All questions of Mock Test: Data Sufficiency for GMAT Exam

In statement 1, b =0, when a < b and b = 0, it implies that a < 0, hence, a is negative and a is not positive. The statement is sufficient.
In statement 2, √a < a, the square root of a number is defined only when in √a  a > 0 . Hence, the statement is  sufficient. Thus,  EACH statement ALONE is sufficient to answer the question asked.

 

Conclusion:

Since both statements independently lead to the determination that the interior angle is 120 degrees, each statement alone is sufficient.

The correct answer is:

EACH statement ALONE is sufficient to answer the question asked.

Explanation:

Let the sides of the square be s, and the length and width of the rectangle be l and w, respectively. We need to find the perimeter of the rectangle, which is 2(l+w).

Statement 1: The perimeter of the square is 24 inches.
This means that s=6 inches. Since the area of the triangle is equal to the area of the rectangle, we have (1/2)(s^2) = lw. So, (1/2)(6^2) = lw, which means lw=18. However, we cannot determine l and w from this statement alone, so it is not sufficient.

Statement 2: The sum of the length and the width is 13 inches.
This means that l+w=13. We cannot determine the area or the perimeter of the rectangle from this statement alone, so it is not sufficient.

Using both statements together, we know that lw=18 and l+w=13. We can solve for l and w using these equations: l=9 and w=4 (or vice versa). Now we can find the perimeter of the rectangle: 2(9+4) = 26 inches. Both statements together are sufficient to answer the question.

We consider 12 and t.
In statement 1, the Least Common multiple of 12 and t is 48 to mean that t is a factor of 48. This mean that, t can take 16, 24 and 48. Hence the statement is not sufficient.
In statement 2, the Greatest Common divisor is 4. Therefore, t can take 4, 8, 16, 28, 32, 40 among others. Thus, the statement is not sufficient.
Combining statements 1and 2, we have 16 featuring in both lists, hence it satisfies the two conditions. Therefore, t =16. Thus BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Before even evaluating the statements, simplify the question. In a more complicated data sufficiency problem, it is likely that some rearranging of the terms will be necessary in order to see the correct answer.
Use the formula for a difference of squares (a2 - b2) = (a + b)(a - b). However, let x2 equal a, meaning a2 = x4.
x4 - y4 = (x2 + y2)(x2 - y2)
Recognize that the expression contains another difference of squares and can be simplified even further.
(x2 + y2)(x2 – y2) = (x2 + y2)(x – y)(x + y)
The question can now be simplified to: "If x and y are distinct positive integers, what is the value of (x2 + y2)(x – y)(x + y)?" If you can find the value of (x2 + y2)(x - y)(x + y) or x4 - y4, you have sufficient data.
Evaluate Statement (1) alone.
Statement (1) says (y2 + x2)(y + x)(x - y) = 240. The information in Statement (1) matches exactly the simplified question. Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
Statement (2) says xy = yx and x > y. In other words, the product of multiplying x together y times equals the product of multiplying y together x times.
The differences in the bases must compensate for the fact that y is being multiplied more times than x (since x > y and y is being multiplied x times while x is being multiplied y times).
4 and 2 are the only numbers that work because only 4 and 2 satisfy the equation n2 = 2n, which is the condition that would be necessary for the equation to hold true.
Observe that this is true: 42 = 24 = 16.
Remember that x > y, so x = 4 and y = 2. Consequently, you know the value of x4 - y4 from Statement (2). So, Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT

In statement 1, |t| > t. But is |-t| = |t|. Hence t can be positive or negative.  When |t| > t, it implies that t is a negative value, hence t < 0. The statement is sufficient.
In statement 2, t² > 0 , but when t is negative or positive, t² > 0 , hence the statement is not sufficient. Thus  Statement (1) ALONE is sufficient but statement (2) is not sufficient.

The volume of the cylinder is given buy v = πr²h.
In statement 1, diameter (d) is increased by 20%, hence the radius is increased by 10%. The new radius is given by 1.1r. The new volume = π(1.1r)²h = 1.21πr²h
Percentage change in volume = (1.21πr²h  - πr²h )/pr²h × 100% = 21%. The statement is sufficient.
In statement 2, height is increased by 21%, hence the new height is 1.21h. The new volume = 1.21πr²h.
Percentage change in volume = (1.21πr²h  - πr²h )/πr²h × 100% = 21%. The statement is sufficient.  Therefore,  EACH statement ALONE is sufficient.

