Questions — OCR (4907 questions)

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OCR C4 2007 January Q7
8 marks Challenging +1.2
The equation of a curve is \(2x^2 + xy + y^2 = 14\). Show that there are two stationary points on the curve and find their coordinates. [8]
OCR C4 2007 January Q8
10 marks Standard +0.3
The parametric equations of a curve are \(x = 2t^2\), \(y = 4t\). Two points on the curve are \(P(2p^2, 4p)\) and \(Q(2q^2, 4q)\).
  1. Show that the gradient of the normal to the curve at \(P\) is \(-p\). [2]
  2. Show that the gradient of the chord joining the points \(P\) and \(Q\) is \(\frac{2}{p + q}\). [2]
  3. The chord \(PQ\) is the normal to the curve at \(P\). Show that \(p^2 + pq + 2 = 0\). [2]
  4. The normal at the point \(R(8, 8)\) meets the curve again at \(S\). The normal at \(S\) meets the curve again at \(T\). Find the coordinates of \(T\). [4]
OCR C4 2007 January Q9
10 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{\sec^2 y}{\cos^2(2x)} \frac{dy}{dx} = 2.$$ [7]
  2. For the particular solution in which \(y = \frac{1}{4}\pi\) when \(x = 0\), find the value of \(y\) when \(x = \frac{1}{8}\pi\). [3]
OCR C4 2007 January Q10
11 marks Standard +0.3
The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5\mathbf{i} + 2\mathbf{j} - 9\mathbf{k}\) and \(4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) respectively.
  1. Find a vector equation for the line \(PQ\). [2]
The position vector of the point \(T\) is \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).
  1. Write down a vector equation for the line \(OT\) and show that \(OT\) is perpendicular to \(PQ\). [4]
It is given that \(OT\) intersects \(PQ\).
  1. Find the position vector of the point of intersection of \(OT\) and \(PQ\). [3]
  2. Hence find the perpendicular distance from \(O\) to \(PQ\), giving your answer in an exact form. [2]
OCR C4 2005 June Q1
4 marks Moderate -0.8
Find the quotient and the remainder when \(x^4 + 3x^3 + 5x^2 + 4x - 1\) is divided by \(x^2 + x + 1\). [4]
OCR C4 2005 June Q2
5 marks Moderate -0.3
Evaluate \(\int_0^{\frac{\pi}{2}} x \cos x dx\), giving your answer in an exact form. [5]
OCR C4 2005 June Q3
7 marks Standard +0.3
The line \(L_1\) passes through the points \((2, -3, 1)\) and \((-1, -2, -4)\). The line \(L_2\) passes through the point \((3, 2, -9)\) and is parallel to the vector \(\mathbf{4i} - \mathbf{4j} + \mathbf{5k}\).
  1. Find an equation for \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [2]
  2. Prove that \(L_1\) and \(L_2\) are skew. [5]
OCR C4 2005 June Q4
7 marks Standard +0.3
  1. Show that the substitution \(x = \tan \theta\) transforms \(\int \frac{1}{(1 + x^2)^2} dx\) to \(\int \cos^2 \theta d\theta\). [3]
  2. Hence find the exact value of \(\int_0^1 \frac{1}{(1 + x^2)^2} dx\). [4]
OCR C4 2005 June Q5
7 marks Moderate -0.3
\(ABCD\) is a parallelogram. The position vectors of \(A\), \(B\) and \(C\) are given respectively by $$\mathbf{a} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}, \quad \mathbf{b} = 3\mathbf{i} - 2\mathbf{j}, \quad \mathbf{c} = \mathbf{i} - \mathbf{j} - 2\mathbf{k}.$$
  1. Find the position vector of \(D\). [3]
  2. Determine, to the nearest degree, the angle \(ABC\). [4]
OCR C4 2005 June Q6
8 marks Standard +0.3
The equation of a curve is \(xy^2 = 2x + 3y\).
  1. Show that \(\frac{dy}{dx} = \frac{2 - y^2}{2xy - 3}\). [5]
  2. Show that the curve has no tangents which are parallel to the \(y\)-axis. [3]
OCR C4 2005 June Q7
10 marks Standard +0.3
A curve is given parametrically by the equations $$x = t^2, \quad y = \frac{1}{t}.$$
  1. Find \(\frac{dy}{dx}\) in terms of \(t\), giving your answer in its simplest form. [3]
  2. Show that the equation of the tangent at the point \(P\left(4, -\frac{1}{4}\right)\) is \(x - 16y = 12\). [3]
  3. Find the value of the parameter at the point where the tangent at \(P\) meets the curve again. [4]
OCR C4 2005 June Q8
11 marks Standard +0.3
  1. Given that \(\frac{3x + 4}{(1 + x)(2 + x)^2} \equiv \frac{A}{1 + x} + \frac{B}{2 + x} + \frac{C}{(2 + x)^2}\), find \(A\), \(B\) and \(C\). [5]
  2. Hence or otherwise expand \(\frac{3x + 4}{(1 + x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
  3. State the set of values of \(x\) for which the expansion in part (ii) is valid. [1]
OCR C4 2005 June Q9
13 marks Standard +0.3
Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of \(20°C\). At time \(t\) minutes later, the temperature of the liquid is \(\theta°C\).
  1. Explain how the information above leads to the differential equation $$\frac{d\theta}{dt} = -k(\theta - 20),$$ where \(k\) is a positive constant. [2]
  2. The liquid is initially at a temperature of \(100°C\). It takes 5 minutes for the liquid to cool from \(100°C\) to \(68°C\). Show that $$\theta = 20 + 80e^{-(\frac{k}{5} \ln \frac{5}{3})t}.$$ [8]
  3. Calculate how much longer it takes for the liquid to cool by a further \(32°C\). [3]
OCR C4 2006 June Q1
4 marks Moderate -0.3
Find the gradient of the curve \(4x^2 + 2xy + y^2 = 12\) at the point \((1, 2)\). [4]
OCR C4 2006 June Q2
7 marks Moderate -0.8
  1. Expand \((1 - 3x)^{-2}\) in ascending powers of \(x\), up to and including the term in \(x^2\). [3]
  2. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 + 2x)^2}{(1 - 3x)^2}\) in ascending powers of \(x\). [4]
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]