All practice questions

Question 521Question →

For the endothermic gaseous reaction $\text{PCl}_5(g) \rightarrow \text{PCl}_3(g) + \text{Cl}_2(g)$ , the standard enthalpy change ( $\Delta H$ ) is $+92\text{ kJ mol}^{-1}$ . The activation energy for the reverse reaction without a catalyst is $58\text{ kJ mol}^{-1}$ . If a catalyst is introduced that lowers the activation energy of the forward reaction by $35\text{ kJ mol}^{-1}$ , what is the activation energy for the catalyzed forward reaction in $\text{kJ mol}^{-1}$ ?

Show answer & explanation

Answer: 115

Answer

The activation energy for the catalyzed forward reaction is

115\text{ kJ mol}^{-1}

.

For an endothermic reaction, the enthalpy change is the difference between the forward and reverse activation energies:

\Delta H = E_{a,\text{forward}} - E_{a,\text{reverse}}

. Rearranging gives the uncatalyzed forward activation energy as

92\text{ kJ mol}^{-1} + 58\text{ kJ mol}^{-1} = 150\text{ kJ mol}^{-1}

. A catalyst lowers the activation energy by

35\text{ kJ mol}^{-1}

, resulting in a catalyzed forward activation energy of

150\text{ kJ mol}^{-1} - 35\text{ kJ mol}^{-1} = 115\text{ kJ mol}^{-1}

.

Step-by-Step Solution

1

Determine the activation energy of the uncatalyzed forward reaction

E_{a,\text{forward, uncatalyzed}} = 150\text{ kJ mol}^{-1}

For an endothermic reaction,

\Delta H = E_{a,\text{forward}} - E_{a,\text{reverse}}

. Substituting

\Delta H = +92\text{ kJ mol}^{-1}

and

E_{a,\text{reverse}} = 58\text{ kJ mol}^{-1}

gives

E_{a,\text{forward, uncatalyzed}} = 92 + 58 = 150\text{ kJ mol}^{-1}

.

2

Apply the effect of the catalyst on the forward activation energy

E_{a,\text{forward, catalyzed}} = 115\text{ kJ mol}^{-1}

The catalyst lowers the energy barrier for the forward reaction by

35\text{ kJ mol}^{-1}

, so

E_{a,\text{forward, catalyzed}} = 150 - 35 = 115\text{ kJ mol}^{-1}

.

Key Concept

Calculation of activation energy for catalyzed and uncatalyzed reactions using enthalpy change and reverse activation energy on an energy profile.

Question 522Question →

An isosceles trapezium has parallel sides of lengths $14\text{ cm}$ and $8\text{ cm}$ . If each of the non-parallel sides has a length of $5\text{ cm}$ , calculate the area of the trapezium in $\text{cm}^2$ .

Show answer & explanation

Answer: 44

Answer

The area of the trapezium is

44\text{ cm}^2

.

Projecting the top base of length

8\text{ cm}

onto the bottom base of length

14\text{ cm}

leaves a difference of

6\text{ cm}

, which is divided equally into two

3\text{ cm}

segments on either side. Using the Pythagorean theorem with the non-parallel side (

5\text{ cm}

) and the base segment (

3\text{ cm}

) gives a height of

4\text{ cm}

. The area is then calculated as

\frac{1}{2} \times (14 + 8) \times 4 = 44\text{ cm}^2

.

Step-by-Step Solution

1

Determine the projection segment length on the longer base

x = \frac{14 - 8}{2} = 3\text{ cm}

Since the trapezium is isosceles, dropping perpendiculars from both ends of the top base creates two identical right-angled triangles at the sides.

2

Calculate the perpendicular height using the Pythagorean theorem

h = \sqrt{5^2 - 3^2} = \sqrt{16} = 4\text{ cm}

The slant side (

5\text{ cm}

), the height (

h

), and the projection segment (

3\text{ cm}

) form a right-angled triangle.

3

Calculate the area of the trapezium

\text{Area} = \frac{1}{2}(14 + 8) \times 4 = 44\text{ cm}^2

The area of a trapezium is given by half the sum of its parallel sides multiplied by its perpendicular height.

