Tüm alıştırma soruları

Soru 7461Soru →

In $\Delta ABC$ , the lengths of the sides are $a = x\text{ cm}$ , $b = (x + 3)\text{ cm}$ , and $c = (x + 2)\text{ cm}$ . If $\angle C = 60^\circ$ , what is the value of $x$ ?

3

5

7

1

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Applying the Cosine Rule

c^2 = a^2 + b^2 - 2ab\cos C

with

a = x

,

b = x + 3

,

c = x + 2

, and

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

gives

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

. Expanding both sides leads to

x^2 + 4x + 4 = x^2 + 3x + 9

. Subtracting

x^2

and isolating

x

yields

x = 5

.

Adım Adım Çözüm

1

Apply the Cosine Rule for side

c

and angle

C

.

c^2 = a^2 + b^2 - 2ab\cos C

The Cosine Rule relates all three sides of a triangle to the cosine of an included angle.

2

Substitute

a = x

,

b = x + 3

,

c = x + 2

, and

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

into the formula.

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

Inserting the given algebraic side lengths and angle allows solving for

x

.

3

Expand both sides of the equation.

x^2 + 4x + 4 = x^2 + (x^2 + 6x + 9) - (x^2 + 3x)

Cancel the factor of

2

with

\frac{1}{2}

and expand

(x+2)^2

and

(x+3)^2

.

4

Simplify the right-hand side.

x^2 + 4x + 4 = x^2 + 3x + 9

Combine like terms:

(x^2 + x^2 - x^2) + (6x - 3x) + 9 = x^2 + 3x + 9

.

5

Subtract

x^2

from both sides and isolate

x

.

4x - 3x = 9 - 4 \implies x = 5

Subtracting

x^2 + 3x + 4

from both sides gives the linear solution

x = 5

.

Anahtar Kavram

Solving for unknown algebraic side lengths using the Cosine Rule.

Soru 7462Soru →

The frequency distribution of scores $x$ obtained by a group of candidates in an aptitude test is given in the table below:

Score ( $x$ )	2	4	6	8	10
Frequency ( $f$ )	3	$k$	7	4	2

If the mean score of the distribution is $5.8$ , find the value of $k$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is

4

.

The mean score of an ungrouped frequency distribution is given by

\bar{x} = \frac{\sum fx}{\sum f}

. Evaluating the sum of frequencies gives

\sum f = 16 + k

and the weighted sum of scores gives

\sum fx = 100 + 4k

. Substituting the given mean of

5.8

, we get

\frac{100 + 4k}{16 + k} = 5.8

. Cross-multiplying yields

100 + 4k = 92.8 + 5.8k

, leading to

1.8k = 7.2

, so

k = 4

.

Adım Adım Çözüm

1

Express total frequency

\sum f

in terms of

k

\sum f = 16 + k

The total number of observations is the sum of all frequencies in the distribution.

2

Calculate the sum of weighted scores

\sum fx

in terms of

k

\sum fx = 100 + 4k

Each score value must be multiplied by its corresponding frequency and summed together.

3

Set up the mean equation using the formula

\bar{x} = \frac{\sum fx}{\sum f}

5.8 = \frac{100 + 4k}{16 + k}

The mean of ungrouped data presented in a frequency table is total sum divided by total frequency.

4

Solve the linear equation for

k

k = 4

Cross-multiplying gives

92.8 + 5.8k = 100 + 4k

, which simplifies to

1.8k = 7.2

, yielding

k = 4

.

Anahtar Kavram

Mean of Ungrouped Data from a Frequency Table

Soru 7463Soru →

Convert the base 2 fractional number $110.101_2$ to its equivalent value in base 10 (decimal). What is the decimal value?

Cevabı ve açıklamayı göster

Cevap: 6.625

Cevap

The decimal value of

110.101_2

is

6.625

.

To convert

110.101_2

to base 10, expand both the integer and fractional parts using powers of 2. The integer component

110_2 = (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 4 + 2 + 0 = 6

. The fractional component

0.101_2 = (1 \times 2^{-1}) + (0 \times 2^{-2}) + (1 \times 2^{-3}) = 0.5 + 0 + 0.125 = 0.625

. Summing these values gives

6 + 0.625 = 6.625

.

