1.02f Solve quadratic equations: including in a function of unknown

448 questions

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OCR MEI C1 Q5
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 Q6
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
  2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
  3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
  4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]
OCR MEI C1 Q6
11 marks Moderate -0.3
  1. \includegraphics{figure_1} Fig. 10 shows a sketch of the graph of \(y = \frac{1}{x}\). Sketch the graph of \(y = \frac{1}{x-2}\), showing clearly the coordinates of any points where it crosses the axes. [3]
  2. Find the value of \(x\) for which \(\frac{1}{x-2} = 5\). [2]
  3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac{1}{x-2}\). Give your answers in the form \(a \pm \sqrt{b}\). Show the position of these points on your graph in part (i). [6]
OCR MEI C1 Q1
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3x^2 + 6x + 10\) and the line \(y = 2 - 4x\). [5]
  2. Write \(3x^2 + 6x + 10\) in the form \(a(x + b)^2 + c\). [4]
  3. Hence or otherwise, show that the graph of \(y = 3x^2 + 6x + 10\) is always above the \(x\)-axis. [2]
OCR MEI C1 Q6
12 marks Moderate -0.8
  1. Write \(4x^2 - 24x + 27\) in the form \(a(x - b)^2 + c\). [4]
  2. State the coordinates of the minimum point on the curve \(y = 4x^2 - 24x + 27\). [2]
  3. Solve the equation \(4x^2 - 24x + 27 = 0\). [3]
  4. Sketch the graph of the curve \(y = 4x^2 - 24x + 27\). [3]
OCR MEI C1 Q4
12 marks Moderate -0.8
Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).
  1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
  2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
  3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
  4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]
OCR MEI C1 Q5
13 marks Moderate -0.8
  1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
  2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
  3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
  4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]
OCR C2 Specimen Q7
9 marks Moderate -0.8
\includegraphics{figure_7} The diagram shows the curves \(y = -3x^2 - 9x + 30\) and \(y = x^2 + 3x - 10\).
  1. Verify that the curves intersect at the points \(A(-5, 0)\) and \(B(2, 0)\). [2]
  2. Show that the area of the shaded region between the curves is given by \(\int_{-5}^{2} (-4x^2 - 12x + 40) dx\). [2]
  3. Hence or otherwise show that the area of the shaded region between the curves is \(228\frac{2}{3}\). [5]
OCR MEI C2 2010 January Q10
11 marks Moderate -0.8
  1. Differentiate \(x^3 - 3x^2 - 9x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x^3 - 3x^2 - 9x\), showing which is the maximum and which the minimum. [6]
  2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis. [3]
  3. Sketch the curve. [2]
Edexcel C2 Q3
7 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) where $$f(x) = 4 + 5x + kx^2 - 2x^3,$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A\), \(B\) and \(C\). Given that \(A\) has coordinates \((-4, 0)\),
  1. show that \(k = -7\), [2]
  2. find the coordinates of \(B\) and \(C\). [5]
OCR C2 Q8
12 marks Standard +0.3
The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]
Using \(x = 6\),
  1. find the common ratio of the series, [1]
  2. find the sum of the first eight terms of the series. [2]
OCR MEI C2 Q3
12 marks Moderate -0.3
The point A has \(x\)-coordinate 5 and lies on the curve \(y = x^2 - 4x + 3\).
  1. Sketch the curve. [2]
  2. Use calculus to find the equation of the tangent to the curve at A. [4]
  3. Show that the equation of the normal to the curve at A is \(x + 6y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. [6]
Edexcel C3 Q3
9 marks Standard +0.3
The function f is even and has domain \(\mathbb{R}\). For \(x \geq 0\), f(x) = \(x^2 - 4ax\), where \(a\) is a positive constant.
  1. In the space below, sketch the curve with equation \(y = \text{f}(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
  2. Find, in terms of \(a\), the value of f(2a) and the value of f(-2a). [2]
Given that \(a = 3\),
  1. use algebra to find the values of \(x\) for which f(x) = 45. [4]
AQA C4 2016 June Q2
5 marks Standard +0.3
By forming and solving a suitable quadratic equation, find the solutions of the equation $$3 \cos 2\theta - 5 \cos \theta + 2 = 0$$ in the interval \(0° < \theta < 360°\), giving your answers to the nearest \(0.1°\). [5 marks]
OCR MEI C4 Q4
16 marks Standard +0.3
\includegraphics{figure_3} In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations $$x = 10 \cos \theta + 5 \cos 2\theta, \quad y = 10 \sin \theta + 5 \sin 2\theta, \quad (0 \leqslant \theta < 2\pi),$$ where \(x\) and \(y\) are in metres.
