All practice questions

Determine the number of integer values of $x$ that satisfy the quadratic inequality $3x^2 - 14x - 5 < 0$ .

Show answer & explanation

Answer: 5

Answer

The total number of integer values satisfying the inequality is 5.

Factoring

3x^2 - 14x - 5 < 0

gives

(3x + 1)(x - 5) < 0

. The critical roots are

x = -\frac{1}{3}

and

x = 5

. Since the parabola opens upward, the expression is negative between the roots, yielding the interval

-\frac{1}{3} < x < 5

. The integer values contained in this interval are

0, 1, 2, 3,

and

4

, which total 5 values.

Step-by-Step Solution

1

Factor the quadratic expression

(3x + 1)(x - 5) < 0

Factoring identifies the critical values where the quadratic expression changes sign.

2

Determine the critical points

x = -\frac{1}{3}

and

x = 5

Setting each factor to zero gives the boundary roots of the equation.

3

Formulate the solution interval

-\frac{1}{3} < x < 5

Since the coefficient of

x^2

is positive, the quadratic curve is below the x-axis strictly between the two roots.

4

List and count the integer solutions

The integers are

0, 1, 2, 3, 4

, yielding 5 integer solutions.

Counting integers strictly greater than

-\frac{1}{3}

and strictly less than

5

.

Key Concept

Solving quadratic inequalities and finding integer solution counts

Estimated Time:1m 30s

Question 622Question →

Find the smallest integer $x$ that satisfies the linear inequality $5 - 3x \le -7$ .

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

x

satisfying the inequality is

4

.

Subtracting 5 from both sides of

5 - 3x \le -7

gives

-3x \le -12

. Dividing both sides by

-3

requires reversing the inequality sign to obtain

x \ge 4

. Therefore, the smallest integer value in the solution set is 4.

Step-by-Step Solution

1

Subtract 5 from both sides of the inequality to isolate the variable term

-3x \le -12

Subtracting a constant from both sides maintains the inequality direction.

2

Divide both sides by

-3

and flip the inequality sign

x \ge 4

Dividing an inequality by a negative number reverses the direction of the inequality symbol.

3

Identify the smallest integer satisfying the condition

4

The solution set contains all real numbers greater than or equal to 4, making 4 the minimum integer value.

Key Concept

Solving Linear Inequalities with Negative Coefficients

Estimated Time:45s

Question 623Question →

If $y = 2\sin(3x) + e^{4x}$ , what is the value of $\frac{dy}{dx}$ at $x = 0$ ?

Show answer & explanation

Answer: 10

Answer

10

Differentiating

y = 2\sin(3x) + e^{4x}

yields

\frac{dy}{dx} = 6\cos(3x) + 4e^{4x}

. Evaluating this expression at

x = 0

gives

6\cos(0) + 4e^{0} = 6(1) + 4(1) = 10

.

Step-by-Step Solution

1

Differentiate each term of

y = 2\sin(3x) + e^{4x}

with respect to

x

.

\frac{dy}{dx} = 6\cos(3x) + 4e^{4x}

Applying the chain rule gives

\frac{d}{dx}[2\sin(3x)] = 2 \times 3\cos(3x) = 6\cos(3x)

and

\frac{d}{dx}[e^{4x}] = 4e^{4x}

.

2

Evaluate the derivative at

x = 0

.

6\cos(0) + 4e^{0} = 6(1) + 4(1) = 10

Substituting

x = 0

gives

\cos(0) = 1

and

e^{0} = 1

.

Key Concept

Differentiation of trigonometric and exponential functions

Question 624Question →

Find the value of $r$ such that $-29 \equiv r \pmod{8}$ , where $0 \le r < 8$ .

Show answer & explanation

Answer: 3

Answer

The value of

r

in standard non-negative remainder form is

3

.

Dividing

-29

by

8

yields a quotient of

-4

and a remainder of

3

, because

8 \times (-4) + 3 = -29

. Since

0 \le 3 < 8

,

3

is the standard non-negative remainder.

Step-by-Step Solution

1

Express

-29

in the form

8q + r

, where

q

is an integer and

0 \le r < 8

.

-29 = 8(-4) + 3

To find the standard remainder modulo

8

, the remainder

r

must satisfy

0 \le r < 8

.

2

Identify the remainder value

r

.

r = 3

Since

3

lies within the required range

[0, 7]

, it is the canonical remainder.