Adım Adım Çözüm

1

Convert the integer part

110_2

to base 10

1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6

Each position to the left of the binary point corresponds to an increasing non-negative power of 2 (

2^0, 2^1, 2^2

).

2

Convert the fractional part

0.101_2

to base 10

1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = \frac{1}{2} + 0 + \frac{1}{8} = 0.5 + 0.125 = 0.625

Each position to the right of the binary point corresponds to a negative power of 2 (

2^{-1}, 2^{-2}, 2^{-3}

).

3

Combine the integer and fractional results

6 + 0.625 = 6.625

The total value in base 10 is the sum of the expanded integer and fractional components.

Anahtar Kavram

Conversion of Fractional Non-Decimal Numbers to Decimal

Soru 7464Soru →

A project team of $5$ members is to be formed from $6$ senior engineers and $4$ junior engineers. If the team must contain strictly more senior engineers than junior engineers, in how many different ways can the team be selected?

180

252

186

3600

Cevabı ve açıklamayı göster

Cevap: $186$

Cevap

186

To form a

5

-member team where senior engineers outnumber junior engineers, we consider three mutually exclusive cases: (5 senior, 0 junior), (4 senior, 1 junior), and (3 senior, 2 junior). Evaluating

^6C_5 \times ^4C_0 = 6

,

^6C_4 \times ^4C_1 = 60

, and

^6C_3 \times ^4C_2 = 120

, and adding them together yields

6 + 60 + 120 = 186

total ways.

Adım Adım Çözüm

1

Identify the conditions and valid cases for team composition

The team must have

5

members. The number of senior engineers must be strictly greater than junior engineers. The valid combinations of (senior, junior) engineers are:

(5, 0)

,

(4, 1)

, and

(3, 2)

.

Any other breakdown (such as 2 senior and 3 junior) would result in senior engineers not outnumbering junior engineers.

2

Calculate the combinations for Case 1 (

5

senior,

0

junior)

^6C_5 \times ^4C_0 = 6 \times 1 = 6

Choosing

5

out of

6

senior engineers and

0

out of

4

junior engineers.

3

Calculate the combinations for Case 2 (

4

senior,

1

junior)

^6C_4 \times ^4C_1 = 15 \times 4 = 60

Choosing

4

out of

6

senior engineers and

1

out of

4

junior engineers.

4

Calculate the combinations for Case 3 (

3

senior,

2

junior)

^6C_3 \times ^4C_2 = 20 \times 6 = 120

Choosing

3

out of

6

senior engineers and

2

out of

4

junior engineers.

5

Sum the combinations across all mutually exclusive cases

Total ways =

6 + 60 + 120 = 186

By the addition principle of counting, the total number of ways is the sum of ways from each disjoint case.

Anahtar Kavram

Conditional combinations using the formula

^nC_r = \frac{n!}{r!(n-r)!}

and the addition principle for disjoint cases.

Soru 7465Soru →

What is the value of the definite integral $\int_{0}^{4} (x + 1) \, dx$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The value of the definite integral is 12.

Integrating

x + 1

yields

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

. Substituting the upper limit

x = 4

gives

\frac{4^2}{2} + 4 = 8 + 4 = 12

. Substituting the lower limit

x = 0

gives

0

. Calculating

12 - 0

gives the final answer of 12.

Adım Adım Çözüm

1

Find the antiderivative of the integrand

f(x) = x + 1

.

\int (x + 1) \, dx = \frac{x^2}{2} + x + C

Apply the power rule of integration

\int x^n \, dx = \frac{x^{n+1}}{n+1}

to

x^1

and

1 = x^0

.

2

Evaluate the antiderivative at the upper bound

x = 4

and lower bound

x = 0

.

F(4) = \frac{4^2}{2} + 4 = \frac{16}{2} + 4 = 12

, and

F(0) = \frac{0^2}{2} + 0 = 0

Substitute the bounds into the antiderivative expression

F(x) = \frac{x^2}{2} + x

.

