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Overview of GMAT Math Question Types and Patterns on the GMAT 02 Sep 2020, 23:10
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[GMAT math practice question]

(Number Properties) n(A) denotes the number of positive divisors of a positive integer A. What is the smallest possible value of x satisfying n(280)÷n(30)·n(x) = 12?

A. 12
B. 14
C. 16
D. 18
E. 20

=>

If n = p^a·q^b·r^c has a prime factorization with different prime numbers p, q, and r, then the number of factors of n is (a + 1)(b + 1)(c + 1).

Since 280 = 2^3·5^1·7^1, we have n(280) = (3 + 1)(1 + 1)(1 + 1) = 4·2·2 = 16.
Since 30 = 2^1·3^1·5^1, we have n(30) = (1 + 1)(1 + 1)(1 + 1) = 2·2·2 = 8.

Then we have n(280)n(x)/n(30) = 16n(x)/8 = 2n(x) = 12 or n(x) = 6.

We have two cases of integers with 6 factors. They are p^2q^1 or p^5 where p and q are different prime numbers since (2 + 1)(1 + 1) = 6 and (5 + 1) = 6.

For x = p^2q^1, when we have p = 2 and q = 3, we have the smallest number 2^2·3^1 = 12.
For x = p^5, when we have p = 2, we have the smallest number 2^5 = 32.
The smallest value of x is 12.

Therefore, A is the correct answer.
Answer: A
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A function satisfies f(xy) = f(x) + f(y) for any positive numbers x and y. We have f(2) = 1. What is the value of f(8)?

A. 1
B. 2
C. 3
D. 4
E. 5
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Solution:

f(4) = f(2·2) = f(2) + f(2) = 1 + 1 = 2.
f(8) = f(2·4) = f(2) + f(4) = 1 + 2 = 3.

Therefore, C is the correct answer.

Answer: C.
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n(A) denotes the number of positive divisors of a positive integer A. What is the smallest possible value of x satisfying \(\frac{n(280)·n(x)}{ n(30)}\) = 12?

A. 12
B. 14
C. 16
D. 18
E. 20
Show more


Solution:


If n = \(p^a\)·\(q^b\)·\(r^c\) has a prime factorization with different prime numbers p, q, and r, then the number of factors of n is (a + 1)(b + 1)(c + 1).

Since 280 = \(2^3\)·\(5^1\)·\(7^1\), we have n(280) = (3 + 1)(1 + 1)(1 + 1) = 4·2·2 = 16.

Since 30 = \(2^1\)·\(3^1\)·\(5^1\), we have n(30) = (1 + 1)(1 + 1)(1 + 1) = 2·2·2 = 8.

Then we have \(\frac{n(280) * n(x)}{n(30)}\) = \(\frac{16n(x)}{8}\) = 2n(x) = 12 or n(x) = 6.

We have two cases of integers with 6 factors. They are \(p^2\)\(q^1\) or \(p^5\) where p and q are different prime numbers since (2 + 1)(1 + 1) = 6 and (5 + 1) = 6.

For x = \(p^2\)\(q^1\), when we have p = 2 and q = 3, we have the smallest number \(2^2\)·\(3^1\) = 12.

For x = \(p^5\), when we have p = 2, we have the smallest number \(2^5\) = 32.

The smallest value of x is 12.

Therefore, A is the correct answer.

Answer: A
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(Algebra) We have \(\frac{x}{2}\) = \(\frac{y}{3}\). What is the value of \(\frac{2x}{(x+y)}\) + \(\frac{3y}{(x-y)}\) + \(\frac{x^2}{(x^2-y^2 )}\)?


Solution:


When we assume \(\frac{x}{2}\) = \(\frac{y}{3}\) = k, we have x = 2k and y = 3k.