Key Concept

Modular Arithmetic and Non-Negative Remainders

Question 625Question →

If $32_x = 23_{10}$ , find the value of the base $x$ .

Show answer & explanation

Answer: 7

Answer

7

Expanding the number

32_x

into base 10 gives

3 \times x + 2 = 3x + 2

. Setting

3x + 2 = 23

leads to

3x = 21

, giving

x = 7

.

Step-by-Step Solution

1

Expand

32_x

in terms of powers of

x

3x + 2

In base

x

, the rightmost digit has a weight of

x^0 = 1

and the next digit to the left has a weight of

x^1 = x

.

2

Equate the expanded form to the given base 10 value and solve for

x

x = 7

Setting

3x + 2 = 23

gives

3x = 21

, which yields

x = 7

upon dividing by 3.

Key Concept

Expansion and conversion from Base x to Base 10

Estimated Time:45s

Question 626Question →

Find the sum of all integer values of $x$ that satisfy the system of inequalities $3x + 1 > 7$ and $x^2 - 4x - 21 \le 0$ .

Show answer & explanation

Answer: 25

Answer

25

Solving

3x + 1 > 7

gives

x > 2

. Solving

x^2 - 4x - 21 \le 0

gives

-3 \le x \le 7

. The combined condition is

2 < x \le 7

. The integer values satisfying this condition are 3, 4, 5, 6, and 7. The sum of these integers is

3 + 4 + 5 + 6 + 7 = 25

.

Step-by-Step Solution

1

Solve the linear inequality

3x + 1 > 7

.

x > 2

Subtracting 1 from both sides gives

3x > 6

, and dividing by 3 yields

x > 2

.

2

Solve the quadratic inequality

x^2 - 4x - 21 \le 0

.

-3 \le x \le 7

Factoring the quadratic expression gives

(x - 7)(x + 3) \le 0

. The roots are

-3

and

7

, and the quadratic curve is non-positive between these roots.

3

Determine the intersection of the solution sets

x > 2

and

-3 \le x \le 7

.

2 < x \le 7

The lower bound is determined by the strict linear constraint

x > 2

, while the upper bound is determined by the inclusive quadratic constraint

x \le 7

.

4

List the integer values of

x

within

2 < x \le 7

and compute their sum.

Integers are

3, 4, 5, 6, 7

; their sum is

25

.

The strict inequality excludes

2

, while the inclusive inequality includes

7

.

Key Concept

System of linear and quadratic inequalities

Question 627Question →

If $2^{3x - 1} = 32$ , find the value of $x$ .

Show answer & explanation

Answer: 2

Answer

The value of

x

is 2.

Rewriting 32 as

2^5

allows equating the exponents

3x - 1 = 5

. Solving for

x

gives

3x = 6

, which simplifies to

x = 2

.

Step-by-Step Solution

1

Express both sides of the equation using a common base of 2

2^{3x - 1} = 2^5

The number 32 can be rewritten in index form as

2^5

.

2

Equate the indices

3x - 1 = 5

If

a^m = a^n

for a non-zero base

a \neq 1

, then

m = n

.

3

Solve the linear equation for

x

x = 2

Adding 1 to both sides yields

3x = 6

, and dividing by 3 gives

x = 2

.

Key Concept

Solving exponential equations by expressing numbers with equal bases

Question 628Question →

Each exterior angle of a regular polygon measures $24^\circ$ . How many sides does this polygon have?

Show answer & explanation

Answer: 15

Answer

The polygon has 15 sides.

The sum of all exterior angles of any convex polygon is always

360^\circ

. Because the polygon is regular, all its exterior angles are equal. Dividing

360^\circ

by the measure of one exterior angle (

24^\circ

) yields exactly 15 sides.

Step-by-Step Solution

1

Identify the formula relating exterior angle and number of sides

The sum of exterior angles of any convex polygon is

360^\circ

. For a regular polygon with

n

sides, each exterior angle equals

\frac{360^\circ}{n}

.

All exterior angles in a regular polygon are equal.

2

Rearrange the formula to solve for the number of sides

n

n = \frac{360^\circ}{\text{Exterior Angle}} = \frac{360^\circ}{24^\circ} = 15

.

Dividing the total sum of exterior angles by the measure of one exterior angle yields the total number of sides.

