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1. A worker suffers a 20% cut in wages. He regains his original pay by obtaining a rise of
(A) 20% (B) 22.50% (C) 25% (D) 27.50 %
2. If $$\mathbf {m}$$ men can do a job in $$\mathbf {d}$$ days , then the number of days in which $$\mathbf {m+r}$$ men can do the job is
(A) $$\mathbf {d+r}$$;
(B) $$\mathbf {{\frac{d}{m}}(m+r)}$$;
(C) $$\mathbf {\frac {d}{m+r}}$$;
(D) $$\mathbf {\dfrac{md}{m+r}}$$.
3. A boy walks from his home to school at 6 km per hour (kmph) . He walks back at 2 kmph. His average speed in kmph, is
(A) 3; (B) 4; (C) 5; (D) ${\sqrt{12}}$.
4. A car travels from P to Q at 30 kilometers per hour (kmph) and returns from Q to P at 40 kmph by the same route. Its average speed, in kmph, is nearest to
(A) 33; (B) 34; (C) 35; (D) 36.
5. A man invests Rs. 10,000 for a year. Of this Rs. 4,000 is invested at the interest rate of 5% per year, Rs. 3,500 at 4% per year and the rest at ${\alpha}$ % per year. His total interest for the year is RS. 500. Then ${\alpha}$ equals
(A) 6.2; (B) 6.3; (C) 6.4; (D) 6.5.
6. Let ${\mathbf {x_{1}, x_{2}, x_{3}}}$ be positive integers such that ${\mathbf {x_i + x_{i+1} = k}}$ for all ${\bf {i}}$, where ${k}$ is a constant. If ${\bf{x_{10}=1}}$, then the value of ${x_{1}}$ is
(A) ${k}$; (B) ${k_{1}}$; (C) ${k+1}$; (D) 1.
7. If ${\bf{a_0 = 1}}$ , ${\bf{a_1 = 1}}$ and ${\bf{a_n = a_{n-1}a_{n-2} +1}}$ for ${\bf{n>1}}$, then
(A) ${\bf{a_{465}}}$ is odd and ${\bf{a_{466}}}$ is even;
(B) ${\bf{a_{465}}}$ is odd and ${\bf{a_{466}}}$ is odd;
(C) ${\bf{a_{465}}}$ is even and ${\bf{a_{466}}}$ is even;
(D) ${\bf{a_{465}}}$ is even and ${\bf{a_{466}}}$ is odd.
8. Two trains of equal length L, travelling at speeds $${v_1}$$ and $${v_2}$$ miles per hour in opposite directions, take T seconds to cross each other. Then L in feet ( 1 mile = 52800 feet ) is
(A) $${\frac {11T}{15(V_1 + V_2)}}$$;
(B) $${\frac {15T}{11(V_1 + V_2)}}$$;
(C) $${\frac {11(V_1 + V_2)T}{15}}$$;
(D) $${\frac {11(V_1 + V_2)}{15T}}$$.
9. A salesman sold two pipes at Rs. 12 each. His profit on one was 20% and the loss on the other was 20%. Then on the whole, he
(A) lost Rs. 1;
(B) gained Rs. 1;
(C) neither gained nor lost;
(D) lost Rs. 2.
10. The value of ${\bf{(256)^{0.16}(16)^{0.18}}}$ is
(A) 4;
(B) 16;
(C) 64;
(D) 256.25.
11. The digit in the unit position of the integer 1! + 2! + 3! + . . . + 99! is
(A) 3;
(B) 0;
(C) 1;
(D) 7.
12. July 3, 1977, was a SUNDAY. Then July 3, 1970, was a
(A) Wednesday
(B) Friday
(C) Sunday
(D) Tuesday.
13. June 10, 1979, was a SUNDAY. Then May 10, 1972, was a
(A) Wednesday;
(B) Thursday;
(C) Tuesday;
(D) Friday.
14. A man started from home at 14:30 hours and drove to a village, arriving there when the village clock indicated 15:15 hours. After staying for 25 minutes (min), he drove back by a different route of length (5/4) times the first route at a rate twice as fast, reaching home at 16:00 hours. As compared to the clock at home, the village clock is
(A) 10 min slow; (B) 5 min slow; (C) 5 min fast; (D) 20 min fast.