\(\frac{2x}{(x+y)}\) + \(\frac{3y}{(x-y)}\) + \(\frac{x^2}{(x^2-y^2 )}\)

= \(\frac{(2 * 2k)}{(2k + 3k)}\) + \(\frac{(3 * 3k)}{(2k - 3k)}\) + \(\frac{(2k)^2}{((2k)^2 - (3k)^2 )}\)

= \(\frac{4k}{5k}\) + \(\frac{9k}{(-k)}\) + \(\frac{4k^2}{(4k^2- 9k^2 ) }\)

= \(\frac{4}{5}\) – 9 - \(\frac{4}{5}\) = -9.

Therefore, C is the correct answer.

Answer: C 
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(Geometry) The figure shows that (BP)is a line passing the center O and (PT)is a tangent line to the circle at point T. If ∠APT = \(20^°\), what is ∠x?


Attachment:
PS Circle & Triangle .jpg
PS Circle & Triangle .jpg [ 7.42 KiB | Viewed 2472 times ]


Solution:

Since PT is tangent to the circle, we have ∠OTP = \(90^°\) and ∠AOT = \(180^°\) - \(20^°\)- \(90^°\) = \(70^°\).

Since the triangle is an isosceles triangle with AO = OT, we have ∠OTA = ∠OAT = ∠x.

Thus ∠x = \(\frac{(180^°- 70^°)}{2}\) = \(55^°\).

Therefore, D is the correct answer.

Answer: D
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Q1. (Algebra)

Attachment:
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PS - Algebra.jpg [ 12.34 KiB | Viewed 2464 times ]

The positive integers from 1 up are arranged, as shown in the table above. Nine numbers in the form of a square are chosen from the arrangement. x is the upper-left most number in the square, and the sum of the 9 numbers is 171. What is the value of x?

A. 10
B. 11
C. 18
D. 19
E. 23
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Solution:

Square form: An Equal number of rows and columns make a square. To form a square of 9 numbers: 3 by 3 square has to be selected.

When we select the first row, it will have 3 numbers as per the table. So,
    We will select the 1st, 2nd, and 3rd numbers.
    The next row selection will give us the 8th, 9th, and 10th numbers.
    The third row will give us the 15th, 16th, and 17th numbers.

Given: x is the upper-left most number, so the 1st number is x = 1, and the sum of the 9 numbers is 171.

=> 1st number: x
=> 2nd number: x + 1
=> 3rd number: x + 2 ….. and so on.

So, we get x + (x + 1) + (x + 2) + (x + 7) + (x + 8) + (x + 9) + (x + 14) + (x + 15) + (x + 16) = 171.

=> 9x + 72 = 171
=> 9x = 99
=> x = 11

Therefore, B is the correct answer.

Answer: B
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A function f(x) = ax + 3 satisfies f(-2) = 5 and 4f(-3) - 3f(2) + f(x) = 9. What is the value of x?

A. 10
B. 13
C. 15
D. 18
E. 20
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Solution:

Solution:

If 4f(-3) - 3f(2) + f(x) = 9, we have to find the value of x.

Given: f(x) = ax + 3 satisfies f(-2) = 5.

Substitute x = -2 into f(x) = ax + 3.

=> f(-2) = a(-2) + 3 = 5
=> -2a + 3 = 5
=> -2a = 5 – 3 = 2
=> a = -1

So, we get f(x) = -x + 3.

Therefore, we get f(-3) = -(-3) + 3 = 6 and f(2) = -2 + 3 = 1.

=> Substituting the values of f(-3) and f(2) into 4f(-3) - 3f(2) + f(x) = 9 gives us
4(6) – 3(1) + f(x) = 9 or 24 – 3 + f(x) = 9.

=> Substituting the value of f(x) = -x + 3 into 24 – 3 + f(x) = 9 gives us
24 - 3 - x + 3 = 9 or x = 24 – 9 = 15.

Therefore, C is the correct answer.

Answer: C
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(Number) n are integers between 1 and 100, inclusive. What is the sum of all n’s?

1) \(2^n\)-1 is a multiple of 5.
2) n is a multiple of 4.
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Solution:

Forget conventional ways of solving math questions.

For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem.

Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the sum of all n’s if n are integers between 1 and 100, inclusive.

Follow the second and the third steps: From the original condition, we have 1 variable (n). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3 Principles and choose D as the most likely answer.

