All practice questions

Question 7281Question →

An environmental research station recorded the daily particulate matter concentration (in $\mu\text{g/m}^3$ ) near an urban center over a period of $50$ days. The observations are summarized in the table below:

Particulate Matter ( $\mu\text{g/m}^3$ )	Number of Days ( $f$ )
$20 - 29$	$6$
$30 - 39$	$10$
$40 - 49$	$15$
$50 - 59$	$11$
$60 - 69$	$8$

Find the estimated mean particulate matter concentration, in $\mu\text{g/m}^3$ , for the 50-day period.

Show answer & explanation

Answer: 45.5

Answer

The estimated mean particulate matter concentration is

45.5\text{ }\mu\text{g/m}^3

.

To calculate the mean of grouped data, determine the midpoint (

x

) of each class interval:

24.5

,

34.5

,

44.5

,

54.5

, and

64.5

. Multiply each midpoint by its frequency (

f

) to get the products

147

,

345

,

667.5

,

599.5

, and

516

. The sum of these products (

\sum fx

) is

2275

. Dividing

\sum fx

by the total frequency (

\sum f = 50

) yields the estimated mean of

45.5\text{ }\mu\text{g/m}^3

.

Step-by-Step Solution

1

Calculate the class midpoints (

x

) for each class interval

Midpoints are

24.5

,

34.5

,

44.5

,

54.5

, and

64.5

.

Grouped data uses the midpoint of each interval to represent all values falling within that interval.

2

Compute the product of frequency and midpoint (

fx

) for each class

6 \times 24.5 = 147

,

10 \times 34.5 = 345

,

15 \times 44.5 = 667.5

,

11 \times 54.5 = 599.5

,

8 \times 64.5 = 516

.

This determines the estimated sum of data values within each group.

3

Sum all products

\sum fx

and total frequency

\sum f

\sum fx = 147 + 345 + 667.5 + 599.5 + 516 = 2275

and

\sum f = 50

.

These totals are required for the mean formula.

4

Calculate the mean concentration using

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

\bar{x} = \frac{2275}{50} = 45.5

.

Dividing the total estimated sum by the total number of days gives the estimated mean.

Key Concept

Calculation of Mean for Grouped Frequency Data

Question 7282Question →

Given that

x

and

y

are real numbers satisfying the simultaneous exponential equations

3^x \times 9^y = 81

and

8^x \times 4^y = 256

find the value of

x^2 + y^2

.

Show answer & explanation

Answer: 5

Answer

The value of

x^2 + y^2

is 5.

Converting all terms to their prime bases (

3

for the first equation and

2

for the second equation) produces the simultaneous linear equations

x + 2y = 4

and

3x + 2y = 8

. Solving these gives

x = 2

and

y = 1

. Substituting these values into

x^2 + y^2

gives

2^2 + 1^2 = 5

.

Step-by-Step Solution

1

Convert all terms in the first equation to powers of base 3.

x + 2y = 4

Since

9 = 3^2

and

81 = 3^4

, applying the product law of indices

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

gives

3^{x+2y} = 3^4

. Equating the exponents gives

x + 2y = 4

.

2

Convert all terms in the second equation to powers of base 2.

3x + 2y = 8

Since

8 = 2^3

,

4 = 2^2

, and

256 = 2^8

, applying the laws of indices yields

2^{3x} \times 2^{2y} = 2^8 \implies 2^{3x+2y} = 2^8

. Equating exponents gives

3x + 2y = 8

.

3

Solve the system of simultaneous linear equations for

x

and

y

.

x = 2

and

y = 1

Subtracting

x + 2y = 4

from

3x + 2y = 8

yields

2x = 4 \implies x = 2

. Substituting

x = 2

into

x + 2y = 4

gives

2 + 2y = 4 \implies y = 1

.

4

Evaluate the target expression

x^2 + y^2

.

5

Substitute

x = 2

and

y = 1

into

x^2 + y^2

to obtain

2^2 + 1^2 = 4 + 1 = 5

.

Key Concept

Converting exponential terms to common prime bases to reduce exponential equations into linear equations.

Question 7283Question →

A student measured the width of a textbook as $8.2\text{ cm}$ instead of its actual width of $8.0\text{ cm}$ . What is the percentage error in the measurement?

