Question 1
Let \( a \) and \( b \) be two sequences such that \( a_n = 13 + 6(n-1) \) and \( b_n = 15 + 7(n-1) \) for all natural numbers \( n \). Then, the largest three-digit integer that is common to both these sequences is:
A. 943
B. 954
C. 967
D. 989
Answer: C. 967
Detailed Solution:
Using properties of arithmetic sequences, we find that the common terms of the two sequences form another arithmetic sequence. Calculating, the largest three-digit common term is 967.
Question 2
A container has 40 liters of milk. Then, 4 liters are removed from the container and replaced with 4 liters of water. This process is repeated until the volume of milk becomes less than that of water. The smallest number of times this process is repeated to achieve this is:
A. 5
B. 6
C. 7
D. 8
Answer: C. 7
Detailed Solution:
Using the formula for repeated removal and replacement, we find that the volume of milk becomes less than water after 7 repetitions.
Question 3
If a person D joins a group of three people, the average weight of the group reduces by \( x \) kg. If another person E joins instead of D, the average increases by \( 2x \) kg. If E’s weight is 12 kg more than D’s, then the value of \( x \) is:
A. 1
B. 2
C. 3
D. 4
Answer: A. 1
Detailed Solution:
Setting up equations for average changes, we solve and find that \( x = 1 \).
Question 4
In a fruit market, there are initially 40% mangoes, 30% bananas, and 30% apples. If 50% of the mangoes, 30 bananas, and 40% of the apples are sold, what is the minimum number of fruits required initially to maintain a balance?
A. 100
B. 120
C. 150
D. 180
Answer: C. 150
Detailed Solution:
By setting up the initial percentages and solving for minimum quantities, the required minimum initial number of fruits is 150.
Question 5
Let \( a_1, a_2, a_3, \ldots \) and \( b_1, b_2, b_3, \ldots \) be arithmetic progressions such that their common differences are prime numbers. If \( a_4 = b_4 \), \( a_7 = b_7 \), and \( b_1 = 0 \), then \( a_1 \) is equal to:
A. 79
B. 83
C. 86
D. 84
Answer: A. 79
Detailed Solution:
Using the given conditions and properties of arithmetic progressions, we find that \( a_1 = 79 \).
