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Numerical Answers (click the questions for workings/explanation)

Question 1:

Question 2: , other roots ,

Question 3:

Question 4:

Question 5:

Question 6:

Question 7:

Question 8:

Question 9:

Question 10:

Sub .

(ii)

Required coordinates:

Comparing the coefficient of ,

Comparing the coefficient of ,

Comparing the coefficient of constant,

Comparing the coefficient of ,

Other roots are .

(bi)

Other

(bii)

Using GC,

Note: they should be drawn at equal lengths of 3 units and 120 degrees apart from each other.

(biii)

They are the coefficients of actually.

Since ABCD is a parallelogram,

(ii)

(iii)

Required angle (1 DP)

(iv)

Midpoint

Distance of to plane is equals to the length of projection of onto the normal.

Distance required

for the expansion to be valid since , thus is not in the domain.

(ii)

(4 DP)

(iii)

Using GC, (4 DP)

A sample size of 30 will be sufficiently large for the manager to approximate the sample mean distribution to a normal distribution by Central Limit Theorem.

The fans should be randomly chosen, i.e., the probability of a fan being chosen should be equal and the fans are chosen independently of each other.

(ii)

Let denote the time to failure, in hours

Let denote the population mean time to failure, in hours.

Let denote the null hypothesis.

Let denote the alternative hypothesis.

(iii)

Let be the required variance.

Under , approximately by Central Limit Theorem since the sample size is sufficiently large.

To not reject at 5% level of significance,

(3 SF)

Let denote the number of left fork bug takes.

(ii)

For the probability to be greatest, is the modal value. This means that and .

.

(iii)

The probability of not entering the black hole .

Bug take 8 steps regardless of which fork it wants.

Required probability (exact)

Or

Required probability

Observe that this is a binomial expansion…

Required probability (exact)

Thus, events and are independent events.

(ii)

If and are mutually exclusive, then .

Note: Students can draw different “designs” so long as you show that A and C does not intersect and they fill up the entire venn diagram.

(iii)

Since and are independent, , . We can find the following information.

Suppose , then

Suppose , then

Minimum

Maximum

(ii)

When .

In order to have , two balls numbered 5 needs to be drawn which is not possible since balls are taken without replacement and there is only one ball numbered 5.

(iii)

The relationships is unlikely to be well modelled by the equation . From the scatter, we observe that as increases, increases at an increasing rate. Thus, a linear model such as would not be suitable.

(ii)

r value for (3 SF)

r value for (3 SF)

Since the r value for is close to 1, is a better model for the relationship between and .

(3 SF)

(iii)

revolutions per minute.

Thus estimate is reliable as is within the given date range and interpolation is a good practice.

(iv)

watts

Thus estimate is not reliable as is not within the given date range and extrapolation is a good practice.

(v)

Replace by .

(3 SF)

(ii)

Let denote the mass of one type of light bulb, in grams.

(3 SF)

(iii)

Let denote the mass of of an empty box, in grams.

(3 SF)

(iv)

(3 SF)

(v)

Using GC, (3 SF)

(vi)

Let denote the mass of a randomly chosen box, each containing a bulb and padding in grams.

(3 SF)