Tips to crack GMAT Math

GMAT Descriptive Statistics : Averages

2012-01-23T17:43:00.002+05:30

Question

If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers?
A. 50
B. 52
C. 49
D. 48
E. 44

Correct Answer : Choice D. 48

Explanation

The average of 5 positive integers is 40. i.e., the sum of these integers = 5*40 = 200

Let the least of these 5 numbers be x.
Then the largest of these 5 numbers will be x + 10.

If we have to maximize the largest of these numbers, we have to minimize all the other numbers.

That is 4 of these numbers are all at the least value possible = x.
So, x + x + x + x + x + 10 = 200
Or x = 38.
So, the largest of these 5 integers is 48.

You could get additional questions on GMAT descriptive statistics on our question bank.

Or you could download and buy an eBook on Descriptive Statistics

Geometry Triangles Data Sufficiency

2012-01-18T19:01:00.002+05:30

Here is a DS question on Geometry on basic properties of triangles.

Is triangle ABC with sides a, b and c acute angled?

1. Triangle with sides a^2, b^2, c^2 has an area of 140 sq cms.

2. Median AD to side BC is equal to altitude AE to side BC.

The correct answer is Choice A. Statement 1 alone is sufficient, while statement 2 is not.

Explanation

From the information given we have to determine if the given triangle is an acute angled triangle.

The information given to us is the measure of the sides of the triangle.

If a, b and c measure the sides of a triangle, and let us say 'a' is the longest side of the triangle, then
1. the triangle is acute angled if a^2 < b^2 + c^2
2. right angled if a^2 = b^2 + c^2 and
3. obtuse angled if a^2 > b^2 + c^2

Now let us evaluate the statements given to us

Statement 1: Triangle with sides a^2, b^2, c^2 has an area of 140 sq cms.
The statement provides us with one valuable information. We can form a triangle with sides a^2, b^2, c^2.

For any triangle we know that sum of two sides is greater than the third side.
So, a^2 < b^2 + c^2

That is the condition to be satisfied for a triangle with sides a, b and c to be an acute angled triangle.
Statement 1 answers in the positive and the information is sufficient.

Statement 2: Median AD to side BC is equal to altitude AE to side BC
We can infer that the triangle is either equilateral or isosceles. An equilateral triangle is definitely an acute angled triangle. However, an isosceles triangle need not be an acute angled triangle.

So, statement 2 is not sufficient.

Statement 1 is sufficient, while statement 2 is not sufficient. Choice A is the correct answer.

Mean, Median - GMAT Descriptive Statistics

2011-11-09T10:10:00.002+05:30

Hi

Questions testing you understanding of the link betwen Mean and Median are frequently tested in the GMAT.

Here is a basic but interesting question on the combo.

Question

Positive integers from 1 to 45, inclusive are placed in 5 groups of 9 each. What is the highest possible average of the medians of these 5 groups?
A. 25
B. 31
C. 15
D. 26
E. 23

Correct Answer
Choice B. 31 is the highest possible average of the medians.

Explanatory Answer
We need to maximize the median in each group in order to maximize the average of all the medians.

The highest possible median is 41 as there should be 4 numbers higher than the median in the group of 9.

So, if we have a group that has a, b, c, d, 41, 42, 43, 44, 45, the median will be 41.
In this set, it is essential not to use any more high values on a, b, c, or d as these do not affect the median.
So, the median of a group that comprises 1, 2, 3, 4, 41, 42, 43, 44, 45 will be 41.

The next group can be 5, 6, 7, 8, 36, 37, 38, 39, 40. The median will be 36.

To maximize medians in all the 5 groups, the medians of the 5 groups will have to be 21, 26, 31, 36 and 41.

The highest possible average of the medians will be the average of these 5 numbers = 31.

GMAT Geometry DS

2011-10-10T19:34:00.000+05:30

Directions

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question
Is triangle ABC obtuse angled?
I) a^2 + b^2 > c^2
II) The center of the circle circumscribing the triangle does not lie inside the triangle

The correct answer is Choice E. Data is insufficient.

Explanatory Answer
In an obtuse angled triangle, if 'c' is the longest side, then c^2 > a^2 + b^2

Statement 1
a^2 + b^2 > c^2.
We have no information about whether c is the longest side in the triangle. Hence, we cannot answer the question. Statement 1 is INSUFFICIENT.

