is parallel to \(y = x\). [5]
Pre-U Pre-U 9794/2 2011 June Q8
15 marks Challenging +1.3
  1. A curve \(C_1\) is defined by the parametric equations $$x = \theta - \sin \theta, \quad y = 1 - \cos \theta,$$ where the parameter \(\theta\) is measured in radians.
    1. Show that \(\frac{dy}{dx} = \cot \frac{1}{2}\theta\), except for certain values of \(\theta\), which should be identified. [5]
    2. Show that the points of intersection of the curve \(C_1\) and the line \(y = x\) are determined by an equation of the form \(\theta = 1 + A \sin(\theta - \alpha)\), where \(A\) and \(\alpha\) are constants to be found, such that \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [4]
    3. Show that the equation found in part (b) has a root between \(\frac{1}{4}\pi\) and \(\pi\). [2]
  2. A curve \(C_2\) is defined by the parametric equations $$x = \theta - \frac{1}{2} \sin \theta, \quad y = 1 - \frac{1}{2} \cos \theta,$$ where the parameter \(\theta\) is measured in radians. Find the y-coordinates of all points on \(C_2\) for which \(\frac{d^2y}{dx^2} = 0\). [4]
Pre-U Pre-U 9794/2 2012 June Q11
15 marks Challenging +1.2
The function f is defined by \(f : t \mapsto 2 \sin t + \cos 2t\) for \(0 \leqslant t < 2\pi\).
  1. Show that \(\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)\). [2]
  2. Determine the range of f. [5]
A curve \(C\) is given parametrically by \(x = 2 \cos t + \sin 2t\), \(y = f(t)\) for \(0 \leqslant t < 2\pi\).
  1. Show that \(x^2 + y^2 = 5 + 4 \sin 3t\). [3]
  2. Deduce that \(C\) lies between two circles centred at the origin, and touches both. [2]
  3. Find the gradient of the tangent to \(C\) at the point at which \(t = 0\). [3]
Pre-U Pre-U 9794/2 Specimen Q8
14 marks Standard +0.8
    1. Find the general solution of the differential equation $$x \frac{dy}{dx} = y(1 + x \cot x),$$ expressing \(y\) in terms of \(x\). [5]
    2. Find the particular solution given that \(y = 1\) when \(x = \frac{1}{2}\pi\). [2]
  2. The real variables \(x\) and \(y\) are related by \(x^2 - y^2 = 2ax - b\), where \(a\) and \(b\) are real constants.
    1. Show that \(\frac{dy}{dx} = 0\) can only be solved for \(x\) and \(y\) if \(b \geqslant a^2\). [5]
    2. Show that \(y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2\). [2]
Edexcel AEA 2011 June Q4
13 marks Challenging +1.2
The curve \(C\) has parametric equations $$x = \cos^2 t$$ $$y = \cos t \sin t$$ where \(0 \leq t < \pi\)
  1. Show that \(C\) is a circle and find its centre and its radius. [5]
% Figure 1 shows a sketch of C with point P, rectangle R with diagonal OP \includegraphics{figure_1} Figure 1 Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \((\cos^2 \alpha, \cos\alpha \sin \alpha)\), \(0 < \alpha < \frac{\pi}{2}\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(OP\) as a diagonal, where \(O\) is the origin.
  1. Show that the area of \(R\) is \(\sin\alpha \cos^3 \alpha\) [1]
  2. Find the maximum area of \(R\), as \(\alpha\) varies. [7]
[Total 13 marks]