1987CEMCQ02

36.
I is wrong.
For example, 1 > -2 but (1)² < (-2)²

II is wrong.
For example, 1 > -2 but 1/1 > 1/-2

III is correct.
Think about the graph of y = 10^x
which is strictly increasing.
For a > b, 10^a > 10^b.

Answer = A

✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀

37.
x log_10(0.1) > log_10(10)

x log_10( 10^(-1) ) > log_10( 10^1 )

x (-1) > 1

x < -1

Answer = E

✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀

43.
Let the distance between A and B is D km.
Recall: Distance = Speed × Time
Than, Time = Distance / Speed

Time for the trip from A to B is D/3 (hours)
Time for the trip from B to A is D/15 (hours)

Total time spent = D/3 + D/15 = 6D/15 = 2D/5

Average Speed
= (Total Distance) / (Total Time)
= (D + D) / (2D/5)
= (2D)/(2D/5)
= 5 (km/h)

Answer = A

✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀

45.

圖片參考：https://s.yimg.com/rk/HA00430218/o/1688923236.png

Answer = E

✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀

50.
Method 1:
Draw the graph of y = sin x
Draw the graph of y = 1 + cos x by shifting y = cos x upward by 1 unit.
You can see there are 2 intersections.


Method 2:
Consider
sin x = 1 + cos x
(sin x)² = (1 + cos x)² ... (*)
sin²x = 1 + 2 cos x + cos²x
1 - cos²x = 1 + 2 cos x + cos²x
2cos²x + 2cosx = 0
cosx(cosx + 1) = 0
cos x = 0 or cos x = -1
x = 90°, 270°, 180°

We need to check the solution because the step in squaring both sides in (*) may create extra solutions.

sin(90°) = 1
1 + cos(90°) = 1, so 90° is a solution

sin(270°) = -1
1 + cos(270°) = 1, so 270° is NOT a solution

sin(180°) = 0
1 + cos(180°) = 0, so 180° is a solution

Two solutions mean two intersections.

Answer = C

✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀ ✿ ❀

54. [Refer to the above graph]
Orange △ ~ purple △ ~ Big △ABC

In terms of areas,
(Orange △) : (purple △) = 1 : 4
(Orange △) : (Big △ABC) = 1 : 9

The required ratio is
1 : (9 - 1 - 4)
= 1 : 4

Answer = C