Algebra and Functions Practice Questions — GMAT — Page 5

Question 81Question →

A car rental agency charges a fixed daily rate plus a constant fee per mile driven. A customer who rented a car for 3 days and drove 150 miles paid a total of $180. Another customer who rented the same model of car for 5 days and drove 200 miles paid a total of$ 270. What is the fixed daily rate, in dollars, charged by the rental agency?

Show answer & explanation

Answer: 30

Answer

The fixed daily rate charged by the rental agency is 30 dollars.

Formulating the equations

3d + 150m = 180

and

5d + 200m = 270

and solving for the daily rate

d

yields

d = 30

.

Step-by-Step Solution

1

Define variables and formulate the system of linear equations.

Let

d

be the fixed daily rate (in dollars) and

m

be the cost per mile driven (in dollars). The scenario translates to:
Equation 1:

3d + 150m = 180

Equation 2:

5d + 200m = 270

Total charge is the linear combination of daily fixed costs and per-mile variable costs.

2

Simplify Equation 1 to express

d

in terms of

m

.

Dividing Equation 1 by 3 yields

d + 50m = 60

, so

d = 60 - 50m

.

Simplifying equations reduces computation error when using substitution.

3

Substitute the expression for

d

into Equation 2 to solve for

m

.

5(60 - 50m) + 200m = 270 \Rightarrow 300 - 250m + 200m = 270 \Rightarrow -50m = -30 \Rightarrow m = 0.60

Substitution eliminates variable

d

, leaving a linear equation in one variable.

4

Calculate the fixed daily rate

d

.

d = 60 - 50(0.60) = 60 - 30 = 30

Substituting

m = 0.60

gives the exact daily fixed rate.

Key Concept

Setting up and solving a system of two linear equations in two variables using elimination or substitution.

Question 82Question →

For all positive real numbers $x$ , the nested radical expression $\sqrt{x\sqrt{x}}$ is mathematically equivalent to $x^{\frac{3}{4}}$ .

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because expressing

\sqrt{x\sqrt{x}}

in fractional exponent form yields

x^{\frac{3}{4}}

for all positive real numbers

x

.

The radical expression

\sqrt{x\sqrt{x}}

simplifies to

x^{\frac{3}{4}}

by systematically applying exponent laws: the inner root gives

x^{\frac{1}{2}}

, multiplying by

x

gives

x^{\frac{3}{2}}

, and applying the outer root raises

x^{\frac{3}{2}}

to the

\frac{1}{2}

power, producing

x^{\frac{3}{4}}

.

Step-by-Step Solution

1

Express the inner radical as a fractional exponent.

\sqrt{x} = x^{\frac{1}{2}}

, making the expression under the outer root

x \cdot x^{\frac{1}{2}}

.

By definition of fractional exponents,

\sqrt[n]{a^m} = a^{\frac{m}{n}}

.

2

Combine the terms inside the outer radical using the product rule for exponents.

x^1 \cdot x^{\frac{1}{2}} = x^{1 + \frac{1}{2}} = x^{\frac{3}{2}}

.

When multiplying terms with the same base, add their exponents.

3

Apply the outer square root as an exponent of

\frac{1}{2}

and simplify using the power rule.

\sqrt{x^{\frac{3}{2}}} = (x^{\frac{3}{2}})^{\frac{1}{2}} = x^{\frac{3}{2} \cdot \frac{1}{2}} = x^{\frac{3}{4}}

.

Raising a power to another power requires multiplying the exponents.

Key Concept

Simplification of nested radical expressions using fractional exponent laws

Question 83Question →

An electronics manufacturing plant produces two types of circuit modules: Module X and Module Y. Producing one Module X requires $3$ minutes on Machine A and $5$ minutes on Machine B. Producing one Module Y requires $4$ minutes on Machine A and $2$ minutes on Machine B. During a certain shift, Machine A was operated for a total of $230$ minutes and Machine B was operated for a total of $220$ minutes. If both machines operated at full capacity with no idle time, how many Module X units were produced?