15. If ${\frac {a+b}{b+c}}$ = ${\frac{c+d}{d+a}}$, then
(A) ${a=c}$;
(B) either ${a=c}$ or ${a+b+c+d = 0}$;
(C) ${a+b+c+d = 0}$;
(D) ${a=c}$ and ${b = d}$
16. The expression ${(1 + q)(1 + q^2)(1 + q^4)(1 + q^8)(1 + q^{16})(1 + q^{32})(1 + q^{64})}$,
Where ${q\ne 1}$, equals
(A) $${\dfrac {1-q^{128}}{1-q}}$$
(B) $${\dfrac {1-q^{64}}{1-q}}$$
(C) $${\dfrac {1-q^(2)^{1+2+…+6}}{1-q}}$$
(D) none of the foregoing expressions.
17. In an election 10% of the voters on the voters’ list did not cast their votes and 60 voters cast their ballot papers blank. There were only two candidates. The winner was supported by 47% of all voters in the list and he got 308 votes more than his rival. The number of voters on the list was
(A) 3600; (B) 6200; (C) 4575; (D) 60.
18. A student took five papers in an examination, where the full marks were the same for each paper. His marks in these papers were in the proportion of 6:7:8:9:10. He obtained (3/5) part of the total full marks. Then the number of papers in which he got more than 50% marks is
(A) 2; (B) 3; (C) 4; (D) 5.
19. Two contestants run a 3-kilometre race along a circular course of length 300 metres. If there speeds are in the ratio of 4:3, how often and where would the winner pass the other? (The initial start-off is not counted as passing.)(A) 4 times; at the starting point.
(B) Twice; at the starting point.
(C) Twice; at a distance of 225 metres from the starting point.
(D) Twice; once at 75 metres and again at 225 metres from the starting point.
20. If ${a, b, c}$ and ${d}$ satisfy the equations${a + 7b + 3c + 5d = 0}$,
${8a + 4b + 6c + 2d = -16}$,
${2a + 6b + 4c + 8d = 16}$,
${5a + 3b + 7c + d = -16}$,then ${(a+b)(b+c)}$ equals
(A) 16; (B) -16; (C) 0; (D) none of the foregoing numbers.
21. Suppose ${x}$ and ${y}$ are positive integers, ${x > y}$, and ${3x + 2y}$ and ${2x + 3y}$ when divided by 5, leave reminders 2 and 3 respectively. It follows that when ${x - y}$ is divided by 5, the remainder necessarily equals
(A) 2; (B) 1; (C) 4; (D) none of the foregoing numbers.
22. The number of different solutions ${(x,y,z)}$ of the equation ${(x+y+z = 10)}$, where each of ${x,y}$ and ${z}$ is a positive integer, is(A) 36; (B) 121; (C) ${10^3-10}$; (D) ${\dbinom{10}{3}}$${\dbinom{10}{2}}$
23. The hands of a clock are observed continuously from 12:45 p.m. onwards. They will be observed to point in the same direction some time between
(A) 1:03 p.m. and 1:04 p.m. ; (B) 1:04 p.m. and 1:05 p.m. ;
(C) 1:05 p.m. and 1:06 p.m. ; (D) 1:06 p.m. and 1:07 p.m.
24. A, B and C are three commodities. A packet containing 5 pieces of A, 3 of B and 7 of C costs Rs. 24.50. A packet containing 2, 1 and 3 of A, B and C respectively, costs Rs. 17.00. The cost of a packet containing 16, 9 and 23 items of A, B and C respectively
(A) is Rs. 55.00; (B) is Rs. 75.50;
(C) is Rs. 100.00; (D) cannot be determined from the given information.
25. Four statements are given below regarding elements and subjects of the set {1,2, {1,2,3}}. Only one of them is correct. Which one is it?
(A) {1,2} ${\in}$ {1,2, {1,2,3}}
(B) {1,2} ${\subseteq}$ {1,2, {1,2,3}}
(C) {1,2,3} ${\subseteq}$ {1,2, {1,2,3}}
(D) 3 ${\in}$ {1,2, {1,2,3}}
26. A collection of non-empty subsets of the set {1, 2, . . . , n} is called a simplex if, whenever a subset S is included in the collection, any non empty subset T of S is also included in the collection. Only one of the following collections of subset of { 1, 2, . . . , n } is a simplex. Which one is it?