Statement 2
The center of the circle circumscribing the triangle does not lie inside the triangle.

> For an acute angled triangle, the center of the circle circumscribing the triangle lies inside the triangle.
> For a right triangle, the center of the circle circumscribing the triangle lies at the mid point of the hypotenuse.
> For an obtuse angled triangle, the center of the circle circumscribing the triangle lies outside the triangle.

From statement 2, we can deduce that the triangle is not an acute angled triangle. It may be a right angled triangle or an obtuse angled triangle. Hence, statement 2 is also INSUFFICIENT.

Combining the two statements, we cannot deduce anything more than what we could deduce using the information from the two statements independently.

Hence, Choice E is the correct answer.

You could get additional GMAT Data Sufficiency Practice questions here.

GMAT Coordinate Geometry

2011-08-23T07:02:00.002+05:30

Here is a question in coordinate geometry. The primary idea tested by this question is one's understanding about equations of straight lines.

Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (8, 0) and the point (0, 12) is the lowest among all elements in set S. How many such points P exist in set S?
A. 1
B. 5
C. 11
D. 8
E. 3

The correct answer is Choice E. 3 points.

Explanatory Answer

The sum of the distances from point P to the other two points will be at its lowest only when point P lies on the line segment joining the points (8, 0) and (0, 12).

The equation of the line segment joining the points (8, 0) and (0, 12) is

Or the equation is 12x + 8y = 96 or 3x + 2y = 24.

We know the elements of set S contain points whose abscissa and ordinate are both natural numbers.

The equation of the line is 3x + 2y = 24 and hence, x will take even values while y will take values that are multiples of 3.

The values are x = 2, y = 9; x = 4, y = 6; x = 6, y = 3.

Hence, there are 3 such points that exist in set S.

Number Properties - Product of factors

2011-07-21T13:50:00.002+05:30

What is the product of all positive factors of 18?
A. 324
B. 5832
C. 39
D. 1521
E. 3042

The correct answer is 5832. Choice B.

Let us first focus on how to find the answer methodically and then understand why the method works.

The How?

Step 1: Find the number of factors for 18.
Step 1a. Express 18 as a product of its prime factors. 18 = 2 * 3^2
Step 1b. Increment the powers of each of the prime factors. The power of 2 is 1. The power of 3 is 2. After incrementing, we get (1 + 1) and (2 + 1)
Step 1c. Number of factors = Product of the incremented numbers = (1 + 1)(2 + 1) = 2 * 3 = 6

Step 2: The product of all factors of 18 = 18^(number of factors/2)
The product = 18^(6/2) = 18^3 = 5832.

The Why?
The factors of 18 are 1, 2, 3, 6, 9 and 18.
Product of the factors = 1*2*3*6*9*18.
We can rewrite this as (1*18)*(2*9)*(3*6) = 18 * 18 * 18.

So, if a positive integer 'n' has 'x' factors, then the product of all its factors = n^(x/2).

What if the number has odd number of factors?
Try the method for a perfect square such as 36 or 16.

GMAT DS : Modulus

2011-04-08T13:42:00.001+05:30

Is xy < 0?

(1) 5|x| + |y| = 0

(2) |x| + 5|y| = 0

Descriptive Statistics - Mean and Median; Divisibility

2010-11-10T10:27:00.002+05:30

How many numbers of 7 consecutive positive integers are divisible by 6?
1) Their average is divisible by 6
2) Their median is divisible by 12

Number Properties - divisibility and remainders

2010-11-08T16:48:00.004+05:30

When a positive integer A is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainders obtained are 3 and 4, respectively. Which of the following is a factor of (A - B)?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

Correct Answer : Choice C

Explanatory Answer
When A is divided by 5, the remainder is 3. So, we can express A = 5x + 3
When B is divided by 5, the remainder is 3. So, we can express B = 5y + 3

So, (A - B) = 5x + 3 - (5y + 3) = 5(x - y). So, (A - B) is a multiple of 5.

Similarly, when A is divided by 7, the remainder is 4. So, we can express A = 7p + 4
When B is divided by 7, the remainder is 4. So, we can express B = 7q + 4

So, (A - B) = 7p + 4 - (7q + 4) = 7(p - q). So, (A - B) is a multiple of 7.