Show answer & explanation

Answer: 30

Answer

30

Formulating equations from the machine times gives

3x + 4y = 230

for Machine A and

5x + 2y = 220

for Machine B. Multiplying the second equation by

2

yields

10x + 4y = 440

. Subtracting

3x + 4y = 230

from

10x + 4y = 440

eliminates

y

and yields

7x = 210

, which solves to

x = 30

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

x

be the number of Module X units produced, and let

y

be the number of Module Y units produced.

Representing unknown quantities with algebraic variables allows for model formulation.

2

Formulate a system of linear equations from the machine time constraints.

Machine A equation:

3x + 4y = 230

Machine B equation:

5x + 2y = 220

The total operational time on each machine equals the sum of times spent producing each module type.

3

Multiply the Machine B equation by

2

to enable elimination of

y

.

2 \times (5x + 2y) = 2 \times 220 \implies 10x + 4y = 440

Matching the coefficient of

y

with the first equation (

4y

) allows elimination via subtraction.

4

Subtract the Machine A equation from the modified Machine B equation and solve for

x

.

(10x + 4y) - (3x + 4y) = 440 - 230 \implies 7x = 210 \implies x = 30

Subtracting eliminates

y

directly, leaving a single linear equation in terms of

x

.

Key Concept

Solving a system of two linear equations in two variables using elimination

Estimated Time:1m 30s

Question 84Question →

If $3(x - 2) + 4 = 2x + 11$ , what is the value of $x$ ?

Show answer & explanation

Answer: 13

Answer

The value of

x

is 13.

Expanding the left side yields

3x - 6 + 4 = 3x - 2

. Equating this to the right side gives

3x - 2 = 2x + 11

. Subtracting

2x

from both sides gives

x - 2 = 11

, and adding

2

to both sides yields

x = 13

.

Step-by-Step Solution

1

Expand the left side of the equation

3x - 6 + 4 = 2x + 11

Apply the distributive property

3(x - 2) = 3x - 6

.

2

Combine like constant terms on the left side

3x - 2 = 2x + 11

Combine

-6

and

+4

to get

-2

.

3

Subtract

2x

from both sides

x - 2 = 11

Move variable terms to one side of the equation.

4

Add 2 to both sides

x = 13

Isolate the variable

x

.

Key Concept

Solving Linear Equations in One Variable

Question 85Question →

If $|2x - 3| = 7$ , which of the following is a possible value of $x$ ?

5

-5

2

10

-4

Show answer & explanation

Answer: $5$

Answer

The value

5

is a valid solution to the given absolute value equation.

The absolute value equation

|2x - 3| = 7

splits into two linear equations:

2x - 3 = 7

and

2x - 3 = -7

. Solving the first equation gives

2x = 10

, so

x = 5

. Solving the second gives

2x = -4

, so

x = -2

. Among the listed options,

5

is the correct solution.

Step-by-Step Solution

1

Set up the two cases for the absolute value equation

|2x - 3| = 7

.

Case 1:

2x - 3 = 7

; Case 2:

2x - 3 = -7

.

An absolute value expression

|u| = c

(where

c \geq 0

) resolves to

u = c

or

u = -c

.

2

Solve Case 1:

2x - 3 = 7

.

2x = 10 \implies x = 5

.

Add 3 to both sides and divide by 2.

3

Solve Case 2:

2x - 3 = -7

.

2x = -4 \implies x = -2

.

Add 3 to both sides and divide by 2.

4

Compare the solutions (

x = 5

and

x = -2

) with the given choices.

The value

5

is present among the options.

Identifies the correct choice matching one of the calculated solutions.

Key Concept

Solving Linear Absolute Value Equations

Estimated Time:45s

Question 86Question →

For all real numbers $a$ and $b$ , the custom operation $\star$ is defined by $a \star b = (a + b)^2 - 3b$ . What is the value of $4 \star 2$ ?

30

24

14

33

18

Show answer & explanation

Answer: 30

Answer

30

Substituting

a = 4

and

b = 2

into the definition

(a + b)^2 - 3b

gives

(4 + 2)^2 - 3(2) = 6^2 - 6 = 36 - 6 = 30

.