(A) The collection of all subsets S with the property that one belongs to S;
(B) The collection of all subsets having exactly 4 elements;
(C) The collection of all non-empty subsets which do not contain any even number;
(D) The collection of all non-empty subsets except for the subset {1}.
27. S is the set whose elements are zero and all even integers, positive and negative. Consider the five operations : [1] addition; [2] subtraction; [3] multiplication; [4] division; and [5] finding the arithmetic mean. Which of these operations when applied to any pair of elements of S, yield only elements of S ?
(A) [1], [2], [3], [4]. (B) [1], [2], [3], [5].
(C) [1], [3], [5]. (D) [1], [2], [3].
28. If X= {1,2,3,4}, Y= {2,3,5,7}, Z= {3,6,8,9}, W= {2,4,8,10}, then
(X ${\triangle}$ Y) ${\triangle}$ (Z ${\triangle}$ W) is
(A) {4,8}; (B) {1,5,6,10}; (C) {1,2,3,5,6,7,9,10}; (D) none of the foregoing sets.
29. If X,Y,Z are any three sets of numbers, then the set of all numbers which belong to exactly two of the sets X,Y,Z is
(A) (X ${\cap}$ Y) ${\cup}$ (Y ${\cap}$ Z) ${\cup}$ (Z ${\cap}$ X);
(B) [(X ${\cup}$ Y) ${\cup}$ Z] – [X ${\triangle}$ Y) ${\triangle}$ Z];
(C)(X ${\triangle}$ Y) ${\cup}$ (Y ${\triangle}$ Z) ${\cup}$ (Z ${\triangle}$ X);
(D) not necessarily any of (A) to (C).
30. For any three sets of P,Q and R, s is an element of (P ${\triangle}$ Q) ${\triangle}$ R if s is in
(A) exactly one of P,Q and R;
(B) at least one of P,Q and R, but not in all three of them at the same time;
(C) exactly two of P,Q and R;
(D) exactly one of P,Q and R or in all the three of them.
31. Let X= {1,2,3,..,10} and P= {1,2,3,4,5}. the number of subsets Q of X such that P ${\triangle}$ Q = {3} is
(A) $2^4-1$; (B) $2^4$; (C) $2^5$; (D) 1.
32. For each positive integer n, consider the set $P_n$ = {1,2,3,..,n}.
Let $Q_1 = P_1$, $Q_2 = P_2$ ${\triangle}$ ${Q_1}$ = {2}, and, in general, $Q_{n+1} = P_{n+1}$ ${\triangle}$ $Q_n$ for n ${\ge}$ 1. Then the number of element in $Q_{2k}$ is
(A) 1; (B) 2k-2; (C) 2k-3; (D) k.
33. For any two sets S and T is defined as the set of all elements that belong to either S or T but not both, that is S ${\Delta}$ T = ( S ${\cup}$ T) – (S ${\cap}$ T). Let A,B and C be sets such that A ${\cap}$ B ${\cap}$ C = ${\Phi}$, and the number of the elements in each of A ${\Delta}$ B, B ${\Delta}$ C and C ${\Delta}$ A equals 100.
Then the number of elements in A ${\cup}$ B ${\cup}$ C equals
(A) 150 (B) 300 (C) 230 (D) 210
34. Let A,B,C and D be finite sets such that ${\mid}$A${\mid}$ < ${\mid}$C${\mid}$ and ${\mid}$B${\mid}$ = ${\mid}$D${\mid}$, where ${\mid}$A${\mid}$ stands for the number of elements in the set A. Then
(A) ${\mid}$A ${\cup}$ B${\mid}$ < ${\mid}$C ${\cup}$ D${\mid}$
(B) ${\mid}$A ${\cup}$ B${\mid}$ ${\le}$ ${\mid}$C ${\cup}$ D${\mid}$ but ${\mid}$A ${\cup}$ B${\mid}$ < ${\mid}$C ${\cup}$ D${\mid}$
(C) ${\mid}$A ${\cup}$ B${\mid}$ < 2${\mid}$C ${\cup}$D ${\mid}$ but ${\mid}$A ${\cup}$ B${\mid}$ ${\le}$ ${\mid}$C ${\cup}$ D${\mid}$
(D) none of the foregoing statements is true.