Combining the two results, we can conclude that (A - B) is a multiple of both 5 and 7.
i.e., (A - B) will be a multiple of 35.

Additional questions on Number Properties and Number Theory

DS - Quadratic, Parabola, and Inequalities

2010-11-08T16:01:00.003+05:30

In xy-plane, Y = ax^2 + bx + c, does the graph intersect with X axis?
1) a > 0
2) c < 0

Number properties - Factorial

2010-11-08T15:53:00.000+05:30

How many trailing zeroes does 60! have?
(A) 12
(B) 9
(C) 15
(D) 14
(E) 13

Permuations and digits

2010-11-08T12:24:00.002+05:30

1. How many six digit positive integers comprising only the digits 1 and 2 exist such that the number is divisible by 3?

For a number to be divisible by 3, the sum of the digits must be divisible by 3.
Case 1: 111111: This can rearrange in only one way.
Case 2: 222222: This can rearrange in only one way.
Case 3: 111222: This can rearrange in 6!/(3! *3!) = 20 ways

The total number of ways = 1 + 1 + 20 = 22.

2. How many five digit positive integers comprising only the digits 1, 2, 3, and 4, each appearing at least once, exist such that the number is divisible by 4?

For a number to be divisible by 4, the last two digits need to be divisible by 4. The last two digits can be 12, 24, 32, and 44.

Case 1: Ending with 12:
(a) The first three digits can be 2, 3, and 4. This can rearrange in 3! = 6 ways
(b) The first three digits can be 1, 3, and 4. This can rearrange in 3! = 6 ways
(c) The first three digits can be 3, 3, and 4. This can rearrange in 3!/2! = 3 ways
(d) The first three digits can be 3, 4, and 4. This can rearrange in 3!/2! = 3 ways
Total = 18 ways.

Case 2: Ending in 24: similar to case 1: 18 ways

Case 3: Ending in 32: Similar to case 1: 18 ways

Case 4: Ending in 44:
The first 3 digits are 1, 2, and 3. This can rearrange in 3! = 6 ways.

TOTAL: = 18 + 18 + 18 + 6 = 60 ways.

Descriptive statistics - Standard Deviation

2010-10-27T17:45:00.004+05:30

If m is the arithmetic mean (average) of 4 integers a, b, c, and d and s is the standard deviation of the 4 integers and s = , then is s > 0?

(1) m > a
(2) a + b + c + d = 0

Solution:

s will be zero only in two instances: (i) when all the elements in the set are the same, or (ii) the set contains only one element, which in this case is not possible. So, we need to check whether a, b, c, and d are the same integers.

Statement (1): m > a
The average will be equal to a, b, c, and d only when a = b = c = d. Since m > a, all the elements in the set cannot be the same, and therefore, s > 0.
SUFFICIENT

Statement (2): a + b + c + d = 0
When a = b = c = d = 0, s = 0
When a = -4, b = 0, c = 0, and d = 4, s > 0
NOT sufficient

Answer: A

Equations - roots and powers

2010-10-27T17:39:00.004+05:30

How many roots does x⁶ –12x⁴ + 32x² = 0 have?
(A) 1
(B) 3
(C) 4
(D) 5
(E) 2

Solution:

x^6 –12(x^4) + 32(x^2) = x^2 (x^4 – 12x^2 + 32) = x^2 (x^2 – 4)(x^2 – 8)

The roots are 0, ±2, and ±2√2: a total of 5.
Please remember that ‘0’ is also a root.
The correct answer is D

Answer to Descriptive Statistics, Prime Numbers, Progression

2010-10-27T17:14:00.003+05:30

The ages of three friends are prime numbers. The sum of the ages is less than 51. If the ages are in Arithmetic Progression (AP) and if at least one of the ages is greater than 10, what is the difference between the maximum possible median and minimum possible median of the ages of the three friends?

(A) 0
(B) 1
(C) 13
(D) 6
(E) 8

The ages are prime numbers in arithmetic progression i.e., they have a common difference. Furthermore, at least one of them is greater than 10.
Let the ages be a, b, and c such that a < b < c.
a + b + c < 51

Since a, b, and c are in AP, a + b + c = 3b
3b < 51 or b < 17

The largest prime number less than 17 is 13.

