Q. In a group of six friends, X is taller than Y but shorter than Z. W is shorter than X but taller than V. U is the shortest. Who is the tallest? (2023)
A.
X
B.
Y
C.
Z
D.
W
Solution
The order of height is U < V < W < X < Z. Therefore, Z is the tallest.
Q. In a harmonic progression, if the first term is 1 and the second term is 1/2, what is the common difference of the corresponding arithmetic progression?
A.
1/2
B.
1/4
C.
1/6
D.
1/8
Solution
The reciprocals are 1 and 2, which are in arithmetic progression with a common difference of 1/2.
Q. In a harmonic progression, if the first term is 2 and the second term is 3, what is the third term?
A.
4
B.
5
C.
6
D.
7
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 3 are 1/2 and 1/3. The common difference is 1/3 - 1/2 = -1/6. The third term's reciprocal will be 1/3 - 1/6 = 1/6, so the third term is 6.
Q. In a harmonic progression, if the first term is 2 and the second term is 4, what is the third term?
A.
1
B.
3
C.
6
D.
8
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 4 are 1/2 and 1/4. The common difference is -1/4. Therefore, the third term's reciprocal is 1/4 - 1/4 = 0, which means the third term is 1.
Q. In a harmonic progression, if the first term is 2 and the second term is 4/3, what is the third term?
A.
1
B.
3/2
C.
2/3
D.
1/2
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of the first two terms are 1/2 and 3/4. The common difference is 1/4, so the reciprocal of the third term is 1/2 + 1/4 = 3/4. Therefore, the third term is 1/(3/4) = 4/3.
Q. In a harmonic progression, if the first term is 3 and the second term is 6, what is the third term?
A.
9
B.
12
C.
15
D.
18
Solution
The reciprocals of the terms are 1/3 and 1/6. The common difference is (1/6 - 1/3) = -1/6. The third term's reciprocal will be 1/6 - 1/6 = 0, which means the third term is 1/12, thus the answer is 12.
Q. In a harmonic progression, if the first term is 3 and the second term is 6, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
4
Solution
The reciprocals of the first two terms are 1/3 and 1/6. The common difference is 1/6 - 1/3 = -1/6, which is incorrect. The correct common difference is 1/3 - 1/6 = 1/6.
Q. In a harmonic progression, if the first term is 4 and the second term is 2, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
4
Solution
The reciprocals of the terms are 1/4 and 1/2. The common difference is 1/2 - 1/4 = 1/4.
Q. In a harmonic progression, if the first term is 4 and the second term is 8, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
4
Solution
The reciprocals are 1/4 and 1/8. The common difference is 1/8 - 1/4 = -1/8, which means the common difference of the corresponding arithmetic progression is 1/8.
Q. In a harmonic progression, if the first term is 4 and the second term is 8, what is the third term?
A.
12
B.
16
C.
20
D.
24
Solution
The reciprocals are 1/4 and 1/8. The common difference is 1/8 - 1/4 = -1/8. The third term's reciprocal will be 1/8 - 1/8 = 0, hence the third term is 16.
Q. In a harmonic progression, if the first term is 5 and the common difference of the corresponding arithmetic progression is 2, what is the second term?
A.
2.5
B.
3.33
C.
4
D.
6
Solution
The first term is 5, and the second term in the harmonic progression corresponds to the reciprocal of the second term in the arithmetic progression, which is 5 + 2 = 7. Thus, the second term is 1/7.
Q. In a harmonic progression, if the first term is 5 and the second term is 10, what is the common difference of the corresponding arithmetic progression?
A.
1
B.
2
C.
3
D.
5
Solution
The reciprocals are 1/5 and 1/10. The common difference is 1/10 - 1/5 = -1/10, which is the difference in the arithmetic progression.
Q. In a harmonic progression, if the first term is a and the second term is b, what is the formula for the nth term?
A.
1/(1/n + 1/a)
B.
1/(1/n + 1/b)
C.
1/(1/a + 1/b)
D.
1/(1/a - 1/b)
Solution
The nth term of a harmonic progression can be expressed as 1/(1/a + (n-1)d) where d is the common difference of the corresponding arithmetic progression.
Q. In a hybrid set scenario, if element x belongs to set A and element y belongs to set B, which of the following can be inferred about their relationship in the hybrid set?
A.
Element x is equal to element y.
B.
Element x and element y are both included in the hybrid set.
C.
Element x cannot be in the hybrid set.
D.
Element y is a subset of element x.
Solution
Both elements x and y can be included in the hybrid set as they belong to their respective sets.
Correct Answer:
B
— Element x and element y are both included in the hybrid set.
Q. In a hybrid set scenario, if element x is in set A and element y is in set B, which of the following can be inferred about their relationship in the hybrid set?
A.
Element x is equal to element y.
B.
Element x and element y are both in the hybrid set.
C.
Element x is not in the hybrid set.
D.
Element y is not in the hybrid set.
Solution
If both elements belong to their respective sets, they will also be included in the hybrid set.
Correct Answer:
B
— Element x and element y are both in the hybrid set.
Q. In a hybrid set scenario, if set C = {a, b, c} and set D = {b, c, d}, which of the following is the correct representation of the symmetric difference between set C and set D? (2023)
A.
{a, d}
B.
{a, b, c, d}
C.
{b, c}
D.
{a, b, c, d, e}
Solution
The symmetric difference includes elements that are in either set but not in both, which results in {a, d}.
Q. In a hybrid set scenario, if set D has 5 elements and set E has 3 elements with 2 elements in common, what is the maximum number of unique elements in the hybrid set?
A.
6
B.
7
C.
8
D.
9
Solution
The maximum number of unique elements is calculated as (5 + 3 - 2) = 6.
