Algebra and Functions Practice Questions — GMAT — Page 7

Question 121Question →

Worker A can complete a specific publishing task alone in $4$ hours, while Worker B can complete the exact same task alone in $6$ hours. Working together continuously at their respective constant individual rates, how many hours will it take both workers to complete one such task?

2.4

5.0

10.0

12.0

2.0

Show answer & explanation

Answer: $2.4$

Answer

The combined time required to complete the task is

2.4

hours.

The correct response calculates each worker's hourly rate of completion (one-fourth and one-sixth of the job per hour), sums them to find a combined rate of five-twelfths of the job per hour, and takes the reciprocal to find that the total time required is

2.4

hours.

Step-by-Step Solution

1

Determine individual work rates per hour.

Worker A's rate is

\frac{1}{4}

of the task per hour; Worker B's rate is

\frac{1}{6}

of the task per hour.

Work rate is defined as the reciprocal of total time required to complete one unit of work.

2

Sum the individual rates to find the combined work rate.

Combined Rate =

\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}

task per hour.

When working together, rates add together linearly.

3

Calculate total time by taking the reciprocal of the combined rate.

Time =

\frac{1}{\text{Combined Rate}} = \frac{12}{5} = 2.4

hours.

Time equals total work (1 unit) divided by the combined rate.

Key Concept

Combined Work Rates

Estimated Time:1m 0s

Question 122Question →

For a certain constant $k$ , the quadratic equation $x^2 - 12x + k = 0$ has two real roots, $r_1$ and $r_2$ . If $r_1^2 + r_2^2 = 94$ , what is the value of $k$ ?

Show answer & explanation

Answer: 25

Answer

25

Applying Vieta's formulas gives

r_1 + r_2 = 12

and

r_1 r_2 = k

. Substituting these into the identity

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

yields

12^2 = 94 + 2k

. Solving

144 = 94 + 2k

gives

2k = 50

, so

k = 25

.

Step-by-Step Solution

1

Apply Vieta's formulas to the given quadratic equation

x^2 - 12x + k = 0

.

The sum of the roots is

r_1 + r_2 = 12

and the product of the roots is

r_1 r_2 = k

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and product is

c/a

.

2

Expand the square of the sum of the roots

(r_1 + r_2)^2

.

(r_1 + r_2)^2 = r_1^2 + r_2^2 + 2r_1 r_2

This algebraic identity connects the sum of roots, sum of squared roots, and product of roots.

3

Substitute

r_1 + r_2 = 12

,

r_1^2 + r_2^2 = 94

, and

r_1 r_2 = k

into the identity.

12^2 = 94 + 2k \implies 144 = 94 + 2k

Replacing terms with known numerical values creates a linear equation in

k

.

4

Isolate and solve for

k

.

2k = 144 - 94 = 50 \implies k = 25

Basic algebraic manipulation yields the exact value of

k

.

Key Concept

Vieta's Formulas and Symmetric Polynomial Identities

Estimated Time:1m 30s

Question 123Question →

The function

f

is defined for all real numbers by

f(x) = \begin{cases} 3x - 1 & \text{if } x \le 4 \\ x^2 - k & \text{if } x > 4 \end{cases}

where

k

is a constant. If

f(f(3)) = 14

, what is the value of

k

?

Show answer & explanation

Answer: 50

Answer

50

To solve for

k

, evaluate the composite function

f(f(3))

step-by-step. First, find

f(3)

. Since

3 \le 4

, we use the first branch of the definition:

f(3) = 3(3) - 1 = 8

. Next, evaluate

f(8)

. Since

8 > 4

, we use the second branch of the definition:

f(8) = 8^2 - k = 64 - k

. Given that

f(f(3)) = 14

, we set

64 - k = 14

, which yields

k = 50

.

Step-by-Step Solution

1

Evaluate the inner function value f(3)

f(3) = 8

Because 3 is less than or equal to 4, the definition f(x) = 3x - 1 applies, yielding f(3) = 3(3) - 1 = 8.

2

Express f(f(3)) in terms of k

f(8) = 64 - k

Since the inner value is 8 and 8 > 4, the piecewise condition specifies using f(x) = x^2 - k, giving f(8) = 8^2 - k = 64 - k.

3

Solve the linear equation for k

k = 50

Setting the calculated expression equal to the given value 14 gives 64 - k = 14, which simplifies to k = 50.

