Mathematics

Each question has 2 possible answers (True or False), so for 10 questions, the total number of different answer sequences is 2^10 = 1024. This ensures no two candidates have identical sequences if the number of candidates is at most 1024.

The grocer wants to gain 10% by selling the mixture at Rs. 68.20 per kg. First, calculate the cost price of the mixture: Cost Price = Selling Price / (1 + Gain %) = 68.20 / 1.10 = Rs. 62 per kg. Using the allegation method to find the ratio of the two teas worth Rs. 60 and Rs. 65 per kg to get a mixture costing Rs. 62 per kg: Ratio = (65 - 62) : (62 - 60) = 3 : 2. Therefore, the grocer must mix the two varieties of tea in the ratio 3:2. This matches option A, confirming that the correct answer is A.

Let the three consecutive odd integers be x, x+2, and x+4. According to the problem, the sum of the last two integers is 13 more than the first integer: (x+2) + (x+4) = x + 13. Simplifying, 2x + 6 = x + 13, so x = 7. Therefore, the integers are 7, 9, and 11, which corresponds to option D. However, checking the sum: 9 + 11 = 20, and 7 + 13 = 20, so the condition holds true for option D, confirming it is correct.

A runs 1 km in 4.5 minutes, so speed of A = 1000/4.5 = 222.22 m/min. B runs 1 km in 5 minutes, so speed of B = 1000/5 = 200 m/min. To finish together, A must give B a start distance 'd' such that time taken by A to run 1000 m equals time taken by B to run (1000 - d) m. So, 1000/222.22 = (1000 - d)/200, solving gives d = 125 m.

Let Arslan have A marbles and Bilal have B marbles. From Arslan's statement: if Bilal gives x marbles to Arslan, both have equal marbles, so A + x = B - x. From Bilal's statement: if Arslan gives twice as many marbles (2x) to Bilal, then Bilal has 30 more than Arslan, so B + 2x = A - 2x + 30. Solving these equations gives x = 4.

The digits given are {2, 3, 5, 1, 7, 9}. To form a four-digit even number without repetition, the last digit must be even. The only even digit available is 2, so the last digit is fixed as 2. For the first digit, we can choose any of the remaining 5 digits (3, 5, 1, 7, 9). For the second digit, we can choose any of the remaining 4 digits, and for the third digit, any of the remaining 3 digits. Therefore, the total number of such numbers is 5 × 4 × 3 × 1 = 60. This matches option A. The original answer D=240 is incorrect, and the explanation in the question was confusing but ultimately supports 60 as the correct count.

Vessel P has 62.5% alcohol, so in 2 litres, alcohol = 2 × 0.625 = 1.25 litres. Vessel Q has 87.5% alcohol, so in 4 litres, alcohol = 4 × 0.875 = 3.5 litres. Total alcohol = 1.25 + 3.5 = 4.75 litres. Total volume = 2 + 4 = 6 litres, so water = 6 - 4.75 = 1.25 litres. The ratio of alcohol to water = 4.75 : 1.25 = 19 : 5. However, the options given do not include 19:5 as a correct ratio for alcohol to water; option D is 19:5 but the question asks for the ratio of alcohol to water, which matches option D. Therefore, the original correct answer D is correct. On rechecking, the ratio 19:5 is correct for alcohol to water. The initial calculation confirms option D is correct.

The total number of groups of 3 boys from 5 boys (A, B, C, D, E) is $\left(\genfrac{}{}{0px}{}{5}{3}\right)=10$. The groups that contain both C and D are those that include C, D, and one of the other boys (A, B, or E), which are 3 groups: ACD, BCD, and CDE. Excluding these 3 groups leaves 10 - 3 = 7 groups that do not contain both C and D together. Therefore, the maximum number of such different groups is 7, which corresponds to option C.

Let the cost of a chair be C and a table be T. From the equations: 2C + 3T = 1300 and 3C + 2T = 1200, solving these simultaneously gives C = 160 and T = 230. The difference T - C = 70, so each table costs Rs.70 more than each chair.

Using the allegation method: The difference between the price of the first rice and the mixture price is 7.20 - 6.30 = 0.90, and the difference between the mixture price and the second rice price is 6.30 - 5.70 = 0.60. The ratio in which the two types of rice should be mixed is the inverse of these differences, i.e., 0.60 : 0.90 = 2 : 3. Therefore, rice at Rs. 7.20 should be mixed with rice at Rs. 5.70 in the ratio 2 : 3, which corresponds to option B.

The word 'NOKIA' has 5 letters. Fixing 'N' at the start and 'A' at the end leaves 3 letters (O, K, I) to arrange in the middle. The number of such arrangements is 3! = 6.

The total number of ways to arrange 6 players in a line is 6! = 720. To find the number of arrangements where two specific players (Asim and Raheem) are together, treat them as a single unit. This gives 5 units to arrange, which can be done in 5! ways. Since Asim and Raheem can switch places within this unit, multiply by 2!. So, the number of arrangements with Asim and Raheem together is 2! × 5! = 240. Therefore, the number of arrangements where Asim and Raheem are never together is 720 - 240 = 480. Hence, the correct answer is 480, which corresponds to option D.

The problem states that a delegation of 5 members is to be formed from 3 ladies and 5 gentlemen, with 2 particular ladies always included. Since these 2 ladies are fixed, we need to select the remaining 3 members from the remaining 1 lady and 5 gentlemen, totaling 6 people. The number of ways to choose 3 members from these 6 is given by the combination formula C(6,3) = 20. Therefore, the correct answer is option A (20).

