Questions — Edexcel (10514 questions)

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Edexcel C3 Q4
6 marks Moderate -0.3
  1. Simplify \(\frac{x^2 + 4x + 3}{x^2 + x}\). [2]
  2. Find the value of \(x\) for which \(\log_2(x^2 + 4x + 3) - \log_2(x^2 + x) = 4\). [4]
Edexcel C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\sec(A + B) \equiv \sec A + \sec B\), for all \(A\) and \(B\)" is false [2]
  2. Prove that $$\tan \theta + \cot \theta = 2\cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]
Edexcel C3 Q6
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} \equiv \tan \theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
Edexcel C3 Q7
10 marks Moderate -0.3
Given that \(y = \log_a x\), \(x > 0\), where \(a\) is a positive constant,
    1. express \(x\) in terms of \(a\) and \(y\), [1]
    2. deduce that \(\ln x = y \ln a\). [1]
  2. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x \ln a}\). [2]
The curve \(C\) has equation \(y = \log_{10} x\), \(x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10. Using the result in part (b),
  1. find an equation for the tangent to \(C\) at \(A\). [4]
The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
  1. Find the exact \(x\)-coordinate of \(B\). [2]
Edexcel C3 Q8
11 marks Standard +0.3
The curve with equation \(y = \ln 3x\) crosses the \(x\)-axis at the point \(P(p, 0)\).
  1. Sketch the graph of \(y = \ln 3x\), showing the exact value of \(p\). [2]
The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.
  1. Show that \(x = q\) is a solution of the equation \(x^2 + \ln 3x = 0\). [4]
  2. Show that the equation in part (b) can be rearranged in the form \(x = \frac{1}{3}e^{-x^2}\). [2]
  3. Use the iteration formula \(x_{n + 1} = \frac{1}{3}e^{-x_n^2}\), with \(x_0 = \frac{1}{3}\), to find \(x_1, x_2, x_3\) and \(x_4\). Hence write down, to 3 decimal places, an approximation for \(q\). [3]
Edexcel C3 Q9
14 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = \text{f}(x)\), \(x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation
  1. \(y = \text{f}^{-1}(x)\), [2]
  2. \(y = 3\text{f}(2x)\). [3]
Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\text{f}: x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$
  1. state
    1. the value of \(c\),
    2. the range of f.
    [3]
  2. Find the value of \(d\), giving your answer to 3 decimal places. [3]
The function g is defined by $$\text{g}: x \mapsto \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$
  1. Find fg(x), giving your answer in its simplest form. [3]
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]
Edexcel C4 Q1
6 marks Standard +0.3
Use integration by parts to find the exact value of \(\int_1^3 x^2 \ln x \, dx\). [6]
Edexcel C4 Q2
12 marks Moderate -0.3
Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V\) m\(^3\). The rate at which the fluid flows, in m\(^3\) min\(^{-1}\), is proportional to the square root of \(V\).
  1. Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation $$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$ [3]
  2. Show that the general solution of the differential equation may be written as $$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$ [4] Given that at time \(t = 0\) the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
  3. find the time, \(T\) minutes, which it takes for the tank to empty. [3]
  4. Find the depth of water in the tank at time \(0.5T\) minutes. [2]
Edexcel C4 Q3
14 marks Standard +0.3
  1. Use the identity for \(\cos(A + B)\) to prove that \(\cos 2A = 2\cos^2 A - 1\). [2]
  2. Use the substitution \(x = 2\sqrt{2} \sin \theta\) to prove that $$\int_2^{\sqrt{6}} \sqrt{(8 - x^2)} \, dx = \frac{1}{3}(\pi + 3\sqrt{3} - 6).$$ [7]
A curve is given by the parametric equations $$x = \sec \theta, \quad y = \ln(1 + \cos 2\theta), \quad 0 \leq \theta < \frac{\pi}{2}.$$
  1. Find an equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{3}\). [5]
Edexcel C4 Q4
11 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{4}{x - 3}\), \(x \neq 3\). The points \(A\) and \(B\) on the curve have \(x\)-coordinates 3.25 and 5 respectively.
  1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
  2. Show that an equation of \(C\) is \(\frac{3y + 4}{y} = 0\), \(y \neq 0\). [1]
The shaded region \(R\) is bounded by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis. The region \(R\) is rotated through 360° about the \(y\)-axis to form a solid shape \(S\).
  1. Find the volume of \(S\), giving your answer in the form \(\pi(a + b \ln c)\), where \(a\), \(b\) and \(c\) are integers. [7]
The solid shape \(S\) is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
  1. show that the volume of the tower is approximately 15500 m\(^3\). [2]