All practice questions

What is the remainder when $17^{17^{17}}$ is divided by $7$ ?

Show answer & explanation

Answer: 5

Answer

The remainder when

17^{17^{17}}

is divided by

7

is

5

.

First, reduce the base modulo

7

:

17 \equiv 3 \pmod 7

, turning the expression into

3^{17^{17}} \pmod 7

. Next, evaluate the pattern of powers of

3 \pmod 7

:

3^1 \equiv 3

,

3^2 \equiv 2

,

3^3 \equiv 6

,

3^4 \equiv 4

,

3^5 \equiv 5

, and

3^6 \equiv 1

, showing a period of

6

. To find which term of the cycle corresponds to the exponent

17^{17}

, evaluate

17^{17} \pmod 6

. Since

17 \equiv -1 \pmod 6

,

17^{17} \equiv (-1)^{17} = -1 \equiv 5 \pmod 6

. Finally, the

5

th term of the cycle gives

3^5 = 243 \equiv 5 \pmod 7

. Therefore, the remainder is

5

.

Step-by-Step Solution

1

Reduce the base modulo 7

17 \equiv 3 \pmod 7

, so

17^{17^{17}} \equiv 3^{17^{17}} \pmod 7

.

Modular arithmetic permits replacing the base of an exponential expression with its remainder upon division by the modulus.

2

Determine the cyclicity period of powers of 3 modulo 7

The remainders of

3^n \pmod 7

repeat in a cycle of length 6:

(3, 2, 6, 4, 5, 1)

.

By Fermat's Little Theorem,

3^6 \equiv 1 \pmod 7

, meaning the sequence of remainders repeats every 6 integer powers.

3

Evaluate the exponent

17^{17}

modulo the period length 6

17 \equiv -1 \pmod 6

, so

17^{17} \equiv (-1)^{17} = -1 \equiv 5 \pmod 6

.

The position within the 6-term cyclicity pattern depends on the exponent modulo 6. Using negative remainders simplifies calculating odd powers of

-1

.

4

Calculate the final remainder using the 5th position in the cyclicity pattern

3^5 = 243 = 7 \times 34 + 5 \equiv 5 \pmod 7

.

Since the exponent leaves a remainder of 5 when divided by 6, the overall remainder corresponds to

3^5 \pmod 7

.

Key Concept

Modular cyclicity and tower exponent remainder reduction

Question 462Question →

If $|x - 5| = 2x - 4$ , what is the value of $x$ ?

3

-1

-1 and 3

1

5

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Solving the absolute value equation yields two candidate values,

x = 3

and

x = -1

. Substituting

x = 3

into the original equation gives

|3 - 5| = 2

and

2(3) - 4 = 2

, which makes the statement true. Therefore,

x = 3

is the only valid solution.

Step-by-Step Solution

1

Set up the two cases for the absolute value equation

|x - 5| = 2x - 4

.

Case 1:

x - 5 = 2x - 4

; Case 2:

x - 5 = -(2x - 4)

.

By definition,

|a| = b

implies

a = b

or

a = -b

when

b \ge 0

.

2

Solve Case 1:

x - 5 = 2x - 4

.

x = -1

.

Subtract

x

and add

4

to both sides.

3

Solve Case 2:

x - 5 = -2x + 4

.

3x = 9 \implies x = 3

.

Add

2x

and add

5

to both sides.

4

Test potential solutions in the original equation to eliminate extraneous roots.

For

x = -1

:

|-1 - 5| = 6 \neq 2(-1) - 4 = -6

(Invalid). For

x = 3

:

|3 - 5| = 2 = 2(3) - 4 = 2

(Valid).

The absolute value of a real number cannot be negative, so the right-hand side

2x - 4

must be non-negative.

Key Concept

Absolute Value Equations and Extraneous Solutions

Question 463Question →

For all real numbers $x$ and $y$ , the custom operation $\odot$ is defined by $x \odot y = x^2 - 2y$ . What is the value of $(3 \odot 2) \odot 4$ ?

17

6

-4

9

-7

Show answer & explanation

Answer: 17

Answer

17

Following the explicit operator rule

x \odot y = x^2 - 2y

, first calculate the inner expression

(3 \odot 2) = 3^2 - 2(2) = 5

. Next, substitute 5 into the outer expression to compute

5 \odot 4 = 5^2 - 2(4) = 25 - 8 = 17

. Thus, the value 17 is correct.

Step-by-Step Solution

1

Evaluate the inner function expression

(3 \odot 2)

using the rule

x \odot y = x^2 - 2y

3 \odot 2 = 3^2 - 2(2) = 9 - 4 = 5

Parentheses dictate that inner custom operations must be evaluated first.

