1.02g Inequalities: linear and quadratic in single variable

420 questions

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AQA AS Paper 1 2021 June Q5
6 marks Moderate -0.3
  1. Sketch the curve $$y = (x - a)^2(3 - x) \quad \text{where } 0 < a < 3$$ indicating the coordinates of the points where the curve and the axes meet. [4 marks] \includegraphics{figure_5}
  2. Hence, solve $$(x - a)^2(3 - x) > 0$$ giving your answer in set notation form. [2 marks]
AQA AS Paper 1 2024 June Q6
4 marks Moderate -0.3
Determine the set of values of \(x\) which satisfy the inequality $$3x^2 + 3x > x + 6$$ Give your answer in exact form using set notation. [4 marks]
AQA AS Paper 1 Specimen Q2
1 marks Moderate -0.8
Consider the two statements, A and B, below. A: \(x^2 - 6x + 8 > 0\) B: \(x > 4\) Choose the most appropriate option below. Circle your answer. [1 mark] \(A \Rightarrow B\) \(A \Leftarrow B\) \(A \Leftrightarrow B\) There is no connection between A and B
AQA AS Paper 2 2018 June Q7
6 marks Moderate -0.8
  1. Express \(2x^2 - 5x + k\) in the form \(a(x - b)^2 + c\) [3 marks]
  2. Find the values of \(k\) for which the curve \(y = 2x^2 - 5x + k\) does not intersect the line \(y = 3\) [3 marks]
AQA AS Paper 2 2020 June Q2
1 marks Easy -1.8
It is given that \(y = \frac{1}{x}\) and \(x < -1\) Determine which statement below fully describes the possible values of \(y\). Tick (\(\checkmark\)) one box. [1 mark] \(-\infty < y < -1\) \(y > -1\) \(-1 < y < 0\) \(y < 0\)
AQA Paper 1 Specimen Q9
8 marks Standard +0.3
A curve has equation \(y = \frac{2x + 3}{4x^2 + 7}\)
    1. Find \(\frac{dy}{dx}\) [2 marks]
    2. Hence show that \(y\) is increasing when \(4x^2 + 12x - 7 < 0\) [4 marks]
  2. Find the values of \(x\) for which \(y\) is increasing. [2 marks]
AQA Paper 2 2018 June Q4
6 marks Standard +0.3
A curve, C, has equation \(y = x^2 - 6x + k\), where \(k\) is a constant. The equation \(x^2 - 6x + k = 0\) has two distinct positive roots.
  1. Sketch C on the axes below. [2 marks]
  2. Find the range of possible values for \(k\). Fully justify your answer. [4 marks]
AQA Paper 2 2024 June Q3
1 marks Easy -1.8
Solve the inequality $$(1 - x)(x - 4) < 0$$ [1 mark] Tick \((\checkmark)\) one box. \(\{x : x < 1\} \cup \{x : x > 4\}\) \(\{x : x < 1\} \cap \{x : x > 4\}\) \(\{x : x < 1\} \cup \{x : x \geq 4\}\) \(\{x : x < 1\} \cap \{x : x \geq 4\}\)
AQA Paper 2 2024 June Q14
3 marks Moderate -0.8
The displacement, \(r\) metres, of a particle at time \(t\) seconds is $$r = 6t - 2t^2$$
  1. Find the value of \(r\) when \(t = 4\) [1 mark]
  2. Determine the range of values of \(t\) for which the displacement is positive. [2 marks]
AQA Paper 3 2019 June Q4
3 marks Moderate -0.8
Sketch the region defined by the inequalities $$y \leq (1 - 2x)(x + 3) \text{ and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\). [3 marks] \includegraphics{figure_4}
AQA Paper 3 2024 June Q7
5 marks Moderate -0.8