Key Concept

Perimeter and Area of Plane Shapes - Area of Isosceles Trapezium

Question 523Question →

Find the value of

x

that satisfies the exponential equation

\frac{4^{x+1} + 4^x + 4^{x-1}}{2^{x+2} + 2^{x+1} + 2^x} = 24

Show answer & explanation

Answer: 5

Answer

5

Factoring

4^{x-1}

from the numerator yields

4^{x-1}(16 + 4 + 1) = 21 \cdot 4^{x-1}

. Factoring

2^x

from the denominator yields

2^x(4 + 2 + 1) = 7 \cdot 2^x

. Dividing the numerical coefficients gives

\frac{21}{7} = 3

. Substituting

4^{x-1} = 2^{2x-2}

into the ratio gives

3 \cdot \frac{2^{2x-2}}{2^x} = 3 \cdot 2^{x-2}

. Setting

3 \cdot 2^{x-2} = 24

leads to

2^{x-2} = 8 = 2^3

, which gives

x - 2 = 3

and therefore

x = 5

.

Step-by-Step Solution

1

Factor out common terms from the numerator and denominator

Numerator:

4^{x-1}(4^2 + 4^1 + 1) = 21 \cdot 4^{x-1}

. Denominator:

2^x(2^2 + 2^1 + 1) = 7 \cdot 2^x

.

Grouping power terms simplifies sums of exponential expressions.

2

Divide the numerator by the denominator and convert bases

\frac{21 \cdot 4^{x-1}}{7 \cdot 2^x} = 3 \cdot \frac{(2^2)^{x-1}}{2^x} = 3 \cdot \frac{2^{2x-2}}{2^x}

Simplifying

\frac{21}{7} = 3

and expressing base 4 in base 2 allows applying laws of indices.

3

Apply the quotient rule of indices:

\frac{a^m}{a^n} = a^{m-n}

3 \cdot 2^{(2x-2) - x} = 3 \cdot 2^{x-2}

Subtracting exponents of like bases simplifies the fractional index expression.

4

Set the simplified expression equal to 24 and solve for

x

3 \cdot 2^{x-2} = 24 \implies 2^{x-2} = 8 = 2^3 \implies x - 2 = 3 \implies x = 5

Equating the exponents when bases are identical yields the linear equation

x - 2 = 3

.

Key Concept

Factoring sums of exponential terms and applying the quotient rule of indices

Question 524Question →

A charged oil droplet of mass $3.2 \times 10^{-15}\text{ kg}$ remains stationary in a vacuum between two horizontal charged plates where there is a uniform vertical electric field of strength $2.0 \times 10^4\text{ N C}^{-1}$ . Taking $g = 10\text{ m s}^{-2}$ and the elementary charge $e = 1.6 \times 10^{-19}\text{ C}$ , determine the number of excess electrons on the droplet.

Show answer & explanation

Answer: 10

Answer

The number of excess electrons on the droplet is 10.

The droplet is in mechanical equilibrium under two equal and opposite forces: the downward gravitational force

W = mg

and the upward electric force

F_e = qE

. Setting

qE = mg

gives

q = \frac{mg}{E} = 1.6 \times 10^{-18}\text{ C}

. By the quantization of charge (

q = Ne

), dividing this charge by the elementary charge

e = 1.6 \times 10^{-19}\text{ C}

gives exactly 10 excess electrons.

Step-by-Step Solution

1

Calculate the gravitational force (weight) acting on the droplet.

W = mg = (3.2 \times 10^{-15}\text{ kg}) \times (10\text{ m s}^{-2}) = 3.2 \times 10^{-14}\text{ N}

.

For stationary equilibrium, weight provides the downward vertical force.

2

Apply the equilibrium condition to find the electric force.

F_e = W = 3.2 \times 10^{-14}\text{ N}

.

The net vertical force must be zero for the droplet to remain suspended.

3

Determine the charge

q

using

F_e = qE

.

q = \frac{F_e}{E} = \frac{3.2 \times 10^{-14}\text{ N}}{2.0 \times 10^4\text{ N C}^{-1}} = 1.6 \times 10^{-18}\text{ C}

.

Electric field strength relates force and charge.

4

Calculate the number of elementary charges using charge quantization

q = Ne

.

N = \frac{q}{e} = \frac{1.6 \times 10^{-18}\text{ C}}{1.6 \times 10^{-19}\text{ C}} = 10

.

Electric charge exists in discrete integer multiples of the elementary charge

e

.

Key Concept

Equilibrium between electrostatic and gravitational forces combined with charge quantization.

Question 525Question →

A curve has the equation $y = \frac{1}{3}x^3 - 2x^2 + 3x + 1$ . What is the positive $x$ -coordinate of the point on the curve where the tangent line is parallel to the line $y = 8x - 5$ ?