2

Substitute the result of step 1 back into the expression to evaluate

5 \odot 4

5 \odot 4 = 5^2 - 2(4) = 25 - 8 = 17

Applying the custom operator rule with

x = 5

and

y = 4

computes the final target value.

Key Concept

Custom Operator Evaluation and Order of Operations

Estimated Time:1m 15s

Question 464Question →

How many distinct 5-letter code words can be formed by rearranging all of the letters in the word $\text{LEVEL}$ ?

15

24

30

60

120

Show answer & explanation

Answer: $30$

Answer

The total number of distinct 5-letter code words that can be formed is 30.

To calculate the number of unique permutations of a set containing repeated items, divide the factorial of the total number of items by the product of the factorials of the counts of each repeated item. In the word

\text{LEVEL}

, there are 5 letters total, with the letter 'L' appearing 2 times and the letter 'E' appearing 2 times. The calculation is

\frac{5!}{2! \times 2!} = \frac{120}{4} = 30

.

Step-by-Step Solution

1

Identify the total number of letters and the counts of identical letters.

The word

\text{LEVEL}

has 5 total letters, where 'L' appears 2 times and 'E' appears 2 times.

Duplicate elements produce identical arrangements if treated as distinct items.

2

Apply the permutation formula for repeated items:

\frac{n!}{n_1! \times n_2! \times \dots}

.

\frac{5!}{2! \times 2!} = \frac{120}{2 \times 2} = \frac{120}{4} = 30

.

Dividing

5!

by

2!

for the repeated 'L's and

2!

for the repeated 'E's eliminates duplicate arrangements.

Key Concept

Permutations with Repetition / Indistinguishable Objects

Question 465Question →

A bakery bakes vanilla cakes and chocolate cakes in a ratio of $3:4$ . If the bakery bakes a total of $35$ cakes in one morning, how many chocolate cakes were baked?

15

20

21

28

45

Show answer & explanation

Answer: $20$

Answer

20 chocolate cakes were baked.

The ratio of vanilla to chocolate cakes is

3:4

, meaning there are

3 + 4 = 7

total equal parts. The fraction of total cakes that are chocolate is

\frac{4}{7}

. Multiplying this fraction by the total of

35

cakes yields

\frac{4}{7} \times 35 = 20

chocolate cakes.

Step-by-Step Solution

1

Determine the total number of ratio parts

Sum of ratio terms:

3 + 4 = 7

total parts.

The given ratio of vanilla to chocolate cakes is

3:4

, which means out of every

7

cakes,

3

are vanilla and

4

are chocolate.

2

Calculate the value of one ratio part

Value per part:

\frac{35}{7} = 5

cakes per part.

Dividing the total count of

35

cakes by the total number of ratio parts gives the multiplier for each part.

3

Compute the total number of chocolate cakes

Chocolate cakes:

4 \times 5 = 20

.

Multiply the chocolate ratio term (

4

) by the value of one part (

5

).

Key Concept

Part-to-Whole Ratio Scaling

Question 466Question →

If $x(x^2 - 7x + 12) = 2(x^2 - 7x + 12)$ , what is the sum of all distinct real values of $x$ that satisfy the equation?

2

5

7

9

14

Show answer & explanation

Answer: 9

Answer

9

To find all solutions to

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

, subtract the right side from the left to obtain

(x - 2)(x^2 - 7x + 12) = 0

. Factoring the quadratic expression gives

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

. Setting each factor to zero reveals three distinct real roots: 2, 3, and 4. The sum of these roots is

2 + 3 + 4 = 9

.

Step-by-Step Solution

1

Rearrange the equation to set one side to zero

x(x^2 - 7x + 12) - 2(x^2 - 7x + 12) = 0

Setting the equation to zero allows the application of the zero-product property without illegally dividing by variable expressions.

2

Factor out the common quadratic expression

(x^2 - 7x + 12)

(x - 2)(x^2 - 7x + 12) = 0

Both terms share the common factor

(x^2 - 7x + 12)

.

3

Factor the quadratic component

(x^2 - 7x + 12)

into linear factors

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

The quadratic factors into

(x - 3)(x - 4)

because

(-3) \times (-4) = 12

and

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

.

4

Apply the zero-product property to find all distinct real roots and compute their sum

Roots are

x = 2

,

x = 3

, and

x = 4

. Sum =

2 + 3 + 4 = 9

.

Setting each linear factor to zero gives the solutions. Summing these solutions yields 9.

