Read each question, consider the options, then reveal the correct answer and explanation.
100 MCQs10 per page5 of 10
MCQ 41Number System
The number 7^12 - 4^12 is exactly divisible by which of the following?
34
36
35
33
Correct answerD — 33
Explanation
For any positive n, x^n - y^n is divisible by x - y, so the expression is divisible by 7 - 4 = 3. Because the exponent 12 is even, x^n - y^n is also divisible by x + y, giving a factor 7 + 4 = 11. Since 3 and 11 are coprime, the expression is divisible by 33.
MCQ 42Mixture Or Alligation
A milk seller mixes free water with milk and sells the mixture at the original cost price of pure milk. In what ratio should water be mixed with milk to earn a profit of 5%?
1:20
1:10
1:5
1:15
Correct answerA — 1:20
Explanation
Let the cost of 1 litre of milk be ₹1. Selling the mixture at ₹1 with a 5% profit means its cost price per litre must be 1/1.05 = 20/21 rupee. By alligation, water:milk = (1−20/21):(20/21−0) = 1/21:20/21 = 1:20. The profit arises because water has no cost.
MCQ 43Ratio And Proportion
Two brothers contribute to a gift in the ratio 3:4. If the second contributes ₹100, what is the total value of the gift?
₹175
₹70
₹105
₹140
Correct answerA — ₹175
Explanation
Four ratio parts equal ₹100, so one part is ₹25. The total contribution is 7 parts, giving 7×₹25=₹175. The source version of this item contains contradictory leftover-savings data, so it has been repaired to preserve the intended ratio-allocation skill.
MCQ 44Approximation
What is the best whole-number approximation to (137 + 17 × 3.99) ÷ (4.02 × 3.98)?
3
5
13
12
Correct answerC — 13
Explanation
Round 3.99, 4.02, and 3.98 to 4. Then the expression is approximately (137+68)/(4×4)=205/16=12.8125, whose nearest whole number is 13. The printed source solution contains an arithmetic error at this item; 13 is the mathematically correct approximation.
MCQ 45Problems Based On Trains
A train travelling at 48 km/h crosses another train of half its length travelling in the opposite direction at 42 km/h in 12 seconds. The first train also crosses a platform in 45 seconds. Find the platform length.
450 m
560 m
400 m
600 m
Correct answerC — 400 m
Explanation
Let the first train length be L and the second L/2. Relative speed=90 km/h=25 m/s. In 12 seconds total length=300 m, so 3L/2=300 and L=200 m. The first train travels 48 km/h=40/3 m/s; in 45 seconds it covers 600 m. Hence platform length=600−200=400 m.
MCQ 46Number System
If N, N + 2, and N + 4 are all prime numbers, how many possible natural-number values of N are there?
1
2
3
4
Correct answerA — 1
Explanation
Among three numbers N, N + 2, and N + 4, one is divisible by 3. If all three are prime, the number divisible by 3 must actually equal 3 itself. This occurs only when N = 3, producing the prime triple 3, 5, 7. Hence there is exactly one possible value.
MCQ 47Ratio And Proportion
Money is divided among P, Q, and R in the ratio 5:6:7, while another amount is divided equally between S and T. If S receives ₹2100 less than P, can Q's share be uniquely determined?
₹2500
₹2000
Cannot be determined
₹1500
Correct answerC — Cannot be determined
Explanation
The first total and the second total are unrelated, and only the difference between S and P is given. Without either total amount or another linking condition, infinitely many values of P and Q are possible. Therefore Q cannot be uniquely determined.
MCQ 48Number System
A prime number is written in the form 3k + 1. Under what condition can it also be written as 6m + 1 for some integer m?
k is even
k is odd
k may be either odd or even
No such representation is possible
Correct answerA — k is even
Explanation
If k is even, write k = 2m. Then 3k + 1 = 6m + 1 directly. If k were odd, then 3k is odd and 3k + 1 would be even; for a prime greater than 3 this is impossible. Thus a prime of the form 3k + 1 must have even k and consequently has the form 6m + 1.
MCQ 49Number System
If b is the greatest perfect-square divisor of a positive integer c, which statement must be true?
c divides b
b and c are coprime
b divides c
b is greater than c
Correct answerC — b divides c
Explanation
By definition, a perfect-square divisor of c must divide c exactly. The greatest such divisor b is therefore still a divisor of c. No general claim can be made that every other divisor of c must divide b; the essential guaranteed relation is b | c.
MCQ 50Work And Wages
Two men and one woman can complete a work in 14 days, while two men and four women can complete it in 8 days. If a man earns ₹90 per day and wages are proportional to efficiency, what should a woman earn per day?
₹48
₹72
₹135
₹60
Correct answerD — ₹60
Explanation
Let one man's and one woman's efficiencies be m and w. Then 14(2m+w)=1 and 8(2m+4w)=1. Equating the total work expressions gives 28m+14w=16m+32w, so 12m=18w and w=3:2. Wages follow the same ratio, so if a man earns ₹90, a woman earns 90×2/3=₹60.