Show answer & explanation

Answer: 5

Answer

The positive

x

-coordinate of the point where the tangent is parallel to the line is

5

.

The gradient of the line

y = 8x - 5

is

8

. Differentiating

y = \frac{1}{3}x^3 - 2x^2 + 3x + 1

gives

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

. Setting

\frac{dy}{dx} = 8

leads to

x^2 - 4x - 5 = 0

, which factors as

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

. The solutions are

x = 5

and

x = -1

. Selecting the positive value gives

x = 5

.

Step-by-Step Solution

1

Find the gradient (slope) of the given straight line.

The line equation is in slope-intercept form

y = mx + c

, where the slope

m = 8

.

Parallel lines have equal slopes, so the gradient of the tangent to the curve must equal 8.

2

Differentiate the curve equation to find the gradient function

\frac{dy}{dx}

.

\frac{dy}{dx} = \frac{d}{dx}\left(\frac{1}{3}x^3 - 2x^2 + 3x + 1\right) = x^2 - 4x + 3

.

The first derivative of a curve represents the gradient of the tangent at any point

x

.

3

Equate the derivative to the slope of the line and solve for

x

.

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

. The roots are

x = 5

and

x = -1

.

Solving the quadratic equation gives all

x

-values where the tangent line has a slope of 8.

4

Select the positive

x

-coordinate as requested.

x = 5

.

The question specifically asks for the positive value among the solutions.

Key Concept

Finding points on a curve where the tangent is parallel to a given line

Question 526Question →

A fruit vendor has $7$ distinct types of fresh fruits on display. A customer wants to buy a gift basket containing exactly $4$ different types of fruits. How many different combinations of fruits can the customer choose?

Show answer & explanation

Answer: 35

Answer

The customer can choose the fruits in 35 different ways.

The number of ways to choose

r = 4

items from

n = 7

distinct items without regard to order is given by

^7C_4 = \frac{7!}{4!3!} = \frac{210}{6} = 35

.

Step-by-Step Solution

1

Identify total elements (

n

) and selected subset size (

r

).

n = 7

and

r = 4

.

Since the selection order inside the fruit basket does not matter, combinations (

nCr

) must be used.

2

Substitute values into the combination formula

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

.

^7C_4 = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!}

.

This evaluates the total ways to choose 4 items from 7 without repetition or ordering.

3

Simplify the factorials and compute the numerical result.

\frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

.

Canceling out

4!

from numerator and denominator gives

(7 \times 6 \times 5) / 6 = 35

.

Key Concept

Combinations (

nCr

)

Estimated Time:45s

Question 527Question →

An aqueous solution of hydrocyanic acid ( $\text{HCN}$ ), a weak monobasic acid, has a concentration of $0.40\text{ mol dm}^{-3}$ at $25^\circ\text{C}$ . Given that the acid dissociation constant ( $K_a$ ) for $\text{HCN}$ is $4.9 \times 10^{-10}\text{ mol dm}^{-3}$ , calculate the hydrogen ion concentration, $[\text{H}^+]$ , in $\text{mol dm}^{-3}$ . Express your answer as the coefficient $A$ in the form $A \times 10^{-5}\text{ mol dm}^{-3}$ .

Show answer & explanation

Answer: 1.4

Answer

The coefficient A is 1.4, which corresponds to a hydrogen ion concentration of

1.4 \times 10^{-5}\text{ mol dm}^{-3}

.

For a weak monobasic acid, the hydrogen ion concentration is determined using the weak acid ionization relationship

[\text{H}^+] = \sqrt{K_a \cdot c}

. Substituting

K_a = 4.9 \times 10^{-10}\text{ mol dm}^{-3}

and

c = 0.40\text{ mol dm}^{-3}

yields

[\text{H}^+] = \sqrt{1.96 \times 10^{-10}} = 1.4 \times 10^{-5}\text{ mol dm}^{-3}

, giving a coefficient of 1.4.

Step-by-Step Solution

1

Write the ionization reaction and equilibrium constant expression

\text{HCN}_{(aq)} \rightleftharpoons \text{H}^+_{(aq)} + \text{CN}^-_{(aq)}

, giving

K_a = \frac{[\text{H}^+][\text{CN}^-]}{[\text{HCN}]}

Hydrocyanic acid is a weak monobasic acid that ionizes partially in water.