Key Concept

Piecewise Function Evaluation and Parameter Resolution

Estimated Time:1m 30s

Question 124Question →

For all non-zero real numbers $a$ and $b$ , the binary operator $\star$ is defined by $a \star b = \frac{a^2 + b}{a}$ . The function $f$ is defined by $f(x) = x \star 16$ for all $x \neq 0$ . If $k$ is a positive real number such that $f(f(k)) = 10$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

By definition of the custom operator,

f(x) = x + \frac{16}{x}

. Evaluating

f(4) = 4 + \frac{16}{4} = 8

and then

f(f(4)) = f(8) = 8 + \frac{16}{8} = 10

confirms that

k = 4

satisfies the condition.

Step-by-Step Solution

1

Express

f(x)

by applying the custom operator definition

f(x) = x \star 16 = \frac{x^2 + 16}{x} = x + \frac{16}{x}

Using

a = x

and

b = 16

in

a \star b = \frac{a^2 + b}{a}

simplifies the function formula.

2

Set

m = f(k)

and solve

f(m) = 10

for

m

m + \frac{16}{m} = 10 \implies m^2 - 10m + 16 = 0 \implies (m - 2)(m - 8) = 0

, yielding

m = 2

or

m = 8

Replacing the inner function evaluation with an intermediate variable creates a manageable quadratic equation.

3

Solve

f(k) = 2

and

f(k) = 8

for positive real values of

k

For

f(k) = 2

:

k^2 - 2k + 16 = 0

has discriminant

(-2)^2 - 4(1)(16) = -60 < 0

, giving no real solutions. For

f(k) = 8

:

k^2 - 8k + 16 = 0 \implies (k - 4)^2 = 0 \implies k = 4

.

Only

m = 8

produces a real value for

k

because

k + \frac{16}{k} \geq 8

for all positive real numbers

k

.

Key Concept

Custom Binary Operators and Nested Function Evaluation

Question 125Question →

If $x$ is a real number such that $x^3 - 3x^2 = 4x - 12$ , what is the sum of all possible values of $x$ ?

-3

0

2

3

5

Show answer & explanation

Answer: $3$

Answer

The sum of all possible values of

x

is 3.

Moving all terms to one side gives

x^3 - 3x^2 - 4x + 12 = 0

. Grouping the terms yields

x^2(x - 3) - 4(x - 3) = 0

, which factors completely into

(x - 2)(x + 2)(x - 3) = 0

. The real solutions are

x = 2

,

x = -2

, and

x = 3

. Adding these values together yields

2 + (-2) + 3 = 3

.

Step-by-Step Solution

1

Rearrange the equation to set it equal to zero.

x^3 - 3x^2 - 4x + 12 = 0

Grouping terms requires having all terms on one side of the equation.

2

Factor by grouping terms in pairs.

x^2(x - 3) - 4(x - 3) = 0 \Rightarrow (x^2 - 4)(x - 3) = 0

Extracting the common factor

(x - 3)

allows reducing the cubic polynomial into a linear factor and a quadratic factor.

3

Factor the difference of squares and set each factor to zero to find all real roots.

(x - 2)(x + 2)(x - 3) = 0 \Rightarrow x = 2, x = -2, x = 3

By the zero-product property, the expression equals zero when any factor is equal to zero.

4

Calculate the sum of all distinct real solutions.

2 + (-2) + 3 = 3

The question asks for the sum of all possible real values of

x

.

Key Concept

Polynomial Factoring by Grouping and Zero-Product Property

Estimated Time:1m 30s

Question 126Question →

What is the sum of all distinct real solutions to the equation $(x^2 - 4)^2 = 9(x - 2)^2$ ?

-4

-2

0

1

3

Show answer & explanation

Answer: $-2$

Answer

The sum of all distinct real solutions is

-2

.

Factoring

(x^2 - 4)^2 = 9(x - 2)^2

gives

(x - 2)^2(x + 2)^2 - 9(x - 2)^2 = 0

. Factoring out

(x - 2)^2

yields

(x - 2)^2[(x + 2)^2 - 9] = 0

, which simplifies to

(x - 2)^2(x - 1)(x + 5) = 0

. The distinct real solutions are

x = 2

,

x = 1

, and

x = -5

. Adding these values together gives

2 + 1 + (-5) = -2

.

