Junior Philippine Institute of Chemical Engineers- Xavier University Department of Academics MODULE 1: PRE-TEST SOLUTIONS General Engineering Principles
John’s father is 5 times older than John and John is twice as old as his sister Alice. In two years time, the sum of their ages will be 58. How old is John now? Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours to paint a similarlysized house. How long would it take the two painters together to paint the house? The average of 10 student scores is 82, but the student mistakenly recorded two values of 79 as 78 and three values of 83 as 85. What is the true average of the set of scores? Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers? Find the domain of 2đ?‘Ľâˆ’8 đ?‘Ľ 2 −16
IN 2 YEARS x+2 5x + 2 0.5x + 2 58
(x+2)+(5x+2)+(0.5x+2) = 58 Solving for x, x = 8
Painter 1 Painter 2
TIME per unit job 12 hr/job 8 hr/job
Total rate =
1 đ?‘—đ?‘œđ?‘? 12 â„Žđ?‘&#x;
+
1 đ?‘—đ?‘œđ?‘? 8 â„Žđ?‘&#x;
JOB per unit time 1/12 job/hr 1/8 job/hr = 0.2083
(-8 – h)2 + (-6 – k)2 = 102 64 + 16h + h2 + 36 + 12k + k2 = 100 16h + 12k = 0
Find the major axis of an ellipse with equation 12x2 + 24y2 - 48x + 144y + 216 = 0.
đ?‘—đ?‘œđ?‘? â„Žđ?‘&#x;
Time for single job by all units = (0.2083 đ?‘—đ?‘œđ?‘? -1 ) = 4.8 hours/job â„Žđ?‘&#x;
Total score = (10 students)(82) = 820 Adjustments
= 2(79-78) = +2 = 3(83-85) = -6
True actual score = 820 + 2 – 6 = 816
What is the slope of a line perpendicular to a second line passing through (5,2) and (9,1)? 3đ?‘Ľâˆ’5 For đ?‘“(đ?‘Ľ) = , 4đ?‘Ľ find đ?‘“ −1 (2)?
True average = 816/10 = 81.6
Let x be the smaller number Let y be the larger number 2y = 3 + 5x 4y + 3x =71
Find the vertical asymptotes of đ?‘Ś = 2đ?‘Ľ . 2 đ?‘Ľ −3đ?‘Ľâˆ’10
The value of 0.25π rad in degrees is: The exact value of sin(15°) is:
Solving simultaneously, use [MODE] – [5] – [1], x, y = {5,14}
(x + 4)(x – 4)
(x + 4)
Finding the zeros of the denominator, (x + 4) = 0 → x = –4 General equation of circle: (x – h)2 + (y – k)2 = r2 Substituting, (6 – h)2 + (8 – k)2 = 102 36 – 12h + h2 + 64 -16k + k2 = 100 -12h – 16k = 0
Solving, (h, k) = (0, 0) General equation of ellipse: (đ?‘Ľâ€“ â„Ž)2 (đ?‘Śâ€“ đ?‘˜)2 + = đ?‘&#x;2 2 đ?‘Ž đ?‘?2 Completing the square of given equation, đ?&#x;? 12x2 + 24y2 - 48x + 144y + 216 = 0 ( ) đ?&#x;?đ?&#x;? 2 2 x + 2y - 4x + 12y + 18 = 0 [(x2 - 4x)+4] + [2(y2 + 6y)+9] = -18+4+18 2 (x – 4x + 4) + 2(y2 + 6y + 9) = 4 (đ?‘Ľâ€“ 2)2 (đ?‘Ś + 3)2 + = 1 4 2 Since a2 = 4 → a = 2 Major axis = 2a = 2(2) → 4 Δđ?‘Ś 1−2 1 Slope of points: → =− Δđ?‘Ľ
9−5
Perpendicularity → m’ = − m’ = −
1 1 4
(− )
4
1
3đ?‘Ľâˆ’5
= The ordinate value for θ = 60° is:
√2 √3 2 2
−
√2 1 2 2
=
√6 4
−
√2 4
Find the sum of the integers from 1 to 100.
