All practice questions

Question 681Question →

Using differentiation from first principles, what is the value of the derivative of the function $f(x) = x^2 + 2x$ at the point where $x = 3$ ?

Show answer & explanation

Answer: 8

Answer

The derivative of

f(x) = x^2 + 2x

evaluated at

x = 3

is 8.

Applying first principles to

f(x) = x^2 + 2x

yields

f'(x) = \lim_{h \to 0} \frac{(x+h)^2 + 2(x+h) - (x^2 + 2x)}{h} = \lim_{h \to 0} (2x + h + 2) = 2x + 2

. Evaluating this derivative at

x = 3

gives

2(3) + 2 = 8

.

Step-by-Step Solution

1

Apply the definition of differentiation from first principles

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

First principles uses the limit of the difference quotient to compute the instantaneous rate of change.

2

Substitute

f(x) = x^2 + 2x

into the difference quotient

\frac{[(x+h)^2 + 2(x+h)] - (x^2 + 2x)}{h} = \frac{2xh + h^2 + 2h}{h}

Expanding terms allows cancellation of non-

h

terms in the numerator.

3

Simplify the fraction by dividing by

h

2x + h + 2

Dividing out

h

removes the indeterminate form

\frac{0}{0}

.

4

Evaluate the limit as

h \to 0

and substitute

x = 3

f'(3) = 2(3) + 2 = 8

Setting

h = 0

yields the derivative function

f'(x) = 2x + 2

, which evaluates to 8 at

x = 3

.

Key Concept

Differentiation from First Principles

Question 682Question →

If $(x - 1)$ is a factor of the polynomial $P(x) = x^3 + 2x^2 - 5x + k$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

is

2

.

According to the Factor Theorem, a linear polynomial

(x - a)

is a factor of

P(x)

if and only if

P(a) = 0

. For the factor

(x - 1)

, setting

x = 1

gives

P(1) = (1)^3 + 2(1)^2 - 5(1) + k = 0

. Simplifying yields

1 + 2 - 5 + k = 0

, which simplifies further to

-2 + k = 0

, giving

k = 2

.

Step-by-Step Solution

1

Apply the Factor Theorem

P(1) = 0

Since

(x - 1)

is a factor, setting

x = 1

makes the polynomial equal to zero.

2

Substitute

x = 1

into

P(x)

1^3 + 2(1)^2 - 5(1) + k = 0

Evaluate the polynomial expression at

x = 1

.

3

Simplify and solve for

k

k = 2

Combine constants

1 + 2 - 5 = -2

and solve

-2 + k = 0

.

Key Concept

Factor Theorem

Question 683Question →

A point $P$ moves in a plane such that its distance from a fixed point $O$ is always $7\text{ cm}$ . What is the diameter, in $\text{cm}$ , of the geometric locus traced out by point $P$ ?

Show answer & explanation

Answer: 14

Answer

The diameter of the locus traced out by point

P

is

14\text{ cm}

.

The locus of a point that maintains a constant distance from a fixed point is a circle. The fixed point

O

is the center of the circle, and the constant distance of

7\text{ cm}

is its radius (

r

). Since the diameter (

D

) of a circle is twice its radius (

D = 2r

), the diameter is

2 \times 7 = 14\text{ cm}

.

Step-by-Step Solution

1

Identify the shape of the geometric locus defined by the condition

A circle centered at point

O

with radius

r = 7\text{ cm}

By definition, the set of all points at a fixed distance from a single point forms a circle.

2

Calculate the diameter using the radius

D = 2 \times 7\text{ cm} = 14\text{ cm}

The diameter of a circle is equal to twice its radius.

Key Concept

Locus of a point at a constant distance from a fixed point

Estimated Time:45s

Question 684Question →

The table below shows the distribution of scores obtained by a group of candidates in an examination:

Score ( $x$ )	2	4	6	8	10
Frequency ( $f$ )	1	2	6	2	1

Calculate the variance of the scores.

Show answer & explanation

Answer: 4

Answer

The variance of the scores is 4.

The mean score is

\bar{x} = 6

. Summing the weighted squared deviations gives

\sum f(x - 6)^2 = 48

. Dividing by the total frequency

N = 12

results in a variance of

\frac{48}{12} = 4

.

