All practice questions

Question 501Question →

If the expression $\frac{\sqrt{48}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{72}}{\sqrt{3} - \sqrt{2}}$ is simplified into the form $m + n\sqrt{6}$ , where $m$ and $n$ are integers, find the value of $m + n$ .

Show answer & explanation

Answer: 26

Answer

The simplified expression is

24 + 2\sqrt{6}

, giving

m = 24

and

n = 2

, so

m + n = 26

.

Simplifying

\sqrt{48}

to

4\sqrt{3}

and

\sqrt{72}

to

6\sqrt{2}

allows rationalization of each fraction by its conjugate. The first fraction becomes

12 - 4\sqrt{6}

and the second becomes

12 + 6\sqrt{6}

. Adding these expressions results in

24 + 2\sqrt{6}

, so

m = 24

and

n = 2

, giving

m + n = 26

.

Step-by-Step Solution

1

Simplify the radical numerators

\sqrt{48} = 4\sqrt{3}

and

\sqrt{72} = 6\sqrt{2}

Decomposing surds into perfect square factors simplifies subsequent algebraic expansion.

2

Rationalize the first term

\frac{4\sqrt{3}}{\sqrt{3} + \sqrt{2}}

12 - 4\sqrt{6}

Multiplying the numerator and denominator by the conjugate

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

removes the surd from the denominator using the difference of squares

(\sqrt{3})^2 - (\sqrt{2})^2 = 1

.

3

Rationalize the second term

\frac{6\sqrt{2}}{\sqrt{3} - \sqrt{2}}

12 + 6\sqrt{6}

Multiplying the numerator and denominator by the conjugate

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

yields a rational denominator of

1

.

4

Combine like surd terms

24 + 2\sqrt{6}

Summing the rational components

(12 + 12 = 24)

and combining similar surd terms

(-4\sqrt{6} + 6\sqrt{6} = 2\sqrt{6})

.

5

Calculate the target sum

m + n

26

Matching coefficients with

m + n\sqrt{6}

gives

m = 24

and

n = 2

, yielding

24 + 2 = 26

.

Key Concept

Rationalization of Binomial Denominators using Conjugates

Question 502Question →

When the polynomial $P(x) = 3x^3 + ax^2 + bx - 10$ is divided by $(x - 2)$ , the remainder is $14$ , and when it is divided by $(x + 1)$ , the remainder is $-16$ . What is the value of $a + b$ ?

Show answer & explanation

Answer: 1

Answer

The value of

a + b

is

1

.

According to the Remainder Theorem, dividing

P(x)

by

(x - 2)

and

(x + 1)

gives

P(2) = 14

and

P(-1) = -16

respectively. Expanding these expressions forms two linear equations:

2a + b = 0

and

a - b = -3

. Solving these simultaneously gives

a = -1

and

b = 2

, so

a + b = 1

.

Step-by-Step Solution

1

Apply the Remainder Theorem for the first divisor

(x - 2)

2a + b = 0

By the Remainder Theorem,

P(2) = 14

. Substituting

x = 2

into

P(x)

yields

3(8) + 4a + 2b - 10 = 14

, which simplifies to

2a + b = 0

.

2

Apply the Remainder Theorem for the second divisor

(x + 1)

a - b = -3

By the Remainder Theorem,

P(-1) = -16

. Substituting

x = -1

into

P(x)

yields

3(-1) + a - b - 10 = -16

, which simplifies to

a - b = -3

.

3

Solve the simultaneous equations for

a

and

b

a = -1

and

b = 2

Adding

2a + b = 0

and

a - b = -3

yields

3a = -3

, giving

a = -1

. Substituting

a = -1

into

2a + b = 0

gives

b = 2

.

4

Calculate the value of

a + b

1

Adding the computed values yields

a + b = -1 + 2 = 1

.

Key Concept

Polynomial Remainder Theorem and Systems of Linear Equations

Estimated Time:2m 0s

Question 503Question →

In how many different ways can $5$ boys and $3$ girls be seated in a straight row such that all $3$ girls must sit together?

Show answer & explanation

Answer: 4320

Answer

The total number of ways to seat

5

boys and

3

girls in a row such that all

3

girls sit together is

4320

.

To arrange

5

boys and

3

girls so that the girls are always together, treat the

3

girls as

1

single unit. Combined with the

5

boys, there are

6

units to arrange in a straight line, which can be done in

6! = 720

ways. Within their group, the

3

girls can be arranged in

3! = 6

ways. By the multiplication principle, the total number of seating arrangements is

720 \times 6 = 4320

.

Step-by-Step Solution

1

Group the restricted items into a single block

The

3

girls form

1

unit. Combined with the

5

boys, there are

5 + 1 = 6

units to arrange.