35. For all subsets A and B of a set X, define the set A*B = (A ${\cap}$ B) ${\cup}$ ((X – A) ${\cap}$ (X-B)).
Then only one of the following statements is true. Which one is it?
(A) A*(X-B) ${\subset}$ A*B and A*(X-B) ${\ne}$ A*B;
(B) A*B = A*(X-B);
(C) A*B ${\subset}$ A*B ${\ne}$ A*(X-B);
(D) X-(A*B) = A*(X-B)
36. Suppose that, A,B,C are sets satisfying (A-B)${\Delta}$(B-C)= A${\Delta}$B. which of the following statements must be true?
(A) A=C (B) A${\cap}$B = B${\cap}$C; (C) A${\cup}$B = B${\cup}$C (D) none of the foregoing statements necessarily true.
37. If ${L_1}$ = {${\alpha^n : n = 0,1,2.....}$} and ${L_2}$ = {${\beta^n : n = 0,1,2.....}$}, then ${L_1}$.${L_2}$ is
(A) ${L_1}$ ${\bigcup}$ ${L_2}$
(B) the language consisting of all words;
(C) { ${\alpha^n}$${\beta^m : n = 0,1,2,...... m = 0,1,2,.....}$}
(D) { ${\alpha^n}$${\beta^n : n = 0,1,2,.....}$}
38. Suppose l is a language which contains the empty word and has the property that whenever P is in L, the word {${\alpha}$.P.{${\beta}$ is also in L. The smallest such L is
(A) { ${\alpha^n}$${\beta^m : n = 0,1,2,...... m = 0,1,2,.....}$}
(B) { ${\alpha^n}$${\beta^n : n = 0,1,2,.....}$}
(C) { ${(\alpha}$${\beta)^n : n = 0,1,2,.....}$}
(D) the language consisting of all possible words.
39. Suppose L is a language which contains the empty word, the word ${\alpha}$ and the word ${\beta}$ and has the property that whenever P and Q are in L, the word P.Q is also in L. The smallest such L is
(A) the language consisting of all possible words;
(B) { ${\alpha^n}$${\beta^n : n = 0,1,2,.....}$}
(C) the language containing precisely the words of the form
${\alpha^n_1}$${\beta^n_1}$ ${\alpha^n_2}$${\beta^n_2}$ …. ${\alpha^n_k}$ ${\beta^n_k}$ ,
where k is any positive integer and ${n_1, n_2,...,n_k}$ are nonnegative integers;
(D) none of the foregoing languages.
40. A relation denoted by ${\gets}$ is defined as follows: For real number x,y,z and w, say that “(x,y) ${\gets}$ (z,w)” if either (i) x < z or (ii) x = z and y > w. If (x,y) ${gets}$ (z,w) and (z,w) ${gets}$ (r,s) then which one of the following is always true?
(A) (y,x) ${\gets}$ (r,s)
(B) (y,x) ${\gets}$ (s,r)
(C) (x,y) ${\gets}$ (s,r)
(D) (x,y) ${\gets}$ (r,s)
41. A subset W of the set of all real numbers is called a ring if the following two conditions are satisfied:
(i) 1 ${in}$ W and
(ii) if a,b ${\in}$ W then a – b ${\in}$ W and ab ${\in}$ W.
Let $$S= \frac{m}{2^n} \mid$$ m and n are integers}
(A) neither S nor T is a ring;
(B) S is a ring and T is not;
(C) T is a ring and S is not;
(D) both S and T are rings.
42. For a real number a, define ${a^+} = max{a,0}$ . for example ${2^+}$ = 2, ${(-3)^+}$ = 0. Then for two real numbers a and b, the equality $(ab)^{+} = (a^{+})(b^{+})$ holds if and only if
(A) both a and b are positive;(B) a and b have the same sign;(C) a=b=0;(D) at least one of a and b is greater than or equals to 0.
43. For any real number x, let [x] denote the largest integer less than or equal to x and <x> = x – [x], that is, the fractional part of x. For arbitrary real numbers x,y and z only one of the following statement is correct. Which one is it?
(A) [x+y+z] = [x]+[y]+[z]
(B) [x+y+z] = [x+y] + [z] = [x] + [y+z] = [x+z] + [y]
(C) < x+y+z > = y+z – [y+z]+ <x>
(D) [x+y+z] = [x+y] + [z+ <y+x>].