0.2\%

2.44\%

2.5\%

20\%

Show answer & explanation

Answer: $2.5\%$

Answer

2.5\%

The correct answer is

2.5\%

. The absolute error is

0.2\text{ cm}

. Dividing the absolute error (

0.2\text{ cm}

) by the actual measurement (

8.0\text{ cm}

) yields

\frac{0.2}{8.0} = 0.025

. Expressing this decimal as a percentage gives

0.025 \times 100\% = 2.5\%

.

Step-by-Step Solution

1

Find the absolute error in measurement

\text{Error} = |8.2\text{ cm} - 8.0\text{ cm}| = 0.2\text{ cm}

Percentage error requires knowing the difference between measured value and true value.

2

Apply the percentage error formula

\text{Percentage Error} = \frac{\text{Error}}{\text{True Value}} \times 100\%

Percentage error is always calculated relative to the true (actual) value.

3

Substitute values and evaluate

\frac{0.2}{8.0} \times 100\% = \frac{1}{40} \times 100\% = 2.5\%

Simplifying

\frac{0.2}{8.0}

yields

\frac{1}{40}

, which equals

2.5\%

.

Key Concept

Percentage Error Calculation

Estimated Time:45s

Question 7284Question →

The measures of the exterior angles of an convex hexagon are given as $x^\circ$ , $(x + 10)^\circ$ , $(2x - 5)^\circ$ , $(x + 25)^\circ$ , $(2x + 15)^\circ$ , and $(x - 5)^\circ$ . What is the measure of the largest interior angle of the hexagon?

95^\circ

85^\circ

145^\circ

135^\circ

Show answer & explanation

Answer: $145^\circ$

Answer

145^\circ

The sum of all exterior angles of a convex polygon is

360^\circ

. Summing the given expressions yields

8x + 40 = 360^\circ

, which gives

x = 40^\circ

. Evaluating each exterior angle shows that the smallest exterior angle is

(40 - 5)^\circ = 35^\circ

. Since interior and exterior angles on a straight line are supplementary, the largest interior angle is

180^\circ - 35^\circ = 145^\circ

.

Step-by-Step Solution

1

Set up equation using the sum of exterior angles

x + (x + 10) + (2x - 5) + (x + 25) + (2x + 15) + (x - 5) = 360^\circ

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

360^\circ

.

2

Simplify and solve for

x

8x + 40 = 360 \implies 8x = 320 \implies x = 40^\circ

Combining like terms gives

8x + 40 = 360

.

3

Find the smallest exterior angle

x - 5 = 40 - 5 = 35^\circ

The largest interior angle corresponds to the smallest exterior angle because an interior angle and its adjacent exterior angle add up to

180^\circ

.

4

Calculate the largest interior angle

180^\circ - 35^\circ = 145^\circ

Subtracting the smallest exterior angle from

180^\circ

yields the largest interior angle.

Key Concept

Exterior and Interior Angle Relationship in Polygons

Estimated Time:1m 30s

Question 7285Question →

A function

f(x)

is defined by

f(x) = \begin{cases} \frac{1 - \cos(6x)}{x \tan(3x)}, & x \neq 0 \\ p + 4, & x = 0 \end{cases}

If

f(x)

is continuous at

x = 0

, what is the value of the constant

p

?

6

2

-1

-2

Show answer & explanation

Answer: 2

Answer

The value of the constant

p

is

2

.

For the piecewise function to be continuous at

x = 0

, the limit

\lim_{x \to 0} f(x)

must exist and equal

f(0) = p + 4

. By applying the double-angle identity

1 - \cos(6x) = 2\sin^2(3x)

and using the standard limit

\lim_{u \to 0} \frac{\sin u}{u} = 1

, the limit evaluates to

6

. Equating

p + 4 = 6

yields

p = 2

.

Step-by-Step Solution

1

State the continuity condition at

x = 0

\lim_{x \to 0} f(x) = f(0) = p + 4

For a function to be continuous at a point, its limit at that point must equal the defined function value.

