Tüm alıştırma soruları

Without using a calculator, evaluate the exact value of the trigonometric expression:

\frac{\sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ}{\tan 45^\circ + \tan^2 60^\circ}

\frac{1}{4}

\frac{1}{2}

\frac{3}{4}

1

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{4}$

Cevap

\frac{1}{4}

Evaluating each trigonometric ratio using standard special angles (

30^\circ, 45^\circ, 60^\circ

) gives a numerator of

\left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} \cdot \frac{1}{2}\right) = \frac{3}{4} + \frac{1}{4} = 1

. The denominator is

\tan 45^\circ + (\tan 60^\circ)^2 = 1 + (\sqrt{3})^2 = 1 + 3 = 4

. Dividing the numerator by the denominator yields

\frac{1}{4}

.

Adım Adım Çözüm

1

Evaluate special angle values for the numerator

\sin 60^\circ = \frac{\sqrt{3}}{2}

,

\cos 30^\circ = \frac{\sqrt{3}}{2}

,

\cos 60^\circ = \frac{1}{2}

, and

\sin 30^\circ = \frac{1}{2}

.

Recall exact surd values for special angles

30^\circ

and

60^\circ

.

2

Simplify the numerator expression

\left(\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} \times \frac{1}{2}\right) = \frac{3}{4} + \frac{1}{4} = 1

.

Multiply exact surd values and add fractions.

3

Evaluate special angle values for the denominator

\tan 45^\circ = 1

and

\tan 60^\circ = \sqrt{3}

, so

\tan^2 60^\circ = (\sqrt{3})^2 = 3

.

Square the exact value of

\tan 60^\circ

.

4

Simplify the denominator and divide the numerator by the denominator

\text{Denominator} = 1 + 3 = 4, \text{ so overall value} = \frac{1}{4}$.

Divide numerator result by denominator result.

Anahtar Kavram

Special Angle Trigonometric Ratios and Exact Value Simplification

Soru 3622Soru →

Evaluate the numerical value of each of the following sequence and series expressions, and arrange the items in ascending order (from smallest to largest value):

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is: the sum to infinity of

18 + 6 + 2 + \dots

(

27

), followed by the 4th term of the geometric progression (

48

), then the 15th term of the arithmetic progression (

59

), and finally the sum of the first 5 terms of the arithmetic progression (

70

).

Evaluating each expression gives values of

27

,

48

,

59

, and

70

. Arranging these from smallest to largest places the sum to infinity (

27

) first, the GP term (

48

) second, the 15th AP term (

59

) third, and the AP sum (

70

) last.

Adım Adım Çözüm

1

Calculate the sum to infinity of the given geometric progression

For

18 + 6 + 2 + \dots

, first term

a = 18

and common ratio

r = \frac{6}{18} = \frac{1}{3}

. Using

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

, we get

S_\infty = \frac{18}{1 - 1/3} = \frac{18}{2/3} = 27

.

The sum to infinity formula for a convergent geometric series is

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

where

|r| < 1

.

2

Calculate the 4th term of the geometric progression

Given

T_2 = ar = 12

and

T_5 = ar^4 = 96

. Dividing

T_5

by

T_2

yields

r^3 = \frac{96}{12} = 8 \implies r = 2

. Thus

a = \frac{12}{2} = 6

. The 4th term is

T_4 = ar^3 = 6 \times 2^3 = 48

.

Using the

n

-th term formula of a GP,

T_n = ar^{n-1}

, to solve simultaneous equations for

a

and

r

.

3

Calculate the 15th term of the arithmetic progression

Given

T_3 = a + 2d = 11

and

T_8 = a + 7d = 31

. Subtracting the equations gives

5d = 20 \implies d = 4

. Substituting back yields

a = 11 - 2(4) = 3

. The 15th term is

T_{15} = a + 14d = 3 + 14(4) = 59

.

Using the

n

-th term formula of an AP,

T_n = a + (n-1)d

, to find the first term

a

and common difference

d

.

4

Calculate the sum of the first 5 terms of the arithmetic progression

With

a = 4

,

d = 5

, and

n = 5

,

S_5 = \frac{5}{2}[2(4) + (5-1)5] = \frac{5}{2}[8 + 20] = \frac{5}{2}(28) = 70

.

Applying the AP sum formula

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

.