2

Apply weak acid approximations

Since

[\text{H}^+] = [\text{CN}^-]

and

K_a

is extremely small,

[\text{HCN}] \approx c = 0.40\text{ mol dm}^{-3}

. Thus,

K_a = \frac{[\text{H}^+]^2}{c}

.

The negligible ionization degree allows the equilibrium concentration of un-ionized acid to be approximated as its initial concentration.

3

Substitute values and solve for

[\text{H}^+]

[\text{H}^+] = \sqrt{K_a \times c} = \sqrt{4.9 \times 10^{-10} \times 0.40} = \sqrt{1.96 \times 10^{-10}} = 1.4 \times 10^{-5}\text{ mol dm}^{-3}

Multiplying

K_a

by the molar concentration gives the square of the hydrogen ion concentration.

4

Determine the coefficient A

A = 1.4

Matching

1.4 \times 10^{-5}\text{ mol dm}^{-3}

to the requested standard scientific notation form

A \times 10^{-5}\text{ mol dm}^{-3}

yields

A = 1.4

.

Key Concept

Weak Acid Ionization Equilibrium and Ka Calculations

Question 528Question →

The daily temperature readings, in degrees Celsius, recorded over four consecutive days are $10$ , $12$ , $14$ , and $16$ . What is the mean deviation of these temperatures?

Show answer & explanation

Answer: 2

Answer

The mean deviation of the temperatures is 2 °C.

To find the mean deviation, first calculate the mean of the data values: (10 + 12 + 14 + 16) / 4 = 13. Next, find the absolute difference of each value from the mean: |10 - 13| = 3, |12 - 13| = 1, |14 - 13| = 1, and |16 - 13| = 3. Finally, average these absolute differences: (3 + 1 + 1 + 3) / 4 = 8 / 4 = 2.

Step-by-Step Solution

1

Calculate the arithmetic mean (average) of the data set

$\bar{x} = \frac{10 + 12 + 14 + 16}{4} = 13$

Mean deviation measures dispersion relative to the mean, so the mean must be calculated first.

2

Compute the absolute deviation of each value from the mean

$|10 - 13| = 3$, $|12 - 13| = 1$, $|14 - 13| = 1$, $|16 - 13| = 3$

Mean deviation requires non-negative distances of each observation from the mean.

3

Sum the absolute deviations and divide by the sample size (N = 4)

$\text{Mean Deviation} = \frac{3 + 1 + 1 + 3}{4} = \frac{8}{4} = 2$

The mean deviation is the average of the absolute deviations.

Key Concept

Mean Deviation for Ungrouped Data

Question 529Question →

If $\log_3 x - 4\log_x 3 = 3$ for $x > 1$ , what is the value of $x$ ?

Show answer & explanation

Answer: 81

Answer

The value of

x

is 81.

Applying the change of base identity

\log_x 3 = \frac{1}{\log_3 x}

transforms the equation into

\log_3 x - \frac{4}{\log_3 x} = 3

. Substituting

u = \log_3 x

yields

u^2 - 3u - 4 = 0

, which factors as

(u - 4)(u + 1) = 0

. Given

x > 1

,

u

must be positive, giving

u = 4

. Converting back to exponential form gives

x = 3^4 = 81

.

Step-by-Step Solution

1

Apply the change of base identity

\log_a b = \frac{1}{\log_b a}

to the term

\log_x 3

.

The term becomes

\frac{1}{\log_3 x}

, so the equation is

\log_3 x - \frac{4}{\log_3 x} = 3

.

To express all logarithmic terms in terms of a single common base.

2

Introduce a substitution variable

u = \log_3 x

.

The equation reduces to

u - \frac{4}{u} = 3

.

Simplifies the equation to a manageable algebraic structure.

3

Multiply the entire equation by

u

and rearrange terms.

u^2 - 3u - 4 = 0

Converts the rational expression into standard quadratic form.

4

Factor the quadratic equation.

(u - 4)(u + 1) = 0 \implies u = 4 \text{ or } u = -1

Finds the candidate values for

u

.

5

Filter out invalid roots based on the domain restriction

x > 1

and solve for

x

.

Since

x > 1

,

\log_3 x > 0

, so

u = 4

. Thus

\log_3 x = 4 \implies x = 3^4 = 81

.

Excludes extraneous solutions and evaluates the final exponentiation.

Key Concept

Logarithmic Change of Base and Quadratic Reduction

Question 530Question →

A football coach records the number of goals scored by a team in six consecutive matches as $1, 3, 4, 6, 7,$ and $9$ . What is the variance of the goals scored?

