Q. If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two numbers if they are both less than 30? (2023)
A.
25
B.
35
C.
40
D.
30
Solution
Let the two numbers be 5a and 5b. Then, LCM(5a, 5b) = 5 * LCM(a, b) = 60, which gives LCM(a, b) = 12. The pairs (3, 4) work, giving numbers 15 and 20, which sum to 35.
Q. If the least common multiple (LCM) of two numbers is 36 and their greatest common divisor (GCD) is 6, what can be inferred about the product of the two numbers?
A.
It is 216.
B.
It is 72.
C.
It is 36.
D.
It is 6.
Solution
The product of two numbers is equal to the product of their LCM and GCD: 36 * 6 = 216.
Q. If the least common multiple (LCM) of two numbers is 60 and their greatest common divisor (GCD) is 5, which of the following pairs could represent these two numbers?
A.
(5, 12)
B.
(10, 30)
C.
(15, 20)
D.
(5, 15)
Solution
The product of the two numbers is equal to the LCM multiplied by the GCD. Thus, 60 * 5 = 300. The pair (15, 20) satisfies this condition since 15 * 20 = 300.
Q. If the least common multiple (LCM) of two numbers is 60 and their greatest common divisor (GCD) is 12, what can be inferred about the product of the two numbers?
A.
It is 720.
B.
It is 60.
C.
It is 12.
D.
It is 5.
Solution
The product of two numbers is equal to the product of their LCM and GCD. Therefore, 60 * 12 = 720.
Q. If the length of a rectangle is increased by 20% and the width is decreased by 10%, what is the percentage change in the area?
A.
8% increase
B.
10% decrease
C.
12% increase
D.
2% decrease
Solution
Let the original length be L and width be W. The new length is 1.2L and the new width is 0.9W. The original area is LW and the new area is (1.2L)(0.9W) = 1.08LW. The percentage change in area is ((1.08LW - LW) / LW) * 100 = 8% increase.
Q. If the length of a river is 120 km and it flows at an average speed of 5 km/h, how long will it take for the river to flow from its source to its mouth? (2021)
A.
20 hours
B.
24 hours
C.
30 hours
D.
36 hours
Solution
Time = Distance / Speed = 120 km / 5 km/h = 24 hours.
Q. If the length of a river is 240 km and it flows at an average speed of 4 km/h, how long will it take for the river to flow from its source to the mouth? (2021)
A.
60 hours
B.
70 hours
C.
80 hours
D.
90 hours
Solution
Time = Distance / Speed = 240 km / 4 km/h = 60 hours.
Q. If the length of a side of a cube is measured as 2.0 ± 0.1 m, what is the maximum possible error in the volume of the cube?
A.
0.8 m³
B.
0.4 m³
C.
0.2 m³
D.
0.1 m³
Solution
Volume V = L³. The maximum error in volume can be calculated using the formula: ΔV = 3L²ΔL. Here, ΔL = 0.1 m, L = 2.0 m, so ΔV = 3(2.0)²(0.1) = 1.2 m³.
Q. If the length of the potentiometer wire is increased while keeping the voltage constant, what will happen to the balance point when measuring a cell's EMF?
A.
It will move to a longer length.
B.
It will move to a shorter length.
C.
It will remain unchanged.
D.
It will become unstable.
Solution
Increasing the length of the potentiometer wire while keeping the voltage constant will increase the balance point length for the same EMF.
Correct Answer:
A
— It will move to a longer length.