5

Order the evaluated quantities from smallest to largest

Comparing numerical results

27 < 48 < 59 < 70

gives the sequence: item 1, item 2, item 4, item 3.

Sorting the calculated values in ascending numerical order.

Anahtar Kavram

Evaluation and comparison of terms and sums in Arithmetic and Geometric Progressions

Tahmini Süre:2m 30s

Soru 3623Soru →

Complete the sentence below by filling in the blanks with the most appropriate prepositions.

Aşağıdaki boşlukları doldurun

The executive council affirmed that the implementation of the proposed fiscal policy remains contingent the approval of the legislature, emphasizing that all budgetary allocations must be commensurate prevailing macroeconomic realities.

Cevabı ve açıklamayı göster

Cevap

The first blank requires 'on' or 'upon', while the second blank requires 'with'.

The adjective 'contingent' takes the dependent preposition 'on' (or 'upon') to indicate that an outcome depends on a specific condition. Furthermore, 'commensurate' strictly collocates with the preposition 'with' when describing proportional correspondence.

Adım Adım Çözüm

1

Determine the dependent preposition that correctly follows the adjective 'contingent'.

The adjective 'contingent' requires 'on' or 'upon' when expressing dependence on a future conditional event or authority.

In standard English usage, the established prepositional pattern is 'contingent on' or 'contingent upon'.

2

Identify the standard prepositional collocation required by the adjective 'commensurate'.

The adjective 'commensurate' mandates the preposition 'with' when denoting equality of proportion, extent, or degree.

Pairing 'commensurate' with prepositions such as 'to' or 'against' is non-standard; the correct idiom is 'commensurate with'.

Anahtar Kavram

Dependent Prepositions and Adjective-Preposition Collocations

Tahmini Süre:1m 30s

Soru 3624Soru →

Complete the sentence below by filling the blank with a word nearest in meaning to 'magnanimous'.

Aşağıdaki boşlukları doldurun

Despite the harsh criticism from his opponents, the leader remained and pardoned them completely.

Cevabı ve açıklamayı göster

Cevap

The word 'forgiving' (or 'generous') correctly completes the sentence as it is nearest in meaning to 'magnanimous'.

The word 'magnanimous' refers to showing a lofty, generous, and forgiving spirit towards others, particularly rivals. Therefore, 'forgiving' or 'generous' is the word nearest in meaning that appropriately completes the context.

Adım Adım Çözüm

1

Analyze the meaning of the target word 'magnanimous' in context.

The sentence describes a leader who pardons opponents despite harsh criticism, indicating a noble and forgiving disposition.

Contextual clues like 'pardoned them completely' indicate high-minded generosity and forgiveness.

2

Identify the word nearest in meaning to fit the blank.

'Forgiving' or 'generous' accurately captures the core meaning of being magnanimous.

Magnanimous describes someone who is generous or forgiving, especially toward a rival or less powerful person.

Anahtar Kavram

Identifying synonyms and words nearest in meaning in contextual sentences.

Tahmini Süre:45s

Soru 3625Soru →

If $2^{2x + 1} - 9(2^x) + 4 = 0$ , what is the product of all real values of $x$ that satisfy the equation?

-2

2

1

-1

Cevabı ve açıklamayı göster

Cevap: $-2$

Cevap

The product of all real values of

x

satisfying the equation is

-2

.

Rewriting

2^{2x+1}

as

2(2^x)^2

allows substitution of

u = 2^x

, giving

2u^2 - 9u + 4 = 0

. Solving for

u

gives

u = \frac{1}{2}

and

u = 4

. Converting back to

x

via

2^x = 2^{-1}

and

2^x = 2^2

yields

x = -1

and

x = 2

. The product of these roots is

(-1) \times 2 = -2

.

Adım Adım Çözüm

1

Apply the law of indices to rewrite the first term.

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

. Thus, the equation becomes

2(2^x)^2 - 9(2^x) + 4 = 0

.

Splitting the index using

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

enables transformation into a quadratic form.

2

Substitute

u = 2^x

and solve the resulting quadratic equation.

2u^2 - 9u + 4 = 0 \implies (2u - 1)(u - 4) = 0 \implies u = \frac{1}{2} \text{ or } u = 4

.

Algebraic substitution simplifies the exponential equation into a standard quadratic equation.

3

Convert the values of

u

back to

x

using index laws.

