1.02d Quadratic functions: graphs and discriminant conditions

Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]

Find the set of values of \(k\) for which the graph of \(y = x^2 + 2kx + 5\) does not intersect the \(x\)-axis. [4]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).

1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]

$$\text{f}(x) = 2x^2 - 4x + 1.$$

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\text{f}(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = \text{f}(x)\). [1]
3. Solve the equation \(\text{f}(x) = 3\), giving your answers in exact form. [3]

\(\text{f}(x) = 4x^2 + 12x + 9\).

1. Determine the number of real roots that exist for the equation \(\text{f}(x) = 0\). [2]
2. Solve the equation \(\text{f}(x) = 8\), giving your answers in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are rational. [4]

Given that the equation $$4x^2 - kx + k - 3 = 0,$$ where \(k\) is a constant, has real roots,

1. show that $$k^2 - 16k + 48 \geq 0, \quad [2]$$
2. find the set of possible values of \(k\), [3]
3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value. [3]

Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]

Find the range of values of \(k\) for which the equation \(2x^2 + kx + 18 = 0\) does not have real roots. [4]

Find the set of values of \(k\) for which the equation \(2x^2 + kx + 2 = 0\) has no real roots. [4]

\includegraphics{figure_2} Fig. 12 shows the graph of \(y = \frac{1}{x-2}\).

1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\). [3]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x-2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x-2}\). [4]

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),

1. find the value of \(c\), [2]
2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]

The functions f and g are defined for all real values of \(x\) by $$\text{f}(x) = x^2 + 4ax + a^2 \text{ and } \text{g}(x) = 4x - 2a,$$ where \(a\) is a positive constant.

1. Find the range of f in terms of \(a\). [4]
2. Given that fg(3) = 69, find the value of \(a\) and hence find the value of \(x\) such that \(\text{g}^{-1}(x) = x\). [6]

\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]