Questions — OCR (4907 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
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]
OCR C4 Q2
5 marks Standard +0.3
A curve has the equation $$x^2 + 2xy^2 + y = 4.$$ Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
OCR C4 Q3
5 marks Standard +0.3
Evaluate $$\int_0^{\frac{\pi}{4}} \sin 2x \cos x \, dx.$$ [5]
OCR C4 Q4
7 marks Standard +0.3
A curve has parametric equations $$x = \cos 2t, \quad y = \cosec t, \quad 0 < t < \frac{\pi}{2}.$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac{1}{2}\).
  1. Find the value of the parameter \(t\) at \(P\). [2]
  2. Show that the tangent to the curve at \(P\) has the equation $$y = 2x + 1.$$ [5]
OCR C4 Q5
8 marks Standard +0.8
  1. Express \(\frac{2 + 20x}{1 + 2x - 8x^2}\) as a sum of partial fractions. [3]
  2. Hence find the series expansion of \(\frac{2 + 20x}{1 + 2x - 8x^2}\), \(|x| < \frac{1}{4}\), in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]
OCR C4 Q6
8 marks Standard +0.3
Use the substitution \(x = 2 \tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]
OCR C4 Q7
9 marks Standard +0.3
A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.
  1. Find the position vector of \(C\). [5]
Given that 1 unit on each coordinate axis represents 200 metres,
  1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]
OCR C4 Q8
12 marks Challenging +1.2
  1. Find \(\int \tan^2 x \, dx\). [3]
  2. Show that $$\int \tan x \, dx = \ln |\sec x| + c,$$ where \(c\) is an arbitrary constant. [4]
\includegraphics{figure_8} The diagram shows part of the curve with equation \(y = x^{\frac{1}{2}} \tan x\). The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac{\pi}{3}\) is rotated through \(360°\) about the \(x\)-axis.
  1. Show that the volume of the solid formed is \(\frac{1}{18}\pi^2(6\sqrt{3} - \pi) - \pi \ln 2\). [5]
OCR C4 Q9
14 marks Standard +0.3
An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.
  1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]
Given that after one week there are 360 insects in the colony,
  1. find the value of \(k\) to 3 significant figures. [2]
Given also that after two and three weeks there are 440 and 600 insects respectively,
  1. comment on suitability of the modelling assumption. [2]
An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25 \cos 0.5t).$$
  1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [3]
  2. Compare the suitability of the two models. [2]
OCR C4 Q1
6 marks Moderate -0.3
Express \(3\cos\theta + 4\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Hence find the range of the function \(f(\theta)\), where $$f(\theta) = 7 + 3\cos\theta + 4\sin\theta \quad \text{for } 0 \leqslant \theta \leqslant 2\pi.$$ Write down the greatest possible value of \(\frac{1}{7 + 3\cos\theta + 4\sin\theta}\). [6]
OCR C4 Q2
7 marks Moderate -0.3
Express \(3\sin x + 2\cos x\) in the form \(R\sin(x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = f(x)\), where $$f(x) = 3\sin x + 2\cos x, \quad 0 \leqslant x \leqslant \pi.$$ [7]