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Date November 2011 Marks available 3 Reference code 11N.1.hl.TZ0.9
Level HL only Paper 1 Time zone TZ0
Command term Determine and Hence Question number 9 Adapted from N/A

Question

Consider the equation \(y{x^2} + (y - 1)x + (y - 1) = 0\).

Find the set of values of y for which this equation has real roots.

[4]
a.

Hence determine the range of the function \(f:x \to \frac{{x + 1}}{{{x^2} + x + 1}}\).

[3]
b.

Explain why f has no inverse.

[1]
c.

Markscheme

for the equation to have real roots

\({(y - 1)^2} - 4y(y - 1) \geqslant 0\)     M1

\( \Rightarrow 3{y^2} - 2y - 1 \leqslant 0\)

(by sign diagram, or algebraic method)     M1

\( - \frac{1}{3} \leqslant y \leqslant 1\)     A1A1

Note: Award first A1 for \( - \frac{1}{3}\) and 1, and second A1 for inequalities. These are independent marks.

 

[4 marks]

a.

\(f:x \to \frac{{x + 1}}{{{x^2} + x + 1}} \Rightarrow x + 1 = y{x^2} + yx + y\)     (M1)

\( \Rightarrow 0 = y{x^2} + (y - 1)x + (y - 1)\)     A1

hence, from (a) range is \( - \frac{1}{3} \leqslant y \leqslant 1\)     A1

[3 marks]

b.

a value for y would lead to 2 values for x from (a)     R1

Note: Do not award R1 if (b) has not been tackled.

 

[1 mark]

c.

Examiners report

(a) The best answered part of the question. The critical points were usually found, but the inequalities were often incorrect. Few candidates were convincing regarding the connection between (a) and (b). This had consequences for (c).

a.

(a) The best answered part of the question. The critical points were usually found, but the inequalities were often incorrect. Few candidates were convincing regarding the connection between (a) and (b). This had consequences for (c).

b.

(a) The best answered part of the question. The critical points were usually found, but the inequalities were often incorrect. Few candidates were convincing regarding the connection between (a) and (b). This had consequences for (c).

c.

Syllabus sections

Topic 2 - Core: Functions and equations » 2.1 » Concept of function \(f:x \mapsto f\left( x \right)\) : domain, range; image (value)

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