44. Suppose that ${x_1}$, …. ${x_n}$ (n>2) are real numbers such that ${x_i}$ = ${-x_{n-i+1}}$ for 1 ${\le}$ i ${\le}$ n. Consider the sum S = ${\sum}$ ${\sum}$ ${\sum}$ ${{x_i}{x_j}{x_k}}$ , where the summations are taken over all i,j,k : 1 ${\le}$ i,j,k ${\le}$ n and i,j,k are all distinct. Then S equals
(A) n! ${x_1}$, ${x_2}$ …. ${x_n}$
(B) (n-3)(n-4)
(C) (n-3)(n-4)(n-5)
(D) none of the foregoing expressions.
45. By an upper bound for a set A of real numbers, we mean any real number x such that every number a in A is smaller than or equal to x. if x is an upper bound for a set A and no number strictly smaller than x is an upper bound for A, then x is called sup A.
Let A and B be two sets of real numbers with x = sup A and y = sup B. Let C be the set of all real numbers of the form a+b where a is in A and b is in B. If z = sup C, then
(A) z < x+y
(B) z > x+y
(C) z = x+y
(D) nothing can be said.
46. There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics and 40 failed in Statistics; and 32 failed in exactly two of these three subjects. Only one student passed in all the three subjects. The number of students failing in all the three subjects is
(A) 12;
(B) 4;
(C) 2;
(D) cannot be determined from the given information.
47. A television station telecasts three types of programs X, Y and Z. A survey gives the following data on television viewing. Among the people interviewed 60% watch program X, 50% watch program Y, 50% watch program Z, 30% watch programs X and Y, 20% watch programs y and Z, 30% watch programs X and Z while 10% do not watch any television program. The percentage of people watching all the three programs X, Y and Z is
(A) 90;
(B) 50;
(C) 10;
(D) 20.
48. In a survey of 100 families, the number of families that read the most recent issues of various magazines was found to be: India Today 42, Sunday 30, New Delhi 28, India Today and Sunday 10, India Today and New Delhi 5, Sunday and New Delhi 8, all three magazines 3. Then the number of families that read none of the three magazines is
(A) 30; (B) 26; (C) 23; (D) 20.
49. In a survey of 100 families, the number of families that read the most recent issues of various magazines was found to be: India Today 42, Sunday 30, New Delhi 28, India Today and Sunday 10, India Today and New Delhi 5, Sunday and New Delhi 8, all three magazines 3. Then the number of families that read either both or none of the two magazines Sunday and India Today is
(A) 48; (B) 38; (C) 72; (D) 58.
50. In a village of 1000 inhabitants, there are three newspapers P, Q and R in circulation. Each of these papers is read by 500 persons. Papers P and Q are read by 250 persons, papers Q and R are read by 250 persons, papers R and P are read by 250 persons. All the three papers are read by 250 persons. Then the number of persons who read no newspaper at all
(A) is 500; (B) is 250 (C) is 0; (D) cannot be determined from the given information.
51. Sixty (60) students appeared in a test consisting of three papers I, II and III. Of these students, 25 passed in paper I, 20 in paper II and 8 in paper III. Further, 42 students passed in at least one of papers I and II, 30 in at least one of papers I and III, 25 in at least one of papers II and III. Only one student passed in all the three papers. Then the number of students who failed in all the papers is
(A) 15; (B) 17; (C) 45; (D) 33.
52. A student studying the weather for d days observed that (i) it rained on 7 days, morning or afternoon; (ii) when it rained in the afternoon, it was clear in the morning; (iii) there were five clear afternoons; and (iv) there were six clear mornings. Then d equals
(A) 7; (B) 11; (C) 10; (D) 9.
53. A club with x numbers is organized into four committees according to the following rules:
(i) Each member belongs to exactly two committees.
(ii) Each pair of committees has exactly one member in common.
Then
(A) x = 4; (B) x = 6; (C) x = 8; (D) x cannot be determined from the given information.