For

u = \frac{1}{2}

,

2^x = 2^{-1} \implies x_1 = -1

. For

u = 4

,

2^x = 2^2 \implies x_2 = 2

.

Equating bases (

2^x = 2^k \implies x = k

) isolates the unknown variable

x

.

4

Compute the product of the solutions

x_1

and

x_2

.

Product =

x_1 \cdot x_2 = (-1) \times 2 = -2

.

The question specifically requests the product of all real values of

x

.

Anahtar Kavram

Solving quadratic-form exponential equations using index laws and variable substitution.

Tahmini Süre:2m 0s

Soru 3626Soru →

In the sentence below, select the option that is MOST NEARLY OPPOSITE IN MEANING to the capitalized word.

After a long day of intensive physical training, the athletes felt LETHARGIC and preferred to rest in their rooms.

Which of the following words is opposite in meaning to LETHARGIC as used in the sentence?

energetic

sluggish

inactive

lethal

Cevabı ve açıklamayı göster

Cevap: energetic

Cevap

The word opposite in meaning to 'lethargic' is 'energetic'.

In the given context, 'lethargic' means lacking energy, feeling slow, or fatigued. The word 'energetic' means active and full of life, making it the exact antonym.

Adım Adım Çözüm

1

Analyze the context of the target word in the sentence.

The word 'lethargic' describes athletes who are tired, lacking energy, and needing rest after intense physical training.

Understanding sentence context establishes the precise operational definition of the target word.

2

Identify the contextual antonym for 'lethargic'.

'Energetic' means full of vigor, activity, and energy, which directly opposes 'lethargic'.

JAMB UTME antonym questions require selecting the word that expresses the opposite condition.

Anahtar Kavram

Identifying contextual antonyms in English sentences

Soru 3627Soru →

A binary operation $\star$ on the set of real numbers is defined by $a \star b = 3a + 2b - 1$ . What is the value of $x$ such that $4 \star x = 25$ ?

6

7

11

14

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is

7

.

Applying the binary operation definition

a \star b = 3a + 2b - 1

with

a = 4

and

b = x

gives

3(4) + 2x - 1 = 25

. Simplifying gives

2x + 11 = 25

, leading to

2x = 14

and

x = 7

.

Adım Adım Çözüm

1

Substitute

a = 4

and

b = x

into the definition of the binary operation

a \star b = 3a + 2b - 1

.

4 \star x = 3(4) + 2x - 1

Applying the given rule for the binary operation.

2

Simplify the expression on the left-hand side.

4 \star x = 12 + 2x - 1 = 2x + 11

Performing basic arithmetic multiplication and addition of constants.

3

Set the simplified expression equal to

25

and solve for

x

.

2x + 11 = 25 \implies 2x = 14 \implies x = 7

Subtracting

11

from both sides and dividing by

2

.

Anahtar Kavram

Evaluation of non-commutative binary operations and solving algebraic equations involving operational rules.

Soru 3628Soru →

Match each vocabulary word in Column A with its precise antonym (word opposite in meaning) in Column B.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Insouciant pairs with Solicitous; Perspicuous pairs with Abstruse; Recalcitrant pairs with Submissive; Magnanimous pairs with Vindictive.

Each word is correctly matched to its exact lexical antonym: Insouciant (carefree) contrasts with Solicitous (anxiously concerned); Perspicuous (clear and lucid) contrasts with Abstruse (deeply obscure); Recalcitrant (defiant and uncooperative) contrasts with Submissive (compliant and yielding); and Magnanimous (generous and forgiving) contrasts with Vindictive (spiteful and vengeful).

Adım Adım Çözüm

1

Analyze the core meaning and emotional tone of each word in Column A.

Insouciant denotes carefree unconcern; Perspicuous denotes lucidity and clarity; Recalcitrant denotes rebellious obstinacy; Magnanimous denotes noble generosity and forgiveness.

Establishing clear lexical definitions is necessary before seeking precise antonymic relationships.

2

Evaluate the terms in Column B to find exact direct opposites for each definition.

Solicitous (deeply concerned) directly opposes Insouciant; Abstruse (hard to understand) directly opposes Perspicuous; Submissive (obedient/compliant) directly opposes Recalcitrant; Vindictive (vengeful/spiteful) directly opposes Magnanimous.

Matching requires selecting words that exhibit maximum semantic contrast.