The graphs with equations $$y = 2 + 3x - 2x^2 \text{ and } x + y = 1$$ are shown in the diagram below. \includegraphics{figure_7} The graphs intersect at the points \(A\) and \(B\) \begin{enumerate}[label=(\alph*)] \item On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities $$0 \leq y \leq 2 + 3x - 2x^2$$ and $$x + y \geq 1$$ [2 marks] \item Find the exact coordinates of \(A\) [3 marks]
Edexcel AS Paper 1 Specimen Q10
4 marks Standard +0.3
The equation \(kx^2 + 4kx + 3 = 0\), where \(k\) is a constant, has no real roots. Prove that $$0 \leqslant k < \frac{3}{4}$$ [4]
OCR PURE Q2
4 marks Easy -1.2
  1. The equation \(x^2 + 3x + k = 0\) has repeated roots. Find the value of the constant \(k\). [2]
  2. Solve the inequality \(6 + x - x^2 > 0\). [2]
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 Q3
1 marks Moderate -0.8
Given \((x - 1)(x - 2)(x - a) < 0\) and \(a > 2\) Find the set of possible values of \(x\). Tick \((\checkmark)\) one box. [1 mark] \(\{x : x < 1\} \cup \{x : 2 < x < a\}\) \(\{x : 1 < x < 2\} \cup \{x : x > a\}\) \(\{x : x < -a\} \cup \{x : -2 < x < -1\}\) \(\{x : -a < x < -2\} \cup \{x : x > -1\}\)
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 1 2023 June Q11
7 marks Standard +0.8
The function f is defined by $$f(x) = 4x^3 - 8x^2 - 51x - 45 \quad (x \in \mathbb{R})$$
    1. Fully factorise \(f(x)\) [2 marks]
    2. Hence, solve the inequality \(f(x) < 0\) [2 marks]
  2. The graph of \(y = f(x)\) is translated by the vector \(\begin{pmatrix} 7 \\ 0 \end{pmatrix}\) The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = g(x)\) Solve the inequality \(g(x) \leq 0\) [3 marks]
AQA Further Paper 2 2019 June Q3
1 marks Moderate -0.8
The set \(A\) is defined by \(A = \{x : -\sqrt{2} < x < 0\} \cup \{x : 0 < x < \sqrt{2}\}\) Which of the inequalities given below has \(A\) as its solution? Circle your answer. [1 mark] \(|x^2 - 1| > 1\) \quad\quad \(|x^2 - 1| \geq 1\) \quad\quad \(|x^2 - 1| < 1\) \quad\quad \(|x^2 - 1| \leq 1\)
AQA Further Paper 2 2020 June Q5
5 marks Standard +0.3
Solve the inequality $$\frac{2x + 3}{x - 1} \leq x + 5$$ [5 marks]
WJEC Unit 1 2022 June Q4
4 marks Moderate -0.3
Solve the inequality \(x^2 + 3x - 6 > 4x - 4\). [4]
WJEC Unit 1 2024 June Q8
4 marks Standard +0.3
Prove that \(x - 10 < x^2 - 5x\) for all real values of \(x\). [4]
SPS SPS FM 2019 Q5
6 marks Standard +0.3
Solve the following inequalities giving your answer in set notation:
  1. \(|4x + 3| < |x - 8|\) [3]
  2. \(\frac{x}{x^2+1} < \frac{1}{2}\) [3]
SPS SPS SM 2020 October Q3
6 marks Moderate -0.3
  1. Write \(3x^2 - 6x + 1\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [2]
  2. Solve \(3x^2 - 6x + 1 \leq 0\), giving your answer in set notation. [4]
SPS SPS SM 2022 October Q8
8 marks Standard +0.3
The equation \(k(3x^2 + 8x + 9) = 2 - 6x\), where \(k\) is a real constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$11k^2 - 30k - 9 > 0$$ [4]
  2. Find the range of possible values for \(k\). [4]
SPS SPS SM Pure 2022 June Q14
6 marks Moderate -0.3
A region, R, is defined by \(x^2 - 8x + 12 \leq y \leq 12 - 2x\)
  1. Sketch a graph to show the region R. Shade the region R.
  2. Find the area of R [6 marks]