Algebra and functions

WMA11Topic 1.13 questions

Indices, surds, quadratics, simultaneous equations, inequalities, curve sketching and transformations.

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Original practice questions written in the style of the exam.

Question 15 marks
The equation $x^2 + kx + (k + 3) = 0$ , where $k$ is a constant, has no real roots.

Find the set of possible values of $k$ .
Mark scheme
- M1 Uses $b^2 - 4ac < 0$ with $a = 1$ , $b = k$ , (1)
Question 26 marks
Solve the simultaneous equations

$x + 2y = 5$ $x^2 + 2y^2 = 11$

c = k + 3

k^2 - 4k - 12 < 0

(1)

M1 Solves

k^2 - 4k - 12 = 0

to find critical values,

(k - 6)(k + 2) = 0

(1)

M1 Chooses the inside region between their critical values (1)

-2 < k < 6

(1)

Mark scheme

M1 Rearranges the linear equation, e.g. $x = 5 - 2y$ , and substitutes into the quadratic (1)
A1 Correct three-term quadratic, e.g. $6y^2 - 20y + 14 = 0$ or $3y^2 - 10y + 7 = 0$ (1)
M1 Solves their quadratic, e.g. $(3y - 7)(y - 1) = 0$ (1)
A1 $y = 1$ and $y = \tfrac{7}{3}$ (1)
M1 Substitutes their $y$ values back into the linear equation to find $x$ (1)
A1 $x = 3, y = 1$ and $x = \tfrac{1}{3}, y = \tfrac{7}{3}$ (1)