### Set 1

Find Solutions at the end of every sectionEach question is followed by two statements. You have to decide whether the information provided in the statements is sufficient for answering the question. |

Mark A | If the question can be answered by using one of the statements alone, but cannot be answered by using the other statements alone. |

Mark B | If the question can be answered by using either statement alone. |

Mark C | If the question can be answered by using both statements together, but cannot be answered by using the either statement alone. |

Mark D | If the question cannot be answered even by using both the statements together. |

1. | If n is an integer, is n even ? | |||

Stmt.(1) | n^{2} -1 is an odd integer. | |||

Stmt.(2) | 3n + 4 is an even integer. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Since n^{2} —1 is odd, n^{2} is even and so n is even; SUFFICIENT. | |||

Expl. of Statement (2) | Since 3n+ 4 is even, 3n is even and so n is even; SUFFICIENT. | |||

Answer | (B) [Each statement alone is sufficient.] |

2. | If n is an integer, is n+ 1 odd ? | |||

Stmt. (1) | n+ 2 is an even integer. | |||

Stmt. (2) | n-1 is an odd integer. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Since n+ 2 is even, n is an even integer, and therefore n+1 would be an odd integer; SUFFICIENT. | |||

Expl. of Statement (2) | Since n-1 is an odd integer, n is an even integer. Therefore n+ 1 would be an odd integer; SUFFICIENT. | |||

Answer | (B) [Each statement alone is sufficient.] |

3. | Is x a negative number ? | |||

Stmt. (1) | 9x > 10x. | |||

Stmt. (2) | x + 3 is positive. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Subtracting 9x from both sides of 9x > lOx gives 0 > x, which expresses the condition that x is negative; SUFFICIENT. | |||

Expl. of Statement (2) | Subtracting 3 from both sides of x+ 3 > 0 gives x > -3, and x > -3 is true for some negative numbers (such as -2 and -1) and for some numbers that aren’t negative (such as 0 and 1); NOT SUFFICIENT. | |||

Answer | (A) [Each statement alone is not sufficient.] |

4. | What is the tens digit of positive integer x ? | |||

Stmt. (1) | x divided by 100 has a remainder of 30. | |||

Stmt. (2) | x divided by 110 has a remainder of 30. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Having a remainder of 30 when x is divided by 100 can only happen if x has a tens digit of 3 and a ones digit of 0, as in 130, 230, 630, and so forth; SUFFICIENT. | |||

Expl. of Statement (2) | When 140 is divided by 110, the quotient is 1 R 30. However, 250 divided by 110 yields a quotient of 2 R 30, and 360 divided by 110 gives a quotient of 3 R 30. Since there is no consistency in the tens digit, more information is needed; NOT SUFFICIENT. | |||

Answer | (A) [Each statement alone is not sufficient.] |

5. | If k is an integer such that 56 < k < 66, what is the value of k ? | |||

Stmt. (1) | If k were divided by 2, the remainder would be 1. | |||

Stmt. (2) | If k + 1 were divided by 3, the remainder would be O. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Determine the value of the integer k, where 56<k< 66 It is given that the remainder is 1 when k is divided by 2, which implies that k is odd Therefore, the value of k can be 57, 59, 61,63, or 65; NOT SUFFICIENT. | |||

Expl. of Statement (2) | It is given that the remainder is 0 when k + 1 is divided by 3, which implies that k + 1 is divisible by 3. Since 56 <k <66 (equivalently, 57 <k+1 <67), the value of k + 1 can be 60, 63, or 66 so the value of k can be 59, 62 or 65; NOT SUFFICIENT. | |||

Taking (1) + (2) Together | Taking (1) and (2) together, 59 and 65 appear in both lists of possible values for k; NOT sufficient. | |||

Answer | (D) [Each statement alone and together is not sufficient.] |

###

Set 2

Find Solutions at the end of every sectionEach question is followed by two statements. You have to decide whether the information provided in the statements is sufficient for answering the question. |

Mark A | If the question can be answered by using one of the statements alone, but cannot be answered by using the other statements alone. |

Mark B | If the question can be answered by using either statement alone. |

Mark C | If the question can be answered by using both statements together, but cannot be answered by using the either statement alone. |