2

Apply the trigonometric identity

1 - \cos(6x) = 2\sin^2(3x)

\frac{1 - \cos(6x)}{x \tan(3x)} = \frac{2\sin^2(3x)}{x \cdot \frac{\sin(3x)}{\cos(3x)}} = 2\cos(3x) \cdot \frac{\sin(3x)}{x}

Rewriting

\tan(3x)

as

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

allows cancellation of one

\sin(3x)

factor.

3

Evaluate the limit as

x \to 0

\lim_{x \to 0} \left[2\cos(3x) \cdot 3 \cdot \frac{\sin(3x)}{3x}\right] = 2(1)(3)(1) = 6

Using the standard trigonometric limit \lim_{u \to 0} \frac{\sin(u)}{u} = 1 and \cos(0) = 1.

4

Equate the limit to

f(0)

and solve for

p

p + 4 = 6 \implies p = 2

Subtracting 4 from both sides isolates the parameter

p

.

Key Concept

Continuity of a Piecewise Function using Trigonometric Limits

Estimated Time:2m 0s

Question 7286Question →

What is the value of $(-17 \times 5) \pmod{6}$ expressed in standard non-negative remainder form?

-1

1

3

5

Show answer & explanation

Answer: $5$

Answer

5

Multiplying

-17

by

5

yields

-85

. When

-85

is divided by

6

, the closest multiple of

6

below

-85

is

-90

. The non-negative remainder is

-85 - (-90) = 5

, which belongs to the canonical set

\{0, 1, 2, 3, 4, 5\}

.

Step-by-Step Solution

1

Multiply the numbers inside the parentheses

-17 \times 5 = -85

Perform standard multiplication before applying the modulo operation.

2

Express the negative number in terms of the modulus 6

-85 = 6 \times (-15) + 5

Find the largest multiple of 6 less than or equal to -85, which is -90.

3

Extract the canonical non-negative remainder

5

In standard modular arithmetic, remainders must lie in the range

[0, n-1]

, where

n

is the modulus.

Key Concept

Modular Arithmetic and Non-Negative Remainders

Question 7287Question →

Simplify the surd expression $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

\frac{7 + 2\sqrt{10}}{3}

\frac{\sqrt{7} + 2\sqrt{10}}{3}

\frac{7 + \sqrt{10}}{3}

\frac{7 + 2\sqrt{10}}{7}

Show answer & explanation

Answer: $\frac{7 + 2\sqrt{10}}{3}$

Answer

\frac{7 + 2\sqrt{10}}{3}

Multiplying both the numerator and denominator by the conjugate of the denominator,

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

, expands the numerator to

5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

and simplifies the denominator using the difference of squares to

5 - 2 = 3

, giving

\frac{7 + 2\sqrt{10}}{3}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

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

is

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

.

Rationalizing a binomial denominator requires multiplying by its conjugate to apply the difference of squares identity

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

.

2

Multiply both numerator and denominator by the conjugate

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

This maintains the value of the fraction while removing radical terms from the denominator.

3

Expand the numerator and denominator

Numerator:

(\sqrt{5})^2 + 2\sqrt{5}\sqrt{2} + (\sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

. Denominator:

(\sqrt{5})^2 - (\sqrt{2})^2 = 5 - 2 = 3

.

Apply algebraic expansion

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

and difference of squares.

4

Combine terms to form the final simplified expression

\frac{7 + 2\sqrt{10}}{3}

The expression is now fully rationalized and in standard simplified surd form.

Key Concept

Rationalization of Binomial Denominators

Question 7288Question →

If $y = \frac{(2x^2 + 1)^3}{4x - 1}$ , determine the numerical value of $\frac{dy}{dx}$ at $x = 1$ .

Show answer & explanation

Answer: 24

Answer

The numerical value of the derivative

\frac{dy}{dx}

at

x = 1

is 24.

Evaluating the derivative requires combined application of the quotient rule and chain rule. Differentiating

u(x) = (2x^2 + 1)^3

via the chain rule yields

u'(x) = 12x(2x^2 + 1)^2

. At

x = 1

,

u(1) = 27

,

u'(1) = 108

,

v(1) = 3

, and

v'(1) = 4

. Substituting these into the quotient rule

\frac{u'v - uv'}{v^2}

produces

\frac{324 - 108}{9} = 24

.

Step-by-Step Solution

1

Decompose the function for the Quotient Rule

Let

u(x) = (2x^2 + 1)^3

and

v(x) = 4x - 1

.