Key Concept

Exterior angle property of regular polygons

Question 629Question →

If $1101_2 + 32_4 = x_5$ , what is the value of $x$ ?

Show answer & explanation

Answer: 102

Answer

The value of

x

is

102

.

To add numbers given in different bases, convert each term to base 10:

1101_2 = 13_{10}

and

32_4 = 14_{10}

. Summing these decimal values gives

13 + 14 = 27_{10}

. Converting

27_{10}

to base 5 by repeated division by 5 gives remainders 2, 0, 1 (from least significant to most significant digit), resulting in

102_5

. Therefore,

x = 102

.

Step-by-Step Solution

1

Convert

1101_2

to decimal (base 10)

13_{10}

Expand in powers of 2:

1(2^3) + 1(2^2) + 0(2^1) + 1(2^0) = 8 + 4 + 0 + 1 = 13

.

2

Convert

32_4

to decimal (base 10)

14_{10}

Expand in powers of 4:

3(4^1) + 2(4^0) = 12 + 2 = 14

.

3

Add the two base-10 numbers together

27_{10}

Perform standard addition:

13 + 14 = 27

.

4

Convert

27_{10}

into base 5

102_5

Divide 27 repeatedly by 5 and record remainders:

27 = 5 \times 5 + 2

,

5 = 1 \times 5 + 0

,

1 = 0 \times 5 + 1

. Reading remainders upward yields

102_5

.

Key Concept

Conversion between different number bases via base 10

Question 630Question →

If $y = e^{2x} \ln(e^x + \sin x)$ , determine the value of $\frac{d^2 y}{dx^2}$ at $x = 0$ .

Show answer & explanation

Answer: 5

Answer

5

Evaluating the second derivative of

y = e^{2x} \ln(e^x + \sin x)

at

x = 0

yields

5

. This is calculated by applying the product, chain, and quotient rules to get

\frac{d^2 y}{dx^2}

, and substituting

x = 0

, where

e^0 = 1

,

\sin 0 = 0

,

\cos 0 = 1

, and

\ln 1 = 0

.

Step-by-Step Solution

1

Differentiate

y = e^{2x} \ln(e^x + \sin x)

with respect to

x

using the product rule.

\frac{dy}{dx} = 2e^{2x} \ln(e^x + \sin x) + e^{2x} \left(\frac{e^x + \cos x}{e^x + \sin x}\right)

The function is composed of

u(x) = e^{2x}

and

v(x) = \ln(e^x + \sin x)

. By chain rule,

v'(x) = \frac{e^x + \cos x}{e^x + \sin x}

.

2

Differentiate the first term

T_1(x) = 2e^{2x} \ln(e^x + \sin x)

to get

T_1'(x)

.

T_1'(x) = 4e^{2x} \ln(e^x + \sin x) + 2e^{2x} \left(\frac{e^x + \cos x}{e^x + \sin x}\right)

Applying the product rule to

2e^{2x}

and

\ln(e^x + \sin x)

.

3

Differentiate the second term

T_2(x) = e^{2x} \left(\frac{e^x + \cos x}{e^x + \sin x}\right)

using the product and quotient rules.

T_2'(x) = 2e^{2x} \left(\frac{e^x + \cos x}{e^x + \sin x}\right) + e^{2x} \left(\frac{(e^x - \sin x)(e^x + \sin x) - (e^x + \cos x)^2}{(e^x + \sin x)^2}\right)

The derivative of

\frac{e^x + \cos x}{e^x + \sin x}

requires the quotient rule.

4

Evaluate

T_1'(0)

and

T_2'(0)

at

x = 0

.

T_1'(0) = 4(1)(0) + 2(1)(2) = 4

, and

T_2'(0) = 2(1)(2) + 1 \left(\frac{(1)(1) - 2^2}{1^2}\right) = 4 - 3 = 1

.

At

x = 0

,

e^0 = 1

,

\sin 0 = 0

,

\cos 0 = 1

, and

\ln(1) = 0

.

5

Sum the evaluated derivative components to find

\frac{d^2 y}{dx^2}\Big|_{x=0}

.

\frac{d^2 y}{dx^2}\Big|_{x=0} = T_1'(0) + T_2'(0) = 4 + 1 = 5

Combining the evaluated terms gives the final numerical value.