Step-by-Step Solution

1

Rewrite

x^2 - 4

as

(x - 2)(x + 2)

and substitute into the left side

[(x - 2)(x + 2)]^2 = (x - 2)^2(x + 2)^2

To identify common factors on both sides of the equation

2

Rearrange the equation and factor out

(x - 2)^2

(x - 2)^2(x + 2)^2 - 9(x - 2)^2 = 0 \implies (x - 2)^2[(x + 2)^2 - 9] = 0

Factoring out common terms avoids losing roots by division

3

Factor the inner quadratic expression using difference of squares

(x + 2)^2 - 9 = (x + 2 - 3)(x + 2 + 3) = (x - 1)(x + 5)

To break down the remaining quadratic into linear factors

4

Identify all distinct real roots and sum them

Roots are

x = 2

,

x = 1

, and

x = -5

. Sum:

2 + 1 + (-5) = -2

To find the sum of all unique real solutions

Key Concept

Polynomial Factoring and Root Preservation

Estimated Time:2m 0s

Question 127Question →

In a geometric sequence of positive numbers, the first term is $3$ and the common ratio is $2$ . What is the value of the 5th term of this sequence?

48

96

1,296

24

16

Show answer & explanation

Answer: $48$

Answer

The 5th term of the sequence is

48

.

The

n

-th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. Substituting

a_1 = 3

,

r = 2

, and

n = 5

gives

a_5 = 3 \cdot 2^4 = 3 \cdot 16 = 48

.

Step-by-Step Solution

1

Identify the formula for the

n

-th term of a geometric sequence.

a_n = a_1 \cdot r^{n-1}

In any geometric sequence, each term is obtained by multiplying the previous term by the common ratio

r

.

2

Substitute the given values into the formula.

a_5 = 3 \cdot 2^{5-1} = 3 \cdot 2^4

The first term

a_1 = 3

, common ratio

r = 2

, and term number

n = 5

.

3

Calculate the exponent and final value.

a_5 = 3 \cdot 16 = 48

Evaluate

2^4 = 16

first according to order of operations, then multiply by 3.

Key Concept

Geometric Sequence

n

-th Term Formula

Question 128Question →

An automated freight transport vessel completes a round trip along a $120$ -mile coastal canal route from Port Alpha to Port Beta and back. On the return trip from Port Beta to Port Alpha, due to opposing currents, the vessel's constant speed was $20$ miles per hour slower than its constant speed on the trip from Port Alpha to Port Beta. If the total travel time for the entire round trip was $5$ hours, what was the average speed of the vessel, in miles per hour, for the entire round trip?

40

48

50

60

30

Show answer & explanation

Answer: 48

Answer

The average speed of the vessel for the entire round trip is 48 miles per hour.

The total distance traveled for the round trip is

120 + 120 = 240

miles. Setting up the time equation

\frac{120}{v} + \frac{120}{v-20} = 5

gives the quadratic equation

v^2 - 68v + 480 = 0

, which factors into

(v-60)(v-8) = 0

. Since speed must be greater than

20

mph for a valid return trip, the outgoing speed is

60

mph and the return speed is

40

mph. Dividing the total distance of

240

miles by the total time of

5

hours yields an overall average speed of

48

miles per hour.

Step-by-Step Solution

1

Define variables and establish time equations for each leg of the journey.

Let

v

be the vessel's speed from Port Alpha to Port Beta in miles per hour. The speed from Port Beta to Port Alpha is

v - 20

mph. Time outgoing is

\frac{120}{v}

hours and time returning is

\frac{120}{v - 20}

hours.

Travel time equals distance divided by rate.

2

Set up and simplify the quadratic equation for total travel time.

\frac{120}{v} + \frac{120}{v - 20} = 5 \implies \frac{24}{v} + \frac{24}{v - 20} = 1 \implies 24(v - 20) + 24v = v(v - 20) \implies v^2 - 68v + 480 = 0

.

The sum of the travel times for both legs must equal the total given trip duration of 5 hours.

3

Solve the quadratic equation to determine the valid speed

v

.

Factoring

(v - 60)(v - 8) = 0

gives roots

v = 60

or

v = 8

. Since the return speed

v - 20

must be positive,

v = 60

mph.

A speed of

v = 8

mph would produce a negative return speed, which is physically invalid.

4

Calculate the average speed for the entire round trip.

Total distance

= 120 + 120 = 240

miles. Total time

= 5

hours. Average speed

= \frac{240}{5} = 48

miles per hour.

Average speed over a multi-leg trip is defined as total distance divided by total time.

Key Concept

Equation Modeling and Average Speed over Multi-Leg Journeys

Estimated Time:2m 0s

Question 129Question →

If $|3 - 2x| = 7$ , what is the product of all possible real values of $x$ ?

-10

-3

3

5

10

Show answer & explanation

Answer: $-10$

Answer

The product of all possible real values of

x

is

-10

.

Solving

|3 - 2x| = 7

yields two equations:

3 - 2x = 7

, which gives

x = -2

, and

3 - 2x = -7

, which gives

x = 5

. Multiplying these two values together gives

(-2) \times 5 = -10

.