Step-by-Step Solution

1

Identify the values assigned to the custom operator variables

a = 4

and

b = 2

The expression

4 \star 2

corresponds to evaluating

a \star b

where

a

is the first operand and

b

is the second operand.

2

Substitute

a = 4

and

b = 2

into the operator definition

(a + b)^2 - 3b

(4 + 2)^2 - 3(2)

Direct substitution into the given algebraic definition.

3

Perform the operations in parentheses and exponentiation

6^2 - 6 = 36 - 6

First add inside the parentheses (

4 + 2 = 6

), then square the sum (

6^2 = 36

), and multiply

3 \times 2 = 6

.

4

Subtract to find the final value

30

Subtracting

6

from

36

yields

30

.

Key Concept

Custom Operator Evaluation

Estimated Time:45s

Question 87Question →

Company A charges a one-time fixed fee of $\$20$ plus $\$5$ per hour to rent a piece of machinery. Company B charges a one-time fixed fee of $\$35$ plus $\$2$ per hour to rent the same machinery. For how many hours of rental will the total cost charged by both companies be equal?

3

5

7

11

15

Show answer & explanation

Answer: 5

Answer

The total cost at both companies will be equal for a rental duration of 5 hours.

The cost charged by Company A is modeled by

20 + 5h

and Company B by

35 + 2h

. Setting these equal yields

20 + 5h = 35 + 2h

, which simplifies to

3h = 15

, giving

h = 5

. Thus, 5 hours is the correct rental duration.

Step-by-Step Solution

1

Define the variable and write total cost equations for each company.

Let

h

represent the number of hours. Company A's cost is

C_A = 20 + 5h

and Company B's cost is

C_B = 35 + 2h

.

Linear cost models combine a fixed base cost with a variable hourly component.

2

Set the two cost equations equal to solve for

h

.

20 + 5h = 35 + 2h

The total costs are equal when

C_A = C_B

.

3

Isolate the variable term

h

on one side of the equation.

5h - 2h = 35 - 20 \Rightarrow 3h = 15 \Rightarrow h = 5

Subtracting

2h

and

20

from both sides simplifies the equation to solve directly for

h

.

Key Concept

Setting up and solving single-variable linear equations from real-world cost contexts.

Estimated Time:45s

Question 88Question →

If the quadratic equation $x^2 - kx + 36 = 0$ has two distinct positive integer roots, $r_1$ and $r_2$ , and $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 19x + m = 0$ , what is the value of $m$ ?

Show answer & explanation

Answer: 70

Answer

The value of

m

is 70.

By applying Vieta's formulas to both quadratic equations, we find that the sum of the original roots is

r_1 + r_2 = 15

and their product is

r_1 r_2 = 36

. Factoring

x^2 - 15x + 36 = (x-3)(x-12) = 0

confirms the original roots are

3

and

12

. The new roots are

3+2=5

and

12+2=14

. Their product

m = 5 \times 14 = 70

.

Step-by-Step Solution

1

Apply Vieta's relations to the first quadratic equation

r_1 + r_2 = k

and

r_1 \cdot r_2 = 36

For any quadratic equation

x^2 + bx + c = 0

, the sum of roots is

-b

and the product of roots is

c

.

2

Apply Vieta's relations to the second quadratic equation

(r_1 + 2) + (r_2 + 2) = 19 \implies r_1 + r_2 = 15

The coefficient of

x

in

x^2 - 19x + m = 0

dictates that the sum of its roots equals

19

.

3

Determine the roots

r_1

and

r_2

and check integer constraints

r_1 = 3

and

r_2 = 12

The integer factors of

36

that sum to

15

are

3

and

12

, satisfying all conditions.

4

Compute the constant term

m

for the transformed equation

m = (3 + 2)(12 + 2) = 5 \times 14 = 70

The constant term

m

equals the product of the transformed roots

(r_1 + 2)

and

(r_2 + 2)

.