Step 1: Key Data from the Question 

Set S has 6 elements.
The elements of set S are distinct.
x and y are prime numbers. Because 2 is already an element in S, both x and y have to be odd.
0 < x < 40 and 0 < y < 40

Step 2: Check Statement I

I. The maximum possible range of the set is greater than 33.
The keyword in this entire statement is maximum. We have to determine whether the maximum value possible for the range exceeds 33.
We know x and y are prime numbers. The largest prime number less than 40 is 37.
If either x or y is 37, the largest number in the set will be 37 and the smallest number is 2.
Therefore, the maximum range of the set will be 37 - 2 = 35. It is greater than 33.

Statement I is true. So, eliminate choices that do not contain I.
Eliminate choice D

Step 3: Check Statement II

II. The median can never be an even number.
There are 6 numbers in the set. Therefore, the median is the arithmetic mean of the 3rd and the 4th term when the numbers are written in ascending or descending order.
The elements are {2, 4, 6, 8, x, y}, where x and y are prime numbers.
If x and y take 3 and 5 as values, the median is 4.5
If x = 3, y = 7 or greater, the median is 5.
If x = 5, y = 7 or greater, the median is 5.5
If x = 7, y = 11 or greater, the median is 6.5
If x = 11 or greater and y = 13 or greater, the median is 7.
It is quite clear that the median is either an odd number or is not an interger. So, the median can never be an even integer.

Statement II is true. Eliminate choices that do not contain II.
Eliminate choices A and C as well.

Step 4: Check Statement III

III. If y = 37, the average of the set will be greater than the median.
If y = 37, the set will be {2, 4, 6, 8, x, 37}, where x is a prime number greater than 2 and less than 37.
The average will be 57 + x657 + x6 = 9.5 + x6x6
If x = 3, median = 5 and average = 10. Average > median.
If x = 5, median = 5.5 and average = 10.33. Average > median
If x = 7, median = 6.5 and average = 10.66. Average > medain
If x = 11 or greater, the median = 7. Average will be definitely greater than 10. So, Average > Median.
It is true that the average is greater than the median if y = 37.

Statement III is also true.
Statements I, II, and III are true.

Choice C is the correct answer.

Deposit (P) = 3000.
Accumulated amount = P + (1 + R/100)n where the variables have their usual meaning for compound interest and
Accumulated amount = P + (P × r/100 × n) where the variables have their usual meaning for simple interest.
In statement 1, r =  4.3 and P =  3000 but we are not given the value of n, hence we cannot find the accumulated amount. Further more, the statement does not give more information about the kind of interest offered, hence, it is not sufficient.
In statement 2, n = 5 and P = 3000 but we are not given the value of r, hence we cannot find the accumulated amount. Furthermore, the statement does not give more information about the kind of interest offered, hence, it is not sufficient.
Combining the two statements, P = 3000, r = 4.5 and n = 5 but, the details given are not sufficient since no specific type on interest is disclosed, therefore, we cannot apply the compound or simple interest formula with accuracy. Thus Statements (1) and (2) TOGETHER are NOT sufficient.

Parallel lines have equal slopes.
In statement 1, if both lines are in the first, second and fourth quadrant then they have a negative slope. This alone is not enough to prove that they are parallel or not, hence, the statement is insufficient.
In statement 2, the y intercepts are 8 and 4 to imply that the equation are of the form y = mx + 8 and y = bx + 4. But this does not enough to determine if the lines area parallel since, notheing is said about the value of m and b, hence, the statement too is insufficient.
Combining the two statements, we have equations having negative gradients, thus, y =-mx + 8 and y = -bx + 4. Since we are not sure of -m = -b, we cannot say that they are parallel on not. Therefore,  Statements (1) and (2) TOGETHER are NOT sufficient.