Step-by-Step Solution

1

Set up two linear equations corresponding to the positive and negative cases of the absolute value expression.

3 - 2x = 7

or

3 - 2x = -7

By definition,

|a| = b

(where

b \ge 0

) implies

a = b

or

a = -b

.

2

Solve the first linear equation for

x

.

-2x = 4 \implies x = -2

Subtract 3 from both sides and divide by

-2

.

3

Solve the second linear equation for

x

.

-2x = -10 \implies x = 5

Subtract 3 from both sides and divide by

-2

.

4

Calculate the product of the two solutions.

(-2) \times 5 = -10

The question asks for the product of all possible real values of

x

.

Key Concept

Absolute Value Equations

Question 130Question →

If $y = 3^k$ , which of the following expressions is equivalent to $9^{k+1}$ for all real values of $k$ ?

9y^2

9y

3y^2

y^2 + 1

y^3

Show answer & explanation

Answer: $9y^2$

Answer

The expression

9y^2

is equivalent to

9^{k+1}

.

Applying the product rule of exponents gives

9^{k+1} = 9^k \cdot 9^1

. Expressing 9 as

3^2

gives

9^k = (3^2)^k = (3^k)^2

. Substituting

y = 3^k

yields

9 \cdot y^2 = 9y^2

.

Step-by-Step Solution

1

Rewrite

9^{k+1}

using the product rule of exponents (

a^{m+n} = a^m \cdot a^n

).

9^{k+1} = 9^k \cdot 9^1 = 9 \cdot 9^k

Separating the exponent addition into a product allows simplification of the constant factor.

2

Express the base 9 in terms of base 3.

9^k = (3^2)^k = (3^k)^2

Since

y = 3^k

, converting base 9 to

3^2

allows substitution of

y

into the expression.

3

Substitute

y = 3^k

into the expression.

9 \cdot (3^k)^2 = 9y^2

Replacing

3^k

with

y

yields the simplified algebraic expression in terms of

y

.

Key Concept

Properties of exponents including power of a power

(a^m)^n = a^{mn}

and product of powers

a^{m+n} = a^m \cdot a^n

.

Estimated Time:45s

Question 131Question →

If $x$ and $y$ are positive real numbers such that $x^2 - y^2 = 105$ and $x + y = 15$ , what is the value of $(x - 2y)^2$ ?

Show answer & explanation

Answer: 9

Answer

The value of

(x - 2y)^2

is 9.

Factoring

x^2 - y^2

into

(x - y)(x + y)

gives

(x - y)(15) = 105

, which simplifies to

x - y = 7

. Solving the system of equations

x + y = 15

and

x - y = 7

gives

x = 11

and

y = 4

. Substituting these values into the target expression

(x - 2y)^2

yields

(11 - 2(4))^2 = 3^2 = 9

.

Step-by-Step Solution

1

Apply the difference of squares factoring identity to

x^2 - y^2

(x - y)(x + y) = 105

The difference of two squares

x^2 - y^2

factors into

(x - y)(x + y)

.

2

Calculate the value of

x - y

x - y = 7

Since

x + y = 15

, dividing 105 by 15 gives

x - y = 7

.

3

Solve the system of equations for

x

and

y

x = 11

and

y = 4

Adding

(x + y) + (x - y) = 15 + 7

yields

2x = 22 \implies x = 11

. Subtracting

(x + y) - (x - y) = 15 - 7

yields

2y = 8 \implies y = 4

.

4

Evaluate the expression

(x - 2y)^2

9

Substitute

x = 11

and

y = 4

into

(x - 2y)^2

to obtain

(11 - 2(4))^2 = (11 - 8)^2 = 3^2 = 9

.

Key Concept

Difference of Squares Factoring and Systems of Equations

Question 132Question →

A private equity firm invested a total of $\$1,000,000$ divided between two portfolio companies, Company A and Company B. During the first year, the value of Company A increased by $x\%$ , while the value of Company B decreased by $x\%$ , where $x > 0$ . At the end of the first year, the value of Company A was $\$720,000$ , and the combined value of both companies was $\$1,040,000$ . If the value of Company B decreased by an additional $x\%$ during the second year relative to its value at the end of the first year, what was the value, in dollars, of Company B at the end of the second year?

Show answer & explanation

Answer: 256000

Answer

The value of Company B at the end of the second year was 256,000 dollars.

By representing the initial investments as algebraic expressions involving the multiplier

k = \frac{x}{100}

and setting up the total initial capital equation

A_0 + B_0 = 1,000,000

, we obtain a quadratic equation in

k

. Solving

(5k-1)^2 = 0

yields

k = 0.2

(

x = 20\%

). Decreasing Company B's Year 1 value of

\$320,000

by

20\%

gives

\$256,000

.

