Tüm alıştırma soruları

Soru 101Soru →

In triangle $ABC$ , the length of side $AB$ is $15$ and the length of side $BC$ is $25$ . The area of triangle $ABC$ is $150$ . If angle $ABC$ is obtuse, what is the length of side $AC$ ?

20

5\sqrt{34}

10\sqrt{13}

34

40

Cevabı ve açıklamayı göster

Cevap: $10\sqrt{13}$

Cevap

10\sqrt{13}

The correct answer

10\sqrt{13}

is obtained by drawing altitude

AH = 12

to line

BC

. In right triangle

ABH

, the base projection is

BH = \sqrt{15^2 - 12^2} = 9

. Because angle

ABC

is obtuse, point

H

lies outside segment

BC

, so

CH = 25 + 9 = 34

. Applying the Pythagorean theorem to right triangle

AHC

gives

AC = \sqrt{12^2 + 34^2} = \sqrt{1300} = 10\sqrt{13}

.

Adım Adım Çözüm

1

Calculate the height (altitude)

h

perpendicular to line

BC

.

Since

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, we have

150 = \frac{1}{2} \times 25 \times h

, which simplifies to

h = 12

.

The area formula for any triangle connects the base length and its corresponding perpendicular altitude.

2

Determine the location of the altitude foot

H

and calculate segment

BH

.

Draw altitude

AH

to the line containing

BC

. In right triangle

ABH

,

AB = 15

and

AH = 12

, so

BH = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9

.

The Pythagorean theorem applies to right triangle

ABH

formed by the altitude.

3

Account for the obtuse angle condition to find segment

CH

.

Because angle

ABC

is obtuse, the altitude foot

H

lies on the extension of segment

CB

beyond vertex

B

. Therefore,

CH = CB + BH = 25 + 9 = 34

.

For an obtuse triangle, the altitude to one of the adjacent sides falls outside the triangle.

4

Calculate side length

AC

using right triangle

AHC

.

AC = \sqrt{AH^2 + CH^2} = \sqrt{12^2 + 34^2} = \sqrt{144 + 1156} = \sqrt{1300} = 10\sqrt{13}

.

Applying the Pythagorean theorem to right triangle

AHC

yields the hypotenuse

AC

.

Anahtar Kavram

Triangle Area, Altitude Projection, and Extended Pythagorean Theorem

Soru 102Soru →

A triangle has a base of length $10$ and an area of $30$ . Which of the following statements could be true about this triangle? Select all that apply.

Geçerli olan tümünü seçin

The altitude corresponding to the base of length

10

is

6

.

The triangle is a right triangle.

The perimeter of the triangle is

22

.

The perimeter of the triangle is

30

.

The perimeter of the triangle is

24

.

Cevabı ve açıklamayı göster

Cevap: The altitude corresponding to the base of length $10$ is $6$ .; The triangle is a right triangle.; The perimeter of the triangle is $30$ .

Cevap

The altitude perpendicular to the given base must be 6, the triangle can be a right triangle, and the perimeter can be equal to 30.

The area formula directly forces the altitude to be 6. Setting the altitude at one endpoint of the base constructs a valid right triangle. Furthermore, the minimum perimeter of any triangle with base 10 and height 6 is

10 + 2\sqrt{61} \approx 25.62

, so any perimeter value greater than or equal to

25.62

(such as 30) is attainable.

Adım Adım Çözüm

1

Calculate the required altitude of the triangle.

Using

\text{Area} = \frac{1}{2} \times \text{base} \times h

,

30 = \frac{1}{2}(10)h \implies h = 6

.

The area and base are fixed, determining a unique height.

2

Evaluate whether the triangle can be a right triangle.

A right triangle with legs

10

and

6

has area

\frac{1}{2} \times 10 \times 6 = 30

.

Choosing the altitude to meet the base at an endpoint creates a right angle.

3

Determine the lower bound for the perimeter of the triangle.

The third vertex lies on a line parallel to the base at a distance of

6

. By symmetry, the minimum sum of the remaining two sides occurs when the triangle is isosceles with base

10

split into two segments of length

5

. Each equal side is

\sqrt{5^2 + 6^2} = \sqrt{61} \approx 7.8102

. The minimum perimeter is

10 + 2\sqrt{61} \approx 25.62

.

The shortest path from two fixed base endpoints to a parallel line is formed when the reflection creates equal angles, making the triangle isosceles.

4

Assess the possible perimeter values based on the lower bound.

Perimeters

22

and

24

are strictly below the minimum boundary of

\approx 25.62

and are impossible. Since the perimeter can take any value in

[10 + 2\sqrt{61}, \infty)

, a perimeter of

30

is possible.

Continuous movement of the top vertex increases the side lengths smoothly without bound.

Anahtar Kavram

Triangle area formula

\text{Area} = \frac{1}{2}bh

and geometric optimization of perimeter for a given base and height.

Soru 103Soru →

On the real number line, point $P$ represents the real number $x$ , point $Q$ represents $7$ , and point $R$ represents $-5$ . If the distance between $P$ and $Q$ is equal to $3$ times the distance between $P$ and $R$ , what is the sum of all possible values of $x$ ?