Key Concept

Vieta's Formulas and Quadratic Root Transformations

Estimated Time:2m 0s

Question 89Question →

For all non-zero real numbers $a$ and $b$ , the custom operator $\Delta$ is defined by $a \Delta b = \frac{a^2 - b^2}{ab}$ . A function $f(x)$ is defined for all $x \neq 0$ and $x \neq -1$ by $f(x) = x \Delta (x + 1)$ . If $k$ is a positive constant such that $f(k) = -\frac{7}{12}$ , what is the value of $(k + 2) \Delta k$ ?

-\frac{16}{15}

\frac{4}{15}

\frac{5}{6}

\frac{16}{15}

\frac{3}{2}

Show answer & explanation

Answer: $\frac{16}{15}$

Answer

The value of

(k + 2) \Delta k

is

\frac{16}{15}

.

Applying the custom operator to

f(x)

yields

f(x) = \frac{x^2 - (x+1)^2}{x(x+1)} = \frac{-2x - 1}{x^2 + x}

. Setting

f(k) = -\frac{7}{12}

gives

\frac{2k + 1}{k^2 + k} = \frac{7}{12}

, which simplifies to

7k^2 - 17k - 12 = 0

. Factoring

(7k + 4)(k - 3) = 0

and selecting the positive root gives

k = 3

. Evaluating

(3+2) \Delta 3 = 5 \Delta 3 = \frac{5^2 - 3^2}{5 \cdot 3} = \frac{16}{15}

.

Step-by-Step Solution

1

Express

f(x)

using the definition of the custom operator

\Delta

.

f(x) = x \Delta (x + 1) = \frac{x^2 - (x + 1)^2}{x(x + 1)} = \frac{x^2 - (x^2 + 2x + 1)}{x(x + 1)} = \frac{-2x - 1}{x^2 + x}

Substitute

a = x

and

b = x + 1

into

a \Delta b = \frac{a^2 - b^2}{ab}

.

2

Set

f(k) = -\frac{7}{12}

and solve for the positive constant

k

.

\frac{-2k - 1}{k^2 + k} = -\frac{7}{12} \implies \frac{2k + 1}{k^2 + k} = \frac{7}{12} \implies 12(2k + 1) = 7(k^2 + k) \implies 7k^2 - 17k - 12 = 0

Cross-multiply and rearrange into a standard quadratic equation.

3

Factor the quadratic equation to find the positive root

k

.

(7k + 4)(k - 3) = 0 \implies k = -\frac{4}{7}

or

k = 3

. Since

k > 0

,

k = 3

.

The question specifies that

k

is a positive constant.

4

Evaluate

(k + 2) \Delta k

for

k = 3

.

(3 + 2) \Delta 3 = 5 \Delta 3 = \frac{5^2 - 3^2}{5 \times 3} = \frac{25 - 9}{15} = \frac{16}{15}

Substitute

a = 5

and

b = 3

into the custom operator definition.

Key Concept

Function definition substitution, non-commutative custom operators, and quadratic solving.

Estimated Time:2m 30s

Question 90Question →

For all real numbers $x$ and $y$ , the custom operator $\Delta$ is defined by $x \Delta y = x^2 - 2y$ . If $k$ is a real number such that $(3 \Delta k) \Delta 4 = 17$ , what is the sum of all possible values of $k$ ?

1

2

8

9

14

Show answer & explanation

Answer: 9

Answer

The sum of all possible values of

k

is 9.

Applying the custom operation rule

x \Delta y = x^2 - 2y

sequentially, the inner evaluation gives

3 \Delta k = 9 - 2k

. Applying the operation again to

(9 - 2k) \Delta 4

gives

(9 - 2k)^2 - 8 = 17

, which simplifies to

(9 - 2k)^2 = 25

. Taking the square root gives two valid solutions for

9 - 2k

:

5

and

-5

. Solving

9 - 2k = 5

yields

k = 2

, and solving

9 - 2k = -5

yields

k = 7

. The sum of these two values is

2 + 7 = 9

.

Step-by-Step Solution

1

Evaluate the inner custom operation

3 \Delta k

.

3 \Delta k = 3^2 - 2k = 9 - 2k

Apply the definition

x \Delta y = x^2 - 2y

with

x = 3

and

y = k

.