54. There were 41 candidates in an examination and each candidate was examined in Algebra, Geometry and Calculus. It was found that 12 candidates failed in Algebra, 7 failed in Geometry and 8 failed in Calculus, 2 in Geometry and Calculus, 3 in calculus and Algebra, 6 in Algebra and Geometry, whereas only 1 failed in all three subjects. Then the number of candidates who passed in all three subjects
(A) is 24; (B) is 2: (C) is 14;
(D) cannot be determined from the given information.
55. In a group of 120 persons there are 80 Bengalis and 40 Gujaratis. Further, 70 persons in the group are Muslims and the remaining Hindus. Then the number of Bengali Muslims in the group is
(A) 30 or more;
(B) exactly 20;
(C) between 15 and 25;
(D) between 20 and 25.
56. In a group of 120 persons there are 70 Bengalees, 35 Gujratis and 15 Maharashtrians. Further, 75 persons in the group are Muslims and the remaining are Hindus. Then the number of Bengali Muslims in the group is
(A) between 10 and 14;
(B) between 15 and 19;
(C) exactly 20;
(D) 25 or more.
57. Four passengers in a compartment of the Delhi – Howrah Rajdhani Express discover that they from an interesting group. Two are lawyers and two are doctor. Two of them speak Bengali and the other two Hindi and no two of the same profession speak the same language. They also discover that two of them are Christians and two Muslims, no two of the same religion are of the same profession and no two of the same religion speak the same language. The Hindi-speaking doctor is a Christian. Then only one of the statements below logically follows. Which one is it?
(A) The Bengali- speaking lawyer is a Muslim.
(B) The Christian lawyer speaks Bengali.
(C) The Bengali- speaking doctor is a Christian.
(D) The Bengali- speaking doctor is a Hindu.
58. In a football league, a particular team played 60 games in a season. The team never lost three games consecutively and never won five games consecutively in that season. If N is the number of games the team won in that season, then N satisfies
(A) 24 ${\le}$ N ${\le}$ 50;
(B) 20 ${\le}$ N ${\le}$ 48;
(C) 12 ${\le}$ N ${\le}$ 40;
(D) 18 ${\le}$ N ${\le}$ 42.
59. A box contains 100 balls of different colours : 28 red, 17 blue, 21 green, 10 white, 12 yellow, 12 black. The smallest number ${n}$ such that any ${n}$ balls drawn from the box will contain at least 15 balls of the same colour, is
(A) 73; (B) 77; (C) 81; (D) 85.
60. Let ${x,y,z,w}$ be positive real number, which satisfy the two conditions that(i) if ${x > y}$ then ${z > w}$; and
(ii) if ${x > z}$ then ${y < w}$.
Then one of the statements given below is a valid conclusion. Which one is it?
(A) If ${x < y}$ then ${z < w}$.
(B) If ${x < w}$.
(C) If ${x > y + z}$ then ${z y + z}$ then ${z > y}$.
(D) If ${x > y + z}$ then ${z y + z}$ then ${z .
61. Consider the statement:
${x(\alpha - x) < y(\alpha - y)}$ for all ${x,y}$ with 0 < ${x}$ < ${y}$ 2}$; (C) if and only if ${\alpha 1}$, such that ${n}$, ${n + 2}$, ${n + 4}$ are prime numbers, is (A) zero; (B) one; (C) infinite; (D) more than one, but finite. 62. In a village, at least 50% of the people read a newspaper. Among those who read a newspaper at the most 25% read more than one paper. Only one of the following statements follows from the statements we have given. Which one is it? (A) At the most 25% read exactly one newspaper. (B) At least 25% read all the newspapers. (C) At the most 37.50% read exactly one newspaper. (D) At least 37.50% read exactly one newspaper. 63. We consider the relation “a person X shakes hand with a person Y”. Obviously, if X shakes hand with Y, then Y shakes hand with X. In a gathering of 99 persons, one of the following statements is always true, considering 0 to be an even number. Which one is it?(A) There is at least one person who shakes hand exactly with an odd number of persons. (B) There is at least one person who shakes hand exactly with an even number of persons. (C) There are even number of persons who shake hand exactly with an even number of persons. (D) None of the foregoing statements. 64. Let P,Q,R,S and T be statements such that if P is true then both Q and S are true, and if both R and S are true then T is false. We then have: (A) If T is true then both P and R must be true. (B) If T is true then both P and R must be false. (C) If T is true then at least one of P and R must be true. (D) If T is true then at least one of P and R must be false. 