All practice questions

Question 7461Question →

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

Show answer & explanation

Answer: 5

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Solving for unknown algebraic side lengths using the Cosine Rule.

Question 7462Question →

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 $k$ .

Show answer & explanation

Answer: 4

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Mean of Ungrouped Data from a Frequency Table

Question 7463Question →

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

Show answer & explanation

Answer: 6.625

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Conversion of Fractional Non-Decimal Numbers to Decimal

Question 7464Question →

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

Show answer & explanation

Answer: $186$

Answer

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.

Step-by-Step Solution

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.

Key Concept

Conditional combinations using the formula

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

and the addition principle for disjoint cases.

Question 7465Question →

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

Show answer & explanation

Answer: 12

Answer

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.

Step-by-Step Solution

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.

Key Concept

Evaluation of definite integrals of simple linear polynomials

Alternative Method

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

.

Estimated Time:45s

Question 7466Question →

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 $5\text{ A}$ ?

Show answer & explanation

Answer: 125

Answer

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}

.

Step-by-Step Solution

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

.

Key Concept

Direct variation involving a squared quantity

Question 7467Question →

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?

Show answer & explanation

Answer: 10

Answer

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\%

.

Step-by-Step Solution

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\%

.

Key Concept

Percentage Increase

Estimated Time:45s

Question 7468Question →

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}

Show answer & explanation

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

Answer

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}

.

Step-by-Step Solution

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.

Key Concept

Definite Integration of Trigonometric Functions

Estimated Time:1m 30s

Question 7469Question →

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

Show answer & explanation

Answer: 69

Answer

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

.

Step-by-Step Solution

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.

Key Concept

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

Question 7470Question →

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

Show answer & explanation

Answer: 3

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Rationalization of Denominators and Surd Conjugates

Question 7471Question →

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.

Show answer & explanation

Answer: 30

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Angle of Elevation and Trigonometric Ratios

Question 7472Question →

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 $\text{kg}$ .

Show answer & explanation

Answer: 16.5

Answer

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}

.

Step-by-Step Solution

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

).

Key Concept

Interquartile Range from Cumulative Frequency Distribution

Question 7473Question →

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

Show answer & explanation

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

Answer

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 $x \to 0$.

Step-by-Step Solution

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 $x = 0$.

Key Concept

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

Question 7474Question →

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

Show answer & explanation

Answer: $6$

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Polynomial Remainder and Factor Theorems

Question 7475Question →

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

Show answer & explanation

Answer: 7

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Indefinite Integration and Initial Value Evaluation

Question 7476Question →

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?

Show answer & explanation

Answer: 25

Answer

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.

Step-by-Step Solution

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}

.

Key Concept

Compound Proportion and Work Efficiency

Question 7477Question →

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 $C = 2A - B$ .

Show answer & explanation

Answer: -15

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Matrix Operations and Determinants

Question 7478Question →

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}

Show answer & explanation

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

Answer

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}

.

Step-by-Step Solution

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.

Key Concept

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

Question 7479Question →

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}

Show answer & explanation

Answer: $64\text{ cm}$

Answer

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}

.

Step-by-Step Solution

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.

Key Concept

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

Estimated Time:1m 30s

Question 7480Question →

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

16

4

2

8

Show answer & explanation

Answer: $16$

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Change of Base Formula for Logarithms

Estimated Time:1m 30s

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