Simplify the question by translating it into algebra.
Let P = the total value of John's parting gift
Let E = the amount each associate contributed
Let N = the number of associates
P = NE = 25E
With this algebraic equation, if you find the value of either P or E, you will know the total value of the parting gift.
Evaluate Statement (1) alone.
Two common ways to evaluate Statement (1) alone:
Statement 1: Method 1
Since the question stated that each person contributed equally, if losing four associates decreased the total value of the parting gift by $200, then the value of each associate's contribution was $50 (=$200/4).
Consequently, P = 25E = 25(50) = $1,250.
Statement 1: Method 2
If four associates leave, there are N - 4 = 25 - 4 = 21 associates.
If the value of the parting gift decreases by $200, its new value will be P - 200.
Taken together, Statement (1) can be translated:
P - 200 = 21E
P = 21E + 200
You now have two unique equations and two variables, which means that Statement (1) is SUFFICIENT.
Although you should not spend time finding the solution on the test, here is the solution.
Equation 1: P = 21E + 200
Equation 2: P = 25E
P = P
25E = 21E + 200
4E = 200
E = $50
P = NE = 25E = 25($50) = $1250
Evaluate Statement (2) alone.
Statement (2) says that $1,225 < P < $1,275. It is crucial to remember that the question stated that "25 associates contribute equally to a parting gift for John in an amount that is an integer." In other words P / 25 must be an integer. Stated differently, P must be a multiple of 25.
There is only one multiple of 25 between 1,225 and 1,275. That number is $1,250. Since there is only one possible value for P, Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

In statement 1, the x and y intercept is 2 and -2 respectively, hence the line passes through (2,0) and (0,-2). The slope is
(-2 - 0)/(2 - 0) = -1.
The equation is y/(x - 2) = -1 hence x = -x + 2.
The statement is sufficient.
In statement 2, the slope is 1. Since we are not given any point where the line is passing, the statement is insufficient. Thus, Statement (1) ALONE is sufficient but statement (2) is not sufficient.

Cost price = $80 is equivalent to 100%.
In statement 1, profit = 30%
= 30/100 × 80 = $24.
Profit = 104 - 80 = $24. The statement is sufficient.
In statement 2, selling price = 104.
selling price = cost price + profit; hence, profit =104 - 80 = $24. The statement is sufficient.

Let A and B represent the number of type A and Type B footballs respectively. Then A + B =  500.
In statement 1, B = 200, hence, from A + B =  500, A =  500 - B = 500 - 200 = $300. Thus 2A = $600. The statement is sufficient.
In statement 2, 2A + 3B = 1200, from A + B =  500,multiplying it by 3, we have 3A + 3B = 1500. Subtracting 2A + 3B = 1200 from 3A + 3B = 1500, we have A = 300; thus 2A = $600. The statement too is sufficient.

The line L passes through (2,8) and (0,0) hence its slope is
slope = (8 - 0)/(2 - 0) = 4.
Since L is perpendicular to the line in question, the product of their slope is -1. Therefore, the slope of the line in question is -1/4.
In statement 1, the line passes the origin, (0,0) and its slope is -1/4 hence its equation is y = -x/4. The statement is sufficient.
In statement 2, line passes through the (2,0.5) and its slope is -1/4. Thus we have (y + 0.5)/(x - 2) = -1/4
y + 1/2 = -x/4 + 1/2
thus  y = -x/4. The statement is sufficient too.
Thus, EACH statement ALONE is sufficient.

Volume = length × width × height
In statement 1, let width be x, length = 2x and height = 4
Volume = x × 2x × 4 = 8x² cubic inches. Since it is in terms of unknown value, x, it is insufficient.
In statement 2, length = 6 inches but the width and height is unknown hence it is not sufficient to determine the volume.
Combining the two statements, length = 2x = 6 hence x = 3 inches.
width = 3 inches and height = 4 inches.
Volume =  3 × 4 × 6 = 72 cubic inches. Thus , BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

4r + 2t = 14
In statement 1, t = 2, substituting for t in 4r + 2t = 14, we have
4r + 4 = 14; 4r = 10; r = 2  1/2. The statement is sufficient.
In statement 2, r > t , thus none of this is assigned a numeric value. Thisa implies that the solution of r in 4r + 2t = 14 will be a set of values. This implies that the statement is not sufficient.