Show answer & explanation

Answer: 7

Answer

The variance of the goals scored is 7.

To find the variance of the goals scored, first calculate the mean:

\bar{x} = \frac{1+3+4+6+7+9}{6} = \frac{30}{6} = 5

. Next, determine the sum of the squared deviations from the mean:

(1-5)^2 + (3-5)^2 + (4-5)^2 + (6-5)^2 + (7-5)^2 + (9-5)^2 = 16 + 4 + 1 + 1 + 4 + 16 = 42

. Dividing this total by the number of data values (

N = 6

) yields

\frac{42}{6} = 7

.

Step-by-Step Solution

1

Calculate the mean of the dataset

\bar{x} = 5

The mean is needed as the reference point for computing deviations.

2

Compute the squared deviation of each data point from the mean

Squared deviations are 16, 4, 1, 1, 4, and 16

Variance measures the average squared distance from the mean.

3

Sum the squared deviations and divide by the number of observations N = 6

Variance = 7

The formula for variance of ungrouped data is \sigma^2 = \frac{\sum (x - \bar{x})^2}{N}.

Key Concept

Variance of Ungrouped Data

Question 531Question →

Calculate the total number of distinct four-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6,$ and $7$ without repetition, such that the resulting number is divisible by either $4$ or $5$ .

Show answer & explanation

Answer: 320

Answer

The total number of such four-digit numbers is

320

.

The total number of valid four-digit numbers is found by adding the count of numbers divisible by

5

(

120

) to the count of numbers divisible by

4

(

200

). Since a number ending in

5

is odd, it cannot be divisible by

4

, making the two conditions mutually exclusive. Thus, the total count is

120 + 200 = 320

.

Step-by-Step Solution

1

Calculate the number of four-digit numbers divisible by

5

.

For a number to be divisible by

5

, its units digit must be

5

(since

0

is not available). There is

1

choice for the units digit. The remaining

3

positions are filled from the remaining

6

available digits in

^{6}P_{3} = 6 \times 5 \times 4 = 120

ways.

Divisibility by

5

requires the last digit to be

5

.

2

Calculate the number of four-digit numbers divisible by

4

.

A number is divisible by

4

if its last two digits form a multiple of

4

. Using distinct digits from

\{1, 2, 3, 4, 5, 6, 7\}

, the valid two-digit endings are

12, 16, 24, 32, 36, 52, 56, 64, 72,

and

76

(

10

valid pairs). For each pair, the first two positions are filled from the remaining

5

digits in

^{5}P_{2} = 5 \times 4 = 20

ways. Thus, total ways =

10 \times 20 = 200

.

Divisibility by

4

depends entirely on the last two digits.

3

Check for overlap (numbers divisible by both

4

and

5

).

A number divisible by

5

must end in

5

, which is an odd digit. All multiples of

4

must end in an even digit. Hence, no number is divisible by both

4

and

5

in this set. The overlap is

0

.

The two events are mutually exclusive.

4

Apply the addition principle of counting.

Total =

120 + 200 - 0 = 320

.

Add the counts of the two mutually exclusive sets.

Key Concept

Restricted Permutations and Mutually Exclusive Events

Estimated Time:1m 30s

Question 532Question →

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

Show answer & explanation

Answer: 32

Answer

The numerical value of

\frac{dy}{dx}

evaluated at

x = 1

is

32

.

Using the product rule together with the chain rule for composite functions, we find

u'(x) = 6(2x - 1)^2

and

v'(x) = 4x(x^2 + 1)

. Evaluating at

x = 1

gives

u(1)=1

,

u'(1)=6

,

v(1)=4

, and

v'(1)=8

. Calculating

\frac{dy}{dx} = u'(1)v(1) + u(1)v'(1) = 6(4) + 1(8) = 32

.

Step-by-Step Solution

1

Set up the product rule for

y = u(x)v(x)

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

and

v(x) = (x^2 + 1)^2

The given function is a product of two composite expressions.

2

Differentiate

u(x)

using the chain rule

u'(x) = 3(2x - 1)^2 \cdot 2 = 6(2x - 1)^2

Differentiating the outer power function and multiplying by the derivative of the inner function

2x - 1

.

3

Differentiate

v(x)

using the chain rule

v'(x) = 2(x^2 + 1) \cdot 2x = 4x(x^2 + 1)

Differentiating the outer power function and multiplying by the derivative of the inner function

x^2 + 1

.