Key Concept

Polynomial Factoring and the Zero-Product Property

Estimated Time:2m 0s

Question 467Question →

What is the sum of all distinct real values of $x$ that satisfy the equation $x(x^2 - 5x + 6) = 2(x^2 - 5x + 6)$ ?

2

3

5

7

9

Show answer & explanation

Answer: 5

Answer

The sum of all distinct real solutions is 5.

To solve the equation

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

, move all terms to the left side to get

x(x^2 - 5x + 6) - 2(x^2 - 5x + 6) = 0

. Factoring out

(x^2 - 5x + 6)

gives

(x - 2)(x^2 - 5x + 6) = 0

. Factoring the quadratic part yields

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

, or

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

. The real roots are

x = 2

and

x = 3

. Taking the sum of these distinct real roots gives

2 + 3 = 5

.

Step-by-Step Solution

1

Move all terms to one side of the equation to avoid dividing by a variable expression.

x(x^2 - 5x + 6) - 2(x^2 - 5x + 6) = 0

Dividing by

(x^2 - 5x + 6)

directly would lose valid roots where

x^2 - 5x + 6 = 0

.

2

Factor out the common quadratic expression

(x^2 - 5x + 6)

.

(x - 2)(x^2 - 5x + 6) = 0

Applying the distributive property isolates the linear factor

(x - 2)

.

3

Factor the quadratic term

x^2 - 5x + 6

into linear factors.

(x - 2)(x - 2)(x - 3) = 0 \implies (x - 2)^2(x - 3) = 0

The quadratic factors into

(x - 2)(x - 3)

because

(-2) + (-3) = -5

and

(-2) \times (-3) = 6

.

4

Identify all distinct real solutions and compute their sum.

Real solutions are

x = 2

and

x = 3

. Distinct sum:

2 + 3 = 5

.

The root

x = 2

has multiplicity 2, but only distinct values must be summed.

Key Concept

Polynomial Factoring and Variable Cancellation Rules

Estimated Time:1m 30s

Question 468Question →

If $x$ is a real number such that $(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1$ , what is the sum of all distinct real values of $x$ that satisfy the equation?

10

12

14

15

9

Show answer & explanation

Answer: 15

Answer

The sum of all distinct real values of

x

that satisfy the equation is 15.

The equation

(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1

holds under three distinct conditions: when the exponent is 0 and the base is non-zero (

x = 4, 5

), when the base is 1 (

x = 1, 4

), and when the base is

-1

provided the exponent is an even integer (

x = 2, 3

, yielding even exponents 6 and 2 respectively). The set of distinct solutions is

\{1, 2, 3, 4, 5\}

, and their sum is 15.

Step-by-Step Solution

1

Analyze the general algebraic conditions under which an expression of the form

u^v = 1

holds for real numbers

u

and

v

.

The equation

u^v = 1

is satisfied in three distinct cases: (1)

v = 0

and

u \neq 0

, (2)

u = 1

, or (3)

u = -1

with

v

being an even integer.

Any non-zero real number raised to the power of 0 equals 1, 1 raised to any power equals 1, and

-1

raised to an even integer power equals 1.

2

Evaluate Case 1 where the exponent

x^2 - 9x + 20 = 0

.

Factoring gives

(x - 4)(x - 5) = 0

, yielding

x = 4

and

x = 5

. Testing the base

x^2 - 5x + 5

at

x = 4

gives

1 \neq 0

, and at

x = 5

gives

5 \neq 0

. Thus

x = 4

and

x = 5

are valid solutions.

Exponent equal to zero produces

u^0 = 1

as long as the base

u \neq 0

.

3

Evaluate Case 2 where the base

x^2 - 5x + 5 = 1

.

Subtracting 1 gives

x^2 - 5x + 4 = 0

, which factors as

(x - 1)(x - 4) = 0

, yielding

x = 1

and

x = 4

. Both are valid.

Base equal to 1 raised to any real exponent equals 1.

4

Evaluate Case 3 where the base

x^2 - 5x + 5 = -1

and check if the exponent is an even integer.

Adding 1 gives

x^2 - 5x + 6 = 0

, which factors as

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

, yielding

x = 2

and

x = 3

. Testing

x = 2

in the exponent gives

2^2 - 9(2) + 20 = 6

(even). Testing

x = 3

gives

3^2 - 9(3) + 20 = 2

(even). Thus

x = 2

and

x = 3

are also valid solutions.

Base

-1

raised to an even integer power equals 1.

5

Combine all unique real values of

x

and compute their sum.