Anahtar Kavram

Antonyms and Words Opposite in Meaning

Soru 3629Soru →

A vessel departs from a harbor $H$ and sails $12\text{ km}$ on a bearing of $040^\circ$ to reach point $A$ . From point $A$ , it changes course and sails $5\text{ km}$ on a bearing of $130^\circ$ to reach point $B$ . What is the bearing of point $B$ from harbor $H$ , to the nearest degree?

063^\circ

073^\circ

017^\circ

107^\circ

Cevabı ve açıklamayı göster

Cevap: $063^\circ$

Cevap

063^\circ

(or

063^\circ

to the nearest degree)

The navigation path forms a right-angled triangle at point

A

because the reverse bearing of

H

from

A

(

220^\circ

) differs from the bearing of

B

from

A

(

130^\circ

) by exactly

90^\circ

. Using the right triangle trigonometric ratio

\tan(\angle AHB) = \frac{5}{12}

, the angle

\angle AHB

is found to be approximately

22.62^\circ

. Adding this angle to the initial bearing of

040^\circ

yields

062.62^\circ

, which rounds to

063^\circ

.

Adım Adım Çözüm

1

Determine the back bearing of

H

from

A

and calculate the interior angle at

A

.

Back bearing of

H

from

A = 040^\circ + 180^\circ = 220^\circ

. Interior angle

\angle HAB = 220^\circ - 130^\circ = 90^\circ

.

Knowing the directions of

AH

and

AB

allows us to find the interior angle of

\triangle HAB

at point

A

.

2

Calculate the interior angle

\angle AHB

using right-triangle trigonometry.

\tan(\angle AHB) = \frac{AB}{HA} = \frac{5}{12} \implies \angle AHB = \arctan\left(\frac{5}{12}\right) \approx 22.62^\circ.

Since

\triangle HAB

has a right angle at

A

, tangent relates the opposite side

AB = 5\text{ km}

to the adjacent side

HA = 12\text{ km}

.

3

Calculate the total bearing of

B

from

H

.

\text{Bearing} = 040^\circ + 22.62^\circ = 062.62^\circ \approx 063^\circ.

The line

HB

lies to the right of line

HA

, so the interior angle

\angle AHB

is added to the initial bearing of line

HA

(

040^\circ

).

Anahtar Kavram

Bearings and Right-Angled Triangles

Tahmini Süre:1m 30s

Soru 3630Soru →

The derived SI unit of force is the newton ( $\text{N}$ ), which can be expressed in terms of fundamental SI base units as $\text{kg}\cdot\text{m}\cdot\text{s}^{x}$ . What is the numerical value of the exponent $x$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The numerical value of the exponent

x

for time in the base SI expression of the newton is

-2

.

Force is defined by Newton's second law as mass times acceleration (

F = ma

). In SI fundamental units, mass is measured in kilograms (

\text{kg}

) and acceleration in meters per second squared (

\text{m}\cdot\text{s}^{-2}

). Combining these yields

\text{1\ N} = \text{1\ kg}\cdot\text{m}\cdot\text{s}^{-2}

. Therefore, the power of time in seconds (

x

) is

-2

.

Adım Adım Çözüm

1

Identify the relationship between force and fundamental quantities

Force is defined as mass multiplied by acceleration (

F = ma

)

This relates force to fundamental quantities of mass, length, and time.

2

Substitute the SI base units of mass and acceleration

Mass unit is

\text{kg}

; acceleration unit is

\text{m}\cdot\text{s}^{-2}

Kilogram, meter, and second are standard fundamental SI units.

3

Determine the exponent of seconds

The combined base unit expression is

\text{kg}\cdot\text{m}\cdot\text{s}^{-2}

, giving

x = -2

Equating the powers of seconds gives

x = -2

.

Anahtar Kavram

Expressing derived units in terms of fundamental SI base units

Soru 3631Soru →

If $(x_1, y_1)$ and $(x_2, y_2)$ are the real solution pairs to the simultaneous equations $x - 2y = 1$ and $x^2 - xy + y^2 = 7$ , what is the value of $x_1 x_2 + y_1 y_2$ ?

-9

-11

-7

-1

Cevabı ve açıklamayı göster

Cevap: $-11$

Cevap

The value of

x_1 x_2 + y_1 y_2

is

-11

.

