Read each question, consider the options, then reveal the correct answer and explanation.
100 MCQs10 per page2 of 10
MCQ 11Number System
For every positive integer n, 7^(6n) - 6^(6n) is guaranteed to be divisible by which of the following numbers?
128
127
129
131
Correct answerB — 127
Explanation
Let A = 7^6 and B = 6^6. Then 7^(6n) - 6^(6n) = A^n - B^n, which is divisible by A - B for every positive integer n. Now 7^6 - 6^6 = 117649 - 46656 = 70993 = 127 x 559. Therefore 127 is a guaranteed factor of the expression for every positive integer n. This uses the standard identity x^n - y^n is divisible by x - y.
MCQ 12Problems Based On Trains
After travelling 80 km, a train meets with an accident and then proceeds at three-fourths of its former speed, arriving 35 minutes late. If the accident had occurred 24 km farther ahead, it would have arrived only 25 minutes late. Find the train's original speed.
48 km/h
50 km/h
30 km/h
55 km/h
Correct answerA — 48 km/h
Explanation
The only difference between the two cases occurs over the 24 km section between the two possible accident points. On that section, the train either moves at its original speed v or at 3v/4, and the time difference is 10 minutes = 1/6 hour. Thus 24/(3v/4)−24/v=1/6. This gives 32/v−24/v=1/6, so 8/v=1/6 and v=48 km/h.
MCQ 13Simple And Decimal Fractions
The numerator of a fraction is increased by 200% and its denominator is increased by 150%. The resulting fraction is 9/35. What was the original fraction?
3/10
2/15
3/16
3/14
Correct answerD — 3/14
Explanation
Let the original fraction be x/y. A 200% increase makes the numerator 3x. A 150% increase makes the denominator 2.5y = 5y/2. Hence (3x)/(5y/2) = 6x/(5y) = 9/35. Solving gives x/y = (9/35)(5/6) = 45/210 = 3/14. This corrects an inconsistency in the source solution, which treated the percentage changes incorrectly.
MCQ 14Number System
Consider the statements: I. To identify primes below 121 by sieving, it is sufficient to eliminate multiples of 2, 3, 5, and 7. II. Every composite number below 121 has a prime factor less than 11. Which option is correct?
Only I
Both I and II
Only II
Neither I nor II
Correct answerB — Both I and II
Explanation
Since sqrt(121) = 11, any composite integer below 121 must have at least one prime factor less than 11. The primes less than 11 are 2, 3, 5, and 7, so eliminating their multiples is sufficient to remove all composites below 121, apart from treating 1 separately as non-prime. Thus both statements are correct.
MCQ 15Word Problems Based On Numbers
A two-digit number has digits whose sum is 10. Reversing its digits decreases the number by 36. What is the product of its two digits?
21
24
36
42
Correct answerA — 21
Explanation
Let the tens digit be x and the units digit be y. Then x + y = 10. The original number is 10x + y and the reversed number is 10y + x. Since reversing decreases the number by 36, (10x + y) - (10y + x) = 36, giving x - y = 4. Solving x + y = 10 and x - y = 4 gives x = 7 and y = 3. Their product is 7 × 3 = 21.
MCQ 16Number System
Consider the statements: I. 7710312401 is divisible by 11. II. 173 is a prime number. Which option is correct?
Only I
Only II
Both I and II
Neither I nor II
Correct answerC — Both I and II
Explanation
For 7710312401, the alternating sum of digits is a multiple of 11, so the number is divisible by 11. To test 173 for primality, it is enough to check prime divisors up to sqrt(173), which is a little above 13. It is not divisible by 2, 3, 5, 7, 11, or 13, so 173 is prime. Therefore both statements are true.
MCQ 17Pipes And Cisterns
Tap A fills a tank in 20 minutes, tap B in 30 minutes, and outlet C empties it in 15 minutes. A, B, and C are opened alternately for one minute each in that order. How long will the tank take to fill?
167 minutes
177⅓ minutes
185 minutes
198 minutes
Correct answerB — 177⅓ minutes
Explanation
Each 3-minute cycle changes the tank by 1/20+1/30−1/15=(3+2−4)/60=1/60. After 59 cycles, 59/60 is filled in 177 minutes. The next minute belongs to A, which fills at 1/20 per minute; filling the remaining 1/60 takes (1/60)/(1/20)=1/3 minute. Total=177⅓ minutes. The printed options omit the exact answer.
MCQ 18Problem Based On Ages
Palash is presently three times Arnav's age. Seven years from now, Palash will be twice Arnav's age. What will be the ratio of Palash's age to Arnav's age 14 years from now?
3:2
5:3
2:1
1:1
Correct answerB — 5:3
Explanation
Let Arnav's present age be x, so Palash is 3x. From 3x+7=2(x+7), we get x=7, so present ages are 7 and 21. Fourteen years from now they will be 21 and 35, so Arnav=35:21=5:3. The printed source solution compared ages from different time points; this repaired version uses the same future date for both.
MCQ 19Word Problems Based On Numbers
In a five-question examination, 5% of candidates answered all five questions and 5% answered none. Of the remaining candidates, 25% answered exactly one question and 20% answered exactly four questions. If 396 candidates answered either two or three questions, how many candidates appeared?
1000
850
900
800
Correct answerD — 800
Explanation
Let the total number of candidates be N. After excluding the 5% who answered all and 5% who answered none, 90% of N remain. Of this remaining group, 25% answered one question and 20% answered four, leaving 55% who answered two or three questions. Thus 0.55 × 0.90N = 396, so 0.495N = 396 and N = 800.
MCQ 20Average
The average weight of three boys P, T, and R is 54 1/3 kg, while the average of another three-boy group sharing some of these boys is 53 kg. Can the average of the five distinct boys be determined without the shared boy's weight?
52.4 kg
53.2 kg
Cannot be determined
53.8 kg
Correct answerC — Cannot be determined
Explanation
The two group totals are known, but when the groups overlap, adding their totals counts the common boy twice. To obtain the total of five distinct boys, the common boy's weight must be subtracted once. Since that individual weight is not supplied, the combined average cannot be determined uniquely.
