Questions — OCR C4 (317 questions)

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OCR C4 2006 June Q3
8 marks Moderate -0.3
  1. Express \(\frac{3 - 2x}{x(3 - x)}\) in partial fractions. [3]
  2. Show that \(\int_1^2 \frac{3 - 2x}{x(3 - x)} dx = 0\). [4]
  3. What does the result of part (ii) indicate about the graph of \(y = \frac{3 - 2x}{x(3 - x)}\) between \(x = 1\) and \(x = 2\)? [1]
OCR C4 2006 June Q4
8 marks Standard +0.3
The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$
  1. Find the angle between \(AB\) and \(AC\). [6]
  2. Find the area of triangle \(ABC\). [2]
OCR C4 2006 June Q5
8 marks Standard +0.3
A forest is burning so that, \(t\) hours after the start of the fire, the area burnt is \(A\) hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to \(A^2\).
  1. Write down a differential equation which models this situation. [2]
  2. After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt. [6]
OCR C4 2006 June Q6
8 marks Standard +0.3
  1. Show that the substitution \(u = e^x + 1\) transforms \(\int \frac{e^{2x}}{e^x + 1} dx\) to \(\int \frac{u - 1}{u} du\). [3]
  2. Hence show that \(\int_0^1 \frac{e^{2x}}{e^x + 1} dx = e - 1 - \ln\left(\frac{e + 1}{2}\right)\). [5]
OCR C4 2006 June Q7
8 marks Standard +0.3
Two lines have vector equations $$\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k} + \lambda(3\mathbf{i} + \mathbf{j} + a\mathbf{k})$$ and $$\mathbf{r} = -8\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu(\mathbf{i} - 2\mathbf{j} - \mathbf{k}),$$ where \(a\) is a constant.
  1. Given that the lines are skew, find the value that \(a\) cannot take. [6]
  2. Given instead that the lines intersect, find the point of intersection. [2]
OCR C4 2006 June Q8
9 marks Standard +0.8
  1. Show that \(\int \cos^2 6x dx = \frac{1}{2}x + \frac{1}{24}\sin 12x + c\). [3]
  2. Hence find the exact value of \(\int_0^{\frac{\pi}{12}} x\cos^2 6x dx\). [6]
OCR C4 2006 June Q9
12 marks Standard +0.3
A curve is given parametrically by the equations $$x = 4\cos t, \quad y = 3\sin t,$$ where \(0 \leq t \leq \frac{1}{2}\pi\).
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [3]
  2. Show that the equation of the tangent at the point \(P\), where \(t = p\), is $$3x\cos p + 4y\sin p = 12.$$ [3]
  3. The tangent at \(P\) meets the \(x\)-axis at \(R\) and the \(y\)-axis at \(S\). \(O\) is the origin. Show that the area of triangle \(ORS\) is \(\frac{6}{\sin 2p}\). [3]
  4. Write down the least possible value of the area of triangle \(ORS\), and give the corresponding value of \(p\). [3]
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]
OCR C4 Q6
10 marks Standard +0.3
Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf{r} = \begin{pmatrix} 1 \\ p \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ -1 \\ q \end{pmatrix},$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter. Given that the point \(A\) with coordinates \((-5, 9, -9)\) lies on \(l\),
  1. find the values of \(p\) and \(q\), [3]
  2. show that the point \(B\) with coordinates \((25, -1, 11)\) also lies on \(l\). [2]
The point \(C\) lies on \(l\) and is such that \(OC\) is perpendicular to \(l\).
  1. Find the coordinates of \(C\). [3]
  2. Find the ratio \(AC : CB\) [2]
OCR C4 Q7
11 marks Standard +0.3
  1. Use the substitution \(x = 2 \sin u\) to evaluate $$\int_0^{\sqrt{3}} \frac{1}{\sqrt{4-x^2}} \, dx.$$ [6]
  2. Evaluate $$\int_0^{\frac{\pi}{2}} x \cos x \, dx.$$ [5]
OCR C4 Q8
12 marks Moderate -0.3
The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
  1. Write down a differential equation connecting \(N\) and \(t\). [1]
Given that initially there are \(N_0\) bacteria present in a culture,
  1. Show that \(N = N_0 e^{kt}\), where \(k\) is a positive constant. [6]
Given also that the number of bacteria present doubles every six hours,
  1. find the value of \(k\), [3]
  2. Find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. [2]
OCR C4 Q1
4 marks Moderate -0.5
Express $$\frac{5x}{(x-4)(x+1)} + \frac{3}{(x-2)(x+1)}$$ as a single fraction in its simplest form. [4]