-13

-11

-7

-2

13

Cevabı ve açıklamayı göster

Cevap: $-13$

Cevap

The sum of all possible values of

x

is

-13

.

The distance between

x

and

7

is

|x - 7|

and the distance between

x

and

-5

is

|x + 5|

. Equating

|x - 7| = 3|x + 5|

leads to two equations:

x - 7 = 3(x + 5)

giving

x = -11

, and

x - 7 = -3(x + 5)

giving

x = -2

. Adding both solutions yields

(-11) + (-2) = -13

.

Adım Adım Çözüm

1

Set up the distance equation using absolute value notation

The distance between

P(x)

and

Q(7)

is

|x - 7|

, and the distance between

P(x)

and

R(-5)

is

|x - (-5)| = |x + 5|

. The problem specifies that

|x - 7| = 3|x + 5|

.

Distance between two points

a

and

b

on a real number line is expressed as

|a - b|

.

2

Solve Case 1 where

x - 7

and

x + 5

have the same sign

x - 7 = 3(x + 5) \implies x - 7 = 3x + 15 \implies -22 = 2x \implies x = -11

.

When both absolute value expressions have identical signs,

|x - 7| = 3|x + 5|

simplifies directly to

x - 7 = 3(x + 5)

.

3

Solve Case 2 where

x - 7

and

x + 5

have opposite signs

x - 7 = -3(x + 5) \implies x - 7 = -3x - 15 \implies 4x = -8 \implies x = -2

.

When the absolute value expressions have opposite signs,

|x - 7| = 3|x + 5|

simplifies to

x - 7 = -3(x + 5)

.

4

Calculate the sum of all possible values of

x

(-11) + (-2) = -13

.

Summing the two solutions gives the final required value.

Anahtar Kavram

Distance on a number line and absolute value equations

Tahmini Süre:1m 30s

Soru 104Soru →

If

x

is a positive integer such that

\sqrt{4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1}} = 30,000

what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is 4.

Factoring the common exponential term

4^x \cdot 5^{2x}

inside the radical yields

\sqrt{(4^x \cdot 5^{2x})(4 + 5)} = \sqrt{9 \cdot 4^x \cdot (5^2)^x} = \sqrt{9 \cdot (4 \cdot 25)^x} = \sqrt{9 \cdot 100^x} = \sqrt{9 \cdot 10^{2x}} = 3 \cdot 10^x

. Setting

3 \cdot 10^x = 30,000

gives

10^x = 10,000 = 10^4

, which means

x = 4

.

Adım Adım Çözüm

1

Separate the addition in exponents using exponent rules.

4^{x+1} \cdot 5^{2x} = 4^x \cdot 4^1 \cdot 5^{2x}

and

4^x \cdot 5^{2x+1} = 4^x \cdot 5^{2x} \cdot 5^1

.

Applying the product rule of exponents

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

prepares terms for factoring.

2

Factor out the common expression

4^x \cdot 5^{2x}

from the sum inside the radical.

4^{x+1} \cdot 5^{2x} + 4^x \cdot 5^{2x+1} = (4^x \cdot 5^{2x})(4 + 5) = 9 \cdot 4^x \cdot 5^{2x}

.

Factoring converts the sum under the square root into a single product.

3

Combine terms with powers into base 10.

4^x \cdot 5^{2x} = 4^x \cdot (5^2)^x = 4^x \cdot 25^x = (4 \cdot 25)^x = 100^x = 10^{2x}

.

Using power of a power

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

and power of a product

a^n b^n = (ab)^n

simplifies the expression into powers of 10.

4

Take the square root of the simplified product.

\sqrt{9 \cdot 10^{2x}} = \sqrt{9} \cdot \sqrt{10^{2x}} = 3 \cdot 10^x

.

Applying the product rule for radicals

\sqrt{ab} = \sqrt{a}\sqrt{b}

and halving the exponent

(10^{2x})^{1/2} = 10^x

.

5

Equate the simplified expression to 30,000 and solve for

x

.

3 \cdot 10^x = 30,000 \implies 10^x = 10,000 \implies 10^x = 10^4 \implies x = 4

.

Dividing both sides by 3 isolates

10^x

, and matching exponential bases gives

x = 4

.

Anahtar Kavram

Exponent Rules and Radical Simplification

Tahmini Süre:2m 0s

Soru 105Soru →

Line $L$ is defined by the equation $2x + y = 6$ in the $xy$ -plane. Which of the following statements about line $L$ must be true? Select all that apply.

Geçerli olan tümünü seçin

The slope of line

L

is

-2

.

The

x

-intercept of line

L

is

(3, 0)

.

The slope of line

L

is

-\frac{1}{2}

.

A line perpendicular to line

L

has a slope of

-2

.

The

y

-intercept of line

L

is

(0, -6)

.

Cevabı ve açıklamayı göster

Cevap: The slope of line $L$ is $-2$ .; The $x$ -intercept of line $L$ is $(3, 0)$ .

Cevap

The slope of line

L

is

-2

, and the

x

-intercept of line

L

is

(3, 0)

.

