Tiêu đề XII - maths - chapter 11 - PROBABILITY (114-124).pdf ✅

114 PROBABILITY NARAYANAGROUP JEE-MAIN SR.MATHS-VOL-II 1. 100 boys are randomly divided into two groups containing 50 boys each. The probability that the two tallest boys are in different groups is 1) 50/99 2) 49/99 3) 25/99 4) 1/2 2. Let x = 33n . The index n is given a positive integral value at random. The probability that the value of x will have 3 in the units place is 1) 1/4 2) 1/2 3) 1/3 4) 1/5 3. Consider all functions that can be defined from the set A = {1, 2, 3} to the set B = {1,2,3,4,5}. A function f (x) is selected at random from these functions. The probability that selected function satisfies f i f j ( ) ( )  for i < j is equal to 1) 6/25 2) 7/25 3) 2/5 4) 12/25 4. Three children are selected at random from a group of 6 boys and 4 girls. It is known that in this group exactly one girl and one boy belong to same parents. The probability that the selected group of children have no blood relations, is equal to 1) 11/15 2) 13/15 3) 14/15 4) 2/15 5. A baised ordinary die is loaded in such a way that probability of getting an even outcome is five times the probability of getting an odd outcome. This die is rolled two times. The probability that the sum of outcomes will be a prime number, is equal to 1) 67/324 2) 63/324 3) 123/324 4) 71/324 6. In a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersect at an interior point of the hexagon is 1) 5 12 2) 7 12 3) 1 26 4) 2 51 7. A sum of money is rounded off to the nearest rupee. The probability that error occurred in rounding off is at least 15 paise is 1) 29 101 2) 29 100 3) 71 101 4) 71 100 8. If the integers ‘m’ and ‘n’ are chosen at random from 1 to 100 then the probability that a number of the form m n 7 7  is divisible by 5 is 1) 1 5 2) 1 7 3) 1 4 4) 1 49 9. If a [ 6,12],   the probability that graph of y = –x2 + 2 (a + 4)x –(3a + 40) is strictly below x-axis is 1) 2 3 2) 1 3 3) 1 2 4) 1 4 10. Seven chits are numbered from 1 to 7. Four are drawn one by one with replacement. The probability that the least number on any selected chit is 5 is 1) 4 2 1 7        2) 4 2 4 7        3) 4 4 3 2 7 7              4) 4 3 7       11. If four whole numbers taken at random are multiplied together, then the probability that the last digit in the product is 1, 3, 7 or 9 is 1) 16 625 2) 32 625 3) 64 625 4) 256 675 12. A natural number x is chosen at random from the first 1000 natural numbers. If [ .] denotes the greatest integer function then the probability that 31 2 3 5 30       x x x x                is 1) 33/1000 2) 34/1000 3) 32/1000 4) 31/1000 13. Given two events A and B, if the odds against A are 2 to 1, and those in favour of A B  are 3 to 1, then 1)   1 1 3 2   P B 2)   1 3 2 4   P B 3)   5 3 12 4   P B 4) 0 1   P B  14. There are ninety cards in a box numbered 10,11,12,........98, 99. Three cards are drawn from the box one by one with replacement. The probability that product of the digits on the cards will be 12 at least once is 1) 3 4 1 5       2) 3 43 45       3) 3 43 1 45       4) 43 45 15. Every evening a student either watches TV or reads a book. The probability of watching TV is 4 5 . If he watches TV, the chance that he will fall a sleep is 3 4 and it is 1 4 when he read a book. On one evening, the student is found to be asleep. The probability that he watched TV is 1) 2 13 2) 12 13 3) 11 13 4) 1 13 LEVEL - III