65. Let P,Q,R and S be four statements such that if P is true then Q is true, if Q is true then R is true and if S is true then one of Q and R is false. Then follows that (A) if S is false then both Q and R are true; (B) if at least one of Q and R is true then S is false; (C) if P is true then S is false; (D) if Q is true then S is true. 66. If A,B,C and D are statements such that if at least one of A and B is true, then at least one of C and D must be true. Further, bothA and C are false. Then (A) if D is false then B is false (B) both B and D are false (C) both B and D are true (D) if D is true then B is true. 67. P,Q and R are statements such that if P is true then at least one of the following is correct : (i) Q is true, (ii) R is not true. Then (A) if both P and Q are true then R is true; (B) if both Q and R are true then P is true; (C) if both P and R are true then Q is true; (D) none of the foregoing statements is correct. 68. It was a hot day and four couples drank together 44 bottles of cold drink. Anita had 2, Biva 3, Chanchala 4, and dipti 5 bottles. Mr. Panikkar drank just as many bottles as his wife, but each of the other men drank more than his wife – Mr. Dube twice, Mr. Narayan three times and Mr. Rao four times as many bottles. then only one of the following statements is correct. Which one is it? (A) Mrs. Panikkar is Chanchala (B) Anita’s husband had 8 bottles (C) Mr. Narayan had 12 bottles (D) Mrs. Rao is Dipti. 69. Every integer of the form ${ (n^3 - n)(n-2)}$, (for n = 3,4,…) is (A) divisible by 6 but not always divisible by 12; (B) divisible by 12 but not always divisible by 24; (C) divisible by 24 but not always divisible by 48; (D) divisible by 9. 70. The number of integers n>1, such that n, n+2, n+4 are all prime numbers , is (A) zero (B) one (c) infinite (D) more than one, but finite. 71. The number of ordered pairs of integers (x,y) satisfying the equation ${x^2+ 6x + y^2 = 4}$ is (A) 2; (B) 4; (C) 6; (D) 8. 72. The number of integer (positive, negative or zero) solutions of xy – 6(x+y) = 0 with ${x \le}$ y is (A) 5; (B) 10; (C) 12; (D) 9. 73. Let P denote the set of all positive integers and S = {(x,y) : x ${\in}$ P, y ${\in}$ P and ${x^2 - y^2 = 666}$. The number of distinct elements in the set S is (A) 0; (B) 1; (C) 2; (D) more than two. 74. If numbers of the form ${3^{4n-2} + 2^{6n-3} + 1}$, where ${n}$ is a positive integer, are divided by 17, the set of all possible remainders is (A) {1} (B) { 0,1} (C) {0, 1, 7} (D) { 1, 7} 75. Consider the sequence: ${a_1 = 101, a_2 = 10101, a_3 = 1010101,}$ and so on. then ${a_k}$ is a composite number (that is, not a prime number) (A) if and only if k ${\ge}$ 2 and 11 divides ${10^{k+1} +1}$ (B) if and only if k ${\ge}$ 2 and 11 divides ${ 10^{k+1} -1}$ (C) if and only if k ${\ge}$ 2 and k-2 is divisible by 3; (D) if and only if k ${\ge}$ 2. 76. Let n be a positive integer. Now consider all numbers of the form ${3^{2n+1}+2^{2n+1}}$. Only one of the following statements is true regarding the last digit of these numbers. Which one is it? (A) It is % for some of these numbers but not for all. (B) It is 5 for all these numbers. (C) It is always 5 for n ${\le}$ 10 and it is 5 for some n>10. (D) It is odd for all of these numbers but not necessarily 5. 77. Which of the following numbers can be expressed as the sum of squares of two integers? (A) 1995; (B) 1999; (C) 2003; (D) none of these integers. 78. If the product of an odd number of odd integers is of the form ${4n + 1}$; then(A) an even number of them must always be of the form ${4n + 1}$; (B) an odd number of them must always be of the form ${4n + 1}$; (C) an odd number of them must always be of the form ${4n + 1}$; (D) none of the above statements is true. 79. The two sequences of numbers { 1, 4, 16, 64, . . .} and { 3, 12, 48, 192, . . .} are mixed as follows: { 1, 3, 4, 12, 16, 48, 64, 192, . . .}. One of the numbers in the mixed series is 1048576. Then the number immediately preceding it is (A) 786432; (B) 262144; (C) 814572; (D) 786516. 80. Let ${(a_1, a_2,a_3,...)}$ be a sequence such that ${a_1 = 2}$ and$latex a_n – a_{n-1} = 2n \$ for all n ${\ge}$ 2. Then ${a_1+a_2+...+a_{20}}$ is
(A) 420 (B) 1750 (C) 3080 (D) 3500
81. The value of ${\Sigma}$ ij, where the summation is over all i and j such that 1 ${\le}$ 10, is
(A) 1320, (B) 2640 (C) 3080 (D) 3500
82. Let ${x_1, x_2,...,x_{100}}$ be hundred integers such that sum of any five of them is 20. Then
(A) the largest ${x_i}$ equals 5;
(B) the smallest ${x_i}$ equals 3;
(C) $x_{17}$ = $x_{83}$
(D) none of the foregoing statements is true.