Let’s look at each condition separately.

Condition (1) tells us that \(2^n\) - 1 is a multiple of 5.

=> Multiples of 5 ends either with 0 or 5 and Powers of 2 ends with 2, 4, 8, or 6.

=> \(2^n\) - 1 will be a multiple of 5 when the power of 2 ends with a 6. This is only possible when 2 is raised to a power of 4 or a multiple of 4. For example: n = 4 = \(2^n\) - 1 = 16 - 1 = 15 (multiple of 5).

In other words, n should be multiple of 4.

From 1 to 100, we have 24 multiples of 4, starting with 4 and ending with 100. Therefore n = 24.

=> The sum of the 24 numbers will be \(\frac{n}{2}\) * [first term + last term]

=> \(\frac{25}{2}\) * [4 + 100]

=> 1,300

Since the answer is unique, the condition is sufficient, according to CMT 2, which states that the number of answers must be one.

Condition (2) tells us that n is a multiple of 4.

=> From 1 to 100, we have 24 multiples of 4, starting with 4 and ending with 100. Therefore n = 24.

The sum of the 24 numbers:

=> \(\frac{n}{2}\) * [first term + last term]

=> \(\frac{25}{2}\) * [4 + 100]

=> 1,300

Since the answer is unique, the condition is sufficient, according to CMT 2, which states that the number of answers must be one.

EACH condition ALONE is sufficient.

Therefore, D is the correct answer.

Also, according to Tip 1, if both the conditions give the same value, the most probable answer is D. It is about 95% likely that D would be the answer when the value of the Condition (1) is equal to the value of the Condition (2).

The answer is D because (1) = (2).

EACH condition ALONE is sufficient.

Therefore, D is the correct answer.

Answer: D
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(Number) n are integers between 1 and 100, inclusive. What is the sum of all n’s?

1) \(2^n\)-1 is a multiple of 5.
2) n is a multiple of 4.
Show more


Solution:

Forget conventional ways of solving math questions.

For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem.

Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the sum of all n’s if n are integers between 1 and 100, inclusive.

Follow the second and the third steps: From the original condition, we have 1 variable (n). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3 Principles and choose D as the most likely answer.

Let’s look at each condition separately.

Condition (1) tells us that \(2^n\) - 1 is a multiple of 5.

=> Multiples of 5 ends either with 0 or 5 and Powers of 2 ends with 2, 4, 8, or 6.

=> \(2^n\) - 1 will be a multiple of 5 when the power of 2 ends with a 6. This is only possible when 2 is raised to a power of 4 or a multiple of 4. For example: n = 4 = \(2^n\) - 1 = 16 - 1 = 15 (multiple of 5).

In other words, n should be multiple of 4.

From 1 to 100, we have 24 multiples of 4, starting with 4 and ending with 100. Therefore n = 24.

=> The sum of the 24 numbers will be \(\frac{n}{2}\) * [first term + last term]

=> \(\frac{25}{2}\) * [4 + 100]

=> 1,300

Since the answer is unique, the condition is sufficient, according to CMT 2, which states that the number of answers must be one.

Condition (2) tells us that n is a multiple of 4.

=> From 1 to 100, we have 24 multiples of 4, starting with 4 and ending with 100. Therefore n = 24.

The sum of the 24 numbers:

=> \(\frac{n}{2}\) * [first term + last term]

=> \(\frac{25}{2}\) * [4 + 100]

=> 1,300

Since the answer is unique, the condition is sufficient, according to CMT 2, which states that the number of answers must be one.

EACH condition ALONE is sufficient.

Therefore, D is the correct answer.

Also, according to Tip 1, if both the conditions give the same value, the most probable answer is D. It is about 95% likely that D would be the answer when the value of the Condition (1) is equal to the value of the Condition (2).

The answer is D because (1) = (2).

EACH condition ALONE is sufficient.

Therefore, D is the correct answer.

Answer: D
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(Solution):

Adam takes 12 hours to paint a certain wall. In 1 hour, Adam will paint \(\frac{1}{12}\) of the wall.