Key Concept

Higher-Order Derivatives of Combined Transcendental Functions

Question 631Question →

The frequency distribution table below shows the daily rainfall (in mm) recorded across $80$ weather monitoring stations during a storm:

Daily Rainfall (mm)	Frequency ( $f$ )
$10 - 19$	$8$
$20 - 29$	$14$
$30 - 39$	$26$
$40 - 49$	$20$
$50 - 59$	$12$

Using linear interpolation from the cumulative frequency distribution (ogive), what is the 75th percentile ( $P_{75}$ ) of the daily rainfall in mm?

Show answer & explanation

Answer: 45.5

Answer

The 75th percentile of the daily rainfall distribution is

45.5\text{ mm}

.

To find the 75th percentile (

P_{75}

), determine the

60^{\text{th}}

cumulative frequency position (

0.75 \times 80 = 60

). The value lies within the

40 - 49

class interval. Applying the lower class boundary

L = 39.5

, preceding cumulative frequency

cf_b = 48

, frequency

f = 20

, and class width

c = 10

, linear interpolation yields

P_{75} = 39.5 + \frac{60 - 48}{20} \times 10 = 45.5\text{ mm}

.

Step-by-Step Solution

1

Calculate cumulative frequencies across all class intervals.

Cumulative frequencies are

8

for

10-19

,

22

for

20-29

,

48

for

30-39

,

68

for

40-49

, and

80

for

50-59

. Total frequency

N = 80

.

Cumulative frequencies are necessary to locate percentile positions on an ogive.

2

Determine the position corresponding to the 75th percentile.

Position

= 0.75 \times 80 = 60^{\text{th}}

cumulative frequency item.

The 75th percentile represents

75\%

of the total sample size.

3

Identify the target class interval parameters containing the 60th observation.

The interval

40 - 49

contains cumulative frequencies from

49

to

68

. Parameters:

L = 39.5

,

c = 10

,

f = 20

,

cf_b = 48

.

Linear interpolation requires the exact boundaries and frequencies of the container class.

4

Apply the percentile interpolation formula

P_{75} = L + \left(\frac{60 - cf_b}{f}\right) \times c

.

P_{75} = 39.5 + \left(\frac{60 - 48}{20}\right) \times 10 = 39.5 + 6 = 45.5\text{ mm}

.

Calculates the exact rainfall value corresponding to the 75th percentile.

Key Concept

Linear interpolation for percentiles using cumulative frequency distribution (ogive)

Estimated Time:1m 30s

Question 632Question →

The table below shows the distribution of masses (in kg) of $50$ bags of cement inspected at a building construction site:

Mass (kg)	Frequency ( $f$ )
$40 - 44$	$5$
$45 - 49$	$12$
$50 - 54$	$18$
$55 - 59$	$15$

Calculate the mean mass of the bags of cement in kg.

Show answer & explanation

Answer: 51.3

Answer

The mean mass of the cement bags is 51.3 kg.

The mean mass of grouped data is computed using the formula

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

, where

x

represents the class midpoints and

f

represents the frequency of each class. The class midpoints are

42

,

47

,

52

, and

57

. Multiplying these midpoints by their respective frequencies yields

210

,

564

,

936

, and

855

. Summing these values gives

\sum fx = 2565

. Dividing by the total frequency

\sum f = 50

produces a mean mass of

\frac{2565}{50} = 51.3\text{ kg}

.

Step-by-Step Solution

1

Determine the midpoint (x) of each class interval

Midpoints are 42, 47, 52, and 57

For grouped data, the class midpoint represents the average value of all observations falling within that class interval.

2

Calculate the product of each midpoint and its corresponding frequency (fx)

Products are 210, 564, 936, and 855

Multiplying the midpoint by frequency gives the total estimated mass contributed by that class interval.

3

Sum all frequencies and all fx products

Total frequency sum = 50, Total product sum = 2565

These totals are required to calculate the weighted average across all intervals.

4

Divide the sum of fx by the total frequency

Mean = 51.3 kg

Applying the formula for grouped mean: Mean = (sum of fx) / (sum of f).

Key Concept

Grouped Data Mean Calculation

Question 633Question →

If $x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ and $y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$ , determine the numerical value of $x^2 + y^2$ .

Show answer & explanation

Answer: 23

Answer

The numerical value of

x^2 + y^2

is

23

.

