Quadratics: Mathematics 9709 (Cambridge International AS & A Level)

The quadratic equation $2x^2 - 8x + c = 0$, where $c$ is a constant, has two distinct real roots.

Which of the following is a correct condition on $c$?

A function is defined by $f(x) = 3x^2 - 12x + 7$ for real $x$.

(a) Express $f(x)$ in the form $a(x - p)^2 + q$, stating the values of the constants $a$, $p$ and $q$. [3]

(b) Write down the coordinates of the minimum point of the graph of $y = f(x)$, explaining how you know this point is a minimum rather than a maximum. [2]

(c) Hence, or otherwise, find the exact solutions of $f(x) = 0$, giving each answer as a single fraction involving a surd. [3]

A curve $C$ has equation $y = x^2 - 5x + 11$, and a line $L$ has equation $y = x + k$, where $k$ is a constant.

(a) Show that the $x$-coordinates of any points where $L$ and $C$ intersect satisfy $x^2 - 6x + (11-k) = 0$. [2]

(b) Given that $L$ is a tangent to $C$, find the value of $k$. [3]

(c) For this value of $k$, find the coordinates of the point where $L$ touches $C$. [3]

(a) Solve the inequality $2x^2 - 5x - 12 \ge 0$. [4]

(b) Hence find the set of values of $x$ for which both $2x^2 - 5x - 12 \ge 0$ and $3x - 1 < 11$. [3]

(a) Using the substitution $u = x^2$, show that the equation $x^4 - 7x^2 - 18 = 0$ can be written as $u^2 - 7u - 18 = 0$. [1]

(b) Solve the equation $u^2 - 7u - 18 = 0$. [2]

(c) Hence find all real values of $x$ satisfying $x^4 - 7x^2 - 18 = 0$, explaining why one of the values of $u$ found in part (b) must be rejected. [3]

The equation $x^2 + kx + 9 = 0$, where $k$ is a constant, has no real roots.

Which of the following gives the complete set of possible values of $k$?

A function is defined by $g(x) = -2x^2 + 12x - 7$ for real $x$.

(a) Express $g(x)$ in the form $-2(x - p)^2 + q$, stating the values of the constants $p$ and $q$. [3]

(b) Write down the coordinates of the maximum point of the graph of $y = g(x)$, explaining how you know this point is a maximum rather than a minimum. [2]

(c) Hence, or otherwise, find the exact solutions of $g(x) = 0$, giving each answer as a single fraction involving a surd. [3]

A ball is thrown vertically upwards. Its height above the ground, $h$ metres, after $t$ seconds is modelled by $h(t) = -5t^2 + 30t, \qquad t \ge 0.$

(a) By factorising $h(t)$, find the two values of $t$ for which $h(t) = 0$, and interpret each value in the context of the ball's flight. [2]

(b) Determine, using the discriminant of an appropriate quadratic equation, whether the ball ever reaches a height of $50$ metres. Justify your answer. [3]

(c) Find the exact times at which the ball's height is $35$ metres, giving each answer in the form $3 \pm \sqrt{n}$. [3]

A rectangle has a perimeter of $48$ m. One side of the rectangle has length $x$ metres.

(a) Show that the area, $A$ m², of the rectangle is given by $A = 24x - x^2$. [2]

(b) Find the set of values of $x$ for which the area is at least $140$ m². [4]

(c) Verify that $x=10$ gives an area of exactly $140$ m², and explain briefly why $x=14$ gives the same area. [2]

(a) Using the substitution $u = \sqrt{x}$, where $x \ge 0$, show that the equation $6x - 5\sqrt{x} - 6 = 0$ can be written as $6u^2 - 5u - 6 = 0$. [2]

(b) Solve the equation $6u^2 - 5u - 6 = 0$. [3]

(c) Hence find the value of $x$ satisfying $6x - 5\sqrt{x} - 6 = 0$, explaining why one of the values of $u$ found in part (b) must be rejected. [3]