Questions — Edexcel C4 (386 questions)

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Edexcel C4 Q5
9 marks Standard +0.3
The gradient at any point \((x, y)\) on a curve is proportional to \(\sqrt{y}\). Given that the curve passes through the point with coordinates \((0, 4)\),
  1. show that the equation of the curve can be written in the form $$2\sqrt{y} = kx + 4,$$ where \(k\) is a positive constant. [5]
Given also that the curve passes through the point with coordinates \((2, 9)\),
  1. find the equation of the curve in the form \(y = \text{f}(x)\). [4]
Edexcel C4 Q6
10 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a vertical cross-section of a vase. The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60°\). When the depth of water in the vase is \(h\) cm, the volume of water in the vase is \(V\) cm\(^3\).
  1. Show that \(V = \frac{1}{9}\pi h^3\). [3]
The vase is initially empty and water is poured in at a constant rate of 120 cm\(^3\) s\(^{-1}\).
  1. Find, to 2 decimal places, the rate at which \(h\) is increasing
    1. when \(h = 6\),
    2. after water has been poured in for 8 seconds. [7]
Edexcel C4 Q7
13 marks Standard +0.3
Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix}\) respectively.
  1. Find a vector equation for the line \(l_1\) which passes through \(A\) and \(B\). [2]
The line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -7 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.$$
  1. Show that lines \(l_1\) and \(l_2\) do not intersect. [5]
  2. Find the position vector of the point \(C\) on \(l_2\) such that \(\angle ABC = 90°\). [6]
Edexcel C4 Q8
14 marks Standard +0.3
$$\text{f}(x) = \frac{x(3x-7)}{(1-x)(1-3x)}, \quad |x| < \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = A + \frac{B}{1-x} + \frac{C}{1-3x}.$$ [4]
  2. Evaluate $$\int_0^{\frac{1}{4}} \text{f}(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational. [5]
  3. Find the series expansion of f(x) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
Edexcel C4 Q1
8 marks Moderate -0.3
The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$
  1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
  2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
  3. Explain why this model would not be appropriate for large values of \(t\). [1]
Edexcel C4 Q2
8 marks Standard +0.8
A curve has the equation $$3x^2 + xy - 2y^2 + 25 = 0.$$ Find an equation for the normal to the curve at the point with coordinates \((1, 4)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]
Edexcel C4 Q3
10 marks Moderate -0.3
  1. Use the substitution \(u = 2 - x^2\) to find $$\int \frac{x}{2 - x^2} \, dx.$$ [4]
  2. Evaluate $$\int_0^{\frac{1}{4}} \sin 3x \cos x \, dx.$$ [6]
Edexcel C4 Q4
11 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x\sqrt{\ln x}\), \(x \geq 1\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
  1. Using the trapezium rule with two intervals of equal width, estimate the area of the shaded region. [4]
The shaded region is rotated through \(360°\) about the \(x\)-axis.
  1. Find the exact volume of the solid formed. [7]
Edexcel C4 Q5
12 marks Standard +0.3
$$f(x) = \frac{5 - 8x}{(1 + 2x)(1 - x)^2}.$$
  1. Express \(f(x)\) in partial fractions. [5]
  2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying each coefficient. [6]
  3. State the set of values of \(x\) for which your expansion is valid. [1]
Edexcel C4 Q6
12 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the curve with parametric equations $$x = t + \sin t, \quad y = \sin t, \quad 0 \leq t \leq \pi.$$
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
  2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis. [3]
  3. Show that the region bounded by the curve and the \(x\)-axis has area 2. [6]
Edexcel C4 Q7
14 marks Standard +0.3
The line \(l_1\) passes through the points \(A\) and \(B\) with position vectors \((\mathbf{3i} + \mathbf{6j} - \mathbf{8k})\) and \((\mathbf{8j} - \mathbf{6k})\) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l_1\). [2]
The line \(l_2\) has vector equation $$\mathbf{r} = (-\mathbf{2i} + \mathbf{10j} + \mathbf{6k}) + \mu(\mathbf{7i} - \mathbf{4j} + \mathbf{6k}),$$ where \(\mu\) is a scalar parameter.
  1. Show that lines \(l_1\) and \(l_2\) intersect. [4]
  2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [2]
The point \(C\) lies on \(l_2\) and is such that \(AC\) is perpendicular to \(AB\).
  1. Find the position vector of \(C\). [6]