Mathematical Methods — Question 20

QCAA 2021, Paper 2 · 3 marks

Q20 · 2021 · Technology-activeComplex unfamiliar3 marks

QUESTION 20 (3 marks)

The random variable

B

is normally distributed with a mean of 0 and a standard deviation of 1. Determine the probability that the quadratic equation

x^2+3x+2B=0

has real roots.

WORKED SOLUTION

3 marks

ANSWER

0.8697

Worked solution

The quadratic has real roots when its discriminant is non-negative:

3^2-4(1)(2B)\ge0,

B\le\frac98

. Using the standard normal distribution,

P\left(B\le\frac98\right)=0.8697.

correctly identifies the need to use the discriminant

[1 mark]

correctly determines the range of values for B

[1 mark]

determines probability

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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