115 PROBABILITY NARAYANAGROUP JEE-MAIN SR.MATHS-VOL-II 16.  is a solution of the equation   n n z z 1   where n n N   2, . Then the probability that  lies on the real axis is 1) 1 n 2) 2 n 3) 1 n 1  4) 2 n 1  17. A critical point x0 of the function f(x) = x3 is selected at random. The probability that f is extremum at x0 is 1) 1 3 2) 2 3 3) 0 4) 1 18. An ellipse of ecentricity 2 2 3 is inscribed in a circle and a point with in a circle is choosen at random. Then the probability that this point lies outside the ellipse is 1) 2 3 2) 8 9 3) 1 3 4) 2 5 19. Two persons A and B agree to meet at a place between 11 to 12 noon. The first one to arrive waits for 20 minutes and then leave. If the time of their arrival be independent and at random, The probability that A and B meet 1) 4 9 2) 5 9 3) 2 9 4) 1 9 20. Two numbers x and y are chosen at random (without replacement ) from amongest the numbers 1,2,3,.....,3n. Then the probability that 3 3 x y  is divisible by 3 is 1) 1 2 2) 1 3 3) 1 4 4) 1 5 21. Two non - negative integers are chosen at randon from the set of non negative integers with replacement. The probability that the sum of their squares is divisible by 10 is 1) 3 50 2) 6 50 3) 9 50 4) 12 50 22. Let F be the set of all 4 digited numbers whose sum is 34. If a number is selected from F, the probability that the selected number is even is 1) 1 10 2) 2 10 3) 3 10 4) 4 10 23. A fair coin is tossed 100 times. The probability of getting tails 1,3, ......, 49 times is 1) 1/2 2) 1/4 3) 1/8 4) 1/16 24. If two numbers selected from 1, 2, 3, ...., 25 and ‘P’ is the probability that difference between them is less than 10 then the value of ‘5P’ is 1) 1 2) 2 3) 3 4) 4 25. If a and b are chosen randomly from the set consisting of numbers 1,2,3,4,5,6 with replacement then the probability that   2/ 0 / 2 6 x x x x Lt a b        is 1) 1 3 2) 1 4 3) 1 9 4) 2 9 26. An experiment has 10 equally likely outcomes. Let A and B be two non empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is 1) 2, 4 or 8 2) 3, 6 or 9 3) 4 or 8 4) 5 or 10 27. In a test of MCQ with 4 choices a student either guess or remember or computes the answer. Probability of making a guess is 1/3, remembering is 1/6 and giving correct answer (when remembered) is 1/8, probability of computing correct answer is 1) 11 24 2) 23 24 3) 23 29 4) 24 29 28. The probabilities of Ramesh using car or scooter or bus or train for going to office are respectively 1 3 2 , , 7 7 7 and 1 7 . The probabilities of his reaching the office late using these modes of transport are respectively 2 1 4 , , 9 9 9 and 1 9 . On one day Ramesh reaches his office on time. Then the probability that he used car on that day is 1) 1 8 2) 1 7 3) 1 6 4) 2 5
116 PROBABILITY NARAYANAGROUP JEE-MAIN SR.MATHS-VOL-II 29. Three dice are thrown, the numbers appearing on them are respectively a, b and c. chance that the r oots of the quadr atic equation ax2 + bx + c = 0 are real is 1) 43 216 2) 23 216 3) 33 216 4) 53 216 30. The probability that the graph of   2 y x a x a       16 8 5 7 5 0, is strictly above the x-axis, If a  20,0 1) 13 20 2) 23 20 3) 3 20 4) 1 19 31. If p q, are chosen randomly with replacement from the set 1,2,3,....,10 the probabililty, that the roots of the equation 2 x px q    0 are real is 1) 3 5 2) 31 50 3) 61 100 4) 29 50 32. If 35 fruits are distributed among 3 patients, then the probability, that no one gets less than 10 is 1) 5 111 2) 7 111 3) 5 222 4) 7 222 33. Eight players 1 2 3 4 5 6 7 8 P P P P P P P P , , , , , , , participate in a boxing ring of a knock out and it is known that if pi and pj fight pi will win (i < j), assume players are paired at random probability P4 reaches finals is. 1) 7 35 2) 4 35 3) 1 2 4) 1 34. The decimal parts of the logarithms of two numbers taken at random are found to six places. what is the chance that the second can be substracted from the first without borrowing? 