The maximum median is 13 (when the ages are 3, 13, and 23 OR 7, 13, and 19).
The minimum median is 7 (when the ages are 3, 7, and 11)

Remember that 3, 5, and 7 is not accepted as atleast one number has to be greater than 10.

The required difference is 13 – 7 = 6.

The corrcet answer is D

Descriptive Statistics, Prime Numbers, Progressions

2010-10-21T11:29:00.003+05:30

This question evaluates your concept and understanding of 3 topics. The core topic tested is that of Descriptive Statistics. Basic understanding of Prime Numbers and Progressions is definitely needed.

The ages of three friends are prime numbers. The sum of the ages is less than 51. If the ages are in Arithmetic Progression (AP) and if at least one of the ages is greater than 10, what is the difference between the maximum possible median and minimum possible median of the ages of the three friends?

(A) 0
(B) 1
(C) 13
(D) 6
(E) 8

Answer to the Number Properties DS question

2010-10-21T11:28:00.004+05:30

Is the two digit positive integer P a prime number?
1) P + 2 and P – 2 are prime.
2) P – 4 and P + 4 are prime.

Statement (1): P + 2 and P – 2 are prime.
One out of 3 consecutive odd integers, (P - 2), P, and (P + 2) will definitely be a multiple of '3'. If (P + 2) and (P - 2) are prime, then P has to be a multiple of '3', which is not prime. The only exception is if the 3 consecutive odd numbers are 3, 5 and 7. However, we are dealing with two digit positive integers.
SUFFICIENT.

Statement (2): P – 4 and P + 4 are prime.
One out of 3 consecutive odd integers, (P - 4), P, and (P + 4) will definitely be a multiple of '3'. If (P + 4) and (P - 4) are prime, then P has to be a multiple of '3', which is not prime. The only exception is if the 3 consecutive odd numbers are 3, 7 and 11. However, we are dealing with two digit positive integers.
SUFFICIENT.

Hence, the correct answer is D.

GMAT Number Properties DS Question

2010-10-08T11:58:00.001+05:30

Number Properties is an often tested area as far as Data Sufficiency is concerned. Here is a simple and interesting question.

Question
Is the two digit positive integer P a prime number?
1) P + 2 and P – 2 are prime.
2) P – 4 and P + 4 are prime.

GMAT Descriptive Statistics

2010-09-12T08:57:00.007+05:30

Descriptive Statistics that includes mean, weighted mean, median, standard deviation and range is a GMAT favorite. At times it becomes "virulent" in its data sufficiency avatar.

Here is a medium difficulty DS question from Descriptive Statistics:

Directions
This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question
What is the median of five integers 20, 22, 26, 30 and 'a'?
1. The mean of the 5 integers is equal to the median of the 5 integers.
2. The range of the 5 integers is greater than 'a'.

The correct answer to this question is B - Statement 2 alone is sufficient to answer this question.

Explanation

From the question stem we know that 'a' is an integer and the median is one of the following 3 values : 22 or 26 or a.

Statement 1 : The mean of the 5 integers is equal to the median of the 5 integers.

The mean of the 5 integers is (20 + 22 + 26 + 30 + a)/5 = (98 + a)/5
As the mean and median are same from statement 1,
When the median is 22, we have 22 = (98 + a)/5 or a = 12.
When the median is 26, we have 26 = (98 + a)/5 or a = 32
When the median is a, we have a = (98 + a)/5 or a = 24.5

However, as 'a' is an integer, it can be either 12 or 32 and hence the median is either 22 or 26.

Statement 1 is NOT sufficient as we get 2 values for the median.

Statement 2: The range of the 5 integers is greater than 'a'.

If 'a' is less than 15, 'a' will always be less than the range. The median will be
22 for all values of 'a' less than 15.

Hence, statement 2 alone is SUFFICIENT to answer the question.

Choice B is the answer.

You can access additional questions on Descriptive Statistics on the GMAT question bank.

If you want a comprehensive material to crack this topic, you can download our GMAT Descriptive Statistics eBook.

4GMAT : Acute Angled Triangles

2010-02-22T12:00:00.002+05:30

Geometry concepts such as basic properties of triangles and how to find if a given triangle is a right triangle or an acute angled triangle or an obtuse angled triangle are often tested in GMAT.

Here is a question using two basic rules about triangles.