Since all

3

girls must sit together, treating them as a single block ensures they are not separated.

2

Calculate the linear arrangements of the combined units

The

6

units can be arranged in

6! = 720

ways.

The number of distinct ways to arrange

n

items in a line is

n!

.

3

Calculate internal arrangements of the girls' block

The

3

girls can be arranged among themselves in

3! = 6

ways.

The

3

girls inside the block are distinct individuals and can swap positions.

4

Apply the fundamental counting principle

Total arrangements =

6! \times 3! = 720 \times 6 = 4320

.

The total number of arrangements is the product of external block arrangements and internal block arrangements.

Key Concept

Permutations with grouping constraints (string method)

Question 504Question →

If $x$ is the smallest positive integer satisfying the modular congruence $2^x \equiv 7 \pmod{11}$ , what is the value of $(3x^2 - 4x + 5) \pmod{11}$ expressed in standard non-negative remainder form?

Show answer & explanation

Answer: 3

Answer

The smallest positive integer exponent satisfying

2^x \equiv 7 \pmod{11}

is

x = 7

. Substituting

x = 7

into

3x^2 - 4x + 5

yields

124

, which simplifies to

3 \pmod{11}

.

Evaluating powers of 2 modulo 11 shows that

2^7 = 128 \equiv 7 \pmod{11}

, giving

x = 7

. Substituting

x = 7

into

3x^2 - 4x + 5

gives

124

, which leaves a remainder of

3

when divided by

11

.

Step-by-Step Solution

1

Find the smallest positive integer exponent

x

satisfying

2^x \equiv 7 \pmod{11}

x = 7

Evaluating consecutive powers of 2 modulo 11 shows

2^1 \equiv 2

,

2^2 \equiv 4

,

2^3 \equiv 8

,

2^4 \equiv 5

,

2^5 \equiv 10

,

2^6 \equiv 9

, and

2^7 \equiv 7

, making

x = 7

the smallest positive integer power.

2

Substitute

x = 7

into the expression

3x^2 - 4x + 5

124

Direct substitution gives

3(7)^2 - 4(7) + 5 = 3(49) - 28 + 5 = 147 - 28 + 5 = 124

.

3

Reduce 124 modulo 11 to standard non-negative remainder form

3

Dividing 124 by 11 yields a quotient of 11 with a remainder of 3 (

124 = 11 \times 11 + 3

).

Key Concept

Modular Exponentiation and Algebraic Evaluation in Modular Arithmetic

Question 505Question →

The mean of five numbers arranged in ascending order is $28$ . The mean of the first three numbers is $22$ , while the mean of the last three numbers is $36$ . Find the median of these five numbers.

Show answer & explanation

Answer: 34

Answer

The median of the five numbers is

34

.

For five numbers ordered from smallest to largest (

x_1, x_2, x_3, x_4, x_5

), the median is the middle value

x_3

. The sum of all five numbers is

5 \times 28 = 140

. The sum of the first three numbers is

x_1 + x_2 + x_3 = 3 \times 22 = 66

, and the sum of the last three numbers is

x_3 + x_4 + x_5 = 3 \times 36 = 108

. Adding these two partial sums gives

(x_1 + x_2 + x_3 + x_4 + x_5) + x_3 = 66 + 108 = 174

. Substituting the overall sum of

140

into the equation yields

140 + x_3 = 174

, which simplifies to

x_3 = 34

.

Step-by-Step Solution

1

Calculate the sum of all five numbers.

Sum of all 5 numbers is

5 \times 28 = 140

.

The total sum of a set of data equals the number of items multiplied by the mean.

2

Calculate the partial sums of the first three and last three numbers.

First three numbers sum to

3 \times 22 = 66

; last three numbers sum to

3 \times 36 = 108

.

Multiplying each sub-group mean by the count of numbers in that sub-group yields the sub-group sum.

3

Set up an equation relating the partial sums to the total sum and the median.

Adding the partial sums counts the third number (median) twice:

66 + 108 = 140 + \text{median}

.

In an ordered set of 5 numbers, the 3rd term is the median and is shared by both the first three and last three elements.

4

Solve for the median.

Median

= 174 - 140 = 34

.

Subtracting the total sum from the combined partial sums isolates the overlapping median value.

Key Concept

Measures of Central Tendency for Ungrouped Data (Relationship between sub-group means, total sum, and median in ordered data)

Question 506Question →

For the matrix $M = \begin{pmatrix} k & 3 & 1 \\ 2 & k & 0 \\ 1 & 5 & 2 \end{pmatrix}$ , the determinant of $M$ is equal to $4$ . What is the positive value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The positive value of

k

is

2

.

