[STRAIGHT OBJECTIVE TYPE] log 4 2 x 4 log. (sin x + cos x) = 10 (A) 24 (B) 36 (C) 20 (D) 12
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1 [STRAIGHT OBJECTIVE TYPE] Q. The equation, ( ) / (A) eactly one real solution (B) two real solutions (C) real solutions (D) no solution. = has : ( n) Q. If 0 sin + 0 cos = and 0 (sin + cos ) = 0 then the value of 'n' is (A) 4 (B) 6 (C) 0 (D) Q. Number of zeros after decimal before a significant figure in (75) 0 is : (use 0 = 0.0 & 0 = 0.477) (A) 0 (B) 9 (C) 8 (D) none 0 Q.4 Number of real solution(s) of the equation = is : (A) eactly four (B) eactly three (C) eactly two (D) eactly one Q.5 The set of all real numbers for which is defined as { > c}. The value of c is (A) 0 (B) (00) 00 (C) (00) 004 (D) (00) 00 Q.6 The number of positive integers, when taken with base 5, have characteristic, is (A) 486 (B) 4 (C) 5 (D) 500 Q.7 Solution set of the inequality, ( + ) 0 is : (A) [ 4, ] (B) [ 4, ) (0, ] (C) (, ) (, ) (D) (, 4) [, ) Q.8 The set of values of satisfying simultaneously the inequalities > 0 is ( 8) ( ) 0 and (A) a unit set (B) an empty set (C) an infinite set (D) a set consisting of eactly two elements. Q.9 Number of ordered pair(s) (, y) simultaneously satisfying the system of equations y y = and = 4 y is/are (A) (B) (C) (D) none 0. Q.0 Let and y be positive numbers such that y y and y 6,then the value of (A) 4 (B) (C) (D) 4 / Q. Let S be the set of ordered triples (, y, z) of real numbers for which 0 ( + y) = z and 0 ( + y ) = z +. Suppose there are real numbers a and b such that for all ordered triples (, y, z) in S we have + y = a 0 z + b 0 z. The value of (a + b) is equal to 5 (A) (B) 9 (C) 5 (D) 4
2 Q. Suppose n be an integer greater than, let a n = c = a 0 + a + a + a + a 4. Then (b c) equals (A) 00 (B) 00 n. Suppose b = a 00 + a + a 4 + a 5 and (C) (D) p p p a b c b c a Q. The epression a p b p c p b c a, wherever defined, simplifies to (A) (B) (C) (D) 4 Q.4 If (sin ) (cos ) ( tan ) ( + tan ) =, then tan is equal to (A) (B) (C) (D) Q.5 Let N = 0 where base of the arithm is 0. The characteristic of the arithm of N to the base, is equal to (A) (B) (C) 4 (D) 5 Q.6 The number of values of satisfying the equation is (A) 0 (B) (C) (D) Q.7 The number of solutions of the equation ln e e ln sin 7 sin cos 7 7 (A) 0 (B) (C) (D) is equal to y Q.8 If ( 0, y 0 ) satisfies the simultaneous equations + y = and then value of ( 0 + y 0 ) is equal to (A) 8 (B) (C) 4 (D) 8 Q.9 Let 'a' denotes the arithm of 0. to the base 0., 'b' denotes the arithm of 4 to the base 8 and 'c' denotes the number whose arithm to the base 0.64 is minus. Then the value of ab c, is (A) lies between 4 and 5. (B) odd composite. (C) odd prime. (D) lies in (, ) Q.0 Number of ordered pair(s) of (, y) satisfying the system of equations, y = 5 and / y = is : (A) one (B) two (C) three (D) four
3 Q. A line = k intersects the graph of y = 5 and the graph of y = 5 ( + 4). The distance between the points of intersection is 0.5. Given k = a b, where a and b are integers, the value of (a + b) is (A) 5 (B) 6 (C) 7 (D) 0 Q. If P is the number of natural numbers whose arithms to the base 0 have the characteristic p and Q is the number of natural numbers arithms of whose reciprocals to the base 0 have the characteristic q then 0 P 0 Q has the value equal to (A) p q + (B) p q (C) p + q (D) p q Q. The solution set of the inequality ( ) + > 0 is (A) [, ] (B) [, 4) (C) (, ] (D) (, 4] [COMPREHENSION TYPE] Paragraph for question nos. 4 to 6 M N = +, where is an integer & [0, ) Q.4 If M & are prime & + M = 7 then the greatest integral value of N is (A) 64 (B) 6 (C) 5 (D) 4 Q.5 If M & are twin prime & + M = 8 then the greatest integral value of N is (A) 64 (B) 65 (C) 78 (D) 79 Q.6 If M & are relative prime & + M = 7 then minimum integral value of N is (A) 5 (B) (C) 6 (D) 8 [MULTIPLE OBJECTIVE TYPE] Q.7 Which of the following real numbers is(are) non-positive? (A) (B) (D) (C) Q.8 If L = r 7 7 r r 0, M = r r and N = r 0 r where p = ( ), then (A) L + M = (B) M + N = 0 (C) L M + N = 6 (D) LMN = 0 r p Q.9 If = 7, then the value of can be equal to : (A) 0 (B) (C) (D) 6 5 7
