18. The area of a triangle is 16*root3. What is the perimeter of the triangle?
1). It is an isosceles triangle
2). Two of the inside angles are 30 degrees.
ANS: (B)
Stmt 1) Given area = 16sqrt(3) and triangle is isosceles. Let b=base and h=height. From given 1/2*b*h=16*sqrt(3). We have 2 variables 1 equation. INSUFF

Stmt 2) Given area = 16sqrt(3) and two inside angles are 30. This implies this triangle is also isosceles, and there is extra information available (angles = 30,30,120). Let b be the base, h be the height and s be the measure of two equal sides. We have following relations
Perimeter = b+2*s, Area = 1/2*b*h, h*tan(30)=b/2, s*cos(30)=b/2 (4 equations and 4 variables). Solvable. SUFF

19. As the figure shows, A is the center of the circle, AB=AC=2, angle BAC is 120 degrees. What is the area of the triangle?
ANS: sqrt(3)
Draw a perpendicular AD to BC, AD bisects BC. Angle B = Angle C = 30.
Height = ABsin 30=1, base = ABcos30+AC cos30 = 2ABcos30=2¡î3.
Area = 1/2*base*height = 1/2*2*sqrt(3) = sqrt(3)

20. The median of the rent of the apartments is 550. What's the average (arithmatic mean) of the rent?
1). The average rent of apartments with rent higher than the median is 890.
2). There are 370 apartments with the rent higher than the median.
ANS: E
Median is basically the middle value when the rents (or items) are arranged in increasing (or decreasing) order. Average rent is (total sum of rents)/(total number of apartments). The question just gives us the median value.
Stmt 1) Average rent of apartments with rent higher than the median is 890 -> Gives gives us an average of upper half. We still have no information about the total number and total sum of apartments. INSUFF
Stmt 2) There are 370 apartments with the rent higher than the median -> Gives just the number of apartments greater than the median. No information about rents. INSUFF
Combining 1 and 2. We can find the total sum of the upper half = 890*370. Still no clue about the sum of rents of houses below the median (lower half) => INSUFF

22. Two cubes each is numbered with 1 to 6 (called dice). Tossing the dices, what is the probability that the sum of the numbers on the top is greater than 9?
ANS: 1/6
Total possibilities = 36
Favorable outcomes = (4,6), (6,4), (5,6), (6,5), (5,5), (6,6) = 6
Probability = 6/36 = 1/6

23. The first term of a geometric sequence is 1/2, and the common ratio is 1/2. The 10th term will fall into which of the following range?
ANS: (C) 0.001 ~ 0.0001
0.1~0.01, 0.01~0.001, 0.001~0.0001,0.0001~0.00001
1st term is 1/2 so 10th term would be (1/2)10 = 1/1024.
(1/1000)>(1/1024)>(1/10000) [just increasing the denominators, fraction decreases] => 0.001>(1/2)10>0.0001. (C)

24. The sides of a triangle is a, b, and c. Is it a right triangle?
1). a^2+b^2=3^2+4^2
2). a^2+c^2=3^2+5^2
ANS: (E)
*Note : a,b and c are NOT given to be integers
Stmt 1) a^2+b^2=3^2+4^2=25 -> Nothing is given about c and its relation with a and b. INSUFF
Stmt 2) a^2+c^2=3^2+5^2=34 -> Nothing is given about b and its relation with a and c. INSUFF
Combining (1) and (2), we have 3 variables (a,b,c) and just 2 equations. There are multiple values possible for a, b and c which satisfy these equations. INSUFF

25. -1

26. A list of consecutive natural numbers contains 9 and 15. Does the list contain number which can be divisible by 8?
1). The list has 10 numbers.
2). 18 is in the list.
ANS: (D)
Stmt 1) The list has 10 numbers -> the list will contain either 8 or 16 (at the minimum). SUFF
Stmt 2) 18 is in the list -> the list contains 16. SUFF
(D)

27. In the xy-plane, line A has slope a and line B has slope b. Are the two lines parallel to each other?
1). 4^a=2^b
2). 2^a=3^b
ANS: (C)
For the 2 lines to be parallel a=b. So basically the question is asking if a=b?
Stmt 1) 4^a = 2^b => 2^2a = 2^b => 2a=b. Cant be sure, because if a=b=0, lines are parallel and for non-zero values of a and b they are not parallel. INSUFF

Stmt 2) if 2^a=3^b then 2^(a/b)=3. Now we know that 2^1=2 and 2^2=4. so the value of (a/b) should lie between 1 and 2 so that 2^(a/b)=3.

Hence there is atleast one more (non-integer) value of a and b (apart from a=b=0) which would satisfy the given equation. Hence statement 2 is not sufficient in itself (gives multiple values of a and b). INSUFF.
Combining (1) and (2), the only possible value for a and b is 0 (Also there are 2 equations and 2 variables). Substitute 2^a=3^b in stmt 1, we get 9^b = 2^b. This is only possible for b=0 (=> a=0). Hence a=b=0 => Parallel. SUFF

(C)

27. In the xy-plane, line A has slope a and line B has slope b. Are the two lines parallel to each other?
1). 4^a=2^b
2). 2^a=3^b
ANS: (C)
For the 2 lines to be parallel a=b. So basically the question is asking if a=b?
Stmt 1) 4^a = 2^b => 2^2a = 2^b => 2a=b. Cant be sure, because if a=b=0, lines are parallel and for non-zero values of a and b they are not parallel. INSUFF
Stmt 2) if 2^a=3^b then 2^(a/b)=3. Now we know that 2^1=2 and 2^2=4. so the value of (a/b) should lie between 1 and 2 so that 2^(a/b)=3.
Hence there is atleast one more (non-integer) value of a and b (apart from a=b=0) which would satisfy the given equation. Hence statement 2 is not sufficient in itself (gives multiple values of a and b). INSUFF.
Combining (1) and (2), the only possible value for a and b is 0 (Also there are 2 equations and 2 variables). Substitute 2^a=3^b in stmt 1, we get 9^b = 2^b. This is only possible for b=0 (=> a=0). Hence a=b=0 => Parallel. SUFF

(C)

28. If x and y are nonzero, is (x^3)(y^5)/(x^2)(y^3)>0
1). x>0
2). y>0
ANS: (A)
The question is basically asking is xy^2>0?
Stmt 1) x>0 => xy^2 is definitely greater than 0 sine y^2 is always greater than 0. SUFF
Stmt 2) y>0 => Doesn’t tell us anything about sign of x. INSUFF
(A)

29. Each segment from A to G has equal distance. Which of the following point represents [-(3^12)] ?
ANS: Point D
Distance between all points is same. Given F as 3^11 and G as 3^12. So distance between 2 points = 3^12 – 3^11 = 3^11*(3-1) = 2*3^11.
So point E = F – 2*3^11 = 3^11 – 2*3^11 = -3^11. Point D = Point E – 2*3^11 = -3^11 – 2*3^11 = -3*3^11 = -1*3^12 = -3^12

31. A ship started off at the rate of a. t hours later, in the same direction, another ship started off at the rate of b. In how many hours the second ship will catch up with the first ship?
ANS: a*t/(b-a)
Method 1: Speed Time Distance
Lets assume that after t1 hrs, 2nd ship catches the first.
Since the 2nd ship catches up, the total distance traveled by second is equal to total distance traveled by the first.
Hence b*t1 = a*t + a*t1 => t1 = a*t/(b-a)
Method 2: Relative speed
Initial distance between them = a*t.
Relative speed of 2nd ship w.r.t 1st ship = b-a.
Time taken to cover the distance = initial distance / relative speed = a*t/(b-a)