4

Apply the product rule formula

\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)

and evaluate at

x = 1

u(1) = 1

,

u'(1) = 6

,

v(1) = 4

,

v'(1) = 8

, giving

\frac{dy}{dx}\Big|_{x=1} = (6)(4) + (1)(8) = 32

Substituting

x = 1

into each individual term simplifies the arithmetic before combining.

Key Concept

Combined application of the Product Rule and Chain Rule

Question 533Question →

Evaluate the definite integral $\int_{0}^{2} (3x^2 + 4) \, dx$ .

Show answer & explanation

Answer: 16

Answer

The value of the definite integral is

16

.

Integrating

3x^2 + 4

with respect to

x

gives the antiderivative

F(x) = x^3 + 4x

. Evaluating this antiderivative at the upper limit

x = 2

yields

2^3 + 4(2) = 16

, and at the lower limit

x = 0

yields

0^3 + 4(0) = 0

. Subtracting the lower boundary value from the upper boundary value gives

16 - 0 = 16

.

Step-by-Step Solution

1

Integrate the polynomial term by term

\int (3x^2 + 4) dx = x^3 + 4x

Apply the power rule of integration \int x^n dx = \frac{x^{n+1}}{n+1} and \int k dx = kx.

2

Apply the fundamental theorem of calculus with limits 0 and 2

[x^3 + 4x]_0^2 = (2^3 + 4(2)) - (0^3 + 4(0)) = 16 - 0 = 16

Evaluate F(b) - F(a) where F(x) is the antiderivative.

Key Concept

Definite Integral Evaluation using the Fundamental Theorem of Calculus

Question 534Question →

Evaluate the trigonometric limit:

\lim_{x \to 0} \frac{\cos(3x) - \cos(x)}{x^2}

What is the numerical value of this limit?

Show answer & explanation

Answer: -4

Answer

The numerical value of the limit is -4.

Using either the sum-to-product identity

\cos(3x) - \cos(x) = -2 \sin(2x) \sin(x)

along with standard limits

\lim_{x \to 0} \frac{\sin(kx)}{x} = k

, or applying L'Hôpital's rule twice on the

\frac{0}{0}

form, yields the exact value

-4

.

Step-by-Step Solution

1

Check the form of the limit by direct substitution

Substituting

x = 0

gives

\frac{\cos(0) - \cos(0)}{0^2} = \frac{1 - 1}{0} = \frac{0}{0}

, an indeterminate form.

Determines whether algebraic transformation or L'Hôpital's rule is required.

2

Transform the numerator using the sum-to-product formula

\cos(3x) - \cos(x) = -2 \sin\left(\frac{3x+x}{2}\right) \sin\left(\frac{3x-x}{2}\right) = -2 \sin(2x) \sin(x)

Converts difference of cosines into product of sines to utilize standard trigonometric limits.

3

Rewrite the fractional expression and apply limit laws

\lim_{x \to 0} \frac{-2 \sin(2x) \sin(x)}{x^2} = -2 \cdot \left(\lim_{x \to 0} \frac{\sin(2x)}{x}\right) \cdot \left(\lim_{x \to 0} \frac{\sin(x)}{x}\right)

Splits

x^2

into

x \cdot x

under each sine function.

4

Evaluate the individual standard limits

\lim_{x \to 0} \frac{\sin(2x)}{x} = 2 \quad \text{and} \quad \lim_{x \to 0} \frac{\sin(x)}{x} = 1

Applies the known fundamental trigonometric limit rule

\lim_{u \to 0} \frac{\sin(au)}{u} = a

.

5

Calculate the final product

-2 \times 2 \times 1 = -4

Combines all factors to reach the evaluated value.

Key Concept

Trigonometric Limits and Indeterminate Forms

Alternative Method

Alternatively, apply L'Hôpital's rule twice. First derivative of numerator over denominator yields

\lim_{x \to 0} \frac{-3\sin(3x) + \sin(x)}{2x}

(still

\frac{0}{0}

). Differentiating a second time yields

\lim_{x \to 0} \frac{-9\cos(3x) + \cos(x)}{2} = \frac{-9(1) + 1}{2} = \frac{-8}{2} = -4

.

Estimated Time:1m 30s

Question 535Question →

Given that $\theta$ is an acute angle satisfying $\sec \theta + \tan \theta = 3$ , what is the exact numerical value of $10 \sin \theta$ ?

Show answer & explanation

Answer: 8

Answer

The numerical value of

10 \sin \theta

is 8.