Time taken = change in time = 30 minutes = 1/2 hours
In statement 1, Initial velocity (v₁) = 20 miles per hour and change in time = 0.5 hours (given)
Final velocity (v₂) = 25 miles per hour.
Acceleration = (v₂ - v₁)/ change in time = (25 - 20)/ 0.5 = 10 miles per hour per hour. Therefore, the statement is sufficient.
In statement 2, Change in velocity = 5 miles per hour and change in time = 0.5 hours (given).
Acceleration = Change in velocity/change in time = 5/0.5 =10 miles per hour per hour. Therefore, the statement is sufficient.
Therefore, EACH statement ALONE is sufficient.

Let cost of pen = P, pencil = Q.
From (1): 3P + 2Q = 80 — (i)
From (2): 2P + 2Q = 70 and Q = 10
Substitute Q = 10 into both equations:
From (2): 2P + 20 = 70 → 2P = 50 → P = 25
From (1): 3P + 20 = 80 → 3P = 60 → P = 20
Conflict in values shows contradiction → So both are needed to cross-verify.

Since s, p and q are interior angles of Isosceles triangle, s + p + q = 180°.
In statement 1, If s = 72°, then p + q + 72 = 180° and
p + q = 180°.
Since we have two unknowns in one equation and we are not sure which angles are base angles, we cannot determine the value of q, hence the statement is not sufficient.
In statement 2, p and q are base angles of the triangle, hence from s + p + q = 180°, we have s + 2q = 180°, but p = q.
But the statement is not sufficient since it does not have any information about s.
Combining the two, we have s + 2q = 180° and s = 72°, we have
72° + 2q = 180°; 2q = 180 - 72 = 108°. q = 54°.
Therefore,  BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Translate the final sentence, which contains the question, into algebra:
"the number of pages indexed per hour by G-Cache" = X
"greater than three times" translates into: >3
"the number of pages indexed by HTML-Compress" = Y

Putting this together:
Was X > 3Y?
Evaluate Statement (1) alone.
Translate the information from Statement (1) into algebra:
X - Y = 1 million
Since the original question states that "both algorithms indexed a positive number of pages per hour", the following inequalities must hold true:
X > 0
Y > 0
Simply knowing that X - Y = 1 million does not provide enough information to determine whether X > 3Y.
This can be seen via an algebraic substitution or by trying different numbers.
Trying Numbers
Let X = 10 and, therefore, Y = 9
10 is NOT > 3(9)
But, let X = 1.1 and, therefore, Y = .1
1.1 IS > 3(.1)
Algebraic Substitution
X - Y = 1 million
X = Y + 1 million
Plug this into the inequality we are trying to solve for:
Was X > 3Y?
Was (Y + 1 million) > 3Y?
Was 1 million > 2Y?
Was 500,000 > Y?
Was Y < 500,000?

Simply knowing that X - Y = 1 million does not provide enough information to determine whether Y < 500,000
Since different legitimate values of Y produce different answers to the question of whether X > 3Y, Statement (1) is not sufficient.
Statement (1) is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Translate the information from Statement (2) into algebra:
400,000 < Y < 1,400,000
We know nothing about the value of X.
If X were 10 million, the answer to the original question was X > 3Y? would be "yes."
If X were 100,000, the answer to the original question was X > 3Y? would be "no."
Since different legitimate values of X and Y produce different answers to the question of whether X > 3Y, Statement (2) is not sufficient.
Statement (2) is NOT SUFFICIENT.
Evaluate Statements (1) and (2) together.
With the information in Statement (1), we concluded that the original question can be boiled down to:
Is Y < 500,000?
Statement (2) says:
400,000 < Y < 1,400,000
Even when combining Statements (1) and (2), we cannot determine whether Y < 500,000
Y could be 450,000 (in which case X = 1,450,000) or Y could be 650,000 (in which case X = 1,650,000). These two different possible values of X and Y would produce different answers to the question "Was Y < 500,000?" Consequently, we would have different answers to the question "Was X > 3Y?"
Statements (1) and (2), even when taken together, are NOT SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT, Statement (2) alone is NOT SUFFICIENT, and Statements (1) and (2), even when taken together, are NOT SUFFICIENT.