Step-by-Step Solution

1

Find the total frequency and calculate the mean score.

Total frequency

N = 12

and mean

\bar{x} = 6

.

The mean is given by

\bar{x} = \frac{\sum f x}{\sum f} = \frac{2 + 8 + 36 + 16 + 10}{12} = \frac{72}{12} = 6

.

2

Determine the sum of squared deviations multiplied by their frequencies.

\sum f(x - \bar{x})^2 = 48.

Evaluating each term:

1(2-6)^2 = 16

,

2(4-6)^2 = 8

,

6(6-6)^2 = 0

,

2(8-6)^2 = 8

, and

1(10-6)^2 = 16

. Summing these yields

16 + 8 + 0 + 8 + 16 = 48

.

3

Compute the variance by dividing the total squared deviations by the total frequency.

Variance = 4.

Variance is

\sigma^2 = \frac{\sum f(x - \bar{x})^2}{\sum f} = \frac{48}{12} = 4

.

Key Concept

Variance of a Frequency Distribution

Question 685Question →

If $\sqrt{28 - 10\sqrt{3}} + \sqrt{7 + 4\sqrt{3}} = k$ , where $k$ is a rational number, find the value of $k$ .

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Answer: 7

Answer

The value of

k

is

7

.

Simplifying each nested surd into standard binomial surd form gives

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

and

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

. Combining these terms cancels out the irrational component

\sqrt{3}

, leaving the integer result

5 + 2 = 7

.

Step-by-Step Solution

1

Simplify the nested surd

\sqrt{28 - 10\sqrt{3}}

into binomial surd form

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

Assuming a form

x - y\sqrt{3}

and squaring both sides gives

x^2 + 3y^2 - 2xy\sqrt{3} = 28 - 10\sqrt{3}

. Matching components leads to

xy = 5

and

x^2 + 3y^2 = 28

, which yields integer values

x = 5

and

y = 1

.

2

Simplify the nested surd

\sqrt{7 + 4\sqrt{3}}

into binomial surd form

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

Assuming a form

u + v\sqrt{3}

and squaring both sides gives

u^2 + 3v^2 + 2uv\sqrt{3} = 7 + 4\sqrt{3}

. Matching components leads to

uv = 2

and

u^2 + 3v^2 = 7

, which yields integer values

u = 2

and

v = 1

.

3

Sum the simplified expressions to calculate

k

k = 7

Summing

(5 - \sqrt{3}) + (2 + \sqrt{3})

results in the irrational parts

-\sqrt{3}

and

\sqrt{3}

cancelling out, leaving

5 + 2 = 7

.

Key Concept

Square Root of a Binomial Surd Expression

Question 686Question →

Three water pipes, $A$ , $B$ , and $C$ , are used to fill a storage tank. The filling rates of pipes $A$ and $B$ are in the ratio $3 : 2$ , while the filling rates of pipes $B$ and $C$ are in the ratio $4 : 3$ . Pipes $A$ and $B$ are opened together for $5\text{ hours}$ , after which pipe $A$ is closed. Pipes $B$ and $C$ are then opened together for $10\text{ hours}$ to completely fill the remaining capacity of the tank. How many hours would pipe $A$ alone take to fill the entire empty tank?

Show answer & explanation

Answer: 20

Answer

Pipe

A

alone will take

20\text{ hours}

to fill the empty tank.

Combining the given ratios

A : B = 3 : 2

and

B : C = 4 : 3

gives

A : B : C = 6 : 4 : 3

. Assigning relative rates of

6r

,

4r

, and

3r

per hour, the work completed in the first phase is

5 \times (6r + 4r) = 50r

, and in the second phase is

10 \times (4r + 3r) = 70r

. The total capacity of the tank is

50r + 70r = 120r

. Dividing the total capacity by pipe

A

's filling rate (

6r

) gives

120r / 6r = 20\text{ hours}

.

Step-by-Step Solution

1

Find the combined ratio of filling rates for pipes A, B, and C

A : B : C = 6 : 4 : 3

Given

A : B = 3 : 2 = 6 : 4

and

B : C = 4 : 3

, aligning the common term

B = 4

yields

A : B : C = 6 : 4 : 3

.