Ben takes 8 hours to paint a certain wall. In 1 hour, Adam will paint \(\frac{1}{8}\) of the wall.

Adam started work and painted alone for 3 hours. Therefore, the amount of work done by Adam is:

=> \(\frac{1}{12}\) * 3 = \(\frac{1}{4}\).


Adam and Ben together can finish (\(\frac{1}{12}\) + \(\frac{1}{8}\) = \(\frac{5}{24}\)) work in 1 hour.

Ben painted for 1 hour alone at the end. Therefore, the amount of work done by Ben is:

=> 1 * \(\frac{1}{8}\) = \(\frac{1}{8}\).

Total work done:

=> \(\frac{1}{4}\) + \(\frac{1}{8}\) = \(\frac{3}{8}\).

Remaining work:

=> 1 - \(\frac{3}{8}\) = \(\frac{5}{8}\) of the work.

This work was done by Adam and Ben together:

=> \(\frac{5}{24}\) work in 1 hour = \(\frac{5}{8}\) the part of work to be completed in 3 hours since (\(\frac{5}{24}\)) * 3 = \(\frac{5}{8}\).

Hence, Adam and Ben worked together for 3 hours.

Therefore, B is the correct answer

Answer B
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PS- The products of x and \(\frac{7}{15}\), as well as of x and 4\(\frac{1}{12}\) are positive integers. What is the smallest possible value of x?


Solution:

We have to find the smallest possible value of x such that x * \(\frac{7}{15}\) and x * 4\(\frac{1}{12}\) are positive integers.


=> x * \(\frac{7}{15}\) = p where p is a positive integer and x * \(\frac{49}{12}\) = q where q is a positive integer.

x = \(\frac{(15p)}{7}\) ---------------(1) and x = \(\frac{12q}{49}\)---------------(2)

Dividing equation (1) by (2), we get:

=> x/x = \(\frac{\frac{15p}{7}}{\frac{12q}{49}}\)

=> 1 = \(\frac{\frac{15p}{7}}{\frac{12q}{49}}\)

=> 1 = \(\frac{(15p)}{7}\) * \(\frac{49}{(12q)}\)

=> 1 = \(\frac{(35p)}{ (4q)}\)

=> \(\frac{q}{p}\) = \(\frac{35}{4}\)

Hence, p = 4 and q = 35.

x = \(\frac{(15 * 4)}{7}\) = \(\frac{60}{7}\)

Therefore, B is the correct answer.

Answer B
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[GMAT math practice question]

\(a\) and \(b\) are real numbers with \((-3x^a)^b = 81x^{12}\) for any value of \(x\). What is the value of \((-6a^4b^3)^2÷9ab^3÷(-2a^2)^2\)?

A. 1570
B. 1578
C. 1720
D. 1728
E. 1950


Solution:

=> Since \((-3)^bx^{ab} = 3^4x^{12}\), we have \(b = 4\) and \(a = 3.\)

Then \((-6a^4b^3)^2 ÷ 9ab^3 ÷ (-2a^2)^2 = 36a^8b^6 ÷ 9ab^3 ÷ 4a^4 = a^3b^3 = 3^3·4^3 = 27·64 = 1728.\)

Therefore, D is the correct answer.

Answer: D
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[GMAT math practice question]

The figure shows the points \(A, B, C, … , G\) and \(∠B = 70°, ∠C = 68°, ∠D = 78°, ∠E = 82°, ∠F = 86°\) and \(∠G = 88°\). What is \(∠A\)?

Attachment:
7.28PS.png
7.28PS.png [ 11.26 KiB | Viewed 2248 times ]

A. \(58°\)

B. \(60° \)

C. \(68°\)

D. \(70°\)

E. \(88°\)


Solution:

\(∠DBE + ∠CEB = ∠BDC + ∠ECD\)

The sum of all the interior angles of quadrilateral \(ACDF\) and triangle \(GBE\) is \(360° + 180° = 540°.\)
The measure of the angle \(∠A\) is
\(540° – ( 70° + 68° + 78° + 82° + 86° + 88° ) = 68°.\)

Therefore, the correct answer is C.
Answer: C
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[GMAT math practice question]

\(△ABC\) and \(△CDE\) are equilateral triangles, as the figure shows. What is the measure of the \(∠x\)?