2

Substitute the inner result into the outer expression

(9 - 2k) \Delta 4 = 17

.

(9 - 2k)^2 - 2(4) = 17 \implies (9 - 2k)^2 - 8 = 17

Apply the definition

x \Delta y = x^2 - 2y

with

x = 9 - 2k

and

y = 4

.

3

Isolate the squared term and solve for

9 - 2k

.

(9 - 2k)^2 = 25 \implies 9 - 2k = 5

or

9 - 2k = -5

Adding 8 to both sides gives 25; taking the square root requires considering both positive and negative roots.

4

Solve each linear equation for

k

and calculate their sum.

Case 1:

9 - 2k = 5 \implies 2k = 4 \implies k = 2

.
Case 2:

9 - 2k = -5 \implies 2k = 14 \implies k = 7

.
Sum =

2 + 7 = 9

.

Solving both equations yields all possible values for

k

.

Key Concept

Custom Operator Evaluation and Quadratic Equation Root Extraction

Estimated Time:2m 0s

Question 91Question →

For all real numbers $p$ and $q$ , the operation $\nabla$ is defined by $p \nabla q = 2p + 3q - pq$ . What is the value of $3 \nabla 4$ ?

5

6

7

18

30

Show answer & explanation

Answer: 6

Answer

The value of

3 \nabla 4

is 6.

Substituting

p = 3

and

q = 4

directly into the expression

2p + 3q - pq

gives

2(3) + 3(4) - (3)(4) = 6 + 12 - 12 = 6

.

Step-by-Step Solution

1

Identify the assigned values for the operator inputs.

p = 3

and

q = 4

.

The expression

3 \nabla 4

matches

p \nabla q

with

p

as the first argument and

q

as the second.

2

Substitute

p = 3

and

q = 4

into the custom operation formula

2p + 3q - pq

.

2(3) + 3(4) - (3)(4)

Direct evaluation of the defined function rule.

3

Perform the multiplication and algebraic addition/subtraction.

6 + 12 - 12 = 6

Simplifying numerical operations gives the final result.

Key Concept

Custom Operator Evaluation

Question 92Question →

Let $S$ be the set of all real solutions to the polynomial equation $(x^2 - 5x + 6)(x^2 - 7x + 12) = (x - 3)^2(x - 2)$ . What is the sum of all elements in set $S$ ?

5

7

8

10

13

Show answer & explanation

Answer: 10

Answer

The sum of all distinct real elements in set

S

is 10.

The correct answer is obtained by completely factoring both quadratic expressions on the left side into

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

. Moving all terms to one side gives

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

. Factoring out

(x - 2)(x - 3)^2

yields

(x - 2)(x - 3)^2(x - 5) = 0

. The distinct solutions are 2, 3, and 5. Their sum is 10.

Step-by-Step Solution

1

Factor the quadratic expressions on the left side of the equation.

x^2 - 5x + 6 = (x - 2)(x - 3)

and

x^2 - 7x + 12 = (x - 3)(x - 4)

. Substituting these into the left side yields

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

.

Factoring simplifies higher-degree polynomial expressions into linear factors.

2

Set up the equation with all terms on one side to avoid dividing by variable expressions.

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

.

Dividing by variable expressions can cause the loss of valid roots.

3

Factor out the common polynomial factor

(x - 2)(x - 3)^2

.

(x - 2)(x - 3)^2 [(x - 4) - 1] = 0

, which simplifies to

(x - 2)(x - 3)^2(x - 5) = 0

.

Factoring allows finding all roots by setting each factor equal to zero.

4

Identify the distinct real solutions forming set

S

and compute their sum.

The roots are

x = 2

,

x = 3

, and

x = 5

. Thus,

S = \{2, 3, 5\}

. The sum of elements in

S

is

2 + 3 + 5 = 10

.

A set contains only unique elements, so repeated roots are counted only once.

Key Concept

Factoring polynomial equations completely without dividing by variable expressions

Estimated Time:2m 0s

Question 93Question →

What is the sum of all real solutions to the equation $x(x - 3) = 10$ ?