2

Express rates in terms of a constant

r

and calculate work done in Phase 1

Work in Phase 1 =

50r

Rates are

A = 6r

,

B = 4r

,

C = 3r

. In

5\text{ hours}

, pipes

A

and

B

fill

5 \times (6r + 4r) = 50r

.

3

Calculate work done in Phase 2

Work in Phase 2 =

70r

In

10\text{ hours}

, pipes

B

and

C

fill

10 \times (4r + 3r) = 70r

.

4

Determine total capacity of the tank

Total Capacity =

120r

Total work required to fill the tank is

50r + 70r = 120r

.

5

Calculate time taken by pipe A alone

20\text{ hours}

Time taken by pipe

A = \frac{\text{Total Capacity}}{\text{Rate of } A} = \frac{120r}{6r} = 20\text{ hours}

.

Key Concept

Compound Ratio and Rates of Work

Question 687Question →

If $\log_3(x + 5) - \log_{\frac{1}{3}}(x - 1) = 3$ , find the real value of $x$ .

Show answer & explanation

Answer: 4

Answer

The real value of

x

is 4.

Using the change of base formula,

\log_{\frac{1}{3}}(x - 1) = -\log_3(x - 1)

. Substituting this back into the equation transforms it into

\log_3(x + 5) + \log_3(x - 1) = 3

. Combining the logarithms using the product property gives

\log_3[(x + 5)(x - 1)] = 3

, which means

(x + 5)(x - 1) = 3^3 = 27

. Expanding leads to

x^2 + 4x - 32 = 0

, factoring into

(x + 8)(x - 4) = 0

. Since logarithms require positive arguments (

x > 1

),

x = -8

is invalid, giving the final answer

x = 4

.

Step-by-Step Solution

1

Apply change of base to express the equation in a single base

\log_{\frac{1}{3}}(x - 1) = \frac{\log_3(x - 1)}{\log_3(1/3)} = -\log_3(x - 1)

Change of base rule

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

with base

c = 3

, where

\log_3(1/3) = -1

.

2

Substitute back into the original equation and combine terms

\log_3(x + 5) - [-\log_3(x - 1)] = \log_3(x + 5) + \log_3(x - 1) = 3

Subtracting a negative logarithm equals adding the positive logarithm.

3

Apply the product rule of logarithms

\log_3[(x + 5)(x - 1)] = 3

Product rule:

\log_b M + \log_b N = \log_b(MN)

.

4

Convert logarithmic equation to quadratic form and solve

(x + 5)(x - 1) = 3^3 = 27 \implies x^2 + 4x - 32 = 0 \implies (x + 8)(x - 4) = 0

Logarithmic definition

\log_b Y = k \implies Y = b^k

.

5

Verify domain constraints

x = 4

Logarithmic domain requires

x - 1 > 0 \implies x > 1

. Therefore,

x = -8

is extraneous.

Key Concept

Logarithmic Change of Base and Algebraic Reduction

Question 688Question →

If $\frac{\sqrt{x} + \sqrt{3}}{\sqrt{x} - \sqrt{3}} = 2$ , find the value of $x$ .

Show answer & explanation

Answer: 27

Answer

The value of

x

is 27.

Cross-multiplying yields

\sqrt{x} + \sqrt{3} = 2\sqrt{x} - 2\sqrt{3}

. Rearranging terms gives

\sqrt{x} = 3\sqrt{3}

. Squaring both sides produces

x = 3^2 \times 3 = 27

.

Step-by-Step Solution

1

Multiply both sides by the denominator

(\sqrt{x} - \sqrt{3})

\sqrt{x} + \sqrt{3} = 2(\sqrt{x} - \sqrt{3})

Clear the rational surd expression to form a linear relation in terms of radicals

2

Expand the terms and isolate

\sqrt{x}

\sqrt{x} = 3\sqrt{3}

Group terms involving

\sqrt{x}

on one side and constant surds on the other side

3

Square both sides of the simplified equation

x = (3\sqrt{3})^2 = 9 \times 3 = 27

Eliminate the radical over

x

to obtain the integer solution

Key Concept

Solving algebraic surd equations using rationalization concepts and properties of radicals

Estimated Time:1m 30s

Question 689Question →

What is the numerical value of $4\sin 30^\circ + 2\cos 60^\circ - \tan 45^\circ$ ?