The word MATERIAL has 8 letters with vowels A, A, E, I (4 vowels) and consonants M, T, R, L (4 consonants). Treating all vowels as a single unit, we have 5 units to arrange (the vowel block + 4 consonants), which can be arranged in 5! = 120 ways. The vowels inside the block can be arranged among themselves in 4!/2! = 12 ways (dividing by 2! for the two identical A's). Therefore, total arrangements = 120 × 12 = 1440. However, the correct calculation is 5! × (4!/2!) = 120 × 12 = 1440, which corresponds to option B. But the original correct_answer was B, so the original answer is correct. On rechecking, the original answer B (1440) is correct. So the correct_answer remains B.

The selling price of the mixture is Rs. 68.20 per kg with a gain of 10%. To find the cost price of the mixture, use the formula: Cost Price = Selling Price / (1 + Gain%) = 68.20 / 1.10 = Rs. 62 per kg. Using the allegation method to mix teas worth Rs. 60 and Rs. 65 to get a mixture costing Rs. 62, the ratio is calculated as (65 - 62) : (62 - 60) = 3 : 2. Therefore, the grocer must mix the teas in the ratio 3:2 to achieve the desired gain.

Let x be the liters of oil at Rs.40 per liter. The total cost of the mixture is (40x + 60*240) and the total volume is (x + 240). The cost per liter of the mixture is Rs.52, so (40x + 14400)/(x + 240) = 52. Solving gives x = 120 liters.

The total number of ways to arrange four distinct books A, B, C, and D is 4! = 24. To find the number of arrangements where A and B are consecutive, treat A and B as a single block. This block and the other two books C and D can be arranged in 3! = 6 ways. Inside the block, A and B can be arranged in 2! = 2 ways. So, the total number of arrangements with A and B together is 2! × 3! = 12. Therefore, the number of arrangements where A and B are never consecutive is 24 - 12 = 12. Hence, the correct answer is option B (12).

The number of ways to select 2 men from 5 is C(5,2) = 10, and the number of ways to select 3 women from 6 is C(6,3) = 20. Multiplying these gives 10 × 20 = 200 ways. However, the options given do not include 200 as the correct answer; option B is 200 but the calculation shows 10 × 20 = 200, so the correct answer is B, not D. Rechecking: C(5,2) = 10, C(6,3) = 20, total = 200, so correct answer is B.

Let the five consecutive even numbers in set x be x, x+2, x+4, x+6, x+8. Their sum is 5x + 20 = 440, so 5x = 420 and x = 84. The least number of set x is 84. Double this is 168. The second least number of the new set is 121 less than 168, so 168 - 121 = 47. Let the new set be five consecutive integers: y, y+1, y+2, y+3, y+4, where y+1 = 47, so y = 46. The sum of the new set is y + (y+1) + (y+2) + (y+3) + (y+4) = 5y + 10 = 5*46 + 10 = 230 + 10 = 240. Therefore, the correct answer is B (240).

The word "THURSDAY" has 8 distinct letters. The number of different words (permutations) that can be formed using all 8 letters is 8! (8 factorial), which equals 40,320.

Using the rule of allegation, the ratio of the quantities of the cheaper rice (Rs. 6 per kg) to the dearer rice (Rs. 8.75 per kg) to obtain a mixture costing Rs. 7.50 per kg is calculated as (8.75 - 7.50) : (7.50 - 6) = 1.25 : 1.5, which simplifies to 5 : 6. Therefore, the correct ratio is 5 : 6, corresponding to option A.

The total number of ways to select 10 representatives from 22 people (12 seniors + 10 juniors) is 22C10. The number of groups with no seniors (only juniors) is 10C10 = 1. Therefore, the number of groups with at least one senior is 22C10 - 1.

Let the original two-digit number be 10y + x, where y is the tens digit and x is the units digit. The sum of the digits of the number obtained by interchanging the digits is x + y = 13. The difference between the original number and the interchanged number is (10y + x) - (10x + y) = 9(y - x) = 45, so y - x = 5. From the two equations, x + y = 13 and y - x = 5, adding them gives 2y = 18, so y = 9. Substituting y = 9 into x + y = 13 gives x = 4. Therefore, the original number is 10y + x = 10*4 + 9 = 49. The interchanged number is 94, and the difference 94 - 49 = 45. The sum of digits of the interchanged number (94) is 9 + 4 = 13. Hence, the original number is 49 (option A).

Let the actual number of children in the school be x. Each child was supposed to get 2 bananas, so total bananas = 2x. On the day, 360 children were absent, so number of children present = x - 360. Each present child got 2 extra bananas, so each got 2 + 2 = 4 bananas. Total bananas distributed = 4(x - 360). Since total bananas remain the same, 2x = 4(x - 360). Solving: 2x = 4x - 1440 => 1440 = 2x => x = 720. Although 720 is not among the options, the correct mathematical solution is 720. The closest option given is 600 (option A), which is likely a typographical error in the options. The verified correct answer is 720 children in the school.

The number of ways to form a group of 5 men and 2 women from 7 men and 3 women is calculated using combinations. The number of ways to choose 5 men out of 7 is C(7,5) = 21, and the number of ways to choose 2 women out of 3 is C(3,2) = 3. Since these choices are independent, multiply them: 21 × 3 = 63. Therefore, the correct answer is 63, which corresponds to option A.