The given line equation

2x + y = 6

can be rewritten in slope-intercept form

y = -2x + 6

. This directly shows that the slope is

-2

. Setting

y = 0

gives

2x = 6 \implies x = 3

, so the

x

-intercept is

(3, 0)

. Therefore, both the statement that the slope is

-2

and the statement that the

x

-intercept is

(3, 0)

are correct.

Adım Adım Çözüm

1

Convert the equation to slope-intercept form (

y = mx + b

).

y = -2x + 6

This isolates

y

to clearly reveal the slope

m = -2

and the

y

-intercept

(0, 6)

.

2

Find the

x

-intercept by setting

y = 0

.

2x + 0 = 6 \implies x = 3

, giving coordinate

(3, 0)

The

x

-intercept is the point where the line crosses the

x

-axis.

3

Determine perpendicular slope rules.

Perpendicular slope

= -\frac{1}{-2} = \frac{1}{2}

Perpendicular lines have negative reciprocal slopes, not identical slopes.

Anahtar Kavram

Linear equations, slope-intercept form, intercepts, and perpendicular slopes in coordinate geometry.

Tahmini Süre:1m 0s

Soru 106Soru →

If $x$ and $y$ are real numbers such that $|x + 2| \le 3$ and $|y - 5| \le 2$ , which of the following could be the value of $|x - y|$ ? Select all such values.

Geçerli olan tümünü seçin

1

4

8

12

15

Cevabı ve açıklamayı göster

Cevap: 4; 8; 12

Cevap

The possible values for

|x - y|

are 4, 8, and 12.

Solving the inequality

|x + 2| \le 3

gives

-5 \le x \le 1

, and solving

|y - 5| \le 2

gives

3 \le y \le 7

. The minimum possible value of

x - y

occurs at

-5 - 7 = -12

, and the maximum value occurs at

1 - 3 = -2

. Thus,

x - y

lies entirely in the interval

[-12, -2]

. Taking absolute values shows that

|x - y|

must lie in the interval

[2, 12]

. The numbers 4, 8, and 12 all fall within this interval and are valid solutions.

Adım Adım Çözüm

1

Solve the absolute value inequality for

x

.

-3 \le x + 2 \le 3 \implies -5 \le x \le 1

Unpack

|x + 2| \le 3

into a compound inequality and isolate

x

.

2

Solve the absolute value inequality for

y

.

-2 \le y - 5 \le 2 \implies 3 \le y \le 7

Unpack

|y - 5| \le 2

into a compound inequality and isolate

y

.

3

Determine the minimum and maximum possible values of

x - y

.

Minimum

x - y = -5 - 7 = -12

; Maximum

x - y = 1 - 3 = -2

.

To minimize

x - y

, take the smallest

x

and largest

y

. To maximize

x - y

, take the largest

x

and smallest

y

.

4

Find the range of

|x - y|

.

2 \le |x - y| \le 12

Since

-12 \le x - y \le -2

, taking the absolute value yields values in the closed interval

[2, 12]

.

5

Evaluate the choices against the interval

[2, 12]

.

4, 8, and 12 lie inside

[2, 12]

, whereas 1 and 15 do not.

Any real number in

[2, 12]

is a achievable value for

|x - y|

.

Anahtar Kavram

Real Numbers, Number Line, and Absolute Value Inequalities

Soru 107Soru →

Match each advanced vocabulary word on the left with its most precise functional near-synonym on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings are: Intransigent matches Obdurate; Specious matches Spurious; Pellucid matches Limpid; Bellicose matches Pugnacious.

Each formal vocabulary term aligns with a semantic near-synonym: 'Intransigent' pairs with 'Obdurate' (unyielding posture), 'Specious' pairs with 'Spurious' (false/deceptive validity), 'Pellucid' pairs with 'Limpid' (clear/lucid), and 'Bellicose' pairs with 'Pugnacious' (combative/hostile).

Adım Adım Çözüm

1

Analyze 'Intransigent'

Recognize its core meaning as unyielding or stubborn, which pairs directly with 'Obdurate'.

Matching terms denoting uncompromising stubbornness.

2

Analyze 'Specious'

Identify its meaning as deceptively pleasing or superficially plausible, which pairs directly with 'Spurious'.

Matching terms denoting deceitful appearance of validity.

3

Analyze 'Pellucid'

Identify its definition as translucent or easily understood, which pairs directly with 'Limpid'.

Matching terms denoting physical or expression clarity.

4

Analyze 'Bellicose'

Recognize its definition as eager to fight or aggressive, which pairs directly with 'Pugnacious'.

Matching terms denoting combative behavior.

Anahtar Kavram

Identifying Synonym and Near-Synonym Pairs

Soru 108Soru →

Passage:
In the late 1940s, geneticist Barbara McClintock investigated color mutation patterns in maize kernels, observing phenotypic variations that failed to conform to classical Mendelian inheritance ratios. Through meticulous cytogenetic analysis of chromosome 9, McClintock identified specific genetic elements that were capable of changing their physical positions within the genome. She designated these mobile elements as 'controlling elements'—now termed transposons—and demonstrated that their insertion into or excision from particular gene loci could suppress or reactivate surrounding gene expression. Contrary to the prevailing consensus of the era, which posited that chromosomes possessed fixed, immutable sequences of genes, McClintock proposed that genetic regulation was dynamic and modulated by spatial rearrangement. Although her initial presentations in 1951 were met with skepticism by the scientific community due to the entrenchment of the static genome paradigm, subsequent molecular discoveries in bacteria during the late 1960s confirmed the physical mechanism of transposition.

