MATHEMATICS 61. If letters of the word KUBER are written in all possible orders and arranged as in a dictionary, then rank of the word KUBER will be: (A) 67 (B) 68 (C) 65 (D) 69 : Alphabetical order of these letters is B, E, K, R, U. Total words starting with B = 4! = 24 Total words starting with E = 4! = 24 Total words starting with KB = 3! = 6 Total words starting with KE = 3! = 6 Total words starting with K = 3! = 6 Next word will be KUBER. Thus rank of the word KUBER = 24 + 24 + 18 + 1 = 67. 62. There are 20 persons among whom are two brothers. The number of ways in which we can arrange them around a circle so that there is exactly one person between the brothers is (A) 19! (B) 2 18! (C) 2! 17! (D) none of these : We can arrange 18 persons around a circle in ( 18-1)! = 17! Ways. Now, there are exactly 18 places where we can arrange the two brothers. Also, the two brothers can be arranged in 2! Ways. Thus, the number of ways of arranging the persons subject to the given condition is (17!)(18)(2!) = 2(18!). Hence (B) is the correct answer.
63. Total number of 4 digit number that are greater than 3000, that can be formed using the digits 1, 2, 3, 4, 5, 6 (no digit is being repeated in any number) is equal to: : (A) 120 (B) 240 (C) 480 (D) 80 Let the formed number is x 1 x 2 x 3 x 4 Clearly, x 1 > 3. Thus total number of such numbers = 4.5.4.3 = 240 64. Two teams are to play a series of 5 matches between them. A match ends in a win or loss or draw for a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecasts correctly for all the matches will contain n people, where n is (A) 81 (B) 243 (C) 486 (D) none of these : The smallest number of people = total number of possible forecasts = total number of possible results = 3 3 3 3 3 = 243. Hence (B) is the correct answer. 65. The number of ways in which 6 men can be arranged in a row so that three particular men are consecutive, is (A) 4 P4 (B) 4 P4 3 P3 (C) 6 P6 3 P3 (D) 3 P3 3 P3 (B)Considering three particular persons as a single group. Number of ways in which these four can be arranged in a row is 4 P 4. Those three can arrange themselves in 3 P 3 ways. So total number of ways = 4 P 4 3 P 3.
66. The total number of three digit numbers, the sum of whose digits is even, is equal to: (A) 450 (B) 350 (C) 250 (D) 325 : Let the number of n = x 1, x 2, x 3. Since x 1 + x 2 + x 3 is even. That means there are following cases : (i) (ii) x 1, x 2, x 3 all are even 4. 5. 5 = 100 ways. x 1 even and x 2, x 3 are odd 4.5.5. = 100 ways (iii) x 1 odd, x 2 even, x 3 odd 5.5.5 = 125 ways (iv) x 1 odd, x 2 even, x 3 odd 5.5.5. = 125 ways 67. Number of ways 6 different flowers can be given to 10 girls, if each can receive any number of flowers is (A) 6 10 (B) 10 6 (C) 60 (D) 10 C6 : (B) Number of ways 10 6. 68. The value of log (0.1 0.01 0.001...) ( 0.05) 20 is 1 (a) 81 (b) 81 1 (c) 20 (d) 20 :. /. /. /
69. If log.04 ( x 1) log0. 2( x 0 1) then x belongs to the interval (a) 1, 2 (b), 2 (c) 2, (d) None of these : log log log log log log 70. If x log b a, y log c b, z log a c, then xyz is : (a) 0 (b) 1 (c) 3 (d) None of these 71. There are 2 identical white balls, 3 identical red balls and 4 green balls of different shades. The number of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the same colour, is: (a) 6 (b) 7 (c) (d) none : 72.. /. /. /. /. / when simplified reduces to: (a) sin x cos x (b) sin (c) sin x cos x (d) sin cos 73. is equal to: Soln: (a) 1 (b) 2 (c) 3/4 (d) none 74. If sin 2 = k, then the value of is equal to (a) (b) (c) (d) Soln: tan cos cot sin
75. If x R, the numbers,, form an A P then a must lie in the interval: (a) [1, 5] (b) [2, 5] (c) [5, 12] (d) [12, ), So the least value of t is 2 Now = 76. One side of an equilateral triangle is 24 cm. The mid-points of its sides are joined to form another triangle whose mid - points are in turn joined to form still another triangle. This process continues indefinitely. Then the sum of the perimeters of all the triangles is (a) 144 cm (b) 212 cm (c) 288 cm (d) none of these 77. If.... upto, then +..... = (a) (b) (c) (d) none of these.... upto. upto /. upto /. upto /. upto /. upto / 78. Suppose A 1, A2, A3,..., A30 are thirty sets each having 5 elements and B 1, B 2,..., Bn are n sets each with 3 elements. Let exactly 10 of the 30 n A i B i 1 j 1 B j ' ' A i s and exactly 9 of the s j = S and each elements of S belongs to. Then n is equal to Soln: (a) 15 (b) 3 (c) 45 (d) None of these 79. If A [ x : f( x) 0] and B [ x : g( x) 0], then A B will be 2 2 (a) [ f ( x)] [ g( x)] 0 (b) f ( x) g( x) (c) ( x) f( x) g (d) None of these
80. Number of solutions of log log is (a) 3 (b) 1 (c) 2 (d) 0 log log log log log log But 2 is not allowed Section- B: Multiple Correct Type: Question No. (81-90): This section contains 10 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE or MORE is correct. 81. Which of the following when simplified, vanishes? (a) (b) log. / log. / (c) log log log (d) log cot log cot log cot log cot 82. Which of the following statement(s) is/are true? (a) log lies between (b) log cos (c) is smaller than 1 (d) log log log ( ) log. ( )/
83. If 2 cos sin, then the value of 4 cos sin is equal to (a) 3 (b) (c) (d) 84. If x = sec tan & y = cosec cot then : (a) x = (b) y = (c) x = (d) xy + x y + 1 = 0
85. If the sides of a right angled triangle are *cos cos cos+ and *sin sin sin+, then the length of the hypotenuse is : (a) 2, cos- (b), cos- (c) 4 cos (d) 4sin 86. Choose the INCORRECT statement(s). (a) sin 82 cos and sin 127 sin97 have the same value. (b) If tan A = & tan B = then tan (A B) must be irrational. (c) The sign of the product sin sin sin is positive. (d) There exists a value of between 0 & 2 which satisfies the equation; sin sin
87. If a, b, c are in H.P., then: (a),, are in H.P. (b) (c) a,, c are in G.P. (d),, are in H.P. :,,,,,,,,, 88. If b, b, b (b ) are three successive terms of a G.P. with common ratio r, the value of r for which the inequality b b b holds is given by (a) r > 3 (b) r < 1 (c) r = 3.5 (d) r = 5.2
89. If f(x) = cos 0 1 sin 0 1, [x] denoting the greatest integer function, then (a) f(0) = 1 (b) f. / It becomes cos4x+sin(-5x) (c) f. / (d) f=0 90. A function f from the set of natural numbers to integers defined by, f (n), when n is odd {, when n is even is: (a) one-one (b) many-one (c) onto (d) into If n is odd we get whole numbers If n is even we get negative integers