Question
If 10, 12 and 'x' are sides of an acute angled triangle, how many integer values of 'x' are possible?
(A) 7
(B) 12
(C) 9
(D) 13
(E) 11

Correct Answer
The correct choice is (C) and the correct answer is 9 values.

Explanatory Answer
Finding the answer to this question requires one to know two rules in geometry.

Rule 1: For an acute angled triangle, the square of the LONGEST side MUST BE LESS than the sum of squares of the other two sides.

Rule 2: For any triangle, sum of any two sides must be greater than the third side.

The sides are 10, 12 and 'x'.

From Rule 2, x can take the following values: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 – A total of 19 values.

When x = 3 or x = 4 or x = 5 or x = 6, the triangle is an OBTUSE angled triangle (Rule 1 is NOT satisfied).

The smallest value of x that satisfies BOTH conditions is 7. (102 + 72 > 122).

The highest value of x that satisfies BOTH conditions is 15. (102 + 122 > 152).

When x = 16 or x = 17 or x = 18 or x = 19 or x = 20 or x = 21, the triangle is an OBTUSE angled triangle (Rule 1 is NOT satisfied).

Hence, the values of x that satisfy both the rules are x = 7, 8, 9, 10, 11, 12, 13, 14, 15. A total of 9 values.

You could access more questions on Geometry at the following location on our website:
http://questionbank.4gmat.com/mba_prep_sample_questions/geometry/index.shtml

GMAT Inequalities, Quadratic Equation

2009-12-30T11:53:00.003+05:30

Inequalities is a big favorite of the GMAT test makers. Inequalities, especially presented as a data sufficiency question that appear in the GMAT test are many a times potential land mines.

Here is an inequality question in the problem solving format

Which of the following is correct if x is a real number and (x - 11)(x - 3) is negative?
A. x^2 + 5x + 6 < 0
B. x^2 + 5x + 6 > 0
C. 5 - x < 0
D. x - 5 < 0
E. 11 - x > 0

The question states that (x - 11)(x - 3) < 0
The solution to this inequality can be obtained as follows:

Case 1: When (x - 11) < 0 and (x - 3) > 0, the product will be negative.
Simplifying, we get x < 11 and x > 3. i.e., 3 < x < 11.

Case 2: When (x - 11) > 0 and (x - 3) < 0, the product will be negative
Simplifying, we get x > 11 and x < 3. This is an impossible solution as x cannot simultaneously be greater than 11 and less than 3. So, infeasible.

So, the solution set to the inequality is 3 < x < 11.

Now, let us look at the options.

Choice (A)
x^2 + 5x + 6 < 0.
Factorizing the quadratic expression, we get (x + 2)(x + 3) < 0.
As we did with the expression in the question, we could get the solution set as -3 < x < -2. We know that the value of x as stated in the question lies between 3 and 11. So, this is not possible.

Choice (B)
x^2 + 5x + 6 > 0.
The solution set will be exactly the opposite as that for choice A. so, x < -3 and x > -2. The question has values of x lying between 3 and 11, which satisfy the condition x > -2.

Hence, choice B is the correct answer.

GMAT DS : Number Theory

2009-11-06T19:37:00.006+05:30

Number Theory is a big hit with the GMAT test setters, especially when setting Data Sufficiency Questions.

Here is a GMAT DS practice question from the Number Theory topic.

Directions
This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Question
When Y is divided by 2, is the remainder 1?
1. (-1)^(Y+2) = -1
2. Y is prime

The given question is an "Is" question. "IS" questions have to answered with an unswerving YES or NO. If your answer to this question is SOMETIMES YES and SOMETIMES NO or in other words MAYBE, then you have not answered the question.

Let us evaluate statement 1.

(-1)^(Y+2) = -1.
(-1)ODD NUMBER = -1

Therefore, Y + 2 is an odd number.
Hence, Y has to be an odd number.

So, when Y is divided by 2, the remainder is 1.

Statement 1 is sufficient.
The answer is either choice (A) or choice (D).

Now let us evaluate the statement 2.

Y is prime

Y could be '2' which is an even number.
So, when Y is divided by 2, the remainder is '0'.

All other prime numbers are odd numbers.
So, when Y is divided by 2, the remainder is '1'.

We cannot conclude is Y is 2 or other prime numbers.

As we are not able to conclude if Y is an even number or an odd number with statement 2, it is not sufficient.

Hence, answer is choice (A ).