Step-by-Step Solution

1

Find the value of Company B at the end of Year 1

Year 1 value of Company B =

1,040,000 - 720,000 = 320,000

dollars

The total combined value of both companies at the end of Year 1 is given as $1,040,000.

2

Formulate algebraic expressions for initial values using rate

k = \frac{x}{100}

A_0 = \frac{720,000}{1+k}

and

B_0 = \frac{320,000}{1-k}

Company A increased by

x\%

so

A_1 = A_0(1+k)

; Company B decreased by

x\%

so

B_1 = B_0(1-k)

.

3

Set up and simplify the quadratic equation for the combined initial investment

\frac{720,000}{1+k} + \frac{320,000}{1-k} = 1,000,000 \implies 25k^2 - 10k + 1 = 0

Dividing by

80,000

yields

\frac{9}{1+k} + \frac{4}{1-k} = 12.5

. Multiplying through by

2(1-k^2)

leads to

2[9(1-k) + 4(1+k)] = 25(1-k^2)

.

4

Solve for

k

and determine percentage

x

(5k - 1)^2 = 0 \implies k = 0.2

, so

x = 20\%

Factoring the perfect square quadratic expression gives a unique solution for

k

.

5

Compute Company B's value at the end of Year 2

320,000 \times (1 - 0.2) = 256,000

dollars

Company B's value decreases by an additional

20\%

of its Year 1 value (

320,000

).

Key Concept

Algebraic Modeling of Rational/Quadratic Equations from Multi-Step Percent Change Scenarios

Estimated Time:2m 30s

Question 133Question →

For a constant $a > 0$ , the quadratic equation $x^2 - ax + (2a + 1) = 0$ has two real roots, $r$ and $s$ . If $r^2 + s^2 = 43$ , what is the value of $a$ ?

Show answer & explanation

Answer: 9

Answer

The value of

a

is 9.

According to Vieta's formulas, the sum of the roots of

x^2 - ax + (2a + 1) = 0

is

r + s = a

and the product of the roots is

rs = 2a + 1

. Using the identity

r^2 + s^2 = (r + s)^2 - 2rs

, substitute the Vieta expressions to obtain

r^2 + s^2 = a^2 - 2(2a + 1) = a^2 - 4a - 2

. Setting this equal to 43 yields

a^2 - 4a - 2 = 43

, which simplifies to

a^2 - 4a - 45 = 0

. Factoring the quadratic gives

(a - 9)(a + 5) = 0

, giving solutions

a = 9

or

a = -5

. Because the problem specifies that

a > 0

,

a

must be 9. Checking the discriminant

\Delta = (-9)^2 - 4(1)(19) = 5 > 0

confirms that two real roots exist.

Step-by-Step Solution

1

Express the sum and product of the roots in terms of

a

using Vieta's formulas.

r + s = a

and

rs = 2a + 1

For a standard quadratic equation

x^2 + bx + c = 0

, the sum of the roots is

-b

and the product of the roots is

c

.

2

Relate

r^2 + s^2

to

(r + s)

and

rs

.

r^2 + s^2 = (r + s)^2 - 2rs = a^2 - 2(2a + 1) = a^2 - 4a - 2

Expanding

(r + s)^2 = r^2 + 2rs + s^2

allows expressing

r^2 + s^2

in terms of known quantities.

3

Substitute

r^2 + s^2 = 43

into the equation and solve for

a

.

a^2 - 4a - 2 = 43 \implies a^2 - 4a - 45 = 0 \implies (a - 9)(a + 5) = 0

Rearranging terms forms a new quadratic equation in terms of

a

.

4

Apply the positivity constraint

a > 0

and verify that the roots are real.

Since

a > 0

,

a = 9

. The discriminant of the original equation is

\Delta = (-9)^2 - 4(1)(19) = 5 > 0

, confirming real roots exist.

The question specifies that

a

is positive and that

r

and

s

are real numbers.

Key Concept

Vieta's Formulas and Quadratic Modeling

Question 134Question →

A delivery truck completes an outbound trip along Route X of length $d$ miles at a constant average speed of $v$ miles per hour. On the return trip along the exact same route, heavy traffic reduces the truck's average speed by $20$ miles per hour, which increases the travel time for the return trip by $2$ hours compared to the outbound trip. Later, the truck completes a trip along Route Y, which has the same distance $d$ as Route X, at an average speed of $(v + 60)$ miles per hour in exactly $2$ hours. What was the truck's average speed $v$ , in miles per hour, on the outbound trip along Route X?