If both x and y are multiples of 11, then both x + y and x - y will be multiples of 11. In other words, if two numbers have a common divisor, their sum and difference retain that divisor.
In case this is hard to conceptualize, consider the following examples:
42 - 18 {both numbers share a common factor of 6}
=(6*7) - (6*3)
=6(7 - 3)
=6(4)
=24 {which is a multiple of 6}

49 + 14 {both numbers share a common factor of 7}
=(7*7) + (7*2)
=7(7+2)
=7*9
=63 {which is a multiple of 7}
However, if x and y are not both multiples of 11, it is possible that x - y is a multiple of 11 while x + y is not a multiple of 11. For example:
68 - 46 = 22 but 68 + 46 = 114, which is not divisible by 11.
The reason x - y is a multiple of 11 but not x + y is that, in this case, x and y are not individually multiples of 11.
Evaluate Statement (1) alone.
Since x-y is a multiple of 22, x-y is a multiple of 11 and of 2 because 22=11*2
If both x and y are multiples of 11, the sum x + y will also be a multiple of 11. Consider the following examples:
44 - 22 = 22 {which is a multiple of 11 and of 22}
44 + 22 = 66 {which is a multiple of 11 and of 22}

88 - 66 = 22 {which is a multiple of 11 and of 22}
88 + 66 = 154 {which is a multiple of 11 and of 22}
However, if x and y are not individually divisible by 11, it is possible that x - y is a multiple of 22 (and 11) while x + y is not a multiple of 11. For example:
78 - 56 = 22 but 78 + 56 = 134 is not a multiple of 11.
Statement (1) alone is NOT SUFFICIENT.
Evaluate Statement (2) alone.
Since the tens digit and the units digit of x are the same, the range of possible values for x includes:
11, 22, 33, 44, 55, 66, 77, 88, 99
Since each of these values is a multiple of 11, x must be a multiple of 11.
Since the tens digit and the units digit of y are the same, the range of possible values for y includes:
11, 22, 33, 44, 55, 66, 77, 88, 99
Since each of these values is a multiple of 11, y must be a multiple of 11.
As demonstrated above, if both x and y are a multiple of 11, we know that both x + y and x - y will be a multiple of 11.
Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is SUFFICIENT.

The parabola y = ax² + bx - 2 faces upwards when a > 0 and downwards when a < 0.
In statement 1, a = b, this does not give the numerical equivalent to a, hence it is not sufficient.
In statement 2, a < 0, hence the parabola faces downwards. Thus the statement is sufficient.

Ratio water : alcohol = 2 : 5.
To imply that water = 2/7 × 14 = 4 cups
Alcohol = 14 - 4 = 10 cups.
If water is increased by 14%, we have
new capacity of water =114/100 × 4 = 4.56 cups.
The new volume of the liquid is 10 + 4.56 cups. Hence the statement is sufficient.
In statement 2, ratio of water : alcohol = 4 : 5 but we are not told the quantity of the liquid added or the quantity of water or alcohol added, hence, the statement is not sufficient.

2x + 1 > 0
In statement 1, a is an integer, when x = -2, 2x + 1 = -3 < 0. When x = 2, 2x + 1 = 5 > 0, hence the statement is not sufficient.
In statement 2, |x| < 1.5 implies that -1.5 < x < 1.5.
When x = 1.4, 2x + 1 = 3.8 > 0. When x = -1.4, 2x + 1 = -1.8 < 0.
Hence, the statement is not sufficient.
Combining the two statements, we have, x, an integer and -1.5 < x < 1.5, considering the more strict condition, -1.5 < x < 1.5, we find that 2x + 1 < 0, when x = -1.4 and 2x + 1 > 0 when x = 1.4.
Therefore, Statements (1) and (2) TOGETHER are NOT sufficient.

a_n = a + (n - 1)d
In statement 1, third term, a₃ = a + (3 - 1)d = 1.428; a + 2d =1.428.
This is one equation with two unknowns hence we cannot determine the value of d. The statement is not sufficient.
In statement 2, a = 1 and a₅ = a + (5 - 1)d = 1.856
Substituting for a, we have 1 + 4d = 1.856; d = 0.214.
The common difference is 0.214, therefore, the statement is sufficient.