Number of diagonals in convex polygon : GMAT

2009-10-24T11:14:00.002+05:30

Any n-sided convex polygon with more than 3 sides will have n(n-3)/2 diagonals.

For instance, let us look at a square. A square has 4 sides and 2 diagonals. Let us apply this formula with n = 4.

We get 4(4-3)/2 = 2 diagonals.

Here is a question on finding the number of diagonals.

If a n-sided convex polygon has 14 diagonals, how many sides does the polygon have?

Any n-sided convex polygon has n(n-3)/2 diagonals.
This polygon has 14 diagonals.
i.e., n(n-3)/2 = 14
Or n(n-3) = 28
Solving for n, we get n = 7.

So, the given polygon has 7 sides.

You can access sample practice questions on Geometry for your GMAT Prep by clicking here

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GMAT PS : Quadratic Equation : Parabola cutting x-axis

2009-07-16T17:07:00.005+05:30

Here is a question testing concepts of Quadratic equation and nature of roots of quadratic equation.

If y = x² + dx + 9 does not cut the x-axis, then which of the following could be a possible value of d?
I. 0
II. -3
III. 9

A. III only
B. II only
C. I and II only
D. II and III only
E. I and III only

Correct Answer : Choice C. Values that 'd' could take are 0 or -3

Explanation
The question states that the curve (parabola) does not cut the x-axis.

If y = x² + dx + 9 cuts the x-axis then, the points at which it cuts the x-axis will be the roots of the quadratic equation x² + dx + 9 = 0.

As any point on the x-axis will be a value on the number line, the roots will be real numbers.

However, if the curve does not cut the x-axis, then roots of the quadratic equation will be imaginary.

For a quadratic equation "ax² + bx + c = 0 to have imaginary roots, the discriminant b² - 4*a*c < 0 (the discriminant should be negative).

In this equation, d² - 36 < 0
Or d² < 36
i.e., -6 < d < 6.

Amongst the values given, d = 0 and d = -3 lie in this range.
Hence, choice C.

GMAT PS QOTW : Number Properties

2009-04-29T13:40:00.002+05:30

This question is an interesting Problem Solving question from the topic number properties.

How many digits does the product of 4¹² and 5²³ contain?
A. 12
B. 13
C. 23
D. 24
E. 35

The correct answer is choice D.

We can rewrite the given numbers as 2²⁴ * 5²³.
i.e., 2 * 2²³ * 5²³
= 2 * 10²³.
This is a 24 digit number.

Number Properties : Data Sufficiency

2009-04-23T17:04:00.003+05:30

Number Properties is an all time favorite with the GMAT test makers. Especially in the DS avtar, number properties questions could be quite potent.

Here is a seemingly innocuous question from Number Properties presented as a DS question.

Question

Is ab positive?
(1) (a+b)^2 < (a-b)^2
(2) a = b

It is an "Is" question. So, the answer has to be a definite YES or a definite NO. It cannot be a MAYBE.

Let us evaluate statement 1.
a^2 + b^2 + 2ab < a^2 + b^2 - 2ab
Simplifying we get, 4ab < 0 or ab < 0.
So, we can convincingly answer that ab is not positive. So, statement 1 is sufficient to answer the question.

The correct answer is either A or D.

Now let us evaluate the statement 2. This is actually the statement that could trick you.

a = b.
So, either both a and b or positive or both a and b are negative. In either case ab is positive.
We will certainly be "tempted" to decide that statement 2 is also sufficient.
The catch is that, both a and b could be 0. In that case ab = 0, which is not positive.
As we are not able to conclude if ab is positive or not with statement 2, it is not sufficient.

So, choice A is the correct answer.

Quadratic Equations : Sum of roots, product of roots

2009-04-08T14:23:00.003+05:30

Quadratic equations are equations of the form ax² + bx + c = 0.

A quadratic equation has two roots. These roots are found either by factorizing the quadratic equation or by using the formula (-b + root (b² - 4ac))/2a and (-b - root (b² - 4ac))/2a

Here is a typical quadratic equation question

If m and n are the roots of the quadratic equation x² - (2 root 5)x - 2 = 0, the value of m² + n² is:

A. 22
B. 24
C. 32
D. 20
E. 18

Correct Answer is Choice B. 24.

Explanation

m and n are roots of the equation.