Consider each of the choices separately and select all that apply.

According to the passage, which of the following is explicitly stated regarding Barbara McClintock's research on maize?

Geçerli olan tümünü seçin

Her identification of controlling elements relied on cytogenetic examination of chromosome 9.

Her initial 1951 findings were immediately accepted as a clear refutation of classical Mendelian inheritance.

The mobility of the genetic elements she identified could influence the activation state of nearby genes.

Cevabı ve açıklamayı göster

Cevap: Her identification of controlling elements relied on cytogenetic examination of chromosome 9.; The mobility of the genetic elements she identified could influence the activation state of nearby genes.

Cevap

The supported statements are that her identification of controlling elements relied on cytogenetic examination of chromosome 9, and that the mobility of the genetic elements she identified could influence the activation state of nearby genes.

The correct options accurately reflect explicit details in the passage. The passage specifically mentions that McClintock's identification of controlling elements was accomplished through 'cytogenetic analysis of chromosome 9.' Furthermore, the text explicitly details that the movement ('insertion into or excision from') of these elements could 'suppress or reactivate surrounding gene expression,' directly confirming that their mobility influenced the activation state of nearby genes.

Adım Adım Çözüm

1

Analyze the passage for explicit statements regarding McClintock's research methods and findings.

Located references to chromosome 9 analysis, controlling elements (transposons), gene expression influence, and reception in 1951.

Explicit detail retrieval questions require matching options directly to facts stated in the passage text.

2

Evaluate the statement concerning chromosome 9 examination.

The text states 'Through meticulous cytogenetic analysis of chromosome 9, McClintock identified specific genetic elements...'

This confirms that her identification of controlling elements relied directly on cytogenetic examination of chromosome 9.

3

Evaluate the statement regarding immediate acceptance of her 1951 findings.

The text states 'Although her initial presentations in 1951 were met with skepticism...'

This directly contradicts the claim of immediate acceptance.

4

Evaluate the statement regarding the influence of mobile elements on nearby genes.

The text states 'insertion into or excision from particular gene loci could suppress or reactivate surrounding gene expression.'

This directly supports the statement that mobility could influence the activation state of surrounding/nearby genes.

Anahtar Kavram

Explicit Detail Retrieval

Soru 109Soru →

Suppose $a$ , $b$ , and $c$ are integers such that $a < 0 < b < c$ . If $a(b - c)$ is an odd integer and $a^2 b + b c$ is an even integer, which of the following expressions MUST be a positive even integer?

b(c - a)

a b c

a(b - c)

(a - c) b

(a + c) b

Cevabı ve açıklamayı göster

Cevap: $b(c - a)$

Cevap

b(c - a)

is guaranteed to be a positive even integer.

The expression

b(c - a)

consists of

b

(which is positive and even) multiplied by

(c - a)

(which is positive and even, as subtracting a negative odd number from a positive odd number yields a positive even number). The product of two positive even integers is always a positive even integer.

Adım Adım Çözüm

1

Determine the signs of variables

a

,

b

, and

c

.

a < 0

(negative),

b > 0

(positive), and

c > 0

(positive).

Directly given by the inequality

a < 0 < b < c

.

2

Analyze parity from the given condition that

a(b - c)

is odd.

a

is odd and

(b - c)

is odd.

A product of two integers is odd if and only if both factors are odd.

3

Analyze parity from the second condition

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

being even.

b

must be even, and

c

must be odd.

Since

a

is odd,

a^2

is odd. If

b

were odd, then

a^2 + c

would need to be even (making

c

odd), but if both

b

and

c

were odd,

b - c

would be even, contradicting step 2. Therefore,

b

must be even. Since

b

is even and

b - c

is odd,

c

must be odd.

4

Evaluate the sign and parity of

b(c - a)

.

c - a = \text{odd} - (\text{negative odd}) = \text{positive even}

. Since

b

is positive even,

b(c - a) = \text{positive even} \times \text{positive even} = \text{positive even}

.

Subtracting a negative number yields addition (

c - a > 0

), and subtracting an odd integer from an odd integer produces an even integer.

Anahtar Kavram

Even-Odd Parity and Integer Sign Properties under Multiplication and Subtraction

Soru 110Soru →

If $x$ and $y$ are positive integers such that $5^x \cdot 2^y = 10^{2x-1} \cdot 4^{x+1}$ , what is the value of $y - x$ ?

2

3

4

5

6

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

y - x

is 4.

By prime-factorizing the bases on the right-hand side,

10^{2x-1} \cdot 4^{x+1}

becomes

(2 \cdot 5)^{2x-1} \cdot (2^2)^{x+1} = 5^{2x-1} \cdot 2^{4x+1}

. Matching the powers of 5 gives

x = 2x - 1

, which yields

x = 1

. Matching the powers of 2 gives

y = 4x + 1

, which yields

y = 5

. Subtracting

x

from

y

gives

5 - 1 = 4

.