In statement 1, since one of the divisor is 7, it implies that the number is a multiple of 7. This allows us to have infinitely many numbers, hence the statement is not sufficient.
In statement 2, the number is divisible by two numbers only. This implies that p is prime since a prime number is divisible by two positive numbers. Since there are infinitely many prime numbers, the statement is not sufficient.
Combining the two statements, we have p a prime number being divisible by 7, hence p must be 7. Therefore, both statements together are sufficient but neither statement alone is sufficient.

In working on this question, it is helpful to remember that a number will be divisible by 9 if the sum of its digits equals 9.
Evaluate Statement (1) alone.
Based upon the information in Statement (1), it is helpful to plug in a few values and see if a pattern emerges:
101 - 19 = -9
102 - 19 = 81; the sum of the digits is 9, which is divisible by 9, meaning the entire expression is divisible by 9
103 - 19 = 981; the sum of the digits is 9 + 8 + 1=18, which is divisible by 9, meaning the entire expression is divisible by 9
104 - 19 = 9981; the sum of the digits is 9(2) + 8 + 1=27, which is divisible by 9, meaning the entire expression is divisible by 9
Notice that, in each instance, the sum of the digits is divisible by 9, meaning the entire expression is divisible by 9.
The pattern that emerges is that there are (n-2) 9s followed by the digit 8 and the digit 1.
The pattern of the sum of the digits of 10n - 19 is 9(n-2) + 9 for all values of n > 1. (For n = 1, the sum is -9, which is also divisible by 9.) This means that the sum of the digits of 10n - 19 is 9(n-1). Since this sum will always be divisible by 9, the entire expression (i.e., 10n - 19) will always be divisible by 9.
Based upon this pattern, Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
Statement (2) says that z + 981 is a multiple of 9. This can be translated into algebra: 9(a constant integer) = z + 981
Divide both sides by 9

Since 981 is divisible by 9 (its digits sum to 18, which is divisible by 9), you can further rewrite Statement (2).

Since an integer minus an integer is an integer, Statement (2) can be rewritten even further. Since z divided by 9 is an integer, z is divisible by 9. Statement (2) is SUFFICIENT.

Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT.

In statement 1, the number is divisible by a prime factor, hence the number can be 2, 4, 6, -2, -4, -6, 8 , 9 . . . . Hence the statement is insufficient.
In statement 2, the number is positive. But there are infinitely many numbers that are positive but not prime, hence, the statement is not sufficient.
Combining the two statements, we have the numbers being positive, thus, 2,4,6,8,9,10,. . . . The numbers are infinitely many, hence Statements (1) and (2) TOGETHER are NOT sufficient.

Base = 12 inches.
In statement 1, hypotenuse = 13 inches. Using Pythagorean theorem, we have height = √(13² - 12²) = 5.
Area = 1/2 × 5 × 12 = 30 square inches. The statement is sufficient.
In statement 2, perpendicular height = 1/2 (base) - 1 = 1/2 × 12 -1 = 5 inches. Base = 12 inches.
Area =1/2 × 5 × 12 = 30 square inches. The statement is sufficient.

In statement 1, 2t + 6s = 8 is one equation is two unknowns, hence we cannot determine the value of t. The statement is insufficient.
In statement 2, t/2 - 2 = -3s/4 can be transformed to t - 4 = -6s/2. But this is an equation with two unknowns hence we cannot determine the value of t. The statement is insufficient.
Combining the two statements, we have  2, t/2 - 2 = -3s/4 which can be transformed to t - 4 = -6s/2 then to 2t - 8 = -6s but this is equal to the equation 2t + 6s = 8, hence we have an equation with two unknowns. Thus, we cannot determine the value of t; the statements (1) and (2) TOGETHER are NOT sufficient.