3

5

7

-3

10

Show answer & explanation

Answer: 3

Answer

The sum of all real solutions to the equation is 3.

Expanding the equation

x(x - 3) = 10

yields

x^2 - 3x - 10 = 0

. Factoring this expression gives

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

, yielding solutions

x = 5

and

x = -2

. The sum of these two real solutions is

5 + (-2) = 3

. Alternatively, applying Vieta's formula directly to

x^2 - 3x - 10 = 0

gives the sum of roots as

-\frac{b}{a} = -\frac{-3}{1} = 3

.

Step-by-Step Solution

1

Expand the left side of the equation.

x^2 - 3x = 10

Multiply

x

across the binomial

(x - 3)

to convert to standard polynomial form.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 3x - 10 = 0

Subtract 10 from both sides so the equation equals zero.

3

Factor the quadratic expression.

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

Find two numbers that multiply to

-10

and add up to

-3

, which are

-5

and

2

.

4

Solve for the roots of the equation.

x = 5

or

x = -2

Apply the zero-product property: set each factor equal to zero.

5

Calculate the sum of the solutions.

Sum

= 5 + (-2) = 3

Add the two real solutions together. Alternatively, by Vieta's formulas, the sum of roots is

-\frac{b}{a} = -\frac{-3}{1} = 3

.

Key Concept

Quadratic Equations and Polynomial Factoring

Estimated Time:1m 0s

Question 94Question →

For all real numbers $x$ and $y$ , the custom operation $\diamondsuit$ is defined by $x \diamondsuit y = 3x - 2y + 1$ . What is the value of $5 \diamondsuit 4$ ?

Show answer & explanation

Answer: 8

Answer

The value of

5 \diamondsuit 4

is 8.

Substituting

x = 5

and

y = 4

into the operation

x \diamondsuit y = 3x - 2y + 1

yields

3(5) - 2(4) + 1 = 15 - 8 + 1 = 8

.

Step-by-Step Solution

1

Substitute the operands into the custom operator rule

5 \diamondsuit 4 = 3(5) - 2(4) + 1

The first term

5

corresponds to

x

and the second term

4

corresponds to

y

in the formula.

2

Simplify the numeric expression

15 - 8 + 1 = 8

Multiply first, then perform subtraction and addition from left to right.

Key Concept

Function Definitions, Evaluation, and Custom Operators

Question 95Question →

For all non-zero real numbers $x$ , the function $f$ is defined by $f(x) = \frac{x^2 - 4}{2x}$ , and the function $g$ is defined by $g(x) = 3x - 1$ . What is the positive value of $a$ for which $f(g(a)) = 0$ ?

Show answer & explanation

Answer: 1

Answer

The positive value of

a

for which

f(g(a)) = 0

is 1.

To solve

f(g(a)) = 0

, first find the values of

y

where

f(y) = 0

. The numerator of

f(y) = \frac{y^2 - 4}{2y}

equals zero when

y^2 - 4 = 0

, yielding

y = 2

and

y = -2

. Setting

g(a) = 3a - 1

equal to these roots gives

3a - 1 = 2 \implies a = 1

and

3a - 1 = -2 \implies a = -1/3

. Since

a

must be positive, the correct value is 1.

Step-by-Step Solution

1

Determine the values of the argument

y

that satisfy

f(y) = 0

.

y = 2

or

y = -2

.

A fraction equals zero when its numerator is zero and its denominator is non-zero. Setting

y^2 - 4 = 0

gives

y = \pm 2

.

2

Substitute

g(a) = 3a - 1

into

y

to solve for

a

.

Solving

3a - 1 = 2

yields

a = 1

; solving

3a - 1 = -2

yields

a = -1/3

.

Setting the expression for

g(a)

equal to each root of

f(y) = 0

identifies all potential values for

a

.

3

Select the value of

a

matching the positivity condition.

a = 1

.

The question explicitly specifies finding the positive value of

a

.

Key Concept

Function composition and evaluation of nested functional equations.

Question 96Question →

For all non-zero real numbers $a$ and $b$ , the binary operator $\diamond$ is defined by $a \diamond b = a - \frac{b}{a}$ . If $k$ is a positive integer such that $(k \diamond 3) \diamond 3 = \frac{1}{2}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The positive integer value of

k

is 3.