40

50

60

80

150

Show answer & explanation

Answer: $60$

Answer

The truck's average outbound speed

v

on Route X was

60

miles per hour.

By setting up time equations for the two legs of Route X, the distance

d

is expressed as

d = \frac{v(v-20)}{10}

. Combining this with the relation for Route Y (

d = 2(v+60)

) yields the quadratic equation

v^2 - 40v - 1200 = 0

. Factoring gives

(v - 60)(v + 20) = 0

, yielding the valid positive speed

v = 60

miles per hour.

Step-by-Step Solution

1

Express the distance

d

in terms of

v

using the time difference on Route X.

d = \frac{v(v - 20)}{10}

The outbound time is

t_1 = \frac{d}{v}

and the return time is

t_2 = \frac{d}{v - 20}

. Given

t_2 - t_1 = 2

, we have

\frac{d}{v - 20} - \frac{d}{v} = 2 \implies \frac{20d}{v(v - 20)} = 2 \implies d = \frac{v(v - 20)}{10}

.

2

Express the distance

d

in terms of

v

using the conditions for Route Y.

d = 2(v + 60)

Route Y has distance

d

, speed

(v + 60)

mph, and takes

2

hours, so

d = \text{rate} \times \text{time} = 2(v + 60)

.

3

Equate the two expressions for

d

to form a quadratic equation in terms of

v

.

v^2 - 40v - 1200 = 0

Setting

\frac{v(v - 20)}{10} = 2(v + 60)

yields

v^2 - 20v = 20v + 1200

, which simplifies to

v^2 - 40v - 1200 = 0

.

4

Solve the quadratic equation for

v

and verify physical feasibility.

v = 60

Factoring

(v - 60)(v + 20) = 0

gives solutions

v = 60

or

v = -20

. Since speed must be positive and greater than

20

mph (to avoid a zero or negative return speed),

v = 60

mph.

Key Concept

Algebraic Modeling of Multi-Leg Motion using Systems of Rational and Quadratic Equations

Question 135Question →

Four sequence-derived values $S_1, S_2, S_3,$ and $S_4$ are defined below. Arrange these values in order from smallest to largest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from smallest to largest is

S_1 < S_3 < S_4 < S_2

(corresponding to

55 < 64 < 72 < 80

).

Computing each value yields

S_1 = 55

,

S_2 = 80

,

S_3 = 64

, and

S_4 = 72

. Arranging these values in ascending order results in

55 < 64 < 72 < 80

, corresponding to

S_1 < S_3 < S_4 < S_2

.

Step-by-Step Solution

1

Calculate the value of

S_1

S_1 = 55

Using the sum formula for an arithmetic sequence

S_n = \frac{n}{2}[2a_1 + (n-1)d]

, we have

S_5 = \frac{5}{2}[2(3) + (5-1)4] = \frac{5}{2}[6 + 16] = 55

.

2

Calculate the value of

S_2

S_2 = 80

Using the sum formula for a finite geometric sequence

S_n = \frac{a_1(r^n - 1)}{r - 1}

, we have

S_4 = \frac{2(3^4 - 1)}{3 - 1} = \frac{2(80)}{2} = 80

.

3

Calculate the value of

S_3

S_3 = 64

Using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n-1)d

, we have

c_{10} = 10 + (10-1)6 = 10 + 54 = 64

.

4

Calculate the value of

S_4

S_4 = 72

Using the sum formula for an infinite geometric series

S_\infty = \frac{a_1}{1 - r}

, we have

S_4 = \frac{48}{1 - 1/3} = \frac{48}{2/3} = 72

.

5

Order the computed values from smallest to largest

S_1 (55) < S_3 (64) < S_4 (72) < S_2 (80)

Comparing the numerical values gives

55 < 64 < 72 < 80

.

Key Concept

Evaluation and comparison of finite and infinite arithmetic and geometric sequence terms and sums

Question 136Question →

For all real numbers $x$ and $y$ , the custom binary operator $\Delta$ is defined by $x \Delta y = x^2 - 5xy + 4y^2$ . The function $f$ is defined by $f(x) = x \Delta 1$ . What is the positive integer value of $k$ such that $f(f(k)) = 0$ ?

Show answer & explanation

Answer: 5

Answer

The positive integer value of

k

is

5

.

Substituting

y = 1

into

x \Delta y = x^2 - 5xy + 4y^2

gives

f(x) = x^2 - 5x + 4

. For

f(f(k)) = 0

, the outer function evaluation requires

f(k)

to be a root of

f(x) = 0

. Solving

x^2 - 5x + 4 = 0

yields roots

1

and

4

. Setting

f(k) = 4

yields

k^2 - 5k = 0

, which has roots

k = 0

and

k = 5

. Since

k

must be a positive integer,

k = 5

. Setting

f(k) = 1

yields irrational values, so

5

is the unique solution.