√đ?&#x;”−√đ?&#x;?
cos đ?‘Ľ sin đ?‘Ľ
Solving simultaneously, a0 = 6 and d = 4 a100 = 6 + 99(4) = 402 By direct input into calculator, 100
∑ đ?‘– = 5050 đ?‘–=1
Evaluate đ?‘Ľ = 3 log 5 √25
√3
+
2
Find the sum of the geometric series 1 1 1 {1, , , ‌ }. 4 16 64
đ?&#x;’
đ??…
From graph, range → [đ?&#x;Ž, ) âˆŞ ( , đ??…] đ?&#x;? đ?&#x;? General sequence term: an = a0 + (n-1)d Term expressions: a5 = a0 + (5-1)d a15 = a0 + (15-1)d 22 = a0 + 4d 62 = a0 + 14d
Alternatively, 1 100 2 99 50 3 97 pairs of 101 4 96 â‹Ž â‹Ž By direct input into calculator, 3 đ?‘Ľ = log 5 √25 2 Solving for x → đ?‘Ľ = , 3 By principle, 3 đ?‘Ľ = log 5 √25 3
Ordinate → y (Abscissa → x) For points in unit circle, (x,y) → (cos θ, sin θ)
cot đ?‘Ľ + tan đ?‘Ľ =
An arithmetic sequence has its 5th term equal to 22 and its 15th term equal to 62. Find its 100th term.
4
Ordinate = sin θ = sin 60° = The identity (cot � + tan �) is equivalent to:
đ??…
√6−√2
→
cos 2 đ?‘Ľ + sin2 đ?‘Ľ 1 = sin đ?‘Ľ cos đ?‘Ľ sin đ?‘Ľ cos đ?‘Ľ = đ??œđ??Źđ??œ đ??ą đ??Źđ??žđ??œ đ??ą
The range of the inverse secant function is:
=4
đ?‘Ś= → 4đ?‘Ľđ?‘Ś = 3đ?‘Ľ − 5 4đ?‘Ľ 4đ?‘Ľđ?‘Ś − 3đ?‘Ľ = −5 → đ?‘Ľ(4đ?‘Ś − 3) = −5 −5 −đ?&#x;“ đ?‘Ľ= → đ?’šâˆ’đ?&#x;? = 4đ?‘Śâˆ’3 đ?&#x;’đ?’™âˆ’đ?&#x;‘ −5 −5 y −1 (2) = = = −đ?&#x;? 4(2) − 3 5 Vertical asymptotes are the roots of the denominator: (đ?‘Ľ 2 − 3đ?‘Ľ − 10) = 0 → (x – 5)(x + 2) = 0 V.A. → x = 5, x = -2 180° 0.25đ?œ‹ ∙ = đ?&#x;’đ?&#x;“° đ?œ‹ By direct input to scientific calculator, sin (15°) =
=
đ?‘š
By principle, sin (15°) = sin (45°-30°) = sin45° cos 30° – cos45° sin30°
Domain is by default all the values of x minus the ones that will make the denominator zero. 2đ?‘Ľâˆ’8 2(đ?‘Ľâˆ’4) 2 → → 2 đ?‘Ľ −16
Find the center of a circle passing through (6,8) and (8,-6) with a radius of 10.
NOW x 5x 0.5x
John Father Alice
3
2
đ?&#x;?
5đ?‘Ľ = √25 = √52 = 53 → đ?’™ = đ?&#x;‘ Converting the series into expressions of integers, 1 1 1 {1, , , ‌ } →{40 , 4−1 , 4−2 , 4−3 ‌ } 4 16 64 Thus, the series can be expressed as ∞
∑ 4−đ?‘– đ?‘–=0
= 0.866
sin đ?‘Ľ cos đ?‘Ľ
The infinite sum cannot be plugged into a calculator, but note that as i increases the actual value decreases. The sum will be obvious with an upper limit of ~10.