Using the standard fundamental identity

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

, we factor it into

(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1

. Substituting the given value

\sec \theta + \tan \theta = 3

gives

\sec \theta - \tan \theta = \frac{1}{3}

. Solving these linear equations yields

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

(so

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

) and

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

. Using

\sin \theta = \tan \theta \cdot \cos \theta

, we find

\sin \theta = \frac{4}{5} = 0.8

. Multiplying by 10 gives the exact result 8.

Step-by-Step Solution

1

Apply the trigonometric identity

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

(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1

Difference of squares factorization links

\sec \theta + \tan \theta

and

\sec \theta - \tan \theta

as reciprocals.

2

Substitute

\sec \theta + \tan \theta = 3

to find

\sec \theta - \tan \theta

\sec \theta - \tan \theta = \frac{1}{3}

Dividing both sides of

3(\sec \theta - \tan \theta) = 1

by 3.

3

Solve the system of equations for

\sec \theta

and

\tan \theta

\sec \theta = \frac{5}{3} \text{ and } \tan \theta = \frac{4}{3}

Adding equations gives

2\sec \theta = \frac{10}{3}

; subtracting gives

2\tan \theta = \frac{8}{3}

.

4

Calculate

\sin \theta

and evaluate

10 \sin \theta

\sin \theta = \frac{4}{5} \implies 10 \sin \theta = 8

Since

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

and

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

,

\sin \theta = \tan \theta \cdot \cos \theta = \frac{4}{5}

.

Key Concept

Trigonometric identities relating secant and tangent ratios

Question 536Question →

An echo sounder on a fishing boat emits an acoustic pulse vertically downward toward the seabed and detects the reflected signal $0.80\text{ s}$ later. If the speed of sound in seawater is $1450\text{ m/s}$ , what is the depth of the sea in meters at this location?

Show answer & explanation

Answer: 580

Answer

The depth of the sea is

580\text{ m}

.

An echo signal travels to the reflecting surface and back, covering twice the depth (

2d = v \times t

). Substituting

v = 1450\text{ m/s}

and

t = 0.80\text{ s}

gives

2d = 1160\text{ m}

, so the seabed depth

d = 580\text{ m}

.

Step-by-Step Solution

1

Identify the given physical quantities

Total travel time

t = 0.80\text{ s}

and speed of sound

v = 1450\text{ m/s}

.

An echo involves sound traveling to the seabed and back, so the recorded time represents a two-way journey.

2

Set up the distance equation for an echo

Total distance traveled by the sound pulse is

2d = v \times t

.

Sound travels to the sea floor and reflects back to the ship, covering a total distance equal to twice the depth.

3

Calculate the depth

d

d = \frac{1450 \times 0.80}{2} = 580\text{ m}

.

Dividing the total round-trip distance by 2 yields the one-way depth.

Key Concept

Calculation of distance using sound echoes in a medium

Question 537Question →

What is the numerical value of the expression $\log_2(\log_3 81) + \log_5\left(\frac{1}{125}\right) + 4^{\log_2 3}$ ?

Show answer & explanation

Answer: 8

Answer

8

Evaluating each term individually: the nested logarithm

\log_2(\log_3 81) = \log_2 4 = 2

; the reciprocal log argument

\log_5(1/125) = -3

; and the exponential log power

4^{\log_2 3} = (2^{\log_2 3})^2 = 3^2 = 9

. Combining these gives

2 - 3 + 9 = 8

.

Step-by-Step Solution

1

Evaluate the first component

\log_2(\log_3 81)

2

Since

81 = 3^4

, the inner expression

\log_3 81 = 4

, and subsequently

\log_2 4 = 2

.

2

Evaluate the second component

\log_5\left(\frac{1}{125}\right)

-3

Since

\frac{1}{125} = 5^{-3}

, applying the power law of logarithms gives

-3

.

3

Evaluate the third component

4^{\log_2 3}

9

Rewrite

4

as

2^2

to obtain

(2^{\log_2 3})^2 = 3^2 = 9

using the fundamental identity

a^{\log_a b} = b

.

4

Sum the results of the three components

8

Calculate

2 + (-3) + 9 = 8

.

Key Concept

Properties of logarithms including change of power, negative exponents, and logarithm exponentiation identities

Question 538Question →

The sum to infinity of a geometric progression (G.P.) with positive terms is $18$ , and the sum of its first two terms is $16$ . What is the first term of the progression?

Show answer & explanation

Answer: 12

Answer

The first term of the progression is 12.

