Questions C3 (1301 questions)

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OCR MEI C3 Q7
6 marks Moderate -0.3
An oil slick is circular with radius \(r\) km and area \(A\) km\(^2\). The radius increases with time at a rate given by \(\frac{dr}{dt} = 0.5\), in kilometres per hour.
  1. Show that \(\frac{dA}{dt} = \pi r\). [4]
  2. Find the rate of increase of the area of the slick at a time when the radius is 6 km. [2]
OCR MEI C3 Q8
18 marks Standard +0.3
Fig. 8 shows the graph of \(y = x\sqrt{1 + x}\). The point P on the curve is on the \(x\)-axis. \includegraphics{figure_8}
  1. Write down the coordinates of P. [1]
  2. Show that \(\frac{dy}{dx} = \frac{3x + 2}{2\sqrt{1 + x}}\). [4]
  3. Hence find the coordinates of the turning point on the curve. What can you say about the gradient of the curve at P? [4]
  4. By using a suitable substitution, show that \(\int_0^0 x\sqrt{1 + x} dx = \int_0^1 \left(u^{\frac{3}{2}} - u^{\frac{1}{2}}\right) du\). Evaluate this integral, giving your answer in an exact form. What does this value represent? [7]
  5. Use your answer to part (ii) to differentiate \(y = x\sqrt{1 + x} \sin 2x\) with respect to \(x\). (You need not simplify your result.) [2]
OCR MEI C3 Q9
18 marks Standard +0.3
The functions f(x) and g(x) are defined by $$f(x) = x^2, \quad g(x) = 2x - 1,$$ for all real values of \(x\).
  1. State the ranges of f(x) and g(x). Explain why f(x) has no inverse. [3]
  2. Find an expression for the inverse function g\(^{-1}\)(x) in terms of \(x\). Sketch the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\) on the same axes. [4]
  3. Find expressions for gf(x) and fg(x). [2]
  4. Solve the equation gf(x) = fg(x). Sketch the graphs of \(y = gf(x)\) and \(y = fg(x)\) on the same axes to illustrate your answer. [4]
  5. Show that the equation f(x + a) = g\(^{-1}\)(x) has no solution if \(a > \frac{1}{4}\). [5]
Edexcel C3 Q1
8 marks Standard +0.3
  1. Find the exact value of \(x\) such that $$3 \arctan (x - 2) + \pi = 0.$$ [3]
  2. Solve, for \(-\pi < \theta < \pi\), the equation $$\cos 2\theta - \sin \theta - 1 = 0,$$ giving your answers in terms of \(\pi\). [5]
Edexcel C3 Q2
9 marks Moderate -0.8
  1. Express $$\frac{4x}{x^2 - 9} - \frac{2}{x + 3}$$ as a single fraction in its simplest form. [4]
  2. Simplify $$\frac{x^3 - 8}{3x^2 - 8x + 4}.$$ [5]
Edexcel C3 Q3
9 marks Moderate -0.3
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\cot x^2\) [2]
  2. \(x^2 e^{-x}\) [3]
  3. \(\frac{\sin x}{3 + 2\cos x}\) [4]
Edexcel C3 Q4
10 marks Standard +0.3
  1. Find, as natural logarithms, the solutions of the equation $$e^{2x} - 8e^x + 15 = 0.$$ [4]
  2. Use proof by contradiction to prove that \(\log_5 3\) is irrational. [6]
Edexcel C3 Q5
12 marks Standard +0.2
The function f is defined by $$f : x \to 3e^{x-1}, \quad x \in \mathbb{R}.$$
  1. State the range of f. [1]
  2. Find an expression for \(f^{-1}(x)\) and state its domain. [4]
The function g is defined by $$g : x \to 5x - 2, \quad x \in \mathbb{R}.$$ Find, in terms of e,
  1. the value of gf(ln 2), [3]
  2. the solution of the equation $$f^{-1}g(x) = 4.$$ [4]
Edexcel C3 Q6
13 marks Standard +0.3
$$f(x) = 2x^2 + 3 \ln (2 - x), \quad x \in \mathbb{R}, \quad x < 2.$$
  1. Show that the equation \(f(x) = 0\) can be written in the form $$x = 2 - e^{kx^2},$$ where \(k\) is a constant to be found. [3]
The root, \(\alpha\), of the equation \(f(x) = 0\) is \(1.9\) correct to \(1\) decimal place.
  1. Use the iteration formula $$x_{n+1} = 2 - e^{kx_n^2},$$ with \(x_0 = 1.9\) and your value of \(k\), to find \(\alpha\) to \(3\) decimal places and justify the accuracy of your answer. [5]
  2. Solve the equation \(f'(x) = 0\). [5]
Edexcel C3 Q7
14 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) which has a maximum point at \((-45, 7)\) and a minimum point at \((135, -1)\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = f(|x|)\),
    2. \(y = 1 + 2f(x)\). [6]
Given that $$f(x) = A + 2\sqrt{2} \cos x^{\circ} - 2\sqrt{2} \sin x^{\circ}, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$ where \(A\) is a constant,
  1. show that f(x) can be expressed in the form $$f(x) = A + R \cos (x + \alpha)^{\circ},$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
  2. state the value of \(A\), [1]
  3. find, to \(1\) decimal place, the \(x\)-coordinates of the points where the curve \(y = f(x)\) crosses the \(x\)-axis. [4]
Edexcel C3 Q1
6 marks Moderate -0.3
\(f(x) \equiv \frac{2x-3}{x-2}\), \(x \in \mathbb{R}\), \(x > 2\).