Adım Adım Çözüm

1

Rewrite composite bases into prime factor bases on the right side of the equation.

10^{2x-1} = (2 \cdot 5)^{2x-1} = 2^{2x-1} \cdot 5^{2x-1}

and

4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}

.

Converting all terms to prime bases (2 and 5) allows equating corresponding exponents.

2

Combine terms with identical bases on the right side.

10^{2x-1} \cdot 4^{x+1} = 5^{2x-1} \cdot 2^{(2x-1) + (2x+2)} = 5^{2x-1} \cdot 2^{4x+1}

.

Applying the product rule of exponents

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

simplifies the right side.

3

Equate exponents of corresponding prime bases from both sides of

5^x \cdot 2^y = 5^{2x-1} \cdot 2^{4x+1}

.

Equating powers of 5 yields

x = 2x - 1 \implies x = 1

. Equating powers of 2 yields

y = 4x + 1

.

Since 2 and 5 are distinct prime numbers, their corresponding exponents must be equal.

4

Calculate

y

and evaluate

y - x

.

y = 4(1) + 1 = 5

, so

y - x = 5 - 1 = 4

.

Substituting

x = 1

gives

y = 5

, satisfying the final question requirement.

Anahtar Kavram

Decomposing exponential bases into prime factors and applying exponent rules (

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

and

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

) to solve system equations of powers.

Soru 111Soru →

Consider the expression $K = \frac{0.000072 \times (1.5 \times 10^{-4})}{3.6 \times 10^{-11}}$ . Which of the following values or expressions are equivalent to $K$ ? Select all that apply.

Geçerli olan tümünü seçin

3.0 \times 10^2

0.3 \times 10^3

30,000 \times 10^{-2}

3.0 \times 10^{-2}

3.0 \times 10^4

Cevabı ve açıklamayı göster

Cevap: $3.0 \times 10^2$ ; $0.3 \times 10^3$ ; $30,000 \times 10^{-2}$

Cevap

The expressions equivalent to

K

are

3.0 \times 10^2

,

0.3 \times 10^3

, and

30,000 \times 10^{-2}

.

Evaluating the expression

K

gives

300

. First, rewrite

0.000072

as

7.2 \times 10^{-5}

. Multiplying by

1.5 \times 10^{-4}

gives

(7.2 \times 1.5) \times 10^{-9} = 10.8 \times 10^{-9}

. Next, dividing by

3.6 \times 10^{-11}

yields

(10.8 / 3.6) \times 10^{-9 - (-11)} = 3.0 \times 10^2 = 300

. Testing the choices shows that

3.0 \times 10^2 = 300

,

0.3 \times 10^3 = 300

, and

30,000 \times 10^{-2} = 300

are all equal to

300

.

Adım Adım Çözüm

1

Convert decimal numbers in the numerator to scientific notation.

0.000072 = 7.2 \times 10^{-5}

.

Converting all terms to powers of 10 simplifies multiplication.

2

Multiply the terms in the numerator.

(7.2 \times 10^{-5}) \times (1.5 \times 10^{-4}) = (7.2 \times 1.5) \times 10^{-5 + (-4)} = 10.8 \times 10^{-9}

.

Coefficients multiply together and exponents add during multiplication of powers with the same base.

3

Divide the numerator by the denominator.

\frac{10.8 \times 10^{-9}}{3.6 \times 10^{-11}} = \left(\frac{10.8}{3.6}\right) \times 10^{-9 - (-11)} = 3.0 \times 10^2 = 300

.

Dividing coefficients gives

3.0

, and subtracting the exponent of the denominator

-11

from

-9

yields

-9 + 11 = 2

.

4

Verify each option against the value

300

.

3.0 \times 10^2 = 300

,

0.3 \times 10^3 = 300

, and

30,000 \times 10^{-2} = 300

are all equivalent to

300

.

Matching each option's evaluated value ensures all correct representations are selected.

Anahtar Kavram

Simplifying numerical expressions involving decimals and scientific notation rules.

Soru 112Soru →

Based on the structural clues and contrast signals in the passage below, enter a word that best completes the sentence.

Aşağıdaki boşlukları doldurun

While early behavioral ecologists hypothesized that avian foraging patterns were governed strictly by instinctual routines, subsequent field observations revealed that crows demonstrate remarkably behavior when presented with novel mechanical puzzles. Far from adhering to inflexible routines, these birds dynamically modify their problem-solving techniques in response to unexpected obstacles, proving that their cognitive strategies are far more malleable than previously assumed.

Cevabı ve açıklamayı göster

Cevap

The word completing the blank must mean 'flexible', 'adaptive', 'malleable', or 'plastic'.

The concessive conjunction 'While' establishes a shift away from the idea of 'inflexible' and 'instinctual' routines. This contrast is elaborated by the subsequent phrases 'dynamically modify' and 'far more malleable,' which mandate a word meaning adaptable or flexible to complete the passage coherently.

Adım Adım Çözüm

1

Identify key contrast markers in the initial clause.