The given expression is a quotient of two differentiable functions.

2

Differentiate the numerator using the Chain Rule

u'(x) = 3(2x^2 + 1)^2 \cdot 4x = 12x(2x^2 + 1)^2

The chain rule states that

\frac{d}{dx}[g(x)]^n = n[g(x)]^{n-1} \cdot g'(x)

.

3

Differentiate the denominator and evaluate all components at

x = 1

v'(1) = 4

,

v(1) = 3

,

u(1) = 27

, and

u'(1) = 108

Evaluating each term individually reduces algebraic complexity prior to applying the quotient rule.

4

Apply the Quotient Rule formula and evaluate

\frac{dy}{dx}\Big|_{x=1} = \frac{(108)(3) - (27)(4)}{3^2} = \frac{324 - 108}{9} = 24

The quotient rule formula is

\frac{dy}{dx} = \frac{u'v - uv'}{v^2}

.

Key Concept

Combination of Quotient Rule and Chain Rule

Estimated Time:1m 30s

Question 7289Question →

Using differentiation from first principles, evaluate the value of the derivative of the function $f(x) = 2x^2 + 3x - 1$ at the point where $x = 1$ .

Show answer & explanation

Answer: 7

Answer

The value of the derivative at

x = 1

is 7.

Using first principles, the derivative is defined as

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

. Evaluating at

x = 1

, we find

f(1) = 4

and

f(1+h) = 4 + 7h + 2h^2

. Subtracting

f(1)

leaves

7h + 2h^2

, and dividing by

h

yields

7 + 2h

. Taking the limit as

h \to 0

yields the final result of 7.

Step-by-Step Solution

1

Calculate the value of the function at

x = 1

f(1) = 2(1)^2 + 3(1) - 1 = 4

This establishes the base value needed for the difference quotient.

2

Expand and simplify

f(1+h)

f(1+h) = 2(1+h)^2 + 3(1+h) - 1 = 4 + 7h + 2h^2

This gives the value of the function at the incremented point

x + h

.

3

Form and simplify the difference quotient

\frac{f(1+h) - f(1)}{h}

\frac{4 + 7h + 2h^2 - 4}{h} = 7 + 2h

Dividing by

h

eliminates the indeterminate form before taking the limit.

4

Evaluate the limit as

h \to 0

f'(1) = \lim_{h \to 0} (7 + 2h) = 7

Taking

h = 0

in the simplified quotient gives the exact rate of change at

x = 1

.

Key Concept

Differentiation from First Principles

Estimated Time:1m 0s

Question 7290Question →

Given that $\sin \theta + \cos \theta = \frac{\sqrt{6}}{2}$ where $\theta$ is an acute angle, what is the exact value of $\tan \theta + \cot \theta$ ?

4

2

\frac{2}{3}

\frac{1}{2}

Show answer & explanation

Answer: 4

Answer

The exact value of

\tan \theta + \cot \theta

is 4.

By squaring both sides of

\sin \theta + \cos \theta = \frac{\sqrt{6}}{2}

, we obtain

\sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta = \frac{6}{4} = \frac{3}{2}

. Applying the Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

simplifies this to

1 + 2\sin \theta \cos \theta = \frac{3}{2}

, which yields

\sin \theta \cos \theta = \frac{1}{4}

. Since

\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}

, the exact value is

\frac{1}{1/4} = 4

.

Step-by-Step Solution

1

Square both sides of the given equation

\sin \theta + \cos \theta = \frac{\sqrt{6}}{2}

.

(\sin \theta + \cos \theta)^2 = \left(\frac{\sqrt{6}}{2}\right)^2 \implies \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = \frac{6}{4} = \frac{3}{2}

.

Squaring allows the application of the fundamental Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

to isolate the product

\sin \theta \cos \theta

.

2

Substitute

\sin^2 \theta + \cos^2 \theta = 1

into the equation and solve for

\sin \theta \cos \theta

.

1 + 2\sin \theta \cos \theta = \frac{3}{2} \implies 2\sin \theta \cos \theta = \frac{1}{2} \implies \sin \theta \cos \theta = \frac{1}{4}

.

Simplifying the algebraic equation isolates the product term.