Using the sum to infinity formula

S_{\infty} = \frac{a}{1 - r} = 18

, we express the first term as

a = 18(1 - r)

. Combining this with the sum of the first two terms

S_2 = a(1 + r) = 16

yields

18(1 - r)(1 + r) = 16 \implies 18(1 - r^2) = 16

. Solving for

r

gives

r^2 = \frac{1}{9}

, so

r = \frac{1}{3}

for a sequence with positive terms. Substituting

r = \frac{1}{3}

back into

a = 18(1 - r)

gives

a = 12

.

Step-by-Step Solution

1

Express the sum to infinity in terms of the first term

a

and common ratio

r

.

a = 18(1 - r)

The sum to infinity formula for a convergent G.P. is

S_{\infty} = \frac{a}{1 - r}

.

2

Write the expression for the sum of the first two terms.

a(1 + r) = 16

The sum of the first two terms is

T_1 + T_2 = a + ar = a(1 + r)

.

3

Substitute

a = 18(1 - r)

into the sum of the first two terms equation.

18(1 - r^2) = 16

Applying the difference of two squares identity

(1 - r)(1 + r) = 1 - r^2

.

4

Solve for the common ratio

r

.

r = \frac{1}{3}

Rearranging gives

1 - r^2 = \frac{8}{9} \implies r^2 = \frac{1}{9}

. Since all terms are positive,

r

must be positive.

5

Calculate the first term

a

.

a = 12

Substitute

r = \frac{1}{3}

into

a = 18(1 - r)

to get

a = 18 \times \frac{2}{3} = 12

.

Key Concept

Geometric Progression sum to infinity and partial sums

Estimated Time:1m 30s

Question 539Question →

The heights (in cm) of a seedling recorded over five consecutive weeks are $9$ , $13$ , $15$ , $17$ , and $21$ . What is the standard deviation of the heights?

Show answer & explanation

Answer: 4

Answer

The standard deviation of the seedling heights is 4 cm.

To find the standard deviation, first compute the mean: (9 + 13 + 15 + 17 + 21) / 5 = 15. Next, calculate the sum of squared deviations: (-6)^2 + (-2)^2 + 0^2 + 2^2 + 6^2 = 36 + 4 + 0 + 4 + 36 = 80. Divide by 5 to find the variance of 16. Finally, taking the square root of 16 gives the standard deviation of 4 cm.

Step-by-Step Solution

1

Calculate the mean of the data values

Mean = 15 cm

The mean is necessary to calculate the deviations of each data point.

2

Calculate the squared deviations from the mean

Squared deviations are 36, 4, 0, 4, and 36 (sum = 80)

Variance measures the average of squared deviations from the mean.

3

Calculate the population variance

Variance = 80 / 5 = 16

Dividing the sum of squared deviations by N gives the variance.

4

Calculate the standard deviation

Standard deviation = sqrt(16) = 4 cm

The standard deviation is the square root of the variance.

Key Concept

Standard Deviation of Ungrouped Data

Question 540Question →

If $50.0\text{ cm}^3$ of a saturated solution of potassium chloride ( $\text{KCl}$ ) contains $14.9\text{ g}$ of the salt at $298\text{ K}$ , what is the solubility of potassium chloride at this temperature in $\text{mol dm}^{-3}$ ? (Molar mass of $\text{KCl} = 74.5\text{ g mol}^{-1}$ )

Show answer & explanation

Answer: 4

Answer

The solubility of potassium chloride at

298\text{ K}

is

4.0\text{ mol dm}^{-3}

.

First, the mass concentration is determined by scaling the

14.9\text{ g}

in

50.0\text{ cm}^3

to

1000\text{ cm}^3

, giving

298.0\text{ g dm}^{-3}

. Dividing

298.0\text{ g dm}^{-3}

by the molar mass of

\text{KCl}

(

74.5\text{ g mol}^{-1}

) yields

4.0\text{ mol dm}^{-3}

.

Step-by-Step Solution

1

Calculate the mass of solute present per cubic decimetre (1000 cm³) of saturated solution.

Mass concentration = 298.0 g dm⁻³

Solubility is expressed relative to 1 dm³ of solution volume.

2

Divide the mass concentration in g dm⁻³ by the molar mass of KCl.

Solubility = 4.0 mol dm⁻³

Molar solubility equals mass concentration divided by molar mass (M).

Key Concept

Solubility Determination in Moles per Decimetre Cubed

Estimated Time:1m 30s

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