The set of distinct real solutions is

\{1, 2, 3, 4, 5\}

. Their sum is

1 + 2 + 3 + 4 + 5 = 15

.

Summing all identified distinct valid roots.

Key Concept

Solving polynomial exponential equations of the form

f(x)^{g(x)} = 1

by systematically testing base-exponent cases.

Estimated Time:2m 0s

Question 469Question →

A community library maintains fiction and non-fiction books in a ratio of $5 : 2$ . If the library has $350$ fiction books, how many non-fiction books are in the library?

100

140

250

700

875

Show answer & explanation

Answer: 140

Answer

There are 140 non-fiction books in the library.

The ratio of fiction to non-fiction books is

5 : 2

. Since there are

350

fiction books, each unit of the ratio represents

\frac{350}{5} = 70

books. Multiplying the non-fiction ratio part (

2

) by

70

gives

2 \times 70 = 140

non-fiction books.

Step-by-Step Solution

1

Set up the ratio of fiction books to non-fiction books.

\frac{\text{Fiction}}{\text{Non-fiction}} = \frac{5}{2}

The problem states that the ratio of fiction to non-fiction is 5 to 2.

2

Substitute the known number of fiction books (350) and let x represent the number of non-fiction books.

\frac{350}{x} = \frac{5}{2}

This establishes a direct proportion between the actual counts and the ratio values.

3

Cross-multiply and solve for x.

5x = 350 \times 2 \implies 5x = 700 \implies x = 140

Solving for x yields the total number of non-fiction books.

Key Concept

Solving ratio word problems using direct proportion

Estimated Time:45s

Question 470Question →

For all real numbers

x

except

0

and

1

, the function

f

satisfies the functional equation

f(x) + 2f\left(\frac{1}{1-x}\right) = 9x + 3

What is the value of

f(2)

?

Show answer & explanation

Answer: 7

Answer

The value of

f(2)

is

7

.

Evaluating the functional equation iteratively at the points of the cycle

x = 2

,

x = -1

, and

x = \frac{1}{2}

produces a system of three linear equations in terms of

f(2)

,

f(-1)

, and

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

. Solving this linear system gives

f(2) = 7

.

Step-by-Step Solution

1

Evaluate the functional equation at

x = 2

f(2) + 2f\left(\frac{1}{1-2}\right) = 9(2) + 3 \implies f(2) + 2f(-1) = 21

This establishes a relationship between

f(2)

and

f(-1)

.

2

Evaluate the functional equation at

x = -1

f(-1) + 2f\left(\frac{1}{1-(-1)}\right) = 9(-1) + 3 \implies f(-1) + 2f\left(\frac{1}{2}\right) = -6

This links

f(-1)

to

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

to build a closed system of transformations under the mapping

g(x) = \frac{1}{1-x}

.

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) + 3 \implies f\left(\frac{1}{2}\right) + 2f(2) = \frac{15}{2}

This completes the cycle since

g\left(\frac{1}{2}\right) = 2

, creating a system of three linear equations with three unknown values.

4

Solve the system of three linear equations for

f(2)

Let

a = f(2)

,

b = f(-1)

, and

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

. From equation (3),

c = \frac{15}{2} - 2a

. Substituting

c

into equation (2) gives

b + 2\left(\frac{15}{2} - 2a\right) = -6 \implies b = 4a - 21

. Substituting

b

into equation (1) gives

a + 2(4a - 21) = 21 \implies 9a - 42 = 21 \implies 9a = 63 \implies a = 7

.

Algebraic substitution yields the exact value of

f(2)

.

Key Concept

Cyclic Functional Equations and System Substitution

Estimated Time:2m 30s

Question 471Question →

If $k$ and $m$ are non-zero real numbers, is $k > m$ ?

(1) $k^2 > m^2$
(2) $\frac{k - m}{m^2 + 1} > 0$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

The target question asks if

k > m

. Evaluating Statement (1) alone:

k^2 > m^2

indicates

|k| > |m|

, which permits cases where

k > m

(e.g.,

k=3, m=2

) and cases where

k < m

(e.g.,

k=-5, m=2

). Thus, Statement (1) is insufficient. Evaluating Statement (2) independently:

\frac{k - m}{m^2 + 1} > 0

. For any real number

m

,

m^2 + 1

is strictly positive. Multiplying both sides of the inequality by

(m^2 + 1)

preserves the inequality sign, yielding

k - m > 0

, which simplifies directly to

k > m

. This yields a definitive 'Yes' answer. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1

Rephrase the target question

The target question asks whether the difference

(k - m)

is strictly positive.

Determining if

k > m

is algebraically equivalent to confirming

k - m > 0

.

