JEE MATH APEX

JEE MAIN TEST – 12

Mathematics Questions (Continued)

1. The relation R = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)\} on the set \{1, 2, 3\} is:
(A) Reflexive, Symmetric but not Transitive
(B) Reflexive, Transitive but not Symmetric
(C) Symmetric, Transitive but not Reflexive
(D) Equivalence relation
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2. A relation on the set A = \{x : |x| < 3, x \in \mathbb{Z}\}, where \mathbb{Z} is the set of integers, is defined by R = \{(x, y) : y = |x|, x \neq -1\}. Then the number of elements in the power set of R is:
(A) 32
(B) 64
(C) 16
(D) 8
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3. Let R be a relation on \mathbb{R}, given by R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number}\}. Then R is:
(A) Reflexive but neither symmetric nor transitive
(B) Reflexive and transitive but not symmetric
(C) Reflexive and symmetric but not transitive
(D) An equivalence relation
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4. Let the mean and variance of 12 observations be \dfrac{9}{2} and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is \dfrac{m}{n}, where m and n are coprime, then m + n is equal to:
(A) 316
(B) 314
(C) 317
(D) 315
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5. The mean and variance of a binomial distribution are \alpha and \dfrac{\alpha}{3} respectively. If P(X = 1) = \dfrac{4}{243}, then P(X = 4 \text{ or } 5) is equal to:
(A) \dfrac{5}{9}
(B) \dfrac{64}{81}
(C) \dfrac{16}{27}
(D) \dfrac{145}{243}
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6. Find the M.D. about median for the following data:
x_i 3 9 17 23 27
f_i 8 10 12 9 5
(A) 8
(B) 6.72
(C) 17
(D) 9.81
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7. The mean of five numbers is 0 and their variance is 2. If three of those numbers are -1, 1, and 2, then the other two numbers are:
(A) -5 and 3
(B) -4 and 2
(C) -3 and 1
(D) -2 and 0
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8. The standard deviation for the scores 1, 2, 3, 4, 5, 6, and 7 is 2. Then the standard deviation of 12, 23, 34, 45, 56, 67, and 78 is:
(A) 2
(B) 4
(C) 22
(D) 11
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9. The mean marks obtained by 300 students in Mathematics are 45. The mean of the top 100 students was 70 and the mean of the last 100 was 20. Then the mean of the remaining 100 students is:
(A) 40
(B) 50
(C) 45
(D) 43
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10. There are 100 families in a society: 40 families buy newspaper A, 30 families buy newspaper B, 30 families buy newspaper C, 10 families buy newspapers A and B, 8 families buy newspapers B and C, 5 families buy newspapers A and C, and 3 families buy newspapers A, B, and C. Then the number of families who do not buy any newspaper is: [Image of Venn diagram 3 sets]
(A) 20
(B) 80
(C) 0
(D) None of these
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11. 5 different balls are placed in 5 different boxes randomly. Find the probability that exactly two boxes remain empty. Given each box can hold any number of balls:
(A) \dfrac{24}{125}
(B) \dfrac{12}{25}
(C) \dfrac{96}{625}
(D) \dfrac{18}{25}
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12. A fair die is thrown up to 20 times. The probability that on the 10th throw, the fourth six appears is:
(A) \dfrac{84 \times 5^6}{6^{10}}
(B) \dfrac{112 \times 5^6}{6^{10}}
(C) \dfrac{84 \times 5^6}{6^{20}}
(D) None
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13. The chances of defective screws in three boxes A, B, C are \dfrac{1}{5}, \dfrac{1}{6}, \dfrac{1}{7} respectively. One box is selected at random and a screw drawn from it at random is found to be defective. Then the probability that it came from box A is:
(A) \dfrac{16}{29}
(B) \dfrac{1}{15}
(C) \dfrac{27}{59}
(D) \dfrac{42}{107}
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14. There are 10 stations between two cities A and B. A train is to stop at three of these 10 stations. The probability that no two of these three stations are consecutive is:
(A) \dfrac{7}{15}
(B) \dfrac{7}{12}
(C) \dfrac{7}{10}
(D) \dfrac{5}{7}
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15. A seven-digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability that the number so formed is divisible by 2 is:
(A) \dfrac{6}{7}
(B) \dfrac{1}{7}
(C) \dfrac{3}{7}
(D) \dfrac{4}{7}
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16. If two events A and B are such that P(A^C) = 0.3, P(B) = 0.4, and P(A \cap B^C) = 0.5, then the value of P\left[ \dfrac{B}{A \cup B^C} \right] is equal to:
(A) \dfrac{1}{4}
(B) \dfrac{1}{2}
(C) \dfrac{3}{4}
(D) \dfrac{3}{8}
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17. If the probability of A failing an examination is \dfrac{1}{5} and that of B is \dfrac{3}{10}, then the probability that either A or B fails is:
(A) \dfrac{11}{25}
(B) \dfrac{1}{2}
(C) \dfrac{19}{50}
(D) None of these
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18. Sixteen players P_1, P_2, P_3, \ldots, P_{16} play in a tournament. If they are grouped into eight pairs, then the probability that P_4 and P_9 are in different groups is equal to:
(A) \dfrac{7}{15}
(B) \dfrac{2}{15}
(C) \dfrac{14}{15}
(D) \dfrac{4}{15}
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19. Two dice are thrown together. The probability that the sum of the numbers appearing on them is 9, given that the number 5 always occurs on the first die, is:
(A) \dfrac{1}{3}
(B) \dfrac{2}{3}
(C) \dfrac{1}{6}
(D) \dfrac{1}{4}
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20. For k = 1, 2, 3, the box B_k contains k red balls and (k + 1) white balls. Let P(B_1) = \dfrac{1}{2}, P(B_2) = \dfrac{1}{3}, and P(B_3) = \dfrac{1}{6}, where P(B_k) represents the probability of selecting box B_k. A box is selected randomly and a ball is drawn. If it is known that the drawn ball is red, then the probability that it has come from box B_2 is:
(A) \dfrac{35}{78}
(B) \dfrac{14}{39}
(C) \dfrac{10}{13}
(D) \dfrac{12}{13}
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21. Total number of equivalence relations defined in the set S = \{a, b, c\} is:
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22. Given the relation R = \{(1, 2), (2, 3)\} on the set A = \{1, 2, 3\}, the minimum number of ordered pairs which, when added to R, make it an equivalence relation is:
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23. Number of ways in which 4 people can be selected out of 10 people sitting in a row such that exactly two are consecutive?
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24. There are 100 tickets numbered 00, 01, 02, \ldots, 99. One ticket is selected at random. Suppose A and B are the sum and product of the digits of the number found on the ticket, and P\left( \dfrac{A = 7}{B = 12} \right), then 2P = ?
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25. A die is thrown 31 times. If the probability of getting 2, 4, or 5 at most 15 times is P, then 2P = ?
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