Even though lines L and M look parallel and angle BAC looks like a right angle, you cannot make these assumptions.
The formula for the area of a triangle is .5bh
Evaluate Statement (1) alone.
Since EC = 2AC, EA = CA, EC = 2(6) = 12 and line AB is an angle bisector of angle EBC. This means that angle ABC = angle ABE. Since we know that angle ABC = 30, we know that angle ABE = 30. Further, since lines L and M are parallel, we know that line AB is perpendicular to line EC, meaning angle BAC is 90.
Since all the interior angles of a triangle must sum to 180:
angle ABC + angle BCA + angle BAC = 180
30 + angle BCA + 90 = 180
angle BCA = 60
Since all the interior angles of a triangle must sum to 180:
angle BCA + angle ABC + angle ABE + angle AEB = 180
60 + 30 + 30 + angle AEB = 180
angle AEB = 60
This means that triangle BCA is an equilateral triangle.
To find the area of triangle BCE, we need the base (= 12 from above) and the height, i.e., line AB. Since we know BC and AC and triangle ABC is a right triangle, we can use the Pythagorean theorem on triangle ABC to find the length of AB.
6² + (AB)² = 12²
AB² = 144 - 36 = 108
AB = 108^1/2
Area = .5bh
Area = .5(12)(108^1/2) = 6*108^1/2
Statement (1) is SUFFICIENT
Evaluate Statement (2) alone.
The sum of the interior angles of any triangle must be 180 degrees.
DCG + GDC + CGD = 180
60 + 30 + CGD = 180
CGD = 90
Triangle CGD is a right triangle.
Using the Pythagorean theorem, DG = 108^1/2
(CG)² + (DG)² = (CD)²
6² + (DG)² = 12²
DG = 108^1/2
At this point, it may be tempting to use DG = 108^1/2 as the height of the triangle BCE, assuming that lines AB and DG are parallel and therefore AB = 108^1/2 is the height of triangle BCE. However, we must show two things before we can use AB = 108^1/2 as the height of triangle BCE: (1) lines L and M are parallel and (2) AB is the height of triangle BCE (i.e., angle BAC is 90 degrees).
Lines L and M must be parallel since angles FDG and CGD are equal and these two angles are alternate interior angles formed by cutting two lines with a transversal. If two alternate interior angles are equal, we know that the two lines that form the angles (lines L and M) when cut by a transversal (line DG) must be parallel.
Since lines L and M are parallel, DG = the height of triangle BCE = 108^1/2. Note that it is not essential to know whether AB is the height of triangle BCE. It is sufficient to know that the height is 108^1/2. To reiterate, we know that the height is 8 since the height of BCE is parallel to line DG, which is 108^1/2.
Since we know both the height (108^1/2) and the base (CE = 12) of triangle BCE, we know that the area is: .5*12*108^1/2 = 6*108^1/2
Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT

Evaluate Statement (1) alone.
Based upon the formula for the average, you know that:
(5 + x2 + 2 + 10x + 3)/5 = -3
x2 + 10x + 5 + 2 + 3 = -15
x2 + 10x + 5 + 2 + 3 + 15 = 0
x2 + 10x + 25 = 0
(x + 5)2 = 0
x = -5
Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
2) The median of 109, -32, -30, 208, -15, x, 10, -43, 7 is -5
Notice that the given list of numbers has 9 numbers. So median must be the 5th number when all numbers are arranged in increasing order. Median is -5 but none of the numbers are -5. Hence x must be -5. Sufficient alone.
Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT

Evaluate Statement (1) alone.
Substitute the value of x from Statement (1) into the equation and manipulate it algebraically.

Since the question says that y is a positive integer, you know that the cube root of y³, which equals y, will also be a positive integer. Statement (1) is SUFFICIENT.
Evaluate Statement (1) alone (Alternative Method).
For the cube root of a number to be an integer, that number must be an integer cubed. Consequently, the simplified version of this question is: "is x + y² equal to an integer cubed?"
Statement (1) can be re-arranged as follows:
x = y³ - y²
y³ = x + y²
Since y is an integer, the cube root of y³, which equals y, will also be an integer.
Since y³ = x + y², the cube root of x + y² will also be an integer. Therefore, the following will always be an integer:

Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
Statement (2) provides minimal information. The question can be written as: "is the following cube root an integer?"

If y = 4, x + y² = 2 + 4² = 18 and the cube root of 18 is not an integer. However, if y = 5, x + y² = 2 + 5² = 27 and the cube root of 27 is an integer. Statement (2) is NOT SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is NOT SUFFICIENT