Show answer & explanation

Answer: 2

Answer

2

Substituting the exact trigonometric values for special angles into the expression gives

4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) - 1 = 2 + 1 - 1 = 2

.

Step-by-Step Solution

1

Recall exact values for special trigonometric angles

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

,

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

, and

\tan 45^\circ = 1

These are standard special angle values derived from standard right-angled triangles.

2

Substitute the values into the original expression

4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) - 1 = 2 + 1 - 1

Direct algebraic substitution.

3

Simplify the resulting numerical expression

2

Combining terms

2 + 1 - 1

gives

2

.

Key Concept

Basic Trigonometric Ratios and Special Angles

Question 690Question →

A solid right pyramid has a square base with a side length of $6\text{ cm}$ and a vertical height of $4\text{ cm}$ . What is the total surface area of the pyramid in square centimeters?

Show answer & explanation

Answer: 96

Answer

The total surface area of the pyramid is

96\text{ cm}^2

.

To determine the total surface area of a square pyramid, sum the base area and the total area of the four triangular faces. The base area is

6 \times 6 = 36\text{ cm}^2

. The slant height of each triangular face is found via the Pythagorean theorem using half the base side length (

3\text{ cm}

) and the vertical height (

4\text{ cm}

), yielding

\sqrt{3^2 + 4^2} = 5\text{ cm}

. The area of one triangular face is

\frac{1}{2} \times 6 \times 5 = 15\text{ cm}^2

, making four faces equal to

60\text{ cm}^2

. Adding the base area gives

36 + 60 = 96\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the area of the square base

Base area =

36\text{ cm}^2

The base is a square of side

6\text{ cm}

, so

\text{Area} = 6^2 = 36\text{ cm}^2

.

2

Find the slant height of each triangular lateral face

Slant height

l = 5\text{ cm}

The slant height forms the hypotenuse of a right-angled triangle inside the pyramid with legs equal to the vertical height (

4\text{ cm}

) and half the base edge (

3\text{ cm}

):

l = \sqrt{4^2 + 3^2} = 5\text{ cm}

.

3

Calculate the combined area of the four triangular faces

Lateral area =

60\text{ cm}^2

Each triangle has base

6\text{ cm}

and height

5\text{ cm}

, giving an area of

\frac{1}{2} \times 6 \times 5 = 15\text{ cm}^2

. For 4 identical faces, the total is

4 \times 15 = 60\text{ cm}^2

.

4

Calculate total surface area

Total Surface Area =

96\text{ cm}^2

Sum the base area and the total lateral area:

36 + 60 = 96\text{ cm}^2

.

Key Concept

Total Surface Area of a Right Pyramid

Question 691Question →

In $\Delta ABC$ , $\sin A = \frac{3}{5}$ , $\sin B = \frac{4}{5}$ , and the side opposite angle $A$ has length $a = 15\text{ cm}$ . What is the length of side $b$ , in centimeters?

Show answer & explanation

Answer: 20

Answer

The length of side

b

is 20 cm.

Using the Sine Rule

\frac{a}{\sin A} = \frac{b}{\sin B}

, substitute

a = 15

,

\sin A = \frac{3}{5}

, and

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

. Evaluating

\frac{15}{3/5}

gives

25

. Multiplying

25

by

\frac{4}{5}

yields

20\text{ cm}

.

Step-by-Step Solution

1

Set up the Sine Rule relationship between sides

a

,

b

and their opposite angles

A

,

B

.

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule relates the side lengths of a triangle to the sines of its angles.

2

Substitute

a = 15

,

\sin A = \frac{3}{5}

, and

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

into the Sine Rule equation.

\frac{15}{3/5} = \frac{b}{4/5}

Direct substitution of known values allows us to solve for the unknown side

b

.

3

Simplify the left side of the equation.

15 \times \frac{5}{3} = 25

Dividing 15 by

\frac{3}{5}

is equivalent to multiplying 15 by

\frac{5}{3}

.

4

Multiply both sides by

\frac{4}{5}

to find

b

.

b = 25 \times \frac{4}{5} = 20\text{ cm}

Isolating

b

gives the final length of side

b

.

Key Concept

Sine Rule

Question 692Question →

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 693Question →

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 694Question →

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 695Question →

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 696Question →

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 697Question →

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 698Question →

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 699Question →

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 700Question →

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

Answer

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