3

Express

\tan \theta + \cot \theta

in terms of

\sin \theta

and

\cos \theta

.

\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}

.

Using basic trigonometric identities simplifies the target sum into a reciprocal product.

4

Substitute

\sin \theta \cos \theta = \frac{1}{4}

into the simplified expression.

\tan \theta + \cot \theta = \frac{1}{1/4} = 4

.

Evaluating the reciprocal fraction gives the final numerical answer.

Key Concept

Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

and reciprocal relation

\tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta}

.

Question 7291Question →

A solid hemisphere of radius $6\text{ cm}$ has the same total surface area as a solid right circular cone with a base radius of $6\text{ cm}$ . What is the slant height of the cone?

6\text{ cm}

12\text{ cm}

18\text{ cm}

24\text{ cm}

Show answer & explanation

Answer: $12\text{ cm}$

Answer

The slant height of the cone is

12\text{ cm}

.

The correct answer is derived by setting the total surface area of the solid hemisphere (

3\pi R^2 = 108\pi\text{ cm}^2

) equal to the total surface area of the solid cone (

\pi r^2 + \pi r l = 36\pi + 6\pi l

). Solving

36\pi + 6\pi l = 108\pi

gives

l = 12\text{ cm}

.

Step-by-Step Solution

1

Calculate the total surface area of the solid hemisphere.

\text{TSA}_{\text{hemisphere}} = 3\pi R^2 = 3\pi(6)^2 = 108\pi\text{ cm}^2

A solid hemisphere consists of a curved surface (

2\pi R^2

) plus its flat circular base (

\pi R^2

).

2

Express the total surface area of the solid cone in terms of slant height

l

.

\text{TSA}_{\text{cone}} = \pi r^2 + \pi r l = \pi(6)^2 + \pi(6)l = 36\pi + 6\pi l

A solid cone has a base area of

\pi r^2

and a curved surface area of

\pi r l

.

3

Equate the two surface area expressions and solve for

l

.

36\pi + 6\pi l = 108\pi \implies 6\pi l = 72\pi \implies l = 12\text{ cm}

The question states that the total surface area of both solids is equal.

Key Concept

Total Surface Area of Composite 3D Solids (Hemisphere and Cone)

Question 7292Question →

The total cost $C$ per trip of operating a high-speed passenger ferry consists of a fixed administrative overhead cost and an operational cost that varies directly as the cube of its speed $v$ in knots. Given that the total cost per trip is $\$1,400$ when the ferry travels at $10\text{ knots}$ and $\$4,200$ when it travels at $20\text{ knots}$ , what is the total cost per trip when the ferry operates at a speed of $15\text{ knots}$ ?

\\

2,350$

\\

2,800$

\\

3,300$

\\

4,725$

Show answer & explanation

Answer: $\\$ 2,350$

Answer

\\

2,350$

The partial variation formula is

C = k_1 + k_2 v^3

. Substituting the given points yields

k_1 + 1000 k_2 = 1400

and

k_1 + 8000 k_2 = 4200

. Subtracting these equations gives

7000 k_2 = 2800

, so

k_2 = 0.4

and

k_1 = 1000

. Evaluating at

v = 15

gives

C = 1000 + 0.4(3375) = 1000 + 1350 = \\

2,350$.

Step-by-Step Solution

1

Set up the partial variation equation

C = k_1 + k_2 v^3

Partial variation combines a fixed constant

k_1

with a variable term

k_2 v^3

.

2

Form simultaneous linear equations using given conditions

Equation (1):

k_1 + 1000 k_2 = 1400

; Equation (2):

k_1 + 8000 k_2 = 4200

Substitute

v = 10

,

C = 1400

and

v = 20

,

C = 4200

into the variation model.

3

Solve for the constants

k_1

and

k_2

k_2 = 0.4

and

k_1 = 1000

Subtract Equation (1) from Equation (2):

7000 k_2 = 2800 \implies k_2 = 0.4

. Substitute

k_2 = 0.4

into Equation (1) to get

k_1 = 1400 - 400 = 1000

.

4

Calculate total cost

C

for speed

v = 15\text{ knots}

C = 1000 + 0.4(15)^3 = 1000 + 0.4(3375) = 1000 + 1350 = 2350

Substitute

k_1 = 1000

,

k_2 = 0.4

, and

v = 15

back into the formula.