Applying the non-commutative custom operator definition

a \diamond b = a - \frac{b}{a}

to the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

requires working from the inside out. Letting

u = k \diamond 3 = k - \frac{3}{k}

, the outer equation becomes

u \diamond 3 = u - \frac{3}{u} = \frac{1}{2}

. Multiplying through by

2u

gives

2u^2 - u - 6 = 0

, which factors as

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

, yielding

u = 2

or

u = -\frac{3}{2}

. Substituting

u = 2

back into

k - \frac{3}{k} = 2

produces

k^2 - 2k - 3 = 0

, which factors as

(k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Testing

u = -\frac{3}{2}

gives

2k^2 + 3k - 6 = 0

, which has no integer solutions.

Step-by-Step Solution

1

Substitute a temporary variable for the inner custom operator expression.

Let

u = k \diamond 3 = k - \frac{3}{k}

.

Simplifies the nested expression

(k \diamond 3) \diamond 3 = \frac{1}{2}

to a single variable operation

u \diamond 3 = \frac{1}{2}

.

2

Apply the operator definition to

u \diamond 3

and solve the resulting quadratic equation for

u

.

u - \frac{3}{u} = \frac{1}{2} \implies 2u^2 - u - 6 = 0 \implies (2u + 3)(u - 2) = 0

, so

u = 2

or

u = -\frac{3}{2}

.

Clearing the denominator

u

transforms the custom operator equation into a standard quadratic equation.

3

Solve for positive integer

k

using the obtained values of

u

.

Setting

k - \frac{3}{k} = 2

yields

k^2 - 2k - 3 = 0 \implies (k - 3)(k + 1) = 0

. Since

k

must be a positive integer,

k = 3

. Setting

k - \frac{3}{k} = -\frac{3}{2}

yields

2k^2 + 3k - 6 = 0

, which has non-integer discriminant

\sqrt{57}

.

Only

u = 2

yields an integer solution for

k

, satisfying all problem constraints.

Key Concept

Custom Operator Evaluation and Nested Functional Equations

Question 97Question →

For all non-zero real numbers $x$ , the function $f$ satisfies the equation $2f(x) + f\left(\frac{1}{x}\right) = 6x$ . What is the value of $f(2)$ ?

Show answer & explanation

Answer: 7

Answer

The value of

f(2)

is 7.

Substituting

x = 2

into the functional equation gives

2f(2) + f\left(\frac{1}{2}\right) = 12

. Substituting

x = \frac{1}{2}

gives

2f\left(\frac{1}{2}\right) + f(2) = 3

. Multiplying the first equation by 2 and subtracting the second equation eliminates

f\left(\frac{1}{2}\right)

, producing

3f(2) = 21

, so

f(2) = 7

.

Step-by-Step Solution

1

Substitute

x = 2

into the functional equation

2f(2) + f\left(\frac{1}{2}\right) = 12

This establishes a linear relation between

f(2)

and

f\left(\frac{1}{2}\right)

.

2

Substitute

x = \frac{1}{2}

into the functional equation

2f\left(\frac{1}{2}\right) + f(2) = 3

This establishes a second linear relation involving the reciprocal input.

3

Eliminate

f\left(\frac{1}{2}\right)

by solving the system of equations

f(2) = 7

Multiplying the first equation by 2 yields

4f(2) + 2f\left(\frac{1}{2}\right) = 24

. Subtracting the second equation gives

3f(2) = 21

, which simplifies to

f(2) = 7

.

Key Concept

Evaluating functional equations by reciprocal variable substitution to form and solve a linear system.

Estimated Time:2m 0s

Question 98Question →

If 2 is a root of the quadratic equation $x^2 - kx + 24 = 0$ , where $k$ is a constant, and the quadratic equation $x^2 - (k + 2)x + m = 0$ has exactly one real solution, what is the value of $m$ ?

Show answer & explanation

Answer: 64

Answer

The value of

m

is 64.