Rationalizing

x

yields

\frac{5 + \sqrt{21}}{2}

and rationalizing

y

yields

\frac{5 - \sqrt{21}}{2}

. The sum

x + y

equals

5

and the product

xy

equals

1

. Substituting these into

x^2 + y^2 = (x + y)^2 - 2xy

gives

5^2 - 2(1) = 23

.

Step-by-Step Solution

1

Rationalize the denominators of

x

and

y

x = \frac{5 + \sqrt{21}}{2}

and

y = \frac{5 - \sqrt{21}}{2}

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

2

Calculate the sum

x + y

and the product

xy

x + y = 5

and

xy = 1

Summing conjugate surd expressions cancels the radical term, and multiplying them applies the difference of two squares.

3

Evaluate

x^2 + y^2

using the identity

(x + y)^2 - 2xy

x^2 + y^2 = 5^2 - 2(1) = 23

Substituting the known sum and product avoids having to square complex surd expressions directly.

Key Concept

Rationalization of binomial denominators and application of symmetric algebraic identities.

Question 634Question →

If $1.41_x = \frac{46}{25}_{10}$ , where $x$ is a positive integer base, find the value of $x$ .

Show answer & explanation

Answer: 5

Answer

The value of the base

x

is 5.

Expanding

1.41_x

yields

1 + \frac{4}{x} + \frac{1}{x^2}

. Setting this equal to

\frac{46}{25}

gives

\frac{4x + 1}{x^2} = \frac{21}{25}

. Cross-multiplying results in the quadratic equation

21x^2 - 100x - 25 = 0

, which factors as

(21x + 5)(x - 5) = 0

. Since a number base must be a positive integer greater than 4,

x = 5

is the only valid solution.

Step-by-Step Solution

1

Expand

1.41_x

using place value powers of

x

.

1 + \frac{4}{x} + \frac{1}{x^2}

Fractional digits to the right of the radix point represent negative powers of the base (

x^{-1}, x^{-2}, \dots

).

2

Equate the expanded form to

\frac{46}{25}

and simplify.

\frac{4x + 1}{x^2} = \frac{21}{25}

Subtracting 1 from both sides isolates the fractional place values.

3

Cross-multiply and solve the quadratic equation

21x^2 - 100x - 25 = 0

.

(21x + 5)(x - 5) = 0 \implies x = 5

A base must be a positive integer strictly greater than any individual digit in the number (digits present are 1 and 4).

Key Concept

Conversion of fractional numbers in non-decimal bases to base 10 and solving polynomial equations in unknown bases.

Question 635Question →

If the surd expression $\frac{12}{\sqrt{6}}$ is simplified to the form $k\sqrt{6}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

is 2.

Multiplying the numerator and denominator of

\frac{12}{\sqrt{6}}

by

\sqrt{6}

yields

\frac{12\sqrt{6}}{6}

. Dividing

12

by

6

simplifies the expression to

2\sqrt{6}

. Matching this with

k\sqrt{6}

gives

k = 2

.

Step-by-Step Solution

1

Rationalize the denominator of

\frac{12}{\sqrt{6}}

by multiplying the numerator and denominator by

\sqrt{6}

.

\frac{12\sqrt{6}}{6}

Multiplying by

\frac{\sqrt{6}}{\sqrt{6}}

removes the radical from the denominator without changing the value of the expression.

2

Divide the integer coefficient in the numerator by the denominator.

2\sqrt{6}

Simplifying

\frac{12}{6}

yields 2.

3

Compare

2\sqrt{6}

with

k\sqrt{6}

to determine the value of

k

.

k = 2

The coefficient of

\sqrt{6}

is 2.

Key Concept

Rationalization of monomial surd denominators

Question 636Question →

If $\frac{\sqrt{50} + \sqrt{18}}{\sqrt{2}} = m$ , what is the value of the integer $m$ ?

Show answer & explanation

Answer: 8

Answer

The value of the integer

m

is 8.

Simplifying

\sqrt{50}

to

5\sqrt{2}

and

\sqrt{18}

to

3\sqrt{2}

gives a numerator of

8\sqrt{2}

. Dividing

8\sqrt{2}

by

\sqrt{2}

cancels out the radical part, yielding the integer 8.

Step-by-Step Solution

1

Simplify the radical expressions in the numerator.

\sqrt{50} = 5\sqrt{2}

and

\sqrt{18} = 3\sqrt{2}

.