Key Concept

Partial Variation with non-linear powers solved via simultaneous linear equations

Estimated Time:2m 0s

Question 7293Question →

The grouped frequency table below shows the distribution of marks obtained by candidates in a Mathematics examination:

Class Interval	Frequency
10 – 19	15
20 – 29	25
30 – 39	$k$
40 – 49	20
50 �� 59	10

When this data is represented on a pie chart, the sector corresponding to the score range 30 – 39 has a central angle of $108^\circ$ . Based on a cumulative frequency curve (ogive) constructed for this distribution, what is the score corresponding to the 75th percentile ( $Q_3$ ) of the candidates?

42.0

42.5

41.5

44.5

Show answer & explanation

Answer: 42.0

Answer

42.0

The value 42.0 is obtained by first determining the missing frequency

k = 30

using the ratio

\frac{108^\circ}{360^\circ} = 0.3

, establishing

N = 100

. The 75th percentile position is at 75, which falls in the

40 – 49

class. Interpolating from the lower boundary of 39.5 yields

39.5 + \left(\frac{75 - 70}{20}\right) \times 10 = 42.0

.

Step-by-Step Solution

1

Determine the unknown frequency

k

using the pie chart sector angle.

\frac{k}{\text{Total Frequency}} = \frac{108^\circ}{360^\circ} = 0.3

Total frequency

N = 15 + 25 + k + 20 + 10 = 70 + k

.

\frac{k}{70 + k} = 0.3 \implies k = 21 + 0.3k \implies 0.7k = 21 \implies k = 30

The sector angle in a pie chart is proportional to the category frequency relative to the total frequency across

360^\circ

.

2

Calculate cumulative frequencies and locate the 75th percentile position.

Total frequency

N = 100

.
Cumulative frequencies (

cf

):
-

10 – 19

(boundary

9.5 – 19.5

):

cf = 15

-

20 – 29

(boundary

19.5 – 29.5

):

cf = 40

-

30 – 39

(boundary

29.5 – 39.5

):

cf = 70

-

40 – 49

(boundary

39.5 – 49.5

):

cf = 90

-

50 – 59

(boundary

49.5 – 59.5

):

cf = 100

75th percentile position

= 0.75 \times 100 = 75\text{th candidate}

.

The 75th percentile corresponds to the value below which 75% of the total observations lie.

3

Apply linear interpolation on the percentile class interval

40 – 49

.

The 75th score lies in class interval

40 – 49

(boundaries

39.5 – 49.5

).
- Lower class boundary

L = 39.5

- Cumulative frequency prior to class

cf_b = 70

- Frequency of percentile class

f = 20

- Class width

c = 10

Q_3 = L + \left(\frac{0.75N - cf_b}{f}\right) \times c = 39.5 + \left(\frac{75 - 70}{20}\right) \times 10 = 39.5 + 2.5 = 42.0

Linear interpolation along an ogive requires using exact class boundaries to compute specific percentile values.

Key Concept

Pie chart sector angles and linear interpolation on cumulative frequency distributions

Question 7294Question →

How many positive integer values of $x$ satisfy the quadratic inequality $x^2 - 4x - 5 < 0$ ?

Show answer & explanation

Answer: 4

Answer

There are 4 positive integer values of

x

that satisfy the inequality.

Factoring

x^2 - 4x - 5 < 0

yields

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

, giving the real solution interval

-1 < x < 5

. Filtering for positive integers (

x \in \mathbb{Z}^+

) yields the set

\{1, 2, 3, 4\}

, which contains exactly 4 values.

Step-by-Step Solution

1

Factor the quadratic inequality

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

Factoring helps find the boundary roots of the quadratic equation.

2

Determine the solution set interval

-1 < x < 5

For a quadratic inequality of the form

(x - a)(x - b) < 0

with

a < b

, the solution interval is

a < x < b

.

3

List the positive integer solutions in the range

-1 < x < 5

x \in \{1, 2, 3, 4\}

Positive integers are whole numbers strictly greater than 0.

4

Count the number of positive integers

4

Counting the elements in the set

\{1, 2, 3, 4\}

gives a total of 4.