We have to find the value of m² + n²

m² + n² = (m + n)² - 2mn

(m + n), the sum of the roots of a quadratic equation of the form ax² + bx + c = 0 is (-b/a)

mn, the product of the roots of the equation = c/a

The sum of the roots of the equation x² - (2 root 5)x - 2 = 0 is (2 root 5).
Product of the roots of the equation = -2.

Hence, (m + n)² - 2mn = (2 root 5)² - 2(-2) = 20 + 4 = 24.

GMAT DS : Geometry, Coordinate Geometry

2009-03-18T19:04:00.002+05:30

Here is a data sufficiency question. It is a question on slopes of lines and tests basic concepts about lines in geometry and coordinate geometry.

Question

Are lines p (with slope m) and q (with slope n) perpendicular to each other?
1. m + 2 = n
2. m + n = 0

Correct Answer: Choice C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Explanatory Answer

If two lines are perpendicular, then the product of the slopes of the two lines will be equal to -1.

In this case, if the product m * n = -1, then the two lines will be perpendicular to each other. If the product is not equal to -1, then they are not perpendicular. We need to assess that conclusively.

Statement 1 m + 2 = n
m could be -1 and n could be 1, in which case the product is -1. Alternatively, m could be 4 and n could be 6 in which case the product is not -1.

As we are not able to conclude using the information in statement 1, it is not sufficient. Choices A and D can be eliminated. We are left with choices B, C or E.

Statement 2 m + n = 0.
m could be -1 and n could be 1 or vice versa. In that case, m * n = -1.
m could be any other number and n could be -m. In that case m * n will not be equal to -1. Hence, statement 2 is also not sufficient. We can eliminate choice B. We are left with choices C or E.

Combining the two statements, we know that m = -n from statement 2. Substituting that in statement 1, we get m + 2 = -m or 2m = -2 or m = -1. Hence, n = 1. Hence, the product m * n = -1.

As the information provided in the two statements is sufficient to answer the question, choice C is the correct answer.

Data Sufficiency : Number Properties

2009-03-13T11:30:00.005+05:30

About a third of the questions that appear on the Quantitative section of the GMAT CAT test are data sufficiency (DS) questions. Here is a DS question from Number properties and number theory. Number properties and number theory is a hot favorite when it comes to setting DS questions.

For the uninitiated amongst us, this is how the instructions to a DS question will look like.

Directions

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Numbers

All numbers used are real numbers.

Figures

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Note: In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question

What is the value of b?

1) a = 3
2) (a-3)(b+2)=0

The correct answer is E. i.e., Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Explanatory Answer
Statement 1: It is obvious that statement 1 is not sufficient. Data given does not provide any information about the value of b. So, we can eliminate choices A and D. So we are down to choices B, C or E.

Statement 2: An equation with two variables 'a' and 'b'. Hence, the information provided is not sufficient.

Note do not make the mistake of solving the equation as either (a - 3) = 0 or (b + 2) = 0 and decide that b = -2. What if only (a - 3) = 0 and (b + 2) was not '0'.

Combining the two statements, we know a = 3, so, (a - 3) = 0. But that leaves the question of what (b + 2) or what 'b' is unanswered.

Hence, choice E is the correct answer.

Data sufficiency questions are usually quite tricky. You can ensure success in cracking these questions only with adequate practice. Any serious aspirant should solve about 250 to 300 data sufficiency questions before taking the actual GMAT test.

Permutation Combination and Probability - Sampling with / without ordering

2005-10-22T10:17:00.000+05:30

The next parameter on which sampling can be classified is Sampling based on whether Ordering (Arrangement) of the elements selected is considered or not.

In this case too, as in the case of sampling with or without replacement let us look at two examples that will help us learn the concept better.

Example 1. In how many ways can a group of students elect a President and a Vice President from 10 contestants if a person cannot hold more than one post?

It is quite obvious that the president can be elected from the 10 contestants in 10 ways and the vice president can be elected from the remaining 9 students in 9 ways. As the group of students elect a president and a vice president, the total number of ways = 10 * 9 = 90.

Let the contestants be recognized by the letters A to J.
Any one of the A to J could have been elected as the president.
Now, let us say, B was elected the president. Then, anyone of the remaining 9 contestants can be elected as vice president. Say, D was elected vice president.