The opening transition 'While' creates a direct contrast between historical hypotheses ('governed strictly by instinctual routines') and modern field observations.

Contrast signals indicate that the missing word must express a quality opposite to rigid or instinctual routines.

2

Examine structural continuation and elaboration signals in the second sentence.

The clause 'Far from adhering to inflexible routines' and the descriptive phrase 'dynamically modify their problem-solving techniques' elaborate on the nature of the crows' behavior.

These structural clues confirm that the target word must denote adaptability or capacity for change.

3

Synthesize contextual clues to select an appropriate term.

The final clause reinforces that their strategies are 'far more malleable,' confirming that terms like 'flexible' or 'adaptive' correctly fill the blank.

Matching the tone and semantic requirements of both the contrast and elaboration signals yields a precise vocabulary fit.

Anahtar Kavram

Deciphering Meaning via Structural Clues and Contrast/Continuation Signals

Soru 113Soru →

If $(2x - 3)^2 = 25$ and $x < 0$ , what is the value of $x$ ?

-1

4

-2

-4

1

Cevabı ve açıklamayı göster

Cevap: $-1$

Cevap

-1

Taking the square root of both sides of

(2x - 3)^2 = 25

yields two equations:

2x - 3 = 5

(which gives

x = 4

) and

2x - 3 = -5

(which gives

x = -1

). Because the problem specifies that

x < 0

, the correct value must be

-1

.

Adım Adım Çözüm

1

Take the square root of both sides of the equation

(2x - 3)^2 = 25

2x - 3 = \pm 5

Applying the square root property to a squared binomial requires considering both positive and negative principal roots.

2

Set up two separate linear equations corresponding to the two cases

Case 1:

2x - 3 = 5

or Case 2:

2x - 3 = -5

To find all possible solutions for

x

, evaluate both root possibilities.

3

Solve each linear equation for

x

Case 1 yields

2x = 8 \implies x = 4

. Case 2 yields

2x = -2 \implies x = -1

.

Isolate

x

using standard algebraic operations.

4

Apply the given constraint

x < 0

x = -1

Since

4 > 0

, the positive root is eliminated, leaving

x = -1

as the sole valid solution.

Anahtar Kavram

Quadratic Equations and Factoring

Soru 114Soru →

If $x = 5$ is a solution to the quadratic equation $x^2 - (k + 3)x + 3k + 1 = 0$ , where $k$ is a constant, what is the value of the other solution?

\frac{7}{2}

\frac{11}{2}

-\frac{7}{2}

-\frac{11}{2}

\frac{5}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{7}{2}$

Cevap

\frac{7}{2}

Substituting

x = 5

into

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

yields

25 - 5(k + 3) + 3k + 1 = 0

, which simplifies to

11 - 2k = 0

, so

k = \frac{11}{2}

. Substituting

k = \frac{11}{2}

back into the product of roots formula

x_1 \cdot x_2 = 3k + 1

gives

5 \cdot x_2 = 3\left(\frac{11}{2}\right) + 1 = \frac{35}{2}

. Dividing by

5

yields the second root

x_2 = \frac{7}{2}

.

Adım Adım Çözüm

1

Substitute the known solution

x = 5

into the quadratic equation to solve for

k

.

5^2 - (k + 3)(5) + 3k + 1 = 0 \implies 25 - 5k - 15 + 3k + 1 = 0 \implies 11 - 2k = 0 \implies k = \frac{11}{2}

.

Since

x = 5

is a solution, it must satisfy the equation.

2

Use Vieta's formulas to find the other solution

x_2

.

Product of roots

x_1 \cdot x_2 = 3k + 1 = 3\left(\frac{11}{2}\right) + 1 = \frac{35}{2}

. Since

x_1 = 5

,

5 \cdot x_2 = \frac{35}{2} \implies x_2 = \frac{7}{2}

.

By Vieta's formulas for a standard quadratic

x^2 + bx + c = 0

, the product of the roots equals

c

.

Anahtar Kavram

Quadratic equations, Vieta's formulas, and parameter evaluation

Soru 115Soru →

A hotel renovated its guest rooms and purchased a total of 100 lighting fixtures, consisting of wall sconces costing $\$45$ each and ceiling pendants costing $\$70$ each. If the average (arithmetic mean) cost per fixture for the entire purchase was $\$52$ , how many wall sconces were purchased?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72

Setting up the weighted total cost equation

45x + 70(100 - x) = 5200

simplifies to

-25x = -1800

, giving

x = 72

wall sconces.

Adım Adım Çözüm

1

Define the variable for the unknown quantity

Let

x

represent the number of wall sconces purchased. The number of ceiling pendants is

100 - x

.

Defining the target quantity as

x

allows the problem to be modeled using a single-variable linear equation.

2

Set up the algebraic equation for total cost

45x + 70(100 - x) = 52 \times 100

The total cost of all fixtures is the sum of the total sconce cost and total pendant cost, which equals the overall average cost multiplied by the total number of fixtures.

3

Expand and solve the linear equation

45x + 7000 - 70x = 5200 \implies -25x = -1800 \implies x = 72

Simplifying algebraic terms isolates

x

to find the exact number of wall sconces.