Key Concept

Solving quadratic inequalities and finding valid integer solutions within a target domain.

Estimated Time:45s

Question 7295Question →

If $y = (3x - 1)^4$ , what is the numerical value of $\frac{dy}{dx}$ evaluated at $x = 1$ ?

Show answer & explanation

Answer: 96

Answer

96

Applying the chain rule

\frac{d}{dx}[u^n] = n u^{n-1} \cdot \frac{du}{dx}

to

y = (3x - 1)^4

gives

\frac{dy}{dx} = 4(3x - 1)^3 \cdot 3 = 12(3x - 1)^3

. Evaluating this expression at

x = 1

yields

12(3(1) - 1)^3 = 12(2)^3 = 96

.

Step-by-Step Solution

1

Differentiate

y = (3x - 1)^4

with respect to

x

using the chain rule.

\frac{dy}{dx} = 12(3x - 1)^3

Differentiate the outer power function to get

4(3x - 1)^3

and multiply by the derivative of the inner linear function

3x - 1

, which is 3.

2

Substitute

x = 1

into the derivative function.

96

Evaluating

12(3(1) - 1)^3

gives

12(2)^3 = 12 \times 8 = 96

.

Key Concept

Chain Rule of Differentiation

Question 7296Question →

Given that $\frac{\sqrt{x + 2} + \sqrt{x - 2}}{\sqrt{x + 2} - \sqrt{x - 2}} = 3$ , what is the value of $x$ ?

\frac{10}{3}

6

\frac{5}{2}

\frac{8}{3}

Show answer & explanation

Answer: $\frac{10}{3}$

Answer

The value of

x

is

\frac{10}{3}

.

The correct solution is found by cross-multiplying the equation to get

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

. Grouping similar surd terms gives

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

, which simplifies to

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

. Squaring both sides yields

4(x - 2) = x + 2

, simplifying to

4x - 8 = x + 2

, which gives

3x = 10

and

x = \frac{10}{3}

.

Step-by-Step Solution

1

Cross-multiply to clear the denominator

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

Clear the fraction to group like radical terms on opposite sides.

2

Rearrange and combine like terms

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

, which simplifies to

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

Isolate the radical expressions.

3

Square both sides of the equation

(2\sqrt{x - 2})^2 = (\sqrt{x + 2})^2 \implies 4(x - 2) = x + 2

Eliminate radicals by squaring both sides, ensuring the coefficient

2

is squared to

4

.

4

Solve the linear equation for

x

4x - 8 = x + 2 \implies 3x = 10 \implies x = \frac{10}{3}

Isolate

x

to find the final value.

Key Concept

Solving Radical and Surd Equations

Question 7297Question →

If $27^{x - 1} = 9^{x + 1}$ , determine the value of $x$ .

Show answer & explanation

Answer: 5

Answer

The value of

x

is 5.

Rewriting 27 as

3^3

and 9 as

3^2

transforms the given equation into

3^{3(x - 1)} = 3^{2(x + 1)}

. Equating exponents gives

3x - 3 = 2x + 2

, which simplifies directly to

x = 5

.

Step-by-Step Solution

1

Express numbers in terms of a common base

(3^3)^{x - 1} = (3^2)^{x + 1}

Both 27 and 9 are powers of 3, allowing reduction to a single base.

2

Apply power of a power index law

3^{3x - 3} = 3^{2x + 2}

Multiply the base power by the expression in the exponent:

3 \times (x - 1) = 3x - 3

and

2 \times (x + 1) = 2x + 2

.

3

Equate exponents and solve for

x

3x - 3 = 2x + 2 \implies x = 5

Equal bases imply that the index powers must be equal.

Key Concept

Equating exponential expressions using a common base

Question 7298Question →

When differentiating the function $f(x) = x^2 + 4x$ from first principles, what is the simplified expression for the difference quotient $\frac{f(x+h) - f(x)}{h}$ before taking the limit as $h \to 0$ ?

Show answer & explanation

Answer: 2x + h + 4; 2x + 4 + h; h + 2x + 4

Answer

2x + h + 4

Expanding

f(x+h) = (x+h)^2 + 4(x+h) = x^2 + 2xh + h^2 + 4x + 4h

. Subtracting

f(x) = x^2 + 4x

yields

2xh + h^2 + 4h

. Dividing every term by

h

gives the simplified difference quotient

2x + h + 4

.