Substituting

x = 2y + 1

into

x^2 - xy + y^2 = 7

gives

3y^2 + 3y - 6 = 0

, which reduces to

y^2 + y - 2 = 0

. The roots are

y = 1

and

y = -2

. Substituting back into the linear expression yields corresponding values

x = 3

and

x = -3

. The two solution pairs are

(3, 1)

and

(-3, -2)

. Calculating

x_1 x_2 + y_1 y_2 = (3)(-3) + (1)(-2)

yields

-11

.

Adım Adım Çözüm

1

Express

x

in terms of

y

from the linear equation.

x = 2y + 1

Isolating variable

x

allows direct substitution into the quadratic equation.

2

Substitute

x = 2y + 1

into the quadratic equation

x^2 - xy + y^2 = 7

.

(2y + 1)^2 - (2y + 1)y + y^2 = 7 \implies 3y^2 + 3y - 6 = 0

This simplifies the system to a single quadratic equation in

y

.

3

Solve the quadratic equation

3y^2 + 3y - 6 = 0

.

y^2 + y - 2 = 0 \implies (y + 2)(y - 1) = 0

, giving

y_1 = 1

and

y_2 = -2

Factoring determines the two possible

y

-coordinates.

4

Determine corresponding

x

-coordinates for each

y

-value.

For

y_1 = 1

,

x_1 = 2(1) + 1 = 3 \implies (3, 1)

. For

y_2 = -2

,

x_2 = 2(-2) + 1 = -3 \implies (-3, -2)

.

Substituting each

y

into

x = 2y + 1

yields the complete solution pairs.

5

Compute

x_1 x_2 + y_1 y_2

.

(3)(-3) + (1)(-2) = -9 - 2 = -11

Evaluates the targeted algebraic expression.

Anahtar Kavram

Solving simultaneous linear and quadratic equations using algebraic substitution.

Soru 3632Soru →

Read the sentence below carefully. Which word nearest in meaning to 'convincing' or 'compelling' best completes the statement?

Aşağıdaki boşlukları doldurun

Despite initial skepticism from the board, the financial analyst presented a case for diversifying the company's investment portfolio.

Cevabı ve açıklamayı göster

Cevap

The word that best completes the sentence is 'cogent', which means clear, logical, and convincing.

The term 'cogent' means clear, logical, and convincingly persuasive. In this context, presenting a cogent case explains how the analyst successfully persuaded the skeptical board.

Adım Adım Çözüm

1

Analyze the context of the sentence

The subject is attempting to overcome 'initial skepticism' from a board by making a strong case for diversifying investments.

Understanding the context clarifies that the required word must denote strength, logical clarity, and persuasive power.

2

Identify the target synonym

The word 'cogent' specifically describes an argument or case that is clear, logical, and compelling.

Fitting 'cogent' into the blank precisely conveys that the analyst's case was convincing enough to address the board's doubts.

Anahtar Kavram

Synonyms and Words Nearest in Meaning

Soru 3633Soru →

The derived SI unit of electric potential, the volt ( $\text{V}$ ), can be expressed in terms of fundamental SI base units as $\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d$ . What is the value of the exponent $c$ ?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The exponent

c

corresponding to the base unit second (

\text{s}

) is

-3

.

Electric potential difference (

V

) is defined as work (

W

) per unit charge (

Q

). In terms of SI base units, work has units of

\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}

and charge has units of

\text{A} \cdot \text{s}

. Dividing work by charge gives

\frac{\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{A} \cdot \text{s}} = \text{kg}^1 \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}

. Equating this to

\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d

yields

c = -3

.

Adım Adım Çözüm

1

Relate electric potential to work and charge

\text{1 V} = \frac{\text{1 J}}{\text{1 C}}

Electric potential difference is defined as work done per unit electric charge.

2

Break down Joules into base SI units

\text{1 J} = \text{1 N} \cdot \text{m} = (\text{1 kg} \cdot \text{m} \cdot \text{s}^{-2}) \cdot \text{m} = \text{1 kg} \cdot \text{m}^2 \cdot \text{s}^{-2}

Work is force multiplied by displacement, and force is mass multiplied by acceleration.

3

Break down Coulombs into base SI units

\text{1 C} = \text{1 A} \cdot \text{s}

Electric charge is electric current multiplied by time.