3

Apply the Fundamental Theorem of Calculus

F(b) - F(a)

.

F(4) - F(0) = 12 - 0 = 12

Subtract the lower bound result from the upper bound result.

Anahtar Kavram

Evaluation of definite integrals of simple linear polynomials

Alternatif Yöntem

The integral

\int_{0}^{4} (x + 1) \, dx

represents the area of a trapezoid under

y = x + 1

from

x = 0

to

x = 4

. The parallel vertical sides are

h_1 = 1

and

h_2 = 5

, with width

b = 4

. Using the area formula for a trapezoid:

\text{Area} = \frac{1+5}{2} \times 4 = 3 \times 4 = 12

.

Tahmini Süre:45s

Soru 7466Soru →

The electric power $P$ dissipated in a resistor varies directly as the square of the current $I$ flowing through it. If a current of $3\text{ A}$ produces a power of $45\text{ W}$ , what is the power dissipated, in watts, when the current is $5\text{ A}$ ?

Cevabı ve açıklamayı göster

Cevap: 125

Cevap

The power dissipated when the current is 5 A is 125 W.

Because electric power varies directly as the square of the current, the formula is

P = k I^2

. Substituting the initial conditions gives

45 = k(3^2) = 9k

, so

k = 5

. Evaluating at

I = 5\text{ A}

gives

P = 5(5^2) = 125\text{ W}

.

Adım Adım Çözüm

1

Set up the variation equation

P = k I^2

Power varies directly as the square of current.

2

Find the constant of variation

k

k = 5

Substitute

P = 45

and

I = 3

into the variation equation:

45 = k(3^2) \implies 45 = 9k \implies k = 5

.

3

Calculate the required power for

I = 5\text{ A}

P = 125\text{ W}

Substitute

k = 5

and

I = 5

into

P = k I^2

:

P = 5(5^2) = 125

.

Anahtar Kavram

Direct variation involving a squared quantity

Soru 7467Soru →

A worker's monthly salary was increased from $\text{N}50,000$ to $\text{N}55,000$ . What is the percentage increase in the worker's salary?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The percentage increase in the worker's salary is

10\%

.

The salary increase is

\text{N}55,000 - \text{N}50,000 = \text{N}5,000

. Evaluating this change relative to the initial salary gives

\frac{5,000}{50,000} \times 100\% = 10\%

.

Adım Adım Çözüm

1

Calculate the absolute increase in salary

Increase =

\text{N}5,000

Subtract the initial salary from the new salary:

\text{N}55,000 - \text{N}50,000 = \text{N}5,000

.

2

Compute the percentage increase relative to the original salary

Percentage Increase =

10\%

Divide the increase by the original value and multiply by

100\%

:

\frac{5,000}{50,000} \times 100\% = 10\%

.

Anahtar Kavram

Percentage Increase

Tahmini Süre:45s

Soru 7468Soru →

What is the value of the definite integral $\int_{0}^{\frac{\pi}{6}} \cos(3x) \, dx$ ?

\frac{1}{3}

1

-\frac{1}{3}

\frac{1}{6}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{3}$

Cevap

The value of the definite integral is

\frac{1}{3}

.

The antiderivative of

\cos(3x)

is

\frac{1}{3}\sin(3x)

. Substituting the upper boundary

x = \frac{\pi}{6}

yields

\frac{1}{3}\sin\left(\frac{\pi}{2}\right) = \frac{1}{3}

, and substituting the lower boundary

x = 0

yields

0

. Subtracting the lower bound evaluation from the upper bound evaluation gives

\frac{1}{3}

.

Adım Adım Çözüm

1

Find the indefinite integral of

\cos(3x)

\int \cos(3x) \, dx = \frac{1}{3}\sin(3x) + C

Using the standard integration rule

\int \cos(kx) \, dx = \frac{1}{k}\sin(kx) + C

.