2

Evaluate Statement (1) independently

Statement (1) states

k^2 > m^2

, which implies

|k| > |m|

. If

k = 3

and

m = 2

, then

9 > 4

and

3 > 2

(Yes). However, if

k = -5

and

m = 2

, then

25 > 4

but

-5 < 2

(No).

Because both 'Yes' and 'No' outcomes are possible, Statement (1) alone is not sufficient.

3

Evaluate Statement (2) independently without carrying over Statement (1)

Statement (2) states

\frac{k - m}{m^2 + 1} > 0

. Since

m

is a real number,

m^2 \ge 0

, which guarantees

m^2 + 1 \ge 1 > 0

. Multiplying the inequality by the strictly positive quantity

(m^2 + 1)

preserves the inequality direction, yielding

k - m > 0

, or

k > m

.

This yields a definitive 'Yes' answer, so Statement (2) alone is sufficient.

Key Concept

Statement Independence in Data Sufficiency

Question 472Question →

An integer $n$ is selected at random from the set $\{1, 2, 3, \dots, 500\}$ . What is the probability that $n$ is a perfect square that leaves a remainder of $1$ when divided by $3$ ?

Show answer & explanation

Answer: 0.03

Answer

0.03

The sample space consists of 500 equally likely outcomes. Perfect squares in this range are generated by integers

k

from 1 to 22. Evaluating

k^2 \pmod 3

shows that

k^2

has a remainder of 1 when divided by 3 for all integers

k

not divisible by 3. Excluding the 7 multiples of 3 from the 22 possible values of

k

leaves 15 favorable integers. Dividing the 15 favorable outcomes by the 500 total outcomes gives a probability of 0.03.

Step-by-Step Solution

1

Determine total possible outcomes.

The total number of integers in the set

\{1, 2, 3, \dots, 500\}

is

N = 500

.

Each integer in the set is equally likely to be selected.

2

Find the range of square roots for perfect squares in the set.

Since

22^2 = 484 \le 500

and

23^2 = 529 > 500

,

n = k^2

for

k \in \{1, 2, \dots, 22\}

.

There are 22 perfect squares in the given sample space.

3

Apply remainder constraints to

k^2

.

A square

k^2

leaves a remainder of

1

when divided by

3

if and only if

k

is not divisible by

3

.

If

k

is a multiple of 3,

k^2

is divisible by 3 (remainder 0). If

k

is not a multiple of 3,

k \equiv 1

or

2 \pmod 3

, so

k^2 \equiv 1 \pmod 3

.

4

Count the number of favorable outcomes.

The multiples of 3 up to 22 are

3, 6, 9, 12, 15, 18, 21

(7 numbers). Thus, there are

22 - 7 = 15

values of

k

that satisfy the condition.

Favorable outcomes correspond exactly to integers

k \in \{1, 2, \dots, 22\}

that are not multiples of 3.

5

Compute the single-event probability.

Probability

= \frac{15}{500} = 0.03

.

The probability of a single event is the number of favorable outcomes divided by the total number of outcomes.

Key Concept

Basic Single-Event Probability with Number Properties

Question 473Question →

If $x$ is a real number such that $x^2 - 5x + 3 = 0$ , what is the value of $x^2 + \frac{9}{x^2}$ ?

Show answer & explanation

Answer: 19

Answer

The value of

x^2 + \frac{9}{x^2}

is 19.

Dividing

x^2 - 5x + 3 = 0

by

x

gives

x + \frac{3}{x} = 5

. Squaring both sides yields

\left(x + \frac{3}{x}\right)^2 = x^2 + 2(x)\left(\frac{3}{x}\right) + \frac{9}{x^2} = 25

, which simplifies to

x^2 + 6 + \frac{9}{x^2} = 25

. Subtracting 6 from both sides yields the final answer of 19.

Step-by-Step Solution

1

Divide the quadratic equation by

x

x + \frac{3}{x} = 5

Since

x = 0

is not a root of

x^2 - 5x + 3 = 0

, dividing by

x

rearranges the terms into

x

and its reciprocal component.

2

Square both sides of

x + \frac{3}{x} = 5

x^2 + 6 + \frac{9}{x^2} = 25

Applying the algebraic identity

(a + b)^2 = a^2 + 2ab + b^2

generates the required expression

x^2 + \frac{9}{x^2}

alongside a constant middle term.

3

Isolate

x^2 + \frac{9}{x^2}

x^2 + \frac{9}{x^2} = 19

Subtracting the constant cross-term 6 from 25 yields the target value.