Expanding the determinant along the second row yields

\det(M) = -2(6-5) + k(2k-1) = 2k^2 - k - 2

. Setting this equal to

4

yields

2k^2 - k - 6 = 0

, which factors into

(2k+3)(k-2)=0

. The positive value is

2

.

Step-by-Step Solution

1

Evaluate the determinant of

M

in terms of

k

using row 2 cofactor expansion

\det(M) = 2k^2 - k - 2

Expanding along the second row gives

-2(6 - 5) + k(2k - 1) - 0 = -2 + 2k^2 - k

.

2

Set the determinant equal to the given value 4 and rearrange into standard quadratic form

2k^2 - k - 6 = 0

Subtracting 4 from both sides gives

2k^2 - k - 6 = 0

.

3

Solve the quadratic equation by factorization

k = -1.5 \text{ or } k = 2

Factoring

(2k + 3)(k - 2) = 0

gives roots

k = -1.5

and

k = 2

.

4

Select the positive root

k = 2

The question specifically asks for the positive value of

k

.

Key Concept

Evaluating a 3x3 matrix determinant using cofactor expansion and solving the resulting quadratic equation for an unknown parameter.

Question 507Question →

A thermometer is calibrated on a custom scale, $X$ , where the ice point ( $0^\circ\text{C}$ ) is marked as $-10^\circ\text{X}$ and the steam point ( $100^\circ\text{C}$ ) is marked as $110^\circ\text{X}$ . What is the reading on this custom scale when a standard Celsius thermometer reads $35^\circ\text{C}$ ?

Show answer & explanation

Answer: 32

Answer

32 °X

Using the linear temperature interpolation formula

\frac{X - \text{LFP}_X}{\text{UFP}_X - \text{LFP}_X} = \frac{C - \text{LFP}_C}{\text{UFP}_C - \text{LFP}_C}

, substituting

\text{LFP}_X = -10

,

\text{UFP}_X = 110

,

C = 35

,

\text{LFP}_C = 0

, and

\text{UFP}_C = 100

gives

\frac{X - (-10)}{110 - (-10)} = \frac{35 - 0}{100 - 0}

. Simplifying gives

\frac{X + 10}{120} = 0.35

, leading to

X + 10 = 42

, so

X = 32^\circ\text{X}

.

Step-by-Step Solution

1

Determine fundamental intervals for both temperature scales

Celsius fundamental interval =

100 - 0 = 100^\circ\text{C}

; Custom scale fundamental interval =

110 - (-10) = 120^\circ\text{X}

Linear temperature scale interpolation requires calculating the total interval between the lower fixed point (LFP) and upper fixed point (UFP).

2

Set up the ratio equation between the two thermometric scales

\frac{X - (-10)}{120} = \frac{35 - 0}{100}

The fractional position of any given temperature relative to its fixed points must be equal on all linear scales.

3

Solve the algebraic equation for

X

X + 10 = 120 \times 0.35 = 42 \implies X = 32^\circ\text{X}

Isolating

X

gives the corresponding reading on the custom temperature scale.

Key Concept

Linear Temperature Scale Conversion and Interpolation

Question 508Question →

Using differentiation from first principles, evaluate the value of the derivative $\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ for the cubic function $f(x) = 2x^3 - 9x^2 + 12x - 5$ at the point $x = 3$ .

Show answer & explanation

Answer: 12

Answer

The derivative evaluated at

x = 3

is equal to 12.

Evaluating the definition of the derivative from first principles for

f(x) = 2x^3 - 9x^2 + 12x - 5

yields

\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = 6x^2 - 18x + 12

. Substituting

x = 3

gives

6(3)^2 - 18(3) + 12 = 54 - 54 + 12 = 12

.

Step-by-Step Solution

1

Set up the difference quotient definition from first principles

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

Differentiation from first principles requires finding the limit of the average rate of change as the increment

h

approaches zero.

2

Substitute

(x+h)

into

f(x) = 2x^3 - 9x^2 + 12x - 5

and expand

f(x+h) = 2x^3 + 6x^2h + 6xh^2 + 2h^3 - 9x^2 - 18xh - 9h^2 + 12x + 12h - 5

Expanding binomial terms

(x+h)^3

and

(x+h)^2

reveals all components involving

h

.

3

Calculate

f(x+h) - f(x)

and factor out

h

f(x+h) - f(x) = h(6x^2 + 6xh + 2h^2 - 18x - 9h + 12)

Terms independent of

h

cancel out completely, isolating

h

as a common factor.

4

Divide by

h

and evaluate the limit as

h \to 0

f'(x) = 6x^2 - 18x + 12

Canceling

h

resolves the

\frac{0}{0}

indeterminate form, allowing direct substitution of

h=0

.

