Questions C4 (1219 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]
OCR C4 Q1
4 marks Moderate -0.5
Find \(\int xe^{3x} dx\). [4]
OCR C4 Q2
4 marks Moderate -0.8
Find the quotient and remainder when \((x^4 + x^3 - 5x^2 - 9)\) is divided by \((x^2 + x - 6)\). [4]
OCR C4 Q3
6 marks Moderate -0.3
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\cot x^2\) [2]
  2. \(\frac{\sin x}{3 + 2\cos x}\) [4]
OCR C4 Q4
7 marks Moderate -0.3
  1. Expand \((1 - 3x)^{-2}, |x| < \frac{1}{3}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [4]
  2. Hence, or otherwise, show that for small \(x\), $$\left(\frac{2-x}{1-3x}\right)^2 \approx 4 + 20x + 85x^2 + 330x^3.$$ [3]
OCR C4 Q5
8 marks Standard +0.3
\includegraphics{figure_5} The diagram shows the curve with parametric equations $$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$ where \(a\) is a positive constant.
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
  1. Show that the area of triangle \(OAB\) is \(a^2\). [5]
OCR C4 Q6
9 marks Standard +0.3
Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.
  1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
  2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]
OCR C4 Q7
9 marks Standard +0.3
At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac{dy}{dt} = -ke^{-0.2t},$$ where \(k\) is a positive constant.
  1. Find an expression for \(y\) in terms of \(k\) and \(t\). [4]
Given that two hours after being filled the depth of water in the tank is 1.6 metres,
  1. find the value of \(k\) to 4 significant figures. [2]
Given also that the hole in the tank is \(h\) cm above the base of the tank,
  1. show that \(h = 79\) to 2 significant figures. [3]
OCR C4 Q8
11 marks Standard +0.3
A curve has the equation $$x^2 - 4xy + 2y^2 = 1.$$
  1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [4]
  2. Show that the tangent to the curve at the point \(P(1, 2)\) has the equation $$3x - 2y + 1 = 0.$$ [3]
The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  1. Find the coordinates of \(Q\). [4]
OCR C4 Q9
14 marks Standard +0.3
  1. Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac{6}{\cos x(2 - \sin x)} dx$$ into the integral $$\int \frac{6}{(1-u^2)(2-u)} du.$$ [4]
  2. Express \(\frac{6}{(1-u^2)(2-u)}\) in partial fractions. [4]
  3. Hence, evaluate $$\int_0^{\pi/6} \frac{6}{\cos x(2 - \sin x)} dx,$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers. [6]
OCR C4 Q1
4 marks Moderate -0.8
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\ln(\cos x)\) [2]
  2. \(x^2 \sin 3x\) [2]
OCR C4 Q2
7 marks Standard +0.3
A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
  2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]
OCR C4 Q3
9 marks Standard +0.3
$$f(x) = 3 - \frac{x-1}{x-3} + \frac{x+11}{2x^2-5x-3}, \quad |x| < \frac{1}{2}.$$
  1. Show that $$f(x) = \frac{4x-1}{2x+1}.$$ [4]
  2. 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]
OCR C4 Q4
9 marks Standard +0.3
A curve has parametric equations $$x = t^3 + 1, \quad y = \frac{2}{t}, \quad t \neq 0.$$
  1. Find an equation for the normal to the curve at the point where \(t = 1\), giving your answer in the form \(y = mx + c\). [6]
  2. Find a cartesian equation for the curve in the form \(y = f(x)\). [3]
OCR C4 Q5
10 marks Standard +0.3
$$f(x) = \frac{15-17x}{(2+x)(1-3x)^2}, \quad x \neq -2, \quad x \neq \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$f(x) = \frac{A}{2+x} + \frac{B}{1-3x} + \frac{C}{(1-3x)^2}.$$ [5]
  2. Find the value of $$\int_{-1}^{0} f(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers. [5]