Mark D | If the question cannot be answered even by using both the statements together. |

1. | What is the value of |x|? | |||

Stmt.(1) | x = -|x| . | |||

Stmt.(2) | x^{2} = 4. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | The absolute value of x, |x|, is always positive or 0, so this only determines that x is negative or 0; NOT SUFFICIENT. | |||

Expl. of Statement (2) | Exactly two values of x (x= ±2) are possible, each of which gives the value 2 for |x| SUFFICIENT. | |||

Answer | (A) [Each statement alone is not sufficient.] |

2. | If x is negative, is x < – 3 ? | |||

Stmt. (1) | x^{2} > 9. | |||

Stmt. (2) | x^{3 }< -9. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Given that x^{2}> 9, it follows that x < -3 or x > 3, a result that can be obtained in a variety of ways. For example, consider the equivalent equation (|x|) >9 that reduces to |x| >3,or consider when the two factors of x^{2} —9 are both positive and when the two factors of x^{2} —9 are both negative, or consider where the graph of the parabola y = x^{2} – 9 is above the x-axis, etc. Since it is also given that x is negative, it follows thatx< -3; SUFFICIENT. | |||

Expl. of Statement (2) | Given that x^{3 }< -9, if x = -4, then x^{3 } = -64, and so x^{3 }<-9 and it is true that x < -3. However, if x = —3, then x = —27, and so x^{3 } < —9, but it is not true that x < —3; NOT SUFFICIENT. | |||

Answer | (A) [Each statement alone is not sufficient.] |

3. | If x and y are integers, is xy even? | |||

Stmt. (1) | x = y + l | |||

Stmt. (2) | x/y is an even integer.. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | Determine if xy is even; Since x and y are consecutive integers, one of these two numbers is even, and hence their product is even. For example, if x is even, then x = 2m for some integer m, and thus xy =(2m)y =(my)(2), which is an integer multiple of 2, so xy is even; SUFFICIENT. | |||

Expl. of Statement (2) | If x /y is even, then x/y= 2n for some integer n, and thus x = 2ny. From this it follows that xy =(2ny)(y) =(ny^{2})(2), which is an integer multiple of 2, so xy is even; SUFFICIENT. | |||

Answer | (B) [Each statement alone is sufficient.] |

4. | Paula and Sandy were among those people who sold raffle tickets to raise money for Club X. If Paula and Sandy sold a total of 100 of the tickets, how many of the tickets did Paula sell? | |||

Stmt. (1) | Sandy sold 2/3 as many of the raffle tickets as Paula did.. | |||

Stmt. (2) | Sandy sold 8 percent of all the raffle tickets sold for Club X.. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | If Paula sold p tickets and Sandy sold s tickets, then p + s = 100. Since Sandy sold 2/3 as many tickets as Paula, s= (2/3)p. The value of p can be determined by solving the two equations simultaneously; SUFFICIENT. | |||

Expl. of Statement (2) | Since the total number of the raffle tickets sold is unknown, the number of tickets that Sandy or Paula sold cannot be determined NOT SUFFICIENT. | |||

Answer | (A) [Each statement alone is not sufficient.] |

5. | What is the number of cans that can be packed in a certain carton?? | |||

Stmt. (1) | The interior volume of this carton is 2,304 cubic inches.. | |||

Stmt. (2) | The exterior of each can is 6 inches high and has a diameter of 4 inches.. | |||

(a) A | (b) B | (c) C | (d) D | |

Expl. of Statement (1) | No information about the size of the cans is given; NOT SUFFICIENT. | |||

Expl. of Statement (2) | No information about the size of the carton is given; NOT SUFFICIENT. | |||

Taking (1) + (2) Together | Taking (1) and (2) together, there is still not enough information to answer the question. If the carton is a rectangular solid that is 1 inch by 1 inch by 2,304 inches and the cans are cylindrical with the given dimensions, then 0 cans can be packed into the carton. However, if the carton is a rectangular solid that is 16 inches by 12 inches by 12 inches and the cans are cylindrical with the given dimensions, then 1 or more cans can be packed into the carton. | |||

Answer | (D) [Each statement alone and together is not sufficient.] |

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