Attachment:
7.27PS.png
7.27PS.png [ 9.36 KiB | Viewed 2239 times ]

A. \(45°\)

B. \(50°\)

C. \(55°\)

D. \(60°\)

E. \(65°\)


Solution:

We have \(AC = BC, CD = CE\) and \(∠BCE = ∠DCE = 60°.\)

Triangles \(ACD\) and \(BCE\) are congruent.

Since \(∠EBC = ∠DAC\), we have \(∠x = ∠ACB = 60°.\)

Therefore, D is the correct answer.
Answer: D
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[GMAT math practice question]

(Ratio) At an entrance examination, the ratio of successful male applicants to successful female applicants is \(5:2\). What is the total number of applicants?

1) The ratio of male applicants to female applicants is \(3:2\).

2) The number of successful applicants is \(140\).
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Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Let x and y be the number of male and female applicants, respectively. Let 5k be the number of successful male applicants and 2k be the number of successful female applicants, giving us x = 5k. Then we have to find the total number of applicants, which is equal to x + y.

Follow the second and the third step: From the original condition, we have 3 variables (x, y, and k). To match the number of variables with the number of equations, we need 3 equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

We know that the ratio of successful male applicants to successful female applicants is 5:2; there are 100 successful male applicants and 40 successful female applicants.

Then the number of male applicants is greater than or equal to 100, and the number of female applicants is greater than or equal to 40.

If the number of male applicants is 120 and that of female applicants is 80, then the total number of applicants is 200.
If the number of male applicants is 150 and that of female applicants is 100, then the total number of applicants is 250.

The answer is not unique, and both conditions 1) and 2) together are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions (1) and (2) together are not sufficient.

Therefore, E is the correct answer.

Answer E


In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
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A function f(x) = ax + 3 satisfies f(-2) = 5 and 4f(-3) - 3f(2) + f(x) = 9. What is the value of x?

A. 10
B. 13
C. 15
D. 18
E. 20
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Solution:

Solution:

If 4f(-3) - 3f(2) + f(x) = 9, we have to find the value of x.

Given: f(x) = ax + 3 satisfies f(-2) = 5.

Substitute x = -2 into f(x) = ax + 3.

=> f(-2) = a(-2) + 3 = 5
=> -2a + 3 = 5
=> -2a = 5 – 3 = 2
=> a = -1

So, we get f(x) = -x + 3.

Therefore, we get f(-3) = -(-3) + 3 = 6 and f(2) = -2 + 3 = 1.

=> Substituting the values of f(-3) and f(2) into 4f(-3) - 3f(2) + f(x) = 9 gives us
4(6) – 3(1) + f(x) = 9 or 24 – 3 + f(x) = 9.

=> Substituting the value of f(x) = -x + 3 into 24 – 3 + f(x) = 9 gives us
24 - 3 - x + 3 = 9 or x = 24 – 9 = 15.

Therefore, C is the correct answer.

Answer: C
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A function satisfies f(xy) = f(x) + f(y) for any positive numbers x and y. We have f(2) = 1. What is the value of f(8)?

A. 1
B. 2
C. 3
D. 4
E. 5
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Solution:

f(4) = f(2·2) = f(2) + f(2) = 1 + 1 = 2.
f(8) = f(2·4) = f(2) + f(4) = 1 + 2 = 3.

Therefore, C is the correct answer.

Answer: C.
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Que: A number 4p25q is divisible by 4 and 9; where p and q are the thousands and units digits, respectively. What is the minimum value of \(\frac{p}{ q}\)

(A) \(\frac{1}{8}\)
(B) \(\frac{1}{7}\)
(C) \(\frac{1}{6}\)
(D) \(\frac{2}{5}\)
(E) \(\frac{5}{2}\)
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Solution: Divisibility property of ‘4’: A number is divisible by ‘4’ when its last two digits are divisible by ‘4’.