Key Concept

Algebraic manipulation of quadratic equations using reciprocal square identities

Estimated Time:2m 0s

Question 474Question →

What is the sum of all real solutions to the polynomial equation

(x^2 - 3x + 1)^2 - 3(x^2 - 3x + 1) + 1 = x ?

Show answer & explanation

Answer: 6

Answer

The sum of all real solutions to the equation is 6.

The equation

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

represents

f(f(x)) = x

for

f(x) = x^2 - 3x + 1

. Expanding and setting to zero yields

x^4 - 6x^3 + 8x^2 + 2x - 1 = 0

. Since solutions to

f(x) = x

(i.e.,

x^2 - 4x + 1 = 0

) also satisfy

f(f(x)) = x

, we can factor out

x^2 - 4x + 1

to get

(x^2 - 4x + 1)(x^2 - 2x - 1) = 0

. Both quadratic factors have positive discriminants (

12

and

8

), ensuring four distinct real roots. By Vieta's formulas, the sum of roots from the first quadratic is 4 and from the second is 2, giving a total sum of 6.

Step-by-Step Solution

1

Express the equation as a polynomial in standard form

Expanding

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

yields

x^4 - 6x^3 + 8x^2 + 2x - 1 = 0

.

Expanding allows the composite expression to be treated as a degree-4 polynomial.

2

Find a quadratic factor using fixed points of the iteration

Let

f(x) = x^2 - 3x + 1

. The equation has the form

f(f(x)) = x

. Any solution to

f(x) = x

must satisfy this equation. Solving

x^2 - 3x + 1 = x

gives

x^2 - 4x + 1 = 0

.

If

f(x) = x

, then

f(f(x)) = f(x) = x

, so

x^2 - 4x + 1

must be a factor of the 4th-degree polynomial.

3

Factor the degree-4 polynomial completely

Polynomial division of

x^4 - 6x^3 + 8x^2 + 2x - 1

by

x^2 - 4x + 1

yields

(x^2 - 4x + 1)(x^2 - 2x - 1) = 0

.

Factoring reduces the 4th-degree equation into two quadratic equations.

4

Verify real roots and calculate their sum

For

x^2 - 4x + 1 = 0

, discriminant

D_1 = 12 > 0

and sum of roots is 4. For

x^2 - 2x - 1 = 0

, discriminant

D_2 = 8 > 0

and sum of roots is 2. The four roots are all real and distinct, giving a total sum of

4 + 2 = 6

.

Vieta's formulas provide the sum of roots for each quadratic factor without needing to compute individual radical expressions.

Key Concept

Polynomial Factoring and Composite Quadratic Equations

Estimated Time:2m 30s

Question 475Question →

An architectural restoration project allocates its workforce across three specialized teams: Masonry, Carpentry, and Glasswork. Initially, the ratio of Masonry workers to Carpentry workers is $5 : 6$ , and the ratio of Carpentry workers to Glasswork workers is $4 : 5$ .

During phase two of the project:
• The number of Masonry workers is increased by $50\%$ .
• The number of Carpentry workers is decreased by $25\%$ .
• The number of Glasswork workers is increased by $X\%$ .

If the new ratio of Masonry workers to the total number of workers across all three teams is $1 : 3$ , what is the value of $X$ ?

Show answer & explanation

Answer: 40

Answer

The value of

X

is 40.

To find

X

, first combine the initial ratios by equating the common Carpentry term: Masonry : Carpentry = 10 : 12 and Carpentry : Glasswork = 12 : 15, yielding a joint ratio of 10 : 12 : 15. Representing the initial teams as

10k

,

12k

, and

15k

, the updated Masonry team is

10k \times 1.5 = 15k

and Carpentry is

12k \times 0.75 = 9k

. Using the part-to-total ratio

\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}

, we set up

\frac{15k}{15k + 9k + \text{New Glasswork}} = \frac{1}{3}

, which yields

\text{New Glasswork} = 21k

. The percentage increase from

15k

to

21k

is

\frac{21k - 15k}{15k} \times 100\% = 40\%

. Thus,

X = 40

.

Step-by-Step Solution

1

Unify the two given ratios into a single three-part ratio.

The ratio Masonry : Carpentry : Glasswork is

10 : 12 : 15

.

Carpentry is common to both given ratios (

5 : 6

and

4 : 5

). The least common multiple of 6 and 4 is 12. Scaling

5 : 6

by 2 yields

10 : 12

, and scaling

4 : 5

by 3 yields

12 : 15

.

2

Define initial workforce quantities using a multiplier

k

.

Initial Masonry =

10k

, Carpentry =

12k

, Glasswork =

15k

.

