1.02k Simplify rational expressions: factorising, cancelling, algebraic division

333 questions

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Edexcel C3 Q2
5 marks Moderate -0.3
The function f is given by \(f: x \mapsto 2 + \frac{3}{x + 2}\), \(x \in \mathbb{R}\), \(x \neq -2\).
  1. Express \(2 + \frac{3}{x + 2}\) as a single fraction. [1]
  2. Find an expression for \(f^{-1}(x)\). [3]
  3. Write down the domain of \(f^{-1}\). [1]
Edexcel C3 Q3
6 marks Moderate -0.3
  1. Express as a fraction in its simplest form $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21}.$$ [3]
  2. Hence solve $$\frac{2}{x - 3} + \frac{13}{x^2 + 4x - 21} = 1.$$ [3]
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 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 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]
OCR MEI C3 Q1
18 marks Standard +0.3
Fig. 9 shows the curve \(y = \frac{x^2}{3x - 1}\). P is a turning point, and the curve has a vertical asymptote \(x = a\). \includegraphics{figure_1}
  1. Write down the value of \(a\). [1]
  2. Show that \(\frac{dy}{dx} = \frac{x(3x - 2)}{(3x - 1)^2}\) [3]
  3. Find the exact coordinates of the turning point P. Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point. [7]
  4. Using the substitution \(u = 3x - 1\), show that \(\int \frac{x^2}{3x - 1} dx = \frac{1}{27} \int \left( u + 2 + \frac{1}{u} \right) du\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac{2}{3}\) and \(x = 1\). [7]
AQA C4 2016 June Q3
8 marks Standard +0.3
  1. Express \(\frac{3 + 13x - 6x^2}{2x - 3}\) in the form \(Ax + B + \frac{C}{2x - 3}\). [4 marks]
  2. Show that \(\int_3^6 \frac{3 + 13x - 6x^2}{2x - 3} \, dx = p + q \ln 3\), where \(p\) and \(q\) are rational numbers. [4 marks]
OCR C4 2007 January Q1
3 marks Moderate -0.8
It is given that $$f(x) = \frac{x^2 + 2x - 24}{x^2 - 4x} \quad \text{for } x \neq 0, x \neq 4.$$ Express \(f(x)\) in its simplest form. [3]
OCR MEI C4 2012 June Q1
5 marks Moderate -0.3
Solve the equation \(\frac{4x}{x+1} - \frac{3}{2x+1} = 1\). [5]
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 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]
AQA FP1 2014 June Q6
10 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).
  1. Write down the equations of all the asymptotes of \(C\). [2 marks]
  2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
    1. Find the \(y\)-coordinate of the stationary point. [1 mark]
    2. Sketch the curve \(C\). [2 marks]
  3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]
AQA FP1 2016 June Q9
11 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).
  1. Write down the equations of the asymptotes of \(C\). [2 marks]
  2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
  3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
  4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]
OCR FP2 2009 January Q9
12 marks Standard +0.8
A curve has equation $$y = \frac{4x - 3a}{2(x^2 + a^2)},$$ where \(a\) is a positive constant.