Therefore, ‘5q’ is divisible by 4.

So, the possible values of ‘q’ are 2 or 6 [both 52 and 56 are divisible by 4]

Divisibility property of ‘9’: A number is divisible by ‘9’ when the sum of all its digits is divisible by ‘9’

Therefore, 4 + p + 2 + 5 + q = 11 + p + q.

So, the possible values of ‘p’ and ‘q’ so that 11 + p + q is divisible by ‘9’are:

For q = 2

=> 11 + p + 2 = 13 + p

=>‘p’ should be 5 [since 13 + 5 = 18 is divisible by 9]

For q = 6

=> 11 + p + 6 = 17 + p
=> ‘p’ should be 1 [since 17 + 1 = 18 is divisible by 9]

So, we have two pairs for ‘p’ and ‘q’: (5, 2) and (1,6)

=> Minimum value of \(\frac{p}{q}\) = \(\frac{1}{6}\)

C is the correct answer.

Answer C
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(Number Properties) p, q, and r are positive integers. What is the value of p + q + r?

1) p, q, and r are prime numbers.
2) The product of p, q, and r is 5 times the sum of p, q, and r.
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Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to determine the value of p + q + r.

Follow the second and the third step: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since we have 3 variables (p, q, and r) and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) together give us that:

Since we have pqr = 5(p + q + r), we assume p = 5.

Then we have 5qr = 5(5 + q + r) or qr = q + r + 5.

Since we have qr – q – r = 5, we have qr – q – r + 1 = 6 (by adding 1 to both sides so that it can easily be factored). Factoring gives us (q - 1)(r - 1) = 6.

Then we have the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2 and r = 7.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3 and r = 4.
If we have q – 1 = 3 and r – 1 = 2, we have q = 4 and r = 3.
If we have q – 1 = 6 and r – 1 = 1, we have q = 7 and r = 2.

However, p, q, and r are prime numbers from condition 1), and their possible numbers are 2, 5, and 7.

Thus, we have p + q + r = 2 + 5 + 7 = 14.

The answer is unique, so both conditions together are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one. So, C seems to be the answer.

However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition 1) tells us that we don’t have a unique solution, obviously.

If p = 2, q = 3, and r = 5, we have p + q + r = 2 + 3 + 5 = 10.
If p = 2, q = 5, and r = 7, we have p + q + r = 2 + 5 + 7 = 14.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that we don’t have a unique solution.

Assume p = 5 and since we have qr – q – r = 5, we have qr – q – r + 1 = 6 or (q - 1)(r - 1) = 6.

Then the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2, r = 7 and p + q + r = 5 + 2 + 7 = 14.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3, r = 4 and p + q + r = 5 + 3 + 4 = 12.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Thus, really, both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.
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A ball thrown up in the air is at a height of h feet, t seconds after it was thrown, where \(h = −3(t − 10)^2 + 250\). What is the height of the ball once it reached its maximum height and then descended for 7 seconds

A) 96 feet
B) 103 feet
C) 164 feet
D) 223 feet
E) 250 feet

Solution: We know that \(h = −3(t − 10)^2 + 250\)

We will first find the value for ‘t’ for which ‘h’ will be maximum.

For ‘h’ to be maximum, \(−3(t − 10)^2\) should be maximum. Since \((t − 10)^2\) is a perfect square, therefore, \((t − 10)^2\) ≥ 0.

But, \(−3(t − 10)^2\) will be ≤ 0 [By the property of reverse inequality]

So, for ‘h’ to be maximum \(−3(t − 10)^2\)= 0

=> \(−3(t − 10)^2\) = 0

=> \((t − 10)^2\) = 0

=> (t − 10) = 0

=> t = 10.

‘7’ seconds after ball has reached maximum height ‘h’ at t = 10 + 7 = 17.

=> \(h = −3(t − 10)^2 + 250\)

=> \(h = −3(17 − 10)^2 + 250\)

=> h = −3 * 49 + 250

=> h = -147 + 250

=> h = 103 feet

Answer B
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