Substituting

x = 2

into

x^2 - kx + 24 = 0

gives

4 - 2k + 24 = 0

, which simplifies to

2k = 28

or

k = 14

. Substituting

k = 14

into

x^2 - (k + 2)x + m = 0

produces

x^2 - 16x + m = 0

. For a quadratic equation

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant

b^2 - 4ac

must equal 0. Therefore,

(-16)^2 - 4(1)(m) = 0

, which means

256 - 4m = 0

, giving

m = 64

.

Step-by-Step Solution

1

Substitute the known root

x = 2

into the equation

x^2 - kx + 24 = 0

.

k = 14

Since

x = 2

is a root of the quadratic equation, evaluating the expression at

x = 2

must equal zero.

2

Substitute

k = 14

into the second quadratic equation

x^2 - (k + 2)x + m = 0

.

x^2 - 16x + m = 0

This determines the linear coefficient of the second quadratic equation.

3

Set the discriminant of

x^2 - 16x + m = 0

to zero.

m = 64

A quadratic equation

ax^2 + bx + c = 0

has exactly one real solution if and only if its discriminant

b^2 - 4ac

equals zero.

Key Concept

Solving quadratic equations via root substitution and applying the discriminant condition for repeated roots.

Estimated Time:1m 30s

Question 99Question →

For all real numbers $x$ except $0$ and $1$ , the function $f(x)$ satisfies the relation $f(x) + 2f\left(\frac{1}{1-x}\right) = 9x$ . What is the value of $f(2)$ ?

Show answer & explanation

Answer: 6

Answer

The value of

f(2)

is

6

.

Evaluating the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x

at the values

x = 2

,

x = -1

, and

x = \frac{1}{2}

creates a system of three linear equations in terms of

f(2)

,

f(-1)

, and

f\left(\frac{1}{2}\right)

. Solving this system yields

9f(2) = 54

, which simplifies directly to

f(2) = 6

.

Step-by-Step Solution

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) \implies f(2) + 2f(-1) = 18

Applying the input

x = 2

creates an equation connecting

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) \implies f(-1) + 2f\left(\frac{1}{2}\right) = -9

Evaluating at the new input

x = -1

generates a second equation connecting

f(-1)

and

f\left(\frac{1}{2}\right)

.

3

Evaluate the functional equation at

x = \frac{1}{2}

f\left(\frac{1}{2}\right) + 2f\left(\frac{1}{1-1/2}\right) = 9\left(\frac{1}{2}\right) \implies f\left(\frac{1}{2}\right) + 2f(2) = 4.5

Evaluating at

x = \frac{1}{2}

completes the cyclic chain by linking back to

f(2)

.

4

Solve the system of three linear equations for

f(2)

From equation 3,

f\left(\frac{1}{2}\right) = 4.5 - 2f(2)

. Substituting into equation 2 yields

f(-1) = 4f(2) - 18

. Substituting into equation 1 gives

f(2) + 2(4f(2) - 18) = 18 \implies 9f(2) = 54 \implies f(2) = 6

.

Eliminating

f\left(\frac{1}{2}\right)

and

f(-1)

isolated

f(2)

to find its exact numerical value.

Key Concept

Cyclic Functional Equations and Substitution Systems

Question 100Question →

If $2x^2 + 5x - 3 = 0$ and $x > 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

x

is 0.5.

Factoring the quadratic equation

2x^2 + 5x - 3 = 0

yields

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

. Setting each linear factor equal to zero gives two possible solutions for

x

:

x = 0.5

and

x = -3

. Because the problem stipulates that

x > 0

, the negative solution is discarded, leaving

x = 0.5

.

Step-by-Step Solution

1

Factor the quadratic expression

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

Splitting the middle term

5x

into

6x - x

allows grouping to factor by grouping.

2

Find the roots of the equation

x = 0.5

or

x = -3

By the zero-product property, if the product of two factors is zero, at least one factor must be zero.

3

Apply the positivity constraint

x > 0

x = 0.5

The root

x = -3

violates the given condition that

x

must be strictly greater than zero.

Key Concept

Solving quadratic equations by factoring and applying domain constraints

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