4 Q.0 The value of satisfying the equation (A) a prime number (C) an even number =, is (B) a composite number (D) an odd number Q. Let L be the number of digits in 40 before decimal and M be the number of zeroes in 40 after decimal before a significant digit, then (A) L + M = 9 (B) L M = (C) L M = (D) L + M = 8 (Use: 0 = 0.477) Q. If =, v = and v 0. = w, then (A) + v + w = 90 (B) + v w = (C) + v + w = 9 (D) + v w = Q. If 4 4, then has (A) one integral solution (C) two irrational solutions 5 (B) two rational solutions (D) no prime solution Q.4 Which of the following vanishes? (A) tan º. tan º. tan º... tan 89º (B) sin º. sin º. sin º... sin 90º (C) (D) tan º + tan º + tan º tan 89º Q.5 The equation 4 ( ) = 0 has : (A) one rational solution (B) one integral solution (C) two real solutions (D) one irrational solution. Q.6 If y = 7 a ( + + a + ) is defined R, then possible integral value(s) of a is/are (A) (B) (C) 4 (D) 5 [MATCH THE COLUMN] Q.7 Column-I Column-II (A) Anti arithm of 0.6 to the base 7 has the value equal to (P) 5 (B) Characteristic of the arithm of 008 to the base is (C) The value of b satisfying the equation, (Q) 7 e b 65 = 0 6 e 0 is (D) Number of naughts after decimal before a significant figure (R) 9 5 comes in the number 6 00, is (S) 0
5 Q.8 Column-I Column-II (A) a If b = 4 and 7, b (P) then the value of (a b 4 ) is equal to (Q) (B) If number of digits in is 'd', and number of cyphers after (R) 6 decimal before a significant figure starts in (0.) 9 is 'c', then (d c) is equal to (C) If N = anti anti ( 96), (S) 6 5 then the characteristic of N to the base, is equal to 5 Q.9 Column-I Column-II (A) If a b then (a + b ) is greater than = 0, (P) (B) If then is coprime with (Q) (C) If p =, then the value of (p + p + ) is less than (R) (D) If 9 =, (S) 4 then the value of is twin prime with (T) 5 [SUBJECTIVE] Q.40 Let N be the number of integers whose arithms to the base 0 have the characteristic 5, and M the number of integers the arithms to the base 0 of whose reciprocals have the characteristic 4. Find ( 0 N 0 M). Q.4 Find the sum of all possible values of satisfying simultaneous the equations = ( ) 4 and (00 ) + (0 ) = 4 +. [Note : Assume base of arithm is 0.] Q.4 If = a, = b then find the value of ab. a b Q.4 If 4 4 7, then find the value of. Q.44 Find the number of integral values of satisfying the equation 4 4 = 9
6 Q.45 If = satisfies the equation 6 ( + ) 6 ( ) =, then find the value of. Q.46 Given ( a) + ( a ) + ( + b ) = (a >, b R) c = d = 4 0 Then find the value of (a + b + c d) Q.47 If sum of the integral values of satisfying the equation is N, then find characteristic of arithm of N to the base 5. Q.48 Let k be the unique positive value satisfying the equation ( 4k) (9k) = 0, then find the value of (7 k). Q.49 If 8 and 4 54, then find the value of. Q.50 Given a = p, 4 b = p and (8) = c p. If c = (p p p + ) ab where N, then find the value of ( ). ANSWER KEY Q. D Q. D Q. C Q.4 B Q.5 B Q.6 D Q.7 B Q.8 A Q.9 A Q.0 A Q. B Q. C Q. A Q.4 D Q.5 B Q.6 A Q.7 C Q.8 D Q.9 D Q.0 B Q. B Q. A Q. B Q.4 D Q.5 C Q.6 C Q.7 A, C, D Q.8 A, B, D Q.9 A, B, C, D Q.0 A, C Q. A, C Q. B, C Q. A, B Q.4 A, B, C, D Q.5 B, C Q.6 B, C, D Q.7 (A) R; (B) S; (C) P; (D) Q Q.8 (A) S, (B) R, (C) Q Q.9 (A) P, Q, R, S ; (B) P, R; (C) S, T; (D) T Q.40 Q.4 0 Q.4 5 Q.4 4 Q.44 Q.45 4 Q.46 Q.47 4 Q Q.49 5 Q
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