  1. Show that \(\frac{\text{d}y}{\text{d}x} = -\frac{\cos \theta + \cos 2\theta}{\sin \theta + \sin 2\theta}\). Verify that \(\frac{\text{d}y}{\text{d}x} = 0\) when \(\theta = \frac{1}{3}\pi\). Hence find the exact coordinates of the highest point A on the path of C. [6]
  2. Express \(x^2 + y^2\) in terms of \(\theta\). Hence show that $$x^2 + y^2 = 125 + 100 \cos \theta.$$ [4]
  3. Using this result, or otherwise, find the greatest and least distances of C from O. [2]
You are given that, at the point B on the path vertically above O, $$2 \cos^2 \theta + 2 \cos \theta - 1 = 0.$$
  1. Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. [4]
OCR MEI M1 Q3
18 marks Standard +0.3
\includegraphics{figure_3} Fig. 7 shows the graph of \(y = \frac{1}{100}(100 + 15x - x^2)\). For \(0 \leq x < 20\), this graph shows the trajectory of a small stone projected from the point Q where \(y\) m is the height of the stone above horizontal ground and \(x\) m is the horizontal displacement of the stone from O. The stone hits the ground at the point R.
  1. Write down the height of Q above the ground. [1]
  2. Find the horizontal distance from O of the highest point of the trajectory and show that this point is \(1.5625\) m above the ground. [5]
  3. Show that the time taken for the stone to fall from its highest point to the ground is \(0.565\) seconds, correct to 3 significant figures. [3]
  4. Show that the horizontal component of the velocity of the stone is \(22.1\text{ms}^{-1}\), correct to 3 significant figures. Deduce the time of flight from Q to R. [5]
  5. Calculate the speed at which the stone hits the ground. [4]
AQA FP1 2016 June Q6
9 marks Standard +0.8
A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).
  1. Show that \(a = 2\). [3 marks]
  2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]
Edexcel AEA 2004 June Q3
11 marks Challenging +1.8
$$f(x) = x^3 - (k+4)x + 2k,$$ where \(k\) is a constant.
  1. Show that, for all values of \(k\), the curve with equation \(y = f(x)\) passes through the point \((2, 0)\). [1]
  2. Find the values of \(k\) for which the equation \(f(x) = 0\) has exactly two distinct roots. [5]
Given that \(k > 0\), that the \(x\)-axis is a tangent to the curve with equation \(y = f(x)\), and that the line \(y = p\) intersects the curve in three distinct points,
  1. find the set of values that \(p\) can take. [5]
Edexcel AEA 2004 June Q6
17 marks Challenging +1.8
$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).
  1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
  2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]
Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),
  1. find the value of \(k\). [2]
  2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]
The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.
  1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
  2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]
AQA AS Paper 1 2020 June Q6
9 marks Moderate -0.3
  1. It is given that $$f(x) = x^3 - x^2 + x - 6$$ Use the factor theorem to show that \((x - 2)\) is a factor of \(f(x)\). [2 marks]
  2. Find the quadratic factor of \(f(x)\). [1 mark]
  3. Hence, show that there is only one real solution to \(f(x) = 0\) [3 marks]
  4. Find the exact value of \(x\) that solves $$e^{3x} - e^{2x} + e^x - 6 = 0$$ [3 marks]
AQA AS Paper 2 2020 June Q8
6 marks Moderate -0.3
  1. Using \(y = 2^{2x}\) as a substitution, show that $$16^x - 2^{(2x+3)} - 9 = 0$$ can be written as $$y^2 - 8y - 9 = 0$$ [2 marks]
  2. Hence, show that the equation $$16^x - 2^{(2x+3)} - 9 = 0$$ has \(x = \log_2 3\) as its only solution. Fully justify your answer. [4 marks]
AQA AS Paper 2 2020 June Q11
11 marks Moderate -0.8
A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\). \includegraphics{figure_11} The path of the jet can be modelled by the equation $$y = -0.0125x^2 + 0.5x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are \((a, 0)\)
    1. Find the value of \(a\). [3 marks]
    2. Find the horizontal distance from \(A\) to the point where the jet hits the ground. [1 mark]
  2. Calculate the maximum vertical height reached by the jet. [4 marks]
  3. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption. [3 marks]
AQA AS Paper 2 2023 June Q6
5 marks Moderate -0.3
A curve has equation $$y = 2x^2 + px + 1$$ A line has equation $$y = 5x - 2$$ Find the set of values of \(p\) for which the line intersects the curve at two distinct points. Give your answer in exact form. [5 marks]
AQA Paper 2 Specimen Q6
5 marks Moderate -0.8
A curve \(C\) has equation \(y = x^2 - 4x + k\), where \(k\) is a constant. It crosses the \(x\)-axis at the points \((2 + \sqrt{5}, 0)\) and \((2 - \sqrt{5}, 0)\)
  1. Find the value of \(k\). [2 marks]
  2. Sketch the curve \(C\), labelling the exact values of all intersections with the axes. [3 marks]
AQA Paper 3 2020 June Q4
7 marks Standard +0.3
\(p(x) = 4x^3 - 15x^2 - 48x - 36\)
  1. Use the factor theorem to prove that \(x - 6\) is a factor of \(p(x)\). [2 marks]
    1. Prove that the graph of \(y = p(x)\) intersects the \(x\)-axis at exactly one point. [4 marks]
    2. State the coordinates of this point of intersection. [1 mark]