  1. Explain why the curve has no asymptotes parallel to the \(y\)-axis. [2]
  2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve. [5]
  3. Find the exact value of \(\int_a^{2a} \frac{4x - 3a}{2(x^2 + a^2)} dx\), showing that it is independent of \(a\). [5]
OCR H240/02 2020 November Q2
4 marks Moderate -0.8
Simplify fully \(\frac{2x^3 + x^2 - 7x - 6}{x^2 - x - 2}\). [4]
AQA Paper 1 2024 June Q3
1 marks Easy -1.2
The expression $$\frac{12x^2 + 3x + 7}{3x - 5}$$ can be written as $$Ax + B + \frac{C}{3x - 5}$$ State the value of \(A\) Circle your answer. [1 mark] $$3 \quad 4 \quad 7 \quad 9$$
AQA Paper 1 Specimen Q4
6 marks Moderate -0.3
\(p(x) = 2x^3 + 7x^2 + 2x - 3\)
  1. Use the factor theorem to prove that \(x + 3\) is a factor of \(p(x)\) [2 marks]
  2. Simplify the expression \(\frac{2x^3 + 7x^2 + 2x - 3}{4x^2 - 1}\), \(x \neq \pm \frac{1}{2}\) [4 marks]
AQA Paper 3 2021 June Q2
1 marks Easy -1.8
Simplify fully $$\frac{(x + 3)(6 - 2x)}{(x - 3)(3 + x)} \quad \text{for } x \neq \pm 3$$ Circle your answer. [1 mark] \(-2\) \quad \(2\) \quad \(\frac{(6 - 2x)}{(x - 3)}\) \quad \(\frac{(2x - 6)}{(x - 3)}\)
AQA Paper 3 2021 June Q6
4 marks Standard +0.3
Given that \(x > 0\) and \(x \neq 25\), fully simplify $$\frac{10 + 5x - 2x^{\frac{1}{2}} - x^{\frac{3}{2}}}{5 - \sqrt{x}}$$ Fully justify your answer. [4 marks]
AQA Further AS Paper 1 2018 June Q13
9 marks Challenging +1.2
The graph of the rational function \(y = f(x)\) intersects the \(x\)-axis exactly once at \((-3, 0)\) The graph has exactly two asymptotes, \(y = 2\) and \(x = -1\)
  1. Find \(f(x)\) [2 marks]
  2. Sketch the graph of the function. [3 marks] \includegraphics{figure_13b}
  3. Find the range of values of \(x\) for which \(f(x) \leq 5\) [4 marks]
AQA Further AS Paper 1 2020 June Q10
8 marks Standard +0.3
  1. Show that the equation $$y = \frac{3x - 5}{2x + 4}$$ can be written in the form $$(x + a)(y + b) = c$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
  2. Write down the equations of the asymptotes of the graph of $$y = \frac{3x - 5}{2x + 4}$$ [2 marks]
  3. Sketch, on the axes provided, the graph of $$y = \frac{3x - 5}{2x + 4}$$ \includegraphics{figure_10} [3 marks]
AQA Further AS Paper 1 2020 June Q14
7 marks Standard +0.8
  1. Given $$\frac{x + 7}{x + 1} \leq x + 1$$ show that $$\frac{(x + a)(x + b)}{x + c} \geq 0$$ where \(a\), \(b\), and \(c\) are integers to be found. [4 marks]
  2. Briefly explain why this statement is incorrect. $$\frac{(x + p)(x + q)}{x + r} \geq 0 \Leftrightarrow (x + p)(x + q)(x + r) \geq 0$$ [1 mark]
  3. Solve $$\frac{x + 7}{x + 1} \leq x + 1$$ [2 marks]
AQA Further Paper 2 2019 June Q2
1 marks Moderate -0.8
Which of the straight lines given below is an asymptote to the curve $$y = \frac{ax^2}{x-1}$$ where \(a\) is a non-zero constant? Circle your answer. [1 mark] \(y = ax + a\) \quad\quad \(y = ax\) \quad\quad \(y = ax - a\) \quad\quad \(y = a\)
AQA Further Paper 2 2024 June Q16
9 marks Challenging +1.2
The function f is defined by $$f(x) = \frac{ax + 5}{x + b}$$ where \(a\) and \(b\) are constants. The graph of \(y = f(x)\) has asymptotes \(x = -2\) and \(y = 3\)
  1. Write down the value of \(a\) and the value of \(b\) [2 marks]
  2. The diagram shows the graph of \(y = f(x)\) and its asymptotes. The shaded region \(R\) is enclosed by the graph of \(y = f(x)\), the \(x\)-axis and the \(y\)-axis. \includegraphics{figure_16}
    1. The shaded region \(R\) is rotated through \(360°\) about the \(x\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [3 marks]
    2. The shaded region \(R\) is rotated through \(360°\) about the \(y\)-axis to form a solid. Find the volume of this solid. Give your answer to three significant figures. [4 marks]
WJEC Unit 3 2023 June Q8
7 marks Standard +0.3
The function \(f\) is defined by \(f(x) = \frac{4x^2 + 12x + 9}{2x^2 + x - 3}\), where \(x > 1\).
  1. Show that \(f(x)\) can be written as \(2 + \frac{5}{x-1}\). [3]
  2. Hence find the exact value of \(\int_3^7 f(x)\,dx\). [4]