4

Substitute base unit expressions into the ratio for electric potential

\text{1 V} = \frac{\text{1 kg} \cdot \text{m}^2 \cdot \text{s}^{-2}}{\text{1 A} \cdot \text{s}} = \text{1 kg}^1 \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}

Combining powers of base units yields the complete fundamental representation.

5

Determine the exponent value

c

c = -3

Comparing

\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c \cdot \text{A}^d

to

\text{kg}^1 \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}

shows that the exponent of

\text{s}

is

-3

.

Anahtar Kavram

Expressing derived units in terms of fundamental SI base units

Soru 3634Soru →

If $\log_9 x + \log_3 x = 3$ , what is the value of $x$ ?

9

3

27

81

Cevabı ve açıklamayı göster

Cevap: $9$

Cevap

The value of

x

is

9

.

Using the change of base formula,

\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{1}{2}\log_3 x

. Adding

\log_3 x

yields

\frac{3}{2}\log_3 x = 3

, which simplifies to

\log_3 x = 2

. Converting to exponential form gives

x = 3^2 = 9

.

Adım Adım Çözüm

1

Express

\log_9 x

in base

3

using the change of base formula.

\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{\log_3 x}{2} = \frac{1}{2}\log_3 x

Converting all logarithmic terms to a common base (base 3) allows combining terms.

2

Substitute

\frac{1}{2}\log_3 x

into the given equation.

\frac{1}{2}\log_3 x + \log_3 x = 3 \implies \frac{3}{2}\log_3 x = 3

Combine like logarithmic terms.

3

Solve for

\log_3 x

.

\log_3 x = 3 \times \frac{2}{3} = 2

Multiply both sides by

\frac{2}{3}

to isolate

\log_3 x

.

4

Convert from logarithmic form to exponential form.

x = 3^2 = 9

By definition,

\log_b a = c

is equivalent to

b^c = a

.

Anahtar Kavram

Logarithms and Change of Base

Soru 3635Soru →

If $\log_4(x + 2) - \log_{16}(x + 2) = 1$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The value of

x

is 14.

Applying the change of base formula converts

\log_{16}(x + 2)

into

\frac{1}{2}\log_4(x + 2)

. The given equation reduces to

\frac{1}{2}\log_4(x + 2) = 1

, which means

\log_4(x + 2) = 2

. Writing this in exponential form yields

x + 2 = 4^2 = 16

, giving

x = 14

.

Adım Adım Çözüm

1

Change the base of

\log_{16}(x + 2)

to base 4

\log_{16}(x + 2) = \frac{\log_4(x + 2)}{\log_4 16} = \frac{1}{2}\log_4(x + 2)

Using the change of base identity

\log_a b = \frac{\log_c b}{\log_c a}

allows all logarithmic terms to share base 4.

2

Substitute and simplify the expression on the left-hand side

\log_4(x + 2) - \frac{1}{2}\log_4(x + 2) = \frac{1}{2}\log_4(x + 2) = 1

Subtracting half of the term from the whole term leaves half of the term.

3

Isolate the logarithmic expression

\log_4(x + 2) = 2

Multiplying both sides of the equation by 2.

4

Convert from logarithmic to index form

x + 2 = 4^2 = 16

By definition,

\log_b a = c \iff b^c = a

.

5

Solve for the variable

x

x = 14

Subtracting 2 from both sides gives the final value of

x

.

Anahtar Kavram

Logarithms and Change of Base

Soru 3636Soru →

If the matrix $A = \begin{pmatrix} 3 & 2 \\ 7 & 5 \end{pmatrix}$ , which matrix represents the inverse $A^{-1}$ ?

\begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

\begin{pmatrix} 5 & 2 \\ 7 & 3 \end{pmatrix}

\begin{pmatrix} -5 & 2 \\ 7 & -3 \end{pmatrix}

\begin{pmatrix} 3 & -2 \\ -7 & 5 \end{pmatrix}

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Cevap: $\begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}$

Cevap

\begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

The correct inverse matrix is computed by evaluating the determinant

\det(A) = 3(5) - 2(7) = 1

and constructing the adjugate matrix by swapping the diagonal elements

3

and

5

while changing the signs of

2

and

7

, resulting in

\begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

.

Adım Adım Çözüm

1

Calculate the determinant of matrix

A

\det(A) = (3)(5) - (2)(7) = 15 - 14 = 1

The inverse requires dividing the adjugate matrix by the determinant of

A

.