83. The smallest positive integer ${n}$ with 24 divisors ( where 1 and ${n}$ are also considered as divisors of ${n}$ ) is
(A) 420; (B) 240; (C) 360; (D) 480;
84. The last digit of ${(2137)^{754}}$ is
(A) 1;
(B) 3;
(C) 7;
(D) 9.59.
85. The smallest integer that produces remainders of 2,4,6 and 1 when divided by 3,5,7 and 11 respectively, is
(A) 104;
(B) 1154;
(C) 419;
(D) none of the foregoing numbers.
86. How many integers n are there such that 2 ${\le}$ n ${\le}$ 1000 and the highest common factor of n and 36 is 1 ?
(A) 166 (B) 332 (C) 361 (D) 416
87. The remainders when ${3^37}$ is divided by 79 is
(A) 78; (B) 1 (C) 2 (D) 35.
88. The remainder when ${4^101}$ is divided by 101 is
(A) 4; (B) 64 (C) 84 (D) 36.
89. The 300 – digit number with all digits equal to 1 is
(A) divisible by neither 37 nor 101;
(B) divisible by 37 but not by 101;
(C) divisible by 101 but not by 37;
(D) divisible by both 37 and 101.
90. The remainder when ${3^12+5^12}$ is divided by 13 is
(A) 1;
(B) 2;
(C) 3;
(D) 4.
91. When $3^{2002}+ 7^{2002}+ 2002$ is divided by 29 the remainder is
(A) 0; (B) 1; (C) 2 (D) 7.
92. Let x = 0.101001000100001…+0.272727…. then x
(A) is irrational;
(B) is rational but ${\sqrt{x}}$ is irrational;
(C) is a root of ${x^2 + 0.27x + 1}$ = 0;
(D) satisfies none of the above properties.
93. The highest power of 18 contained in ${\dbinom{50}{25}}$ is
(A) 3; (B) O; (C) 1; (D) 2.
94. The number of divisors of 2700 including 1 and 2700 equals
(A) 12 (B) 16 (C) 36 (D) 18
95. The number of different factors of 1800 equals
(A) 12 (B) 210 (C) 36 (D) 18.
96. The number of different factors of 3003 is
(A) 2; (B) 15; (C) 7; (D) 16.
97. The number of divisors of 6000, where 1 and 6000 are also considered as divisors of 6000, is
(A) 40; (B) 50; (C) 60 (D) 30
98. The number of positive integers which divide 240 (where both 1 and 240 are also considered as divisors) is
(A) 18; (B) 20; (C) 30; (D) 24.
99. The sum of all the positive divisors of 1800 (including 1 and 1800) is
(A) 7201;
(B) 6045;
(C) 5040;
(D) 4017.
100. Let ${d_1}$, ${d_2}$,…., ${d_k}$ be all the factors of a positive integer n including 1 and n.
Suppose ${{d_1}+{d_2}+...+{d_k}}$ = 72. Then the value of${\dfrac{1}{d_1}}$ + ${\dfrac{1}{d_2}}$ +…+ ${\dfrac{1}{d_k}}$
(A) is ${\dfrac{k^2}{72}}$
(B) is ${\dfrac{72}{k}}$
(C) is ${\dfrac{72}{n}}$
(D) cannot be computed from the given information.