2

Apply the upper limit of integration

x = \frac{\pi}{6}

\frac{1}{3}\sin\left(3 \cdot \frac{\pi}{6}\right) = \frac{1}{3}\sin\left(\frac{\pi}{2}\right) = \frac{1}{3}(1) = \frac{1}{3}

Substituting the upper limit into the antiderivative.

3

Apply the lower limit of integration

x = 0

and subtract

\frac{1}{3} - \frac{1}{3}\sin(0) = \frac{1}{3} - 0 = \frac{1}{3}

Evaluating the antiderivative at the limits according to the Fundamental Theorem of Calculus.

Anahtar Kavram

Definite Integration of Trigonometric Functions

Tahmini Süre:1m 30s

Soru 7469Soru →

Convert the base 5 number $234_5$ to a base 10 (decimal) number. What is the value in base 10?

Cevabı ve açıklamayı göster

Cevap: 69

Cevap

The decimal (base 10) equivalent of

234_5

is 69.

Expanding

234_5

gives

(2 \times 5^2) + (3 \times 5^1) + (4 \times 5^0) = 50 + 15 + 4 = 69

.

Adım Adım Çözüm

1

Write out the positional expansion of

234_5

using powers of 5 starting from position 0 on the right.

234_5 = (2 \times 5^2) + (3 \times 5^1) + (4 \times 5^0)

Converting from any non-decimal base to base 10 involves multiplying each digit by its corresponding positional base power.

2

Evaluate the powers of 5 and multiply by the respective digits.

2 \times 25 = 50

,

3 \times 5 = 15

,

4 \times 1 = 4

Apply basic arithmetic exponents:

5^2=25

,

5^1=5

,

5^0=1

.

3

Sum all calculated values together.

50 + 15 + 4 = 69

Adding the positional values yields the total value in base 10.

Anahtar Kavram

Converting a non-decimal number to base 10 using expansion by powers of the base.

Soru 7470Soru →

If $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} = p + q\sqrt{3}$ , where $p$ and $q$ are rational numbers, what is the value of $p + q$ ?

1

2

3

5

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

p + q

is 3.

Multiplying both numerator and denominator by the conjugate

(\sqrt{6} + \sqrt{2})

yields

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

. Matching terms with

p + q\sqrt{3}

gives

p = 2

and

q = 1

, so

p + q = 3

.

Adım Adım Çözüm

1

Multiply the numerator and denominator by the conjugate of the denominator,

(\sqrt{6} + \sqrt{2})

.

\frac{(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})}

To eliminate radicals from the denominator.

2

Expand both numerator and denominator.

Numerator:

6 + 2\sqrt{12} + 2 = 8 + 4\sqrt{3}

. Denominator:

6 - 2 = 4

.

Using algebraic expansion

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

and difference of two squares.

3

Simplify the resulting fraction.

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

Dividing each term in the numerator by 4.

4

Equate to

p + q\sqrt{3}

to determine

p

and

q

, then find

p + q

.

p = 2

,

q = 1 \implies p + q = 2 + 1 = 3

.

Comparing rational and irrational parts separately.

Anahtar Kavram

Rationalization of Denominators and Surd Conjugates

Soru 7471Soru →

A vertical flagpole of height $15\text{ m}$ standing on level ground casts a shadow of length $15\sqrt{3}\text{ m}$ . Calculate the angle of elevation of the sun in degrees.

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The angle of elevation of the sun is 30 degrees.

The tangent of the angle of elevation is given by the height divided by the shadow length,

\tan\theta = \frac{15}{15\sqrt{3}} = \frac{1}{\sqrt{3}}

, which corresponds to an angle of

30^\circ

.

Adım Adım Çözüm

1

Formulate the trigonometric relationship using the right triangle formed by the flagpole, the shadow, and the sunlight ray.

\tan\theta = \frac{\text{height of flagpole}}{\text{length of shadow}} = \frac{15}{15\sqrt{3}}

The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side.

2

Simplify the ratio and evaluate the angle.

\tan\theta = \frac{1}{\sqrt{3}} \implies \theta = 30^\circ

The standard acute angle whose tangent value is

\frac{1}{\sqrt{3}}

is

30^\circ

.