1) 6 1 2       2) 6 11 20       3) 6 11 19       4) 6 10 19       35. A three digited number is written down at random. The probability that it will have two and only two consecutive identical place values is 1) 2 81 2) 3 50 3) 9 50 4) 7 50 36. India plays two matches each with west indies and Australia. In any match, the probabilities. India getting points 0, 1 and 2 are 0.45, 0.05, 0.50 respectively. Assuming that the out are independent , the probability of india getting atleast 7 points is 1) 0.8750 2) 0.0875 3) 0.0625 4) 0.0 37. A box contain N coins, m of which are fair and rest are biased. the probability of getting a head when a fair coin is tossed is 1/2, while it is 2/3 when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. the probability that the coin drawn is fair if 1) 8 8 m N m 2) 8 m N m 3) 9 8 m N m 4) 9 8 N N m 38. Four numbers are multiplied together .Then the probability that the product will be divisible by 5 or 10 is———— 1) 369 625 2) 399 625 3) 125 625 4) 133 625 39. A signal which can be green or red with prob- ability 4 5       and 1 5       respectively is received by the station A and Transmitted to B. The probability each station receive signal cor- rectly is 3 4      . If the signal received by B is green, then the probability original signal was green is 1) 3/5 2) 6/7 3) 20/23 4) 9/20 40. Let w 1 be a cube root of unity. A fair die is rolled three times. If 1 2 r r, and 3r are the numbers obtained on the die, then the probability that 1 2 3 0 r r r w w w    is (AIEEE 2010) 1) 1/18 2) 1/9 3) 2/9 4) 1/36 41. A car is parked by a driver amongst 25 cars in a row, not at either end. When he returns he finds that 10 places are empty. The probability that both the neighbouring places of drivers car are vacant is 1) 9/92 2) 15/92 3) 21/92 4) 27/92
117 PROBABILITY NARAYANAGROUP JEE-MAIN SR.MATHS-VOL-II 42. If four dice are thrown together, then the probability that the sum of the numbers appearing on them is 13 is 1) 5 216 2) 11 216 3) 35 324 4) 11 432 LEVEL - III - KEY 1) 1 2) 1 3) 2 4) 3 5) 4 6) 1 7) 4 8) 3 9) 3 10) 3 11) 1 12)1 13) 3 14) 3 15) 2 16)3 17) 3 18) 1 19) 2 20) 2 21) 3 22) 3 23) 2 24) 3 25) 3 26) 4 27) 4 28) 2 29) 1 30) 1 31) 2 32) 4 33) 2 34) 2 35) 3 36) 2 37) 3 38) 1 39) 3 40) 3 41) 2 42) 3 LEVEL - III - HINTS 1. n (S) = 100! 50!50!2! n (A) = 2.98! 49!49!2! P(A) = 50 99 2. x = 33n ; only four digits may be at the units place i.e., 1, 3, 7, 9; n (S) = 4 Let A be the event of getting 3 at its units place n (A) = 1 ; P(A) = 1/4 3. n (S) = 53 = 125 A function f : A B such that f f f (1) (2) (3)   falls in one of the following four categories Case 1 : f (1) < f (2) < f (3) No. of trails in this case 5C3 Case 2 : f (1) = f (2) < f (3) No. of trails in this case 5C2 Case 3 : f (1) < f (2) = f (3) No. of trails in this case 5C2 Case 4 : f (1) = f (2) = f (3) No. of trails in this case 5C1 No.of desired trails n (E) = 35 ; P (E) = 7/25 4. { (B1 , G1 ) (B2 , G2 ) (B3 , G3 ) (B4 , G4 ) B5 , B6 } Req. Probability = 4 1 10 3 8 4 1 1 15 C C     1 1 1 5  5. Let A be the event of getting even no and B be the event of gettin an odd number on a die. P(A) = 5 18 P(B) = 1 18 P (2 or 3 or 5 or 7 or 11) = 1 1 1 5 1 5 1 5 1 5 . 2 . 4 . 6 . 2 . 18 18 18 18 18 18 18 18 18 18     71 324  6. 6 c2 – 4 c2 = 9; n(S) = 9 c2 = 36, no. of selections of four vertices = 6c4 = 15; P(E) = 5 12 7. Error is at least 15 paise when we round off 15P, 16P.........49P, 50P, 51P.......85P, P (E) = 71 100 8. 1 2 3 4 7 7, 7 49,7 343,7 2401     From each of value of m in 1,2,3,4,.....,100 there are 25 values of n Req. prob = 100 25 1 100 100 4    9. The total length of the interval = 12 – (– 6)=18. If graph of y = –x2 + 2 (a + 4)x – (3a+40) is entirely below x-axis, the value of discriminant of the above quadratic expression must be negative a < 0 D < 0 f (x) <0 X -axis 4 (a + 4)2 – 4(–1) (-(3a+40))< 0 a2 + 5a – 24 < 0(a + 8) (a – 3) < 0-8 < a < 3 but a[–6,12] –6 < a < 3 for event to happen length of interval = 3 – ( – 6) = 9 Hence of required probability = 9 1/ 2 18  10. 4 4 4 n(s)  7 , n(A)  3  2 11. 4 4 n(s) 10 , n(A)  4 12. ‘x’ should be a multiple of the L.C.M of 2,3,5 (i.e., 30)  n (A) = 33 13. P A : P A 2 :1      ; P A B : P A B 3 :1        P A B P A P B P A B                5 3 P B 12 4   14. 3 86 ( ) 1 90 P A        3 43 1 45       