  1. Find the range of \(f\). [2]
  2. Show that \(f(f(x) = x\) for all \(x > 2\). [3]
  3. Hence, write down an expression for \(f^{-1}(x)\). [1]
Edexcel C3 Q2
7 marks Moderate -0.8
Solve each equation, giving your answers in exact form.
  1. \(e^{4x-3} = 2\) [3]
  2. \(\ln (2y - 1) = 1 + \ln (3 - y)\) [4]
Edexcel C3 Q3
8 marks Standard +0.3
The curve \(C\) has the equation \(y = 2e^x - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate \(1\).
  1. Find an equation for the tangent to \(C\) at \(P\). [4]
The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).
  1. Show that the area of triangle \(OQR\), where \(O\) is the origin, is \(\frac{9}{3-e}\). [4]
Edexcel C3 Q4
9 marks Standard +0.3
  1. Express $$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)}$$ as a single fraction in its simplest form. [5]
  2. Hence, show that the equation $$\frac{x-10}{(x-3)(x+4)} - \frac{x-8}{(x-3)(2x-1)} = 1$$ has no real roots. [4]
Edexcel C3 Q5
9 marks Challenging +1.2
Find the values of \(x\) in the interval \(-180 < x < 180\) for which $$\tan (x + 45)^{\circ} - \tan x^{\circ} = 4,$$ giving your answers to 1 decimal place. [9]
Edexcel C3 Q6
10 marks Standard +0.8
  1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [6]
  2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]
Edexcel C3 Q7
12 marks Standard +0.3
  1. Use the identity $$\cos (A + B) = \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin^2 \frac{x}{2}.$$ [3]
  2. Prove that, for \(\sin x \neq 0\), $$\frac{1 - \cos x}{\sin x} \equiv \tan \frac{x}{2}.$$ [3]
  3. Find the values of \(x\) in the interval \(0 \leq x \leq 360^{\circ}\) for which $$\frac{1 - \cos x}{\sin x} = 2 \sec^2 \frac{x}{2} - 5,$$ giving your answers to 1 decimal place where appropriate. [6]
Edexcel C3 Q8
14 marks Standard +0.3
A curve has the equation \(y = (2x + 3)e^{-x}\).
  1. Find the exact coordinates of the stationary point of the curve. [4]
The curve crosses the \(y\)-axis at the point \(P\).
  1. Find an equation for the normal to the curve at \(P\). [2]
The normal to the curve at \(P\) meets the curve again at \(Q\).
  1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
  2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
  3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]
OCR C3 Q1
4 marks Moderate -0.3
Evaluate $$\int_2^6 \sqrt{3x-2} \, dx.$$ [4]
OCR C3 Q2
5 marks Moderate -0.8
Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\frac{6}{\sqrt{2x-7}}\) [2]
  2. \(x^2 e^{-x}\) [3]
OCR C3 Q3
7 marks Standard +0.8
  1. Prove the identity $$\sqrt{2} \cos (x + 45)° + 2 \cos (x - 30)° \equiv (1 + \sqrt{3}) \cos x°.$$ [4]
  2. Hence, find the exact value of \(\cos 75°\) in terms of surds. [3]
OCR C3 Q4
8 marks Standard +0.3
$$\text{f}(x) = x^2 + 5x - 2 \sec x, \quad x \in \mathbb{R}, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}.$$
  1. Show that the equation \(\text{f}(x) = 0\) has a root, \(\alpha\), such that \(1 < \alpha < 1.5\) [2]
  2. Show that a suitable rearrangement of the equation \(\text{f}(x) = 0\) leads to the iterative formula $$x_{n+1} = \cos^{-1} \left( \frac{2}{x_n^2 + 5x_n} \right).$$ [3]
  3. Use the iterative formula in part (ii) with a starting value of 1.25 to find \(\alpha\) correct to 3 decimal places. You should show the result of each iteration. [3]
OCR C3 Q5
8 marks Standard +0.0
The function f is defined by $$\text{f}(x) \equiv 2 + \ln (3x - 2), \quad x \in \mathbb{R}, \quad x > \frac{2}{3}.$$
  1. Find the exact value of \(\text{f}(1)\). [2]
  2. Find an equation for the tangent to the curve \(y = \text{f}(x)\) at the point where \(x = 1\). [4]
  3. Find an expression for \(\text{f}^{-1}(x)\). [2]
OCR C3 Q6
9 marks Standard +0.8
  1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]
OCR C3 Q7
9 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with equation \(y = 2x - e^{\frac{1}{2}x}\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).
  1. Find the area of the shaded region, giving your answer in terms of e. [4]
The shaded region is rotated through four right angles about the \(x\)-axis.
  1. Using Simpson's rule with two strips, estimate the volume of the solid formed. [5]