Factor out perfect square numbers from within each radical.

2

Sum the simplified surds in the numerator.

5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}

.

Surds with identical radicands are like terms and can be added by adding their coefficients.

3

Divide the numerator by the denominator.

\frac{8\sqrt{2}}{\sqrt{2}} = 8

.

Cancel the common factor of

\sqrt{2}

present in both numerator and denominator.

Key Concept

Simplification and division of surds

Question 637Question →

The set of four numbers $\{x, 6, 8, 10\}$ has a variance of $5$ . Given that $x < 6$ , find the value of $x$ .

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

By expressing the mean as

\bar{x} = \frac{x+24}{4}

and setting the sum of squared deviations divided by 4 equal to 5, we arrive at the quadratic equation

x^2 - 16x + 48 = 0

. Solving this equation gives two possible values,

x = 4

and

x = 12

. Applying the restriction

x < 6

uniquely identifies

x = 4

as the correct solution.

Step-by-Step Solution

1

Calculate the mean of the dataset in terms of

x

\bar{x} = \frac{x + 24}{4}

The mean of a dataset is the sum of all values divided by the total number of items.

2

Write the variance equation using the formula

\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N}

\frac{(x - \bar{x})^2 + (6 - \bar{x})^2 + (8 - \bar{x})^2 + (10 - \bar{x})^2}{4} = 5

The given variance is 5 and the number of observations

N = 4

.

3

Substitute

\bar{x} = \frac{x + 24}{4}

and expand the squared deviations

\frac{3}{16}x^2 - 3x + 14 = 5 \implies x^2 - 16x + 48 = 0

Expanding the squared terms and multiplying through by

\frac{16}{3}

yields a standard quadratic equation.

4

Solve the quadratic equation

x^2 - 16x + 48 = 0

for

x

(x - 4)(x - 12) = 0 \implies x = 4 \text{ or } x = 12

Factoring the quadratic equation gives two possible roots.

5

Apply the given condition

x < 6

x = 4

Since

x < 6

, we reject

x = 12

and select

x = 4

.

Key Concept

Variance of Ungrouped Data and Algebraic Problem Solving

Question 638Question →

The frequency distribution table below shows the mass, in grams, of $50$ industrial steel bearings measured during a precision manufacturing audit:

Mass (g)	Frequency ( $f$ )
$10 - 19$	$5$
$20 - 29$	$12$
$30 - 39$	$x$
$40 - 49$	$y$
$50 - 59$	$8$

If the mean mass of the bearings is $34.7\text{ g}$ , what is the value of the missing frequency $x$ ?

Show answer & explanation

Answer: 18

Answer

18

The value of

x

is 18 because setting up the total frequency sum gives

x + y = 25

, and using class midpoints to compute the mean yields

1915 - 10x = 1735

, which solves to

x = 18

.

Step-by-Step Solution

1

Express the relationship between the missing frequencies using total frequency.

x + y = 25

or

y = 25 - x

The total number of industrial steel bearings is 50, so

5 + 12 + x + y + 8 = 50

.

2

Determine the midpoint (

m

) of each class interval.

Midpoints are 14.5, 24.5, 34.5, 44.5, and 54.5 respectively.

The class midpoint is calculated as

\frac{\text{lower limit} + \text{upper limit}}{2}

.

3

Formulate the equation for the sum of products of frequencies and midpoints.

\sum fm = 802.5 + 34.5x + 44.5y

Multiply each class midpoint by its corresponding frequency and sum the results.

4

Substitute

y = 25 - x

and solve for

x

using the mean formula.

x = 18

Setting

\frac{1915 - 10x}{50} = 34.7

yields

1915 - 10x = 1735

, which gives

10x = 180

and thus

x = 18

.

Key Concept

Measures of Central Tendency for Grouped Data - Mean with Unknown Frequencies

Estimated Time:3m 0s

Question 639Question →

A composite plane figure is formed from a rectangle $ABCD$ measuring $20\text{ cm}$ by $14\text{ cm}$ . A semicircle with diameter $BC = 14\text{ cm}$ is attached externally along side $BC$ . At the opposite end, an isosceles triangle $AED$ with base $AD = 14\text{ cm}$ and perpendicular height $6\text{ cm}$ is cut out from the interior of the rectangle. Taking $\pi = \frac{22}{7}$ , what is the total area of the figure in $\text{cm}^2$ ?

