1.02d Quadratic functions: graphs and discriminant conditions

307 questions

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SPS SPS FM 2024 October Q2
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS SM 2024 October Q4
6 marks Moderate -0.3
The quadratic equation \(kx^2 + 2kx + 2k = 3x - 1\), where \(k\) is a constant, has no real roots.
  1. Show that \(k\) satisfies the inequality $$4k^2 + 16k - 9 > 0.$$ [4]
  2. Hence find the set of possible values of \(k\). Give your answer in set notation. [2]
SPS SPS SM 2024 October Q4
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]
SPS SPS SM 2025 October Q9
4 marks Moderate -0.8
  1. Show that the equation \(x^2 + kx - k^2 = 0\) has real roots for all real values of \(k\). [2]
  2. Show that the roots of the equation \(x^2 + kx - k^2 = 0\) are \(\left(\frac{-1 \pm \sqrt{5}}{2}\right)k\). [2]
SPS SPS SM 2025 November Q3
5 marks Standard +0.3
The equation \(kx^2 + 4x + (5 - k) = 0\), where \(k\) is a constant, has 2 different real solutions for \(x\). Find the set of possible values of \(k\). Write your answer using set notation. [5]
OCR H240/02 2017 Specimen Q3
9 marks Moderate -0.8
A publisher has to choose the price at which to sell a certain new book. The total profit, \(£t\), that the publisher will make depends on the price, \(£p\). He decides to use a model that includes the following assumptions. • If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small. • If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small. The graphs below show two possible models. \includegraphics{figure_3}
  1. Explain how model A is inconsistent with one of the assumptions given above. [1]
  2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k(12p - p^2)\), and find the value of the constant \(k\). [2]
  3. The publisher needs to make a total profit of at least £6400. Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie. [4]
  4. Comment briefly on how realistic model B may be in the following cases. • \(p = 0\) • \(p = 12.1\) [2]
Pre-U Pre-U 9795/2 2014 June Q11
10 marks Challenging +1.8
It is given that the trajectory of a projectile which is launched with speed \(V\) at an angle \(\alpha\) above the horizontal has equation $$y = x\tan\alpha - \frac{gx^2}{2V^2}(1 + \tan^2\alpha),$$ where the point of projection is the origin, and the \(x\)- and \(y\)-axes are horizontal and vertically upwards respectively.
  1. Express the above equation as a quadratic equation in \(\tan\alpha\) and deduce that the boundary of all accessible points for this projectile has equation $$y = \frac{1}{2gV^2}(V^4 - g^2x^2).$$ [4]
  2. A stone is thrown with speed \(\sqrt{gh}\) from the top of a vertical tower, of height \(h\), which stands on horizontal ground. Find
    1. the maximum distance, from the foot of the tower, at which the stone can land, [3]
    2. the angle of elevation at which the stone must be thrown to achieve this maximum distance. [3]