Step-by-Step Solution

1

Substitute

y = 1

into the operator definition to find

f(x)

.

f(x) = x^2 - 5x + 4

f(x) = x \Delta 1 = x^2 - 5x(1) + 4(1)^2

.

2

Set

f(u) = 0

for

u = f(k)

and solve for

u

.

u = 1

or

u = 4

Factoring

u^2 - 5u + 4 = 0

yields

(u - 1)(u - 4) = 0

.

3

Solve

f(k) = 4

for

k

.

k = 0

or

k = 5

k^2 - 5k + 4 = 4 \implies k^2 - 5k = 0 \implies k(k - 5) = 0

.

4

Solve

f(k) = 1

for

k

and check for integer solutions.

k = \frac{5 \pm \sqrt{13}}{2}

(irrational roots)

k^2 - 5k + 4 = 1 \implies k^2 - 5k + 3 = 0

.

5

Select the positive integer solution.

k = 5

The value

k = 0

is not positive, and the roots from

f(k) = 1

are not integers.

Key Concept

Custom operator evaluation, composite/nested functions, and quadratic root analysis.

Question 137Question →

If $x$ is a real number satisfying the equation $2x^2 - 7x + 3 = 0$ and $y$ is a real number satisfying the equation $y^2 + 5y - 14 = 0$ , what is the maximum possible value of $\frac{x}{y}$ ?

\frac{3}{2}

\frac{3}{7}

\frac{1}{4}

-\frac{3}{7}

6

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

The maximum possible value of

\frac{x}{y}

is

\frac{3}{2}

.

Factoring

2x^2 - 7x + 3 = 0

gives

(2x - 1)(x - 3) = 0

, so

x \in \left\{\frac{1}{2}, 3\right\}

. Factoring

y^2 + 5y - 14 = 0

gives

(y + 7)(y - 2) = 0

, so

y \in \{-7, 2\}

. The four possible values for

\frac{x}{y}

are

\frac{3}{2}

,

-\frac{3}{7}

,

\frac{1}{4}

, and

-\frac{1}{14}

. The greatest value among these is

\frac{3}{2}

.

Step-by-Step Solution

1

Solve the quadratic equation for

x

by factoring.

Factor

2x^2 - 7x + 3 = 0

as

(2x - 1)(x - 3) = 0

, yielding solutions

x = \frac{1}{2}

and

x = 3

.

Finding all valid real roots of the first equation determines possible values for the numerator.

2

Solve the quadratic equation for

y

by factoring.

Factor

y^2 + 5y - 14 = 0

as

(y + 7)(y - 2) = 0

, yielding solutions

y = -7

and

y = 2

.

Finding all valid real roots of the second equation determines possible values for the denominator.

3

Evaluate all possible combinations for the ratio

\frac{x}{y}

.

The possible ratios are

\frac{3}{2} = 1.5

,

\frac{3}{-7} = -\frac{3}{7}

,

\frac{1/2}{2} = \frac{1}{4} = 0.25

, and

\frac{1/2}{-7} = -\frac{1}{14}

.

Testing all root pairs ensures we identify the maximum overall value.

4

Select the maximum value among the evaluated ratios.

The largest value is

\frac{3}{2}

.

Comparing

1.5

,

-0.428

,

0.25

, and

-0.071

shows

1.5

is the greatest.

Key Concept

Solving quadratic equations via polynomial factoring and optimizing rational expressions over discrete solution sets.

Estimated Time:1m 30s

Question 138Question →

Let $S$ be the set of all real numbers $x$ that satisfy the inequality $x^2 - 6x \leq 0$ . How many integer values of $k$ are there such that the equation $||x - 4| - |x + 2|| = k$ has at least one solution $x \in S$ ?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

Solving the quadratic inequality

x^2 - 6x \leq 0

gives the domain

S = [0, 6]

. Over this closed interval, the function

g(x) = ||x - 4| - |x + 2||

is continuous and attains its minimum value of 0 at

x = 1

and its maximum value of 6 at

x = 4

(and throughout

[4, 6]

). By the Intermediate Value Theorem,

g(x)

takes on all real values in the interval

[0, 6]

. The integer values of

k

for which

g(x) = k

has a solution in

S

are

0, 1, 2, 3, 4, 5,

and

6

, making a total of 7 integers.