Anahtar Kavram

Angle of Elevation and Trigonometric Ratios

Soru 7472Soru →

The table below details the cumulative frequency distribution of the masses, in $\text{kg}$ , of $80$ agricultural packages recorded during an export inspection:

Mass Boundary ( $\text{kg}$ )	Cumulative Frequency
$\leq 19.5$	$8$
$\leq 29.5$	$24$
$\leq 39.5$	$52$
$\leq 49.5$	$72$
$\leq 59.5$	$80$

Using linear interpolation on the cumulative frequency data, determine the interquartile range of the package masses in $\text{kg}$ .

Cevabı ve açıklamayı göster

Cevap: 16.5

Cevap

The interquartile range of the package masses is

16.5\text{ kg}

.

The interquartile range measures the spread of the middle

50\%

of the data and is computed as

Q_3 - Q_1 = 43.5 - 27.0 = 16.5\text{ kg}

.

Adım Adım Çözüm

1

Find the lower quartile (

Q_1

)

Q_1 = 27.0\text{ kg}

The rank for

Q_1

is

\frac{1}{4} \times 80 = 20

, which lies in the

19.5 - 29.5

boundary interval with cumulative frequency increasing from

8

to

24

.

2

Find the upper quartile (

Q_3

)

Q_3 = 43.5\text{ kg}

The rank for

Q_3

is

\frac{3}{4} \times 80 = 60

, which lies in the

39.5 - 49.5

boundary interval with cumulative frequency increasing from

52

to

72

.

3

Calculate the Interquartile Range

\text{IQR} = 16.5\text{ kg}

Interquartile Range is the difference between the upper quartile (

Q_3

) and lower quartile (

Q_1

).

Anahtar Kavram

Interquartile Range from Cumulative Frequency Distribution

Soru 7473Soru →

What is the numerical value of the limit

\lim_{x \to 0} \frac{\sqrt{4 + x} - \sqrt{4 - x}}{x}

?

\frac{1}{2}

0

2

1

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{2}$

Cevap

The numerical value of the limit is $\frac{1}{2}$.

Rationalizing the numerator by multiplying with its conjugate gives $\frac{2x}{x(\sqrt{4+x} + \sqrt{4-x})}$. Canceling $x$ yields $\frac{2}{\sqrt{4+x} + \sqrt{4-x}}$, which evaluates to $\frac{2}{4} = \frac{1}{2}$ as $x \to 0$.

Adım Adım Çözüm

1

Identify the indeterminate form

Substituting $x = 0$ directly yields $\frac{\sqrt{4} - \sqrt{4}}{0} = \frac{0}{0}$, which is an indeterminate form requiring rationalization.

Direct evaluation cannot give the true limit value when an indeterminate form is encountered.

2

Rationalize the numerator

Multiply the numerator and denominator by the conjugate $\sqrt{4 + x} + \sqrt{4 - x}$:

\frac{(\sqrt{4 + x} - \sqrt{4 - x})(\sqrt{4 + x} + \sqrt{4 - x})}{x(\sqrt{4 + x} + \sqrt{4 - x})} = \frac{(4 + x) - (4 - x)}{x(\sqrt{4 + x} + \sqrt{4 - x})}

The difference of squares identity $(a - b)(a + b) = a^2 - b^2$ eliminates the radical signs in the numerator.

3

Simplify the numerator and cancel common terms

$\frac{4 + x - 4 + x}{x(\sqrt{4 + x} + \sqrt{4 - x})} = \frac{2x}{x(\sqrt{4 + x} + \sqrt{4 - x})} = \frac{2}{\sqrt{4 + x} + \sqrt{4 - x}}$

Canceling the common factor $x$ eliminates the zero-causing term in the denominator.

4

Evaluate the simplified limit as $x \to 0$

$\frac{2}{\sqrt{4 + 0} + \sqrt{4 - 0}} = \frac{2}{2 + 2} = \frac{2}{4} = \frac{1}{2}$

Direct substitution is now valid since the expression is continuous at $x = 0$.