Using an algebraic variable preserves the proportional relationships across all teams.

3

Calculate the updated worker counts for Masonry and Carpentry.

New Masonry =

15k

, New Carpentry =

9k

.

Masonry increases by

50\%

(

10k \times 1.50 = 15k

). Carpentry decreases by

25\%

(

12k \times 0.75 = 9k

).

4

Set up the part-to-total ratio equation to solve for the new Glasswork team size.

New Glasswork worker count =

21k

.

The total workforce is

15k + 9k + \text{New Glasswork} = 24k + \text{New Glasswork}

. Given

\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}

, we have

\frac{15k}{24k + \text{New Glasswork}} = \frac{1}{3} \implies 45k = 24k + \text{New Glasswork} \implies \text{New Glasswork} = 21k

.

5

Compute the percentage increase

X

for the Glasswork team.

X = 40

.

The initial Glasswork count was

15k

and the new count is

21k

. The percentage increase is

\frac{21k - 15k}{15k} \times 100 = \frac{6}{15} \times 100 = 40\%

.

Key Concept

Combining compound ratios and solving multi-step part-to-whole proportion equations.

Estimated Time:2m 30s

Question 476Question →

A container holds $20$ liters of a solution that is $15\%$ acid by volume. If $5$ liters of pure water are added to the solution, what is the percentage concentration of acid in the resulting mixture?

Show answer & explanation

Answer: 12

Answer

The concentration of acid in the resulting mixture is 12%.

The correct concentration is found by taking the invariant amount of pure acid (

3

liters) and dividing it by the total updated volume of the liquid (

25

liters), which yields

\frac{3}{25} = 0.12

, or

12\%

.

Step-by-Step Solution

1

Determine the amount of solute (pure acid) in the original mixture.

Amount of acid =

0.15 \times 20 = 3

liters.

The solution initially consists of

15\%

acid out of a total volume of

20

liters.

2

Determine the total volume of the new solution after dilution.

New total volume =

20 + 5 = 25

liters.

Adding

5

liters of pure water increases the total liquid volume without adding any additional acid.

3

Compute the new percentage concentration.

New concentration =

\frac{3}{25} \times 100\% = 12\%

.

The concentration is the ratio of pure solute to total solution volume expressed as a percentage.

Key Concept

Dilution of a mixture by adding pure solvent

Question 477Question →

If $p$ and $q$ are non-zero real numbers such that $p \neq q$ , is $\frac{p^3 q - p q^3}{p - q} > 0$ ?

(1) $pq > 0$
(2) $p + q > 0$

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Factoring the numerator gives

p^3 q - p q^3 = pq(p-q)(p+q)

. Since

p \neq q

, dividing by

(p-q)

yields

pq(p+q)

. To determine whether

pq(p+q) > 0

, we need the sign of the product

pq(p+q)

. Statement (1) tells us

pq > 0

, which is insufficient by itself because

p+q

could be positive or negative. Statement (2) tells us

p+q > 0

, which is insufficient by itself because

pq

could be positive or negative. Together, both terms

pq

and

p+q

are positive, so their product must be positive, providing a definitive 'Yes'.

Step-by-Step Solution

1

Rephrase the target question stem algebraically by factoring the numerator.

\frac{p^3 q - p q^3}{p - q} = \frac{p q (p^2 - q^2)}{p - q} = \frac{p q (p - q)(p + q)}{p - q} = pq(p + q)

. The target question simplifies to: 'Is

pq(p+q) > 0

?'

Simplifying the stem isolates the specific product whose sign determines sufficiency.

2

Evaluate Statement (1):

pq > 0

.

If

p = 1, q = 1

, then

pq = 1 > 0

and

pq(p+q) = 1(2) = 2 > 0

(Yes). If

p = -1, q = -1

, then

pq = 1 > 0

and

pq(p+q) = 1(-2) = -2 < 0

(No).

Knowing

pq > 0

leaves the sign of

p+q

unknown, so Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2):

p + q > 0

.

If

p = 2, q = 1

, then

p+q = 3 > 0

and

pq(p+q) = 2(3) = 6 > 0

(Yes). If

p = 3, q = -1

, then

p+q = 2 > 0

and

pq(p+q) = (-3)(2) = -6 < 0

(No).

Knowing

p+q > 0

leaves the sign of

pq

unknown, so Statement (2) alone is NOT sufficient.

4

Evaluate Statement (1) and Statement (2) together.

From Statement (1),

pq > 0

. From Statement (2),

p+q > 0

. Therefore,

pq(p+q) > 0

as it is the product of two strictly positive real numbers.