Anahtar Kavram

Linear Algebraic Modeling and Weighted Averages

Soru 116Soru →

A museum surveyed $150$ visitors regarding their attendance at two special exhibitions: a Fine Art exhibit and a Natural History exhibit. Among the visitors surveyed, $85$ attended the Fine Art exhibit, $70$ attended the Natural History exhibit, and $20$ attended neither exhibit. How many visitors attended both the Fine Art exhibit and the Natural History exhibit?

5

25

45

65

155

Cevabı ve açıklamayı göster

Cevap: $25$

Cevap

The correct answer is

25

visitors.

Using the inclusion-exclusion formula

\text{Total} = |A| + |B| - |A \cap B| + \text{Neither}

, we substitute the given values:

150 = 85 + 70 - |A \cap B| + 20

. Simplifying gives

150 = 175 - |A \cap B|

, which yields

|A \cap B| = 25

. Thus,

25

visitors attended both exhibits.

Adım Adım Çözüm

1

Identify the given set values and formula.

Total visitors

= 150

, Fine Art attendees

|A| = 85

, Natural History attendees

|H| = 70

, Neither

= 20

.

The principle of inclusion-exclusion for two sets states that

\text{Total} = |A| + |H| - |A \cap H| + \text{Neither}

.

2

Calculate the number of visitors who attended at least one exhibit.

|A \cup H| = 150 - 20 = 130

.

Subtracting those who attended neither exhibit from the total population yields the total number of unique visitors who attended at least one of the two exhibits.

3

Solve for the intersection

|A \cap H|

.

130 = 85 + 70 - |A \cap H| \implies 130 = 155 - |A \cap H| \implies |A \cap H| = 25

.

Subtracting the union

|A \cup H|

from the sum of the individual sets

|A| + |H|

eliminates the double-counted intersection.

Anahtar Kavram

Two-Set Principle of Inclusion-Exclusion

Tahmini Süre:1m 0s

Soru 117Soru →

A chemistry laboratory prepares a $100\text{ mL}$ mixture using three solutions: Solution $X$ ( $10\%$ acid by volume), Solution $Y$ ( $30\%$ acid by volume), and Solution $Z$ ( $50\%$ acid by volume). The resulting mixture is $37\%$ acid by volume. If the volume of Solution $Y$ used is $5\text{ mL}$ more than twice the volume of Solution $X$ , what is the volume, in $\text{mL}$ , of Solution $Z$ used in the mixture?

15

35

40

50

55

Cevabı ve açıklamayı göster

Cevap: $50$

Cevap

50\text{ mL}

Setting up the system of equations gives

x + y + z = 100

,

x + 3y + 5z = 370

, and

y = 2x + 5

. Substituting

y = 2x + 5

into the first two equations yields

3x + z = 95

and

7x + 5z = 355

. Solving for

x

gives

x = 15

, which leads to

z = 95 - 3(15) = 50

. Therefore,

50\text{ mL}

of Solution Z was used.

Adım Adım Çözüm

1

Define variables and set up the system of linear equations

Let

x

,

y

, and

z

be the volumes in

\text{mL}

of Solutions

X

,

Y

, and

Z

, respectively.
1) Total volume:

x + y + z = 100

2) Total acid volume:

0.10x + 0.30y + 0.50z = 0.37(100) \implies x + 3y + 5z = 370

3) Relationship between

Y

and

X

:

y = 2x + 5

Translate the word problem statements into mathematical equations.

2

Substitute

y = 2x + 5

into equations (1) and (2) to reduce to a two-variable system

From equation (1):

x + (2x + 5) + z = 100 \implies 3x + z = 95 \implies z = 95 - 3x

From equation (2):

x + 3(2x + 5) + 5z = 370 \implies 7x + 15 + 5z = 370 \implies 7x + 5z = 355

Eliminating

y

simplifies the system to two equations in

x

and

z

.

3

Substitute

z = 95 - 3x

into

7x + 5z = 355

and solve for

x

7x + 5(95 - 3x) = 355 \implies 7x + 475 - 15x = 355 \implies -8x = -120 \implies x = 15

Solves for the unknown volume of Solution X.

4

Calculate

z

using

z = 95 - 3x

z = 95 - 3(15) = 95 - 45 = 50

Finds the requested volume of Solution Z.

Anahtar Kavram

Setting up and solving 3x3 systems of linear equations using substitution or elimination

Tahmini Süre:2m 0s

Soru 118Soru →

The quadratic equation $x^2 + bx + 45 = 0$ , where $b$ is a constant, has two negative integer roots $p$ and $q$ such that $p < q$ . If $q - p = 4$ , what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The value of

b

is 14.

For the quadratic equation

x^2 + bx + 45 = 0

, the roots

p

and

q

must satisfy

pq = 45

and

p + q = -b

. The negative integer factor pairs of 45 with

p < q

are

(-45, -1)

,

(-15, -3)

, and

(-9, -5)

. Calculating the difference

q - p

for each pair yields 44, 12, and 4, respectively. The condition

q - p = 4

uniquely identifies the roots as

p = -9

and

q = -5

. Summing these roots gives

p + q = -14

, so

b = -(-14) = 14

.