Anahtar Kavram

Evaluation of algebraic limits of indeterminate form 0/0 using radical rationalization.

Soru 7474Soru →

The polynomial $P(x) = 2x^3 + ax^2 + bx + 6$ leaves a remainder of $12$ when divided by $(x - 1)$ and has $(x + 3)$ as a factor. What is the value of $a - b$ ?

6

8

4

-28

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The value of

a - b

is

6

.

Applying the Remainder Theorem with

P(1) = 12

gives

a + b = 4

. Applying the Factor Theorem with

P(-3) = 0

gives

3a - b = 16

. Solving these simultaneous linear equations gives

a = 5

and

b = -1

. Evaluating

a - b = 5 - (-1)

gives

6

.

Adım Adım Çözüm

1

Apply the Remainder Theorem for the divisor

(x - 1)

a + b = 4

According to the Remainder Theorem,

P(1) = 12

. Substituting

x = 1

gives

2(1)^3 + a(1)^2 + b(1) + 6 = 12 \implies a + b + 8 = 12 \implies a + b = 4

.

2

Apply the Factor Theorem for the factor

(x + 3)

3a - b = 16

According to the Factor Theorem,

P(-3) = 0

. Substituting

x = -3

gives

2(-3)^3 + a(-3)^2 + b(-3) + 6 = 0 \implies -54 + 9a - 3b + 6 = 0 \implies 9a - 3b = 48 \implies 3a - b = 16

.

3

Solve the system of linear equations for

a

and

b

a = 5

and

b = -1

Adding

(a + b = 4)

and

(3a - b = 16)

yields

4a = 20 \implies a = 5

. Substituting

a = 5

into

a + b = 4

gives

b = -1

.

4

Calculate the target value

a - b

a - b = 6

Subtracting

b

from

a

gives

5 - (-1) = 5 + 1 = 6

.

Anahtar Kavram

Polynomial Remainder and Factor Theorems

Soru 7475Soru →

If $y = \int (6x - 5\sin(x)) \, dx$ and $y = 12$ when $x = 0$ , what is the value of the constant of integration $C$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Integrating

6x - 5\sin(x)

yields

3x^2 + 5\cos(x) + C

. Substituting

x = 0

and

y = 12

gives

12 = 3(0)^2 + 5(1) + C = 5 + C

, which gives

C = 7

.

Adım Adım Çözüm

1

Integrate each term of

6x - 5\sin(x)

with respect to

x

y = 3x^2 + 5\cos(x) + C

The integral of

6x

is

3x^2

, and the integral of

-5\sin(x)

is

+5\cos(x)

.

2

Substitute the initial conditions

x = 0

and

y = 12

to calculate

C

12 = 3(0)^2 + 5\cos(0) + C \implies 12 = 5 + C \implies C = 7

Since

\cos(0) = 1

, the right-hand side reduces to

5 + C

.

Anahtar Kavram

Indefinite Integration and Initial Value Evaluation

Soru 7476Soru →

A contractor estimates that $20$ workers, working $8\text{ hours}$ per day, can complete a road construction project in $30\text{ days}$ . All workers work at the same constant rate. After working for $10\text{ days}$ , $4$ workers leave the site. To make up for the loss, the daily working duration for each remaining worker is increased to $10\text{ hours}$ per day. However, due to fatigue, the work efficiency of each remaining worker drops by $20\%$ . How many additional days will be required to complete the remaining work?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

It will take 25 additional days to complete the remaining work.

The total project requires

4,800

worker-hours (

20 \times 8 \times 30

). In the first

10

days,

1,600

worker-hours are completed, leaving

3,200

worker-hours. After

4

workers leave,

16

workers remain. Working

10

hours a day at

80\%

efficiency, the

16

workers produce

16 \times 10 \times 0.8 = 128

effective worker-hours each day. Dividing the remaining

3,200

worker-hours by

128

gives exactly

25

additional days.