Step-by-Step Solution

1

Solve the quadratic inequality to define the set

S

.

x^2 - 6x \leq 0 \iff x(x - 6) \leq 0 \iff 0 \leq x \leq 6

. Thus,

S = [0, 6]

.

The solution set of

x(x-6) \leq 0

lies between the roots

x = 0

and

x = 6

inclusive.

2

Analyze the inner function

f(x) = |x - 4| - |x + 2|

for

x \in [0, 6]

.

Critical points of absolute values occur at

x = -2

and

x = 4

. Within

[0, 6]

, we split at

x = 4

.

The signs of

(x - 4)

and

(x + 2)

determine how the absolute value bars simplify.

3

Evaluate

g(x) = |f(x)|

on the sub-interval

[0, 4]

.

For

0 \leq x \leq 4

:

|x - 4| = 4 - x

and

|x + 2| = x + 2

. Thus,

f(x) = 2 - 2x

and

g(x) = |2 - 2x|

.

Evaluating

g(x)

at key points gives

g(0) = 2

,

g(1) = 0

, and

g(4) = 6

. By continuity,

g(x)

covers all values in

[0, 6]

on this interval.

4

Evaluate

g(x)

on the sub-interval

[4, 6]

.

For

4 \leq x \leq 6

:

|x - 4| = x - 4

and

|x + 2| = x + 2

. Thus,

f(x) = -6

and

g(x) = |-6| = 6

.

g(x)

remains constant at 6 for

x \in [4, 6]

.

5

Determine the range of

g(x)

on

S

and count the integer values of

k

.

The range of

g(x)

for

x \in [0, 6]

is

[0, 6]

. The integer values in this interval are

0, 1, 2, 3, 4, 5, 6

.

The equation

g(x) = k

has a solution in

S

if and only if

k

lies in the range of

g(x)

over

S

. There are

6 - 0 + 1 = 7

such integers.

Key Concept

Absolute value functions case evaluation and finding the range over a restricted domain.

Question 139Question →

If $|3x + 4| = 19$ , what is the positive value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The positive value of

x

is 5.

Solving the equation

|3x + 4| = 19

yields two cases:

3x + 4 = 19

, giving

x = 5

, and

3x + 4 = -19

, giving

x = -\frac{23}{3}

. The positive value among these solutions is 5.

Step-by-Step Solution

1

Split the absolute value equation into two linear equations.

3x + 4 = 19

or

3x + 4 = -19

The equation

|a| = b

for

b \ge 0

implies

a = b

or

a = -b

.

2

Solve for

x

in both cases.

x = 5

or

x = -\frac{23}{3}

Subtract 4 from both sides and divide by 3.

3

Select the value satisfying the problem constraint.

5

The problem asks specifically for the positive value of

x

.

Key Concept

Solving Linear Absolute Value Equations

Question 140Question →

For all non-zero real numbers $x$ and $y$ , the binary operator $\diamondsuit$ is defined by $x \diamondsuit y = \frac{x}{y} - \frac{y}{x}$ . If the function $g$ is defined by $g(x) = x \diamondsuit 2$ for all non-zero real numbers $x$ , what is the value of $g(g(4))$ ?

-\frac{7}{12}

\frac{7}{12}

\frac{3}{2}

-\frac{15}{4}

0

Show answer & explanation

Answer: $-\frac{7}{12}$

Answer

-\frac{7}{12}

Evaluating the inner expression

g(4)

yields

4 \diamondsuit 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

. Evaluating

g\left(\frac{3}{2}\right)

yields

\frac{3/2}{2} - \frac{2}{3/2} = \frac{3}{4} - \frac{4}{3} = -\frac{7}{12}

, which is the correct final value.

Step-by-Step Solution

1

Evaluate the inner function expression

g(4)

using the given definition

g(x) = x \diamondsuit 2

.

g(4) = 4 \diamondsuit 2 = \frac{4}{2} - \frac{2}{4} = 2 - \frac{1}{2} = \frac{3}{2}

.

Nested function evaluations require evaluating the innermost expression first.

2

Substitute the result

g(4) = \frac{3}{2}

into the outer function to find

g\left(\frac{3}{2}\right)

.

g\left(\frac{3}{2}\right) = \frac{3}{2} \diamondsuit 2 = \frac{\frac{3}{2}}{2} - \frac{2}{\frac{3}{2}}

.

The output of the inner function becomes the input for the outer function.

3

Simplify the resulting fractions and subtract them.

\frac{3}{4} - \frac{4}{3} = \frac{9 - 16}{12} = -\frac{7}{12}

.

Finding a common denominator allows direct subtraction of fractions.

Key Concept

Nested Function Evaluation with Custom Operators

Estimated Time:1m 30s

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