Combining both statements provides a definitive 'Yes' answer to the simplified target question.

Key Concept

Question Stem Simplification and Target Rephrasing

Question 478Question →

If $x$ satisfies the equation $|8x + 13| = 10x + 5$ , what is the sum of all valid real solutions for $x$ ?

4

3

-1

-4

5

Show answer & explanation

Answer: 4

Answer

The sum of all valid real solutions for

x

is 4.

The solution

x = 4

is the only value of

x

that satisfies

|8x + 13| = 10x + 5

because evaluating both sides yields

45 = 45

. The alternative algebraic candidate,

x = -1

, produces a right-hand side of

-5

, which is impossible for an absolute value expression. Thus, the sum of all valid real solutions is simply 4.

Step-by-Step Solution

1

Split the absolute value equation into two linear case equations

Case 1:

8x + 13 = 10x + 5

; Case 2:

8x + 13 = -(10x + 5)

By definition of absolute value,

|u| = v

implies

u = v

or

u = -v

, provided

v \ge 0

.

2

Solve Case 1 for

x

2x = 8 \implies x = 4

Subtract

8x

and

5

from both sides to isolate

x

.

3

Solve Case 2 for

x

8x + 13 = -10x - 5 \implies 18x = -18 \implies x = -1

Expand the right-hand side and combine like terms to isolate

x

.

4

Test candidate solutions in the original equation to eliminate extraneous roots

For

x = 4

:

|8(4)+13| = |45| = 45

and

10(4)+5 = 45

(Valid). For

x = -1

:

|8(-1)+13| = |5| = 5

and

10(-1)+5 = -5

(Extraneous).

An absolute value output cannot equal a negative number.

5

Calculate the sum of all valid solutions

The only valid solution is

x = 4

, so the sum is

4

.

Only valid roots must be summed.

Key Concept

Absolute Value Equations and Checking for Extraneous Solutions

Question 479Question →

Set $S$ consists of $n$ consecutive integers. The product of the smallest and largest integers in $S$ is $-144$ . If the sum of all elements in $S$ is $91$ , what is the value of $n$ ?

Show answer & explanation

Answer: 26

Answer

The number of consecutive integers

n

in Set

S

is 26.

By combining the consecutive set sum formula

n(a + b) = 182

with the term span relation

b - a = n - 1

and the product constraint

ab = -144

, testing the factors of

182

uniquely determines

n = 26

, corresponding to the set of consecutive integers from

-9

to

16

.

Step-by-Step Solution

1

Relate the number of elements

n

to the smallest element

a

and largest element

b

.

b - a = n - 1

In any set of

n

consecutive integers, the distance between the maximum and minimum elements is

n - 1

.

2

Use the sum formula for an evenly spaced set.

n(a + b) = 182

The sum is

\text{Sum} = n \times \frac{a + b}{2} = 91

, which simplifies to

n(a + b) = 182

.

3

Factor

182

to find integer solutions satisfying

a \cdot b = -144

.

n = 26

,

a = -9

,

b = 16

For

n = 26

,

a + b = 7

and

b - a = 25

, giving

a = -9

and

b = 16

. Their product is

(-9)(16) = -144

.

Key Concept

Properties of consecutive integer sets, average/median sum formula, and term boundary indexing

Question 480Question →

If $x$ is a real number such that $\sqrt[4]{3^x + 3^x + 3^x} = 27$ , what is the value of $x$ ?

Show answer & explanation

Answer: 11

Answer

The value of

x

is 11.

Summing three terms of

3^x

yields

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

. Expressing the fourth root of

3^{x+1}

as a fractional power gives

3^{\frac{x+1}{4}}

. Equating this to

27 = 3^3

results in

\frac{x+1}{4} = 3

, leading directly to

x = 11

.

Step-by-Step Solution

1

Combine the repeated addition terms inside the fourth root.

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

Adding three identical terms

3^x

is equivalent to multiplying

3^x

by 3, which increments the exponent by 1.

2

Rewrite the fourth root expression using a fractional exponent.

\sqrt[4]{3^{x+1}} = 3^{\frac{x+1}{4}}

Applying the property

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

.

3

Express the constant on the right side of the equation with base 3.

27 = 3^3

Rewriting both sides of the equation with the same base allows equating exponents.

4

Equate the exponents and solve for

x

.

\frac{x+1}{4} = 3 \implies x + 1 = 12 \implies x = 11

Since the bases are equal, the powers must be equal.

Key Concept

Combining like exponential terms and equating fractional exponents

Estimated Time:1m 30s

Answer

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