Show answer & explanation

Answer: 315

Answer

The total area of the composite figure is 315 cm².

The area of the complete shape is found by adding the area of the external semicircle (

77\text{ cm}^2

) to the area of the rectangle (

280\text{ cm}^2

) and subtracting the area of the removed triangle (

42\text{ cm}^2

), yielding a net area of

315\text{ cm}^2

.

Step-by-Step Solution

1

Calculate the area of the rectangular portion

ABCD

Area of rectangle

= 20\text{ cm} \times 14\text{ cm} = 280\text{ cm}^2

The main section of the figure before modifications is a rectangle with length

20\text{ cm}

and width

14\text{ cm}

.

2

Calculate the area of the externally attached semicircle

Area of semicircle

= \frac{1}{2} \times \frac{22}{7} \times 7^2 = 77\text{ cm}^2

The diameter of the semicircle is

BC = 14\text{ cm}

, giving a radius

r = 7\text{ cm}

. Area is given by

\frac{1}{2}\pi r^2

.

3

Calculate the area of the removed internal triangle

AED

Area of triangle

= \frac{1}{2} \times 14\text{ cm} \times 6\text{ cm} = 42\text{ cm}^2

The triangle has base

AD = 14\text{ cm}

and height

6\text{ cm}

. Area is calculated as

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

.

4

Combine the areas to find the total remaining area of the figure

Total Area

= 280\text{ cm}^2 + 77\text{ cm}^2 - 42\text{ cm}^2 = 315\text{ cm}^2

The external semicircle increases the total area while the internal cut-out triangle reduces it.

Key Concept

Area of Composite Plane Figures

Estimated Time:2m 0s

Question 640Question →

Two parallel lines, $L_1$ and $L_2$ , are situated in a plane with $L_1$ above $L_2$ . A regular polygon of $n$ sides has one of its sides, $AB$ , lying entirely on line $L_1$ . An adjacent side, $BC$ , extends downwards into the region between $L_1$ and $L_2$ . A line segment $BD$ is drawn from vertex $B$ perpendicular to line $L_2$ , meeting $L_2$ at point $D$ . The segment $BD$ lies inside the interior angle $\angle ABC$ of the polygon and divides $\angle ABC$ into two angles, $\angle ABD$ and $\angle DBC$ , such that $\angle ABD : \angle DBC = 3 : 2$ . What is the total number of diagonals of this regular polygon?

Show answer & explanation

Answer: 54

Answer

54

Because line L_1 is parallel to line L_2 and BD is perpendicular to L_2, BD is also perpendicular to L_1, making angle ABD equal to 90°. Based on the 3:2 ratio, angle DBC equals 60°, giving a total interior angle of 150°. Solving (n - 2) × 180° / n = 150° yields n = 12 sides. Substituting n = 12 into the diagonal formula n(n - 3) / 2 gives 54 diagonals.

Step-by-Step Solution

1

Determine the measure of angle ABD using parallel line properties

\angle ABD = 90^\circ

Because line L_1 is parallel to line L_2 and line segment BD is perpendicular to L_2, BD is also perpendicular to L_1.

2

Calculate the measure of angle DBC from the given angle ratio

\angle DBC = 60^\circ

With \angle ABD : \angle DBC = 3 : 2 and \angle ABD = 90^\circ, setting 3 ratio units equal to 90^\circ gives 1 unit = 30^\circ, so 2 units = 60^\circ.

3

Find the total interior angle measure of the regular polygon

\angle ABC = 150^\circ

The interior angle \angle ABC is the sum of \angle ABD and \angle DBC, which gives 90^\circ + 60^\circ = 150^\circ.

4

Determine the number of sides n of the regular polygon

n = 12

Using the regular polygon interior angle formula \frac{(n - 2) \times 180^\circ}{n} = 150^\circ, solving for n yields 180n - 360 = 150n \implies 30n = 360 \implies n = 12.

5

Calculate the total number of diagonals

54

Applying the polygon diagonal formula D = \frac{n(n - 3)}{2} with n = 12 gives D = \frac{12 \times (12 - 3)}{2} = \frac{12 \times 9}{2} = 54.

Key Concept

Combining parallel line perpendicularity properties, angle partitioning, regular polygon interior angle measure, and diagonal count calculation

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