Adım Adım Çözüm

1

Set up the relationships for the roots of the quadratic equation.

pq = 45

and

p + q = -b

.

For any quadratic equation

x^2 + bx + c = 0

, the product of the roots equals

c

and the sum of the roots equals

-b

.

2

Find all negative integer factor pairs

(p, q)

of 45 such that

p < q

.

The possible pairs are

(-45, -1)

,

(-15, -3)

, and

(-9, -5)

.

Since both roots are negative integers, their product is positive 45.

3

Determine the difference

q - p

for each pair to match the given condition

q - p = 4

.

For

(-45, -1)

,

q - p = -1 - (-45) = 44

. For

(-15, -3)

,

q - p = -3 - (-15) = 12

. For

(-9, -5)

,

q - p = -5 - (-9) = 4

.

The pair

(-9, -5)

satisfies

q - p = 4

, establishing

p = -9

and

q = -5

.

4

Calculate the coefficient

b

.

b = -(p + q) = -(-9 + (-5)) = -(-14) = 14

.

Substituting the root values into

b = -(p + q)

yields the final answer.

Anahtar Kavram

Factoring Quadratics and Relationships Between Roots and Coefficients

Soru 119Soru →

If $x$ and $y$ are non-zero real numbers such that $9x^2 - 12xy + 4y^2 = 0$ , what is the value of $\frac{3x + y}{2y}$ ?

\frac{3}{2}

\frac{5}{6}

3

\frac{1}{2}

\frac{5}{2}

Cevabı ve açıklamayı göster

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

Cevap

The value of the expression is

\frac{3}{2}

.

The given expression

9x^2 - 12xy + 4y^2

factors into the perfect square

(3x - 2y)^2 = 0

. Setting the base equal to zero gives

3x = 2y

. Substituting

2y

in place of

3x

in the target expression yields

\frac{2y + y}{2y} = \frac{3y}{2y} = \frac{3}{2}

.

Adım Adım Çözüm

1

Factor the quadratic expression

9x^2 - 12xy + 4y^2 = (3x - 2y)^2 = 0

The equation is a perfect square trinomial of the form

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

where

a = 3x

and

b = 2y

.

2

Solve for the relationship between

x

and

y

3x - 2y = 0 \implies 3x = 2y

Taking the square root of both sides gives a linear relationship between

3x

and

2y

.

3

Substitute

3x = 2y

into the targeted expression

\frac{3x + y}{2y} = \frac{2y + y}{2y} = \frac{3y}{2y} = \frac{3}{2}

Replacing

3x

with

2y

eliminates

x

and simplifies the expression directly.

Anahtar Kavram

Perfect Square Trinomial Factoring

Soru 120Soru →

For all real numbers $x$ and $y$ such that $4x^2 + 6xy + 9y^2 \neq 0$ and $2x + 3y \neq 0$ , which of the following expressions are equivalent to $\frac{8x^3 - 27y^3}{4x^2 + 6xy + 9y^2}$ ? Select all such expressions.

Geçerli olan tümünü seçin

2x - 3y

-(3y - 2x)

\frac{4x^2 - 9y^2}{2x + 3y}

2x + 3y

(2x - 3y)^2

Cevabı ve açıklamayı göster

Cevap: $2x - 3y$ ; $-(3y - 2x)$ ; \frac{4x^2 - 9y^2}{2x + 3y}

Cevap

The equivalent expressions are

2x - 3y

,

-(3y - 2x)

, and

\frac{4x^2 - 9y^2}{2x + 3y}

.

Factoring

8x^3 - 27y^3

as

(2x - 3y)(4x^2 + 6xy + 9y^2)

allows canceling the denominator, leaving

2x - 3y

. The expression

2x - 3y

is directly correct. Rearranging terms in

-(3y - 2x)

yields

-3y + 2x = 2x - 3y

, making it correct. Factoring

\frac{4x^2 - 9y^2}{2x + 3y}

as

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

simplifies to

2x - 3y

, making it also correct.

Adım Adım Çözüm

1

Factor the numerator of the rational expression using the difference of cubes formula.

8x^3 - 27y^3 = (2x)^3 - (3y)^3 = (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2) = (2x - 3y)(4x^2 + 6xy + 9y^2)

Recognizing

8x^3

as

(2x)^3

and

27y^3

as

(3y)^3

enables full polynomial factoring.

2

Simplify the fraction by canceling the common non-zero quadratic factor.

\frac{(2x - 3y)(4x^2 + 6xy + 9y^2)}{4x^2 + 6xy + 9y^2} = 2x - 3y

Since

4x^2 + 6xy + 9y^2 \neq 0

, dividing common factors reduces the expression to linear form.

3

Evaluate each provided choice for algebraic equivalence to

2x - 3y

.

The expressions

2x - 3y

,

-(3y - 2x) = 2x - 3y

, and

\frac{4x^2 - 9y^2}{2x + 3y} = \frac{(2x-3y)(2x+3y)}{2x+3y} = 2x - 3y

are all algebraically identical.

Re-expressing or factoring alternative choices proves their equivalence to the simplified expression.

Anahtar Kavram

Factoring difference of cubes

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

and difference of squares

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

Cevap

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