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How much copper and how much magnesium should be added to 200lbs of an alloy containing 20% copper and 40% magnesium to produce an alloy containing 24% copper and 50% magnesium?
The sum of the digits of a three-digit number is 11. If the first and last digits are interchanged, the new number is greater than the original number by 594. If the tens and the units digits are interchanged, the resulting number is greater than the original by 36. Find the original number.
A man wishes to fence in a rectangular lot. If he uses material costing P17/meter for the front of the lot and material costing P14/meter for the other three sides, the fence will cost P8,900. If he uses the cheaper material for all four sides, the fence will cost P8,540. Find the dimensions of the lot.
A man rows 12 miles downstream on a river in 2 hrs. He requires another 3 hrs. to row back to his starting position. Find the speed of the current and the rate the man can row in still water.
Two pipes empty a vat in 1hr and 12 min. When one pipe is used to fill and one to empty the vat, it is filled in 3 hrs. How long would it take each pipe to empty the vat if only one pipe is open?
It is claimed that the average weight of babies at birth is 3.4 kg. The average weight of the random
sample of 30 newly born babies was determined. It was found out that the average weight was 3.1 kg.
Is there a reason to believe that the average weight of babies at birth is not 3.4 kg.? Assume that the
population standard deviation is 1.1 kg. Use the 0.05 level of significance
1) What are the null hypotheses and alternative hypotheses of the research scenario?
2) What is the level of significance?
3) What is the test statistic?
4) What is the critical value in the scenario?
5) What could be the conclusion?
The Head of the Math Department announced that the mean score of Grade 11 students in the first
midterm examination in Statistics and Probability was 89 and the standard deviation was 12. One
student believed that the mean score was less than this, randomly selected 34 students and computed
their mean score. She obtained a mean score of 85. At 0.01 of level of significance, test the student’s
belief.
1) What are the null hypotheses and alternative hypotheses of the research scenario?
2) What is the level of significance?
3) What is the test statistic?
4) What is the critical value in the scenario?
5) What could be the conclusion?
Question 3 [10 marks]
Which of the following statements is true about the given function?
(1) The Intermediate Value Theorem does not hold between x = −1 and x = 1.
(2) The function has two roots in the interval [−2, 4].
(3) It will take at least 15 iterations of the bisection method to approximate the root between
x = −2/3 and x = 2 correct to 10−4
.
(4) It will take no more than 14 iterations for the bisection method to converge to the root between
x = −2/3 and x = 2 correct to 10−5
.
(5) The function has at least one singular point.
Question 2 [10 marks]
Which of the following statements about the given function is FALSE?
(1) the graph of the function has one point of inflection and has two relative extrema.
(2) The function has no absolute extremum.
(3) The graph of the function has two roots in the interval [0, 3.5]
(4) The function has a point of inflection point at x = 0.
(5) The function has a relative maximum point at x =
−2
3
and relative minimum point at x = 2.
Question 1 [10 marks]
To have an idea on whether we should apply the bisection method to determine the root of f(x) = 0
in a given interval, we may
(1) draw the graph of f(x) and observe the graphs then conclude
(2) check if f(x) and f
(x) are continuous then conclude
(3) apply the function f(x) to the endpoints of the given interval and check the sign of the corre-
sponding outputs.
(4) check if f(x) has a critical point in the given interval
(5) check if f(x) has a point of inflection in the given interval.
(For Questions 2 to 5)
Consider the function f(x) = x
3 − 2x
2 − 4x + 2.
#339914 on Dec 2023