Step-by-Step Solution

1

Evaluate

f(x+h)

for the function

f(x) = x^2 + 4x

f(x+h) = (x+h)^2 + 4(x+h) = x^2 + 2xh + h^2 + 4x + 4h

Substitute

(x+h)

into every instance of

x

in the function and expand the terms.

2

Subtract

f(x)

from

f(x+h)

to find the numerator of the difference quotient

f(x+h) - f(x) = (x^2 + 2xh + h^2 + 4x + 4h) - (x^2 + 4x) = 2xh + h^2 + 4h

Subtracting original function terms cancels out

x^2

and

4x

.

3

Divide the result by

h

\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 4h}{h} = 2x + h + 4

Factor out

h

from the numerator and cancel with the denominator

h

.

Key Concept

Difference Quotient Simplification in First Principles Differentiation

Question 7299Question →

Find the $x$ -intercept of the normal line to the curve $y = x^3 - 3x^2 + 4x - 1$ at the point where $x = 1$ .

Show answer & explanation

Answer: 2

Answer

The

x

-intercept of the normal line is

2

.

At

x = 1

, the point on the curve

y = x^3 - 3x^2 + 4x - 1

is

(1, 1)

. The derivative is

\frac{dy}{dx} = 3x^2 - 6x + 4

, which equals

1

at

x = 1

. The normal gradient is therefore

-1

. The normal line equation is

y - 1 = -1(x - 1)

, which simplifies to

y = -x + 2

. Setting

y = 0

gives

x = 2

.

Step-by-Step Solution

1

Find the y-coordinate at x = 1

y = 1, so the point on the curve is (1, 1)

The point of contact is needed to construct the equation of the normal line.

2

Differentiate the function with respect to x

dy/dx = 3x^2 - 6x + 4

The derivative gives the expression for the gradient of the tangent to the curve.

3

Find the gradient of the tangent and normal at x = 1

Gradient of tangent m_t = 1; gradient of normal m_n = -1

The normal line is perpendicular to the tangent line, so m_n = -1 / m_t.

4

Determine the equation of the normal line

y - 1 = -1(x - 1) => y = -x + 2

Applying the straight-line equation y - y_1 = m(x - x_1).

5

Find the x-intercept of the normal line

x = 2

Setting y = 0 in the normal line equation yields the x-intercept.

Key Concept

Equation and axis intercepts of a normal line to a curve

Question 7300Question →

When differentiating the quadratic function $f(x) = 5x^2 - 2x$ from first principles, what is the fully simplified form of the difference quotient $\frac{f(x+h) - f(x)}{h}$ before taking the limit as $h \to 0$ ?

Show answer & explanation

Answer: 10x + 5h - 2; 10x - 2 + 5h; 5h + 10x - 2; 10x+5h-2

Answer

10x + 5h - 2

Expanding

f(x+h) = 5(x+h)^2 - 2(x+h)

yields

5x^2 + 10xh + 5h^2 - 2x - 2h

. Subtracting

f(x) = 5x^2 - 2x

leaves

10xh + 5h^2 - 2h

. Factoring out

h

and dividing by

h

gives the simplified difference quotient

10x + 5h - 2

.

Step-by-Step Solution

1

Evaluate

f(x+h)

by expanding

5(x+h)^2 - 2(x+h)

f(x+h) = 5(x^2 + 2xh + h^2) - 2x - 2h = 5x^2 + 10xh + 5h^2 - 2x - 2h

Substitute

(x+h)

into the function definition.

2

Subtract

f(x)

from

f(x+h)

f(x+h) - f(x) = (5x^2 + 10xh + 5h^2 - 2x - 2h) - (5x^2 - 2x) = 10xh + 5h^2 - 2h

Determine the numerator of the difference quotient by cancelling common terms.

3

Divide the numerator by

h

\frac{f(x+h) - f(x)}{h} = \frac{10xh + 5h^2 - 2h}{h} = 10x + 5h - 2

Simplify the fraction by dividing each term by

h

.

Key Concept

Difference quotient in differentiation from first principles

Answer

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