You will realize that the 90 outcomes include the case of "B" being the president and "D" being the vice president and "B" being the vice president and "D" being the president.

i.e., for B and D being the two contestants who were elected, there are two possibilities - BD or DB and therefore, this is an example of sampling with Ordering

GMAT Tips - Do not skip questions

2005-04-16T15:10:00.002+05:30

Here is an insight from the official GMAT® test organizers - GMAC® - The graduate Management Admissions Council.

The GMAC® briefing to the test-preparation community on the transition of the GMAT® exam to new vendors in March - April 2005 had GMAC® Chief Psychometrician Larry Rudner fielded a host of questions about the underpinnings of the computer algorithm that scores the test. He emphasized that candidates must complete every question on the GMAT® exam; skipping a question is not an option, and test takers are best advised to make an educated guess if they are unsure of an answer.

To read the full text of the news, you can visit the following URL
http://www.gmac.com/gmac/VirtualLibrary/Publications/GMNews/2005/MarchApril/GMACTestPrepBriefing.htm

You can also visit the following URL to get an insight into how some students cracked the GMAT and secured admissions from top B Schools.
How I Cracked GMAT?

Permutation Combination and Probability - Sampling with / without replacement

2004-11-17T22:27:00.000+05:30

Hi

These two topics are invariably considered by many GMAT aspirants to be tougher than a whole lot of other topics in the quant section of the GMAT test.

The biggest advantage with a Permutation Combination or a Probability question is that many a times you will not have the need to do any calculation at all. Most answers will be either mentioned in terms of factorials or as 3C2 or 4P3 format.

Having said that, the main catch with these two topics is that you need to think a little differently while attempting questions from these two topics as compared to say a question from profit and loss or speed time and distance. Once you have managed to figure that difference, it is one of the easiest topics to work on.

The common question that many of my students seem to be coming back to me with is whether the given question is a combination question or a permutation question?

Let me see how that can be resolved with the help of the following examples on permutation and combination.

To begin with let us introduce the term sampling. A sampling is an activity of picking up (selecting) none or more or all items from a group of items.

For e.g. let us consider the act of selecting 3 students from a class of 10 students to attend a seminar. This act of selecting 3 students from 10 students is commonly referred to in the Permutation Combination (Combinatorics / counting methods) parlance as sampling.

Sampling can be classified broadly in terms of two parameters.

1. Sampling with replacement or sampling without replacement
2. Sampling with ordering or sampling without ordering (arrangement or without arrangement)

In the example provided above, we have to choose 3 different students from 10 students. After the 1st student is selected, we have only 9 students in the class to choose the 2nd student. After the 2nd student is selected, we have only 8 students in the class to choose the 3rd student. In effect, we had done 3 samplings - 1 for choosing the 1st student, 2 for choosing the 2nd student and the last one for selecting the third student.

Well as you would have noticed, in the above example the number of students available after each round of sampling kept decreasing. i.e., while sampling the second time, we only had 9 students from whom we could select one student. This kind of sampling where we did not replace the outcome of the earlier sampling back is know as "Sampling without replacement".

Let us take a look at another example where we will replace the outcome of the earlier samplings back so that it (they) is (are) available for the subsequent stages of sampling. Let us say we are interested in writing a 3-digit positive integer. The first place or the hundreds position can take any of the 9 values from 1 to 9 (it cannot include "0"). The second position or the tens place can take any of the 10 values 0 to 9 and the third place or the units digit can be any of the 10 values 0 to 9.

Take for instance, a number like 242. The digit "2" appears both in the hundreds as well as the units place. Writing the 3 digit number actually involved 3 samplings. The first for the hundreds place, second for the tens place and finally for the units place. As the number 2 was available to be used again when the sampling was done for the tens and the units place, this is an example of sampling with replacement.

Similar examples can be found while forming words using the alphabets of a language. Take for example the word "Permutation". The letter "t" has been used twice. It is again an example of Sampling with replacement.

Tips to crack GMAT Math

GMAT Descriptive Statistics : Averages

If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers? A. 50B. 52C. 49D. 48E. 44

Correct Answer : Choice D. 48

Explanation

You could get additional questions on GMAT descriptive statistics on our question bank.

Or you could download and buy an eBook on Descriptive Statistics

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Permutation Combination and Probability - Sampling with / without replacement

If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers?
A. 50
B. 52
C. 49
D. 48
E. 44