Adım Adım Çözüm

1

Calculate total work units required for the entire project

Total work =

4800\text{ worker-hours}

Work rate is proportional to workers multiplied by total hours worked.

2

Calculate completed work and remaining work

Work done =

1600\text{ worker-hours}

, Remaining work =

3200\text{ worker-hours}

Subtracting completed work from total work gives the remaining required effort.

3

Determine the effective rate per day after conditions change

Effective daily output =

128\text{ worker-hours/day}

16 remaining workers working 10 hours daily at 80% efficiency yield

16 \times 10 \times 0.8 = 128

worker-hours per day.

4

Divide remaining work by the new effective daily rate

Number of additional days =

25\text{ days}

Dividing

3200\text{ worker-hours}

by

128\text{ worker-hours/day}

gives

25\text{ days}

.

Anahtar Kavram

Compound Proportion and Work Efficiency

Soru 7477Soru →

Given the matrices $A = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & 5 \end{pmatrix}$ , find the determinant of the matrix $C = 2A - B$ .

Cevabı ve açıklamayı göster

Cevap: -15

Cevap

The determinant of the matrix

C = 2A - B

is

-15

.

Scalar multiplication gives

2A = \begin{pmatrix} 4 & 2 \\ 8 & 6 \end{pmatrix}

. Subtracting

B

yields

C = \begin{pmatrix} 3 & 3 \\ 6 & 1 \end{pmatrix}

. Evaluating the determinant gives

\det(C) = (3)(1) - (3)(6) = 3 - 18 = -15

.

Adım Adım Çözüm

1

Multiply matrix

A

by scalar

2

2A = \begin{pmatrix} 4 & 2 \\ 8 & 6 \end{pmatrix}

Scalar multiplication requires multiplying each entry of matrix

A

by

2

.

2

Subtract matrix

B

from matrix

2A

entry-wise to find matrix

C

C = \begin{pmatrix} 4 - 1 & 2 - (-1) \\ 8 - 2 & 6 - 5 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 6 & 1 \end{pmatrix}

Subtract corresponding entries of matrix

B

from

2A

.

3

Calculate the determinant of matrix

C

\det(C) = (3)(1) - (3)(6) = 3 - 18 = -15

The determinant of a

2 \times 2

matrix

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

is

ad - bc

.

Anahtar Kavram

Matrix Operations and Determinants

Soru 7478Soru →

If $\sin \theta = \frac{4}{5}$ , where $\theta$ is an acute angle, what is the exact value of the expression $\frac{\sec^2 \theta - 1}{\cot \theta + \csc \theta}$ ?

\frac{8}{9}

\frac{16}{27}

\frac{8}{25}

\frac{4}{3}

Cevabı ve açıklamayı göster

Cevap: $\frac{8}{9}$

Cevap

The exact value of the expression is

\frac{8}{9}

.

Given an acute angle

\theta

with

\sin \theta = \frac{4}{5}

, the adjacent side is

3

. This gives

\tan \theta = \frac{4}{3}

,

\cot \theta = \frac{3}{4}

, and

\csc \theta = \frac{5}{4}

. Using the identity

\sec^2 \theta - 1 = \tan^2 \theta

, the numerator is

\left(\frac{4}{3}\right)^2 = \frac{16}{9}

. The denominator

\cot \theta + \csc \theta = \frac{3}{4} + \frac{5}{4} = 2

. Dividing

\frac{16}{9}

by

2

yields

\frac{8}{9}

.

Adım Adım Çözüm

1

Determine all required trigonometric ratios for the acute angle

\theta

.

Given

\sin \theta = \frac{4}{5}

, the opposite side is

4

and the hypotenuse is

5

. By the Pythagorean theorem, the adjacent side is

\sqrt{5^2 - 4^2} = 3

. Therefore,

\cos \theta = \frac{3}{5}

,

\tan \theta = \frac{4}{3}

,

\cot \theta = \frac{3}{4}

, and

\csc \theta = \frac{5}{4}

.

Defining the right-triangle side lengths allows direct evaluation of all six trigonometric ratios.

2

Simplify the numerator using standard trigonometric identities.