2

Find the adjugate of matrix

A

\text{adj}(A) = \begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

For a 2x2 matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

, swap the main diagonal elements (

a

and

d

) and change the signs of the off-diagonal elements (

b

and

c

).

3

Compute

A^{-1} = \frac{1}{\det(A)} \text{adj}(A)

A^{-1} = \frac{1}{1} \begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix} = \begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

Multiply the adjugate matrix by the reciprocal of the determinant.

Anahtar Kavram

2x2 Matrix Inversion

Tahmini Süre:1m 0s

Soru 3637Soru →

Which of the following physical properties of a metal is directly attributed to the presence of a mobile 'sea' of delocalized valence electrons?

High electrical conductivity in the solid state

High solubility in polar solvents such as water

Low melting and boiling points due to weak intermolecular forces

Conduction of electricity only when molten or dissolved in aqueous solution

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Cevap: High electrical conductivity in the solid state

Cevap

High electrical conductivity in the solid state

High electrical conductivity in the solid state is a direct consequence of delocalized valence electrons moving freely throughout the metallic crystal lattice when an electric potential difference is applied.

Adım Adım Çözüm

1

Identify the essential structural feature of metallic bonding.

Metallic bonding consists of a giant lattice of positive metal ions surrounded by a sea of mobile, delocalized valence electrons.

Understanding the presence of free-moving valence electrons explains the physical behavior of metals.

2

Relate delocalized electrons to electrical conduction.

When an electric field is applied, the mobile electrons drift towards the positive potential, carrying electric charge through the solid metal.

Electrical conduction requires mobile charge carriers, which in metals are the delocalized valence electrons present even in the solid state.

Anahtar Kavram

Metallic bonding and electrical conductivity due to delocalized electrons

Tahmini Süre:1m 0s

Soru 3638Soru →

Which of the following classes of vertebrates possesses a heart composed of two atria and a single ventricle?

Amphibia

Pisces

Mammalia

Aves

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Cevap: Amphibia

Cevap

Amphibia

Amphibians possess a three-chambered heart consisting of two atria (left and right) and a single ventricle.

Adım Adım Çözüm

1

Recall the heart chamber configuration for vertebrate classes.

Pisces have 2 chambers, Amphibia have 3 chambers, Reptiles have 3 (or partially divided 4) chambers, while Aves and Mammalia have 4 chambers.

Heart structure complexity evolved progressively across vertebrate groups.

2

Match the specific 3-chamber structure (two atria and one ventricle) to the target class.

Amphibians possess two atria receiving blood and a single undivided ventricle.

Oxygenated blood from skin/lungs and deoxygenated blood from the body enter separate atria before pumping through a shared ventricle.

Anahtar Kavram

Comparative Vertebrate Heart Structure

Soru 3639Soru →

In a diploid cell analyzed during interphase, a specific sequence of nucleotides that governs seed coat structure is situated at a precise, fixed position along a pair of morphologically matching chromosomes. An individual organism carries two different functional variants of this sequence at that position, resulting in different protein products. Which of the following correctly identifies the genetic terms for the matching chromosomes, the specific position, and the alternative sequence variants?

The matching chromosomes are homologous chromosomes, the fixed position is the locus, and the alternative sequence variants are alleles.

The matching chromosomes are sister chromatids, the fixed position is the locus, and the alternative sequence variants are genotypes.

The matching chromosomes are homologous chromosomes, the fixed position is the allele, and the alternative sequence variants are continuous phenotypes.

The matching chromosomes are homologous chromosomes, the fixed position is the locus, and the alternative sequence variants are codominant traits regardless of their expression.

Cevabı ve açıklamayı göster

Cevap: The matching chromosomes are homologous chromosomes, the fixed position is the locus, and the alternative sequence variants are alleles.

Cevap

The matching chromosomes are homologous chromosomes, the fixed position is the locus, and the alternative sequence variants are alleles.

In diploid genetics terminology, matching maternal and paternal chromosome pairs are termed homologous chromosomes. The physical coordinate or position of a gene along the chromosome is called the locus, and the alternative versions of the DNA sequence at that locus are called alleles.

Adım Adım Çözüm

1

Identify the relationship between morphologically matching chromosome pairs in a diploid organism.