\sec^2 \theta - 1 = \tan^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9}

.

Applying the fundamental identity

1 + \tan^2 \theta = \sec^2 \theta

simplifies the numerator.

3

Evaluate the denominator.

\cot \theta + \csc \theta = \frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2

.

Adding fractional values with a common denominator.

4

Divide the numerator by the denominator.

\frac{16/9}{2} = \frac{8}{9}

.

Dividing the simplified numerator by the simplified denominator yields the final result.

Anahtar Kavram

Evaluation of trigonometric expressions using fundamental identities and right-triangle ratio definitions.

Soru 7479Soru →

A composite plane figure is formed by constructing a semicircle externally on one side of a square of side length $14\text{ cm}$ . What is the perimeter of the resulting figure? (Take $\pi = \frac{22}{7}$ )

64\text{ cm}

78\text{ cm}

86\text{ cm}

50\text{ cm}

Cevabı ve açıklamayı göster

Cevap: $64\text{ cm}$

Cevap

64\text{ cm}

The outer perimeter of the composite shape consists of three straight edges of the square and one curved semicircular arc. Three sides of length

14\text{ cm}

yield

42\text{ cm}

. The arc length of a semicircle with radius

7\text{ cm}

is

\pi r = \frac{22}{7} \times 7 = 22\text{ cm}

. Adding these together gives

42\text{ cm} + 22\text{ cm} = 64\text{ cm}

.

Adım Adım Çözüm

1

Identify the exposed straight sides of the square contributing to the perimeter.

The square has 4 sides, but 1 side is attached to the semicircle inside the figure. Thus, 3 sides are on the outer boundary:

3 \times 14\text{ cm} = 42\text{ cm}

.

Perimeter only includes the outer boundary of a composite shape.

2

Calculate the radius and arc length of the attached semicircle.

The diameter of the semicircle is equal to the side length of the square (

d = 14\text{ cm}

), so radius

r = 7\text{ cm}

. Semicircular arc length

= \pi r = \frac{22}{7} \times 7 = 22\text{ cm}

.

The curved boundary is half of the total circle circumference.

3

Sum the outer straight boundaries and the curved boundary.

Total perimeter

= 42\text{ cm} + 22\text{ cm} = 64\text{ cm}

.

Combining all outer segment lengths gives the total perimeter.

Anahtar Kavram

Perimeter of composite plane figures involving straight line segments and circular arcs.

Tahmini Süre:1m 30s

Soru 7480Soru →

If $\log_2 x + \log_4 x + \log_{16} x = 7$ , what is the value of $x$ ?

16

4

2

8

Cevabı ve açıklamayı göster

Cevap: $16$

Cevap

16

Using the change of base property,

\log_4 x = \frac{1}{2}\log_2 x

and

\log_{16} x = \frac{1}{4}\log_2 x

. Combining like terms yields

\left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x

. Setting

\frac{7}{4}\log_2 x = 7

gives

\log_2 x = 4

, which leads to

x = 2^4 = 16

.

Adım Adım Çözüm

1

Apply the change of base formula to express all logarithmic terms in base 2.

\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{1}{2}\log_2 x

and

\log_{16} x = \frac{\log_2 x}{\log_2 16} = \frac{1}{4}\log_2 x

.

Logarithms with different bases must be converted to a common base to combine them.

2

Substitute the transformed terms back into the equation and factor out

\log_2 x

.

\log_2 x + \frac{1}{2}\log_2 x + \frac{1}{4}\log_2 x = \left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x.

Factoring allows summing the coefficients of

\log_2 x

.

3

Solve for

\log_2 x

by equating the simplified expression to 7.

\frac{7}{4}\log_2 x = 7 \implies \log_2 x = 7 \times \frac{4}{7} = 4.

Isolating the logarithmic term gives its numerical value.

4

Convert the logarithmic equation to exponential form to solve for

x

.

x = 2^4 = 16.

By definition,

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

.

Anahtar Kavram

Change of Base Formula for Logarithms

Tahmini Süre:1m 30s

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