Morphologically matching pairs carrying genes for identical traits (one inherited from each parent) are defined as homologous chromosomes.

Diploid organisms inherit one chromosome set from each parent, forming homologous pairs.

2

Determine the term for the fixed physical location of a gene on a chromosome.

The specific physical position occupied by a gene on a chromosome is its locus (plural: loci).

Locus specifically designates physical mapping location on chromosomal DNA.

3

Determine the term for non-identical nucleotide variants of the gene situated at that location.

Alternative forms or variants of a gene occupying the same locus are defined as alleles.

Alleles account for genetic variations in traits.

Anahtar Kavram

Basic Genetics Terminology: Homologous Chromosomes, Locus, and Alleles

Tahmini Süre:1m 30s

Soru 3640Soru →

An investigation comparing the anatomical and physiological adaptations of three poikilothermic vertebrates—the Nile tilapia (Oreochromis niloticus), the adult African toad (Sclerophrys regularis), and the rainbow lizard (Agama agama)—evaluated their cardiac chamber arrangement and primary nitrogenous excretory product. Which of the following combinations correctly identifies both features for each organism in sequence?

Nile tilapia: 2 chambers and ammonia; Adult African toad: 3 chambers and urea; Rainbow lizard: 3 chambers (incomplete ventricular septum) and uric acid

Nile tilapia: 3 chambers and ammonia; Adult African toad: 2 chambers and urea; Rainbow lizard: 3 chambers (incomplete ventricular septum) and uric acid

Nile tilapia: 2 chambers and urea; Adult African toad: 3 chambers and uric acid; Rainbow lizard: 4 fully separated chambers and ammonia

Nile tilapia: 2 chambers and ammonia; Adult African toad: 3 chambers and uric acid; Rainbow lizard: 3 chambers (incomplete ventricular septum) and urea

Cevabı ve açıklamayı göster

Cevap: Nile tilapia: 2 chambers and ammonia; Adult African toad: 3 chambers and urea; Rainbow lizard: 3 chambers (incomplete ventricular septum) and uric acid

Cevap

The correct combination is Nile tilapia: 2 chambers and ammonia; Adult African toad: 3 chambers and urea; Rainbow lizard: 3 chambers (incomplete ventricular septum) and uric acid.

The correct response accurately synthesizes the cardiac structural evolution and excretory physiological adaptations across the three poikilothermic classes. Bony fishes such as the Nile tilapia possess a 2-chambered heart and practice ammonotelism (excreting ammonia). Adult amphibians such as the African toad have evolved a 3-chambered heart (2 atria, 1 ventricle) and practice ureotelism (excreting urea). Non-crocodilian reptiles like the rainbow lizard feature a 3-chambered heart with an incomplete inter-ventricular septum and practice uricotelism (excreting insoluble uric acid to conserve body water).

Adım Adım Çözüm

1

Analyze cardiac chamber architecture across the three poikilothermic vertebrate classes.

Pisces (Nile tilapia) possess a 2-chambered heart (1 atrium, 1 ventricle) with single circulation. Adult Amphibia (African toad) possess a 3-chambered heart (2 atria, 1 ventricle) with double circulation. Non-crocodilian Reptilia (rainbow lizard) possess a 3-chambered heart with a partially divided ventricle.

Evolutionary trends in poikilothermic vertebrates display a transition from single systemic-branchial circulation to incomplete double circulation.

2

Evaluate the primary nitrogenous excretory product for each vertebrate class based on habitat and water conservation needs.

Aquatic fishes (ammonotelic) excrete highly soluble and toxic ammonia directly into surrounding water. Terrestrial adult amphibians (ureotelic) excrete less toxic urea. Terrestrial reptiles (uricotelic) excrete insoluble uric acid paste to minimize water loss.

Nitrogenous waste elimination adaptations directly correlate with water availability in each organism's ecological niche.

3

Synthesize the combined heart chamber count and excretory product for all three representative species in sequence.

Nile tilapia = 2 chambers, ammonia; Adult African toad = 3 chambers, urea; Rainbow lizard = 3 chambers (partially divided ventricle), uric acid.

Matching all criteria simultaneously identifies the single accurate response.

Anahtar Kavram

Comparative cardiac anatomy and nitrogenous waste excretion across poikilothermic vertebrate classes (Pisces, Amphibia, Reptilia)

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