5

Substitute

x = 3

into

f'(x)

f'(3) = 6(3)^2 - 18(3) + 12 = 12

Evaluating at

x = 3

gives the numerical value of the instantaneous rate of change at that specific point.

Key Concept

Differentiation from First Principles

Question 509Question →

Given the function $y = \frac{(x^2 + 1)^3}{2x - 3}$ , what is the numerical value of $\frac{dy}{dx}$ evaluated at $x = 2$ ?

Show answer & explanation

Answer: 50

Answer

The numerical value of

\frac{dy}{dx}

at

x = 2

is 50.

Applying the Quotient Rule

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

along with the Chain Rule to find

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

, and evaluating all components at

x = 2

gives

u(2) = 125

,

u'(2) = 300

,

v(2) = 1

, and

v'(2) = 2

. Substituting these values yields

\frac{(300)(1) - (125)(2)}{1^2} = 50

.

Step-by-Step Solution

1

Set up the Quotient Rule components

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

and

v(x) = 2x - 3

The given expression is a quotient of two functions requiring

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

.

2

Differentiate

u(x)

using the Chain Rule and

v(x)

using basic power rules

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

and

v'(x) = 2

Differentiating the outer power 3 gives

3(x^2 + 1)^2

, and multiplying by the derivative of the inner function

(2x)

gives

6x(x^2 + 1)^2

.

3

Evaluate

u(2)

,

u'(2)

,

v(2)

, and

v'(2)

u(2) = 125

,

u'(2) = 300

,

v(2) = 1

, and

v'(2) = 2

Substituting

x = 2

into each function and derivative simplifies calculation of the overall derivative.

4

Substitute values into the Quotient Rule formula

\left.\frac{dy}{dx}\right|_{x=2} = \frac{(300)(1) - (125)(2)}{(1)^2} = 50

Evaluating

\frac{u'(2)v(2) - u(2)v'(2)}{[v(2)]^2}

yields the exact numerical result.

Key Concept

Combined Application of Quotient Rule and Chain Rule

Question 510Question →

If $43_x + 56_x = 121_x$ , where $x$ represents a positive integer base, find the value of $x$ .

Show answer & explanation

Answer: 8

Answer

The value of the base

x

is 8.

Expanding each base

x

number into polynomial form gives

(4x + 3) + (5x + 6) = x^2 + 2x + 1

. Simplifying yields the quadratic equation

x^2 - 7x - 8 = 0

, which factors as

(x - 8)(x + 1) = 0

. Since a number base must be a positive integer greater than any digit present in the problem (

x > 6

),

x = 8

.

Step-by-Step Solution

1

Convert all base

x

numbers into base 10 algebraic expressions.

43_x = 4x + 3

,

56_x = 5x + 6

, and

121_x = x^2 + 2x + 1

.

Place-value expansion expresses numbers in base

x

as polynomials in

x

.

2

Set up the algebraic equation corresponding to the addition.

(4x + 3) + (5x + 6) = x^2 + 2x + 1 \implies 9x + 9 = x^2 + 2x + 1

.

The sum of the left-hand terms equals the right-hand term.

3

Rearrange into standard quadratic form and factor.

x^2 - 7x - 8 = 0 \implies (x - 8)(x + 1) = 0

.

Moving all terms to one side allows solving for the roots of the quadratic equation.

4

Determine the valid base value.

x = 8

.

Number bases must be positive integers greater than all individual digits present in the expression (

x > 6

).

Key Concept

Unknown base equations and expansion

Question 511Question →

If the surd expression $\sqrt{75} - \sqrt{12} + \frac{6}{\sqrt{3}}$ is simplified to the form $k\sqrt{3}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is 5.

Simplifying

\sqrt{75}

yields

5\sqrt{3}

, simplifying

\sqrt{12}

yields

2\sqrt{3}

, and rationalizing

\frac{6}{\sqrt{3}}

gives

2\sqrt{3}

. Adding these together gives

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

. Equating

5\sqrt{3}

to

k\sqrt{3}

yields

k = 5

.

Step-by-Step Solution

1

Simplify the individual square roots.

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}

and

\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

.

Factoring out perfect square numbers allows surds to be written in basic radical form.

2

Rationalize the fractional surd term

\frac{6}{\sqrt{3}}

.

\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}

.

Multiplying numerator and denominator by

\sqrt{3}

eliminates the radical from the denominator.

3

Combine like surd terms.

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

.

Like surds share the same radical factor and can be added or subtracted algebraically.

4

Equate the simplified expression to

k\sqrt{3}

.

k = 5

.

Comparing coefficients of

\sqrt{3}

reveals the value of

k

.

Key Concept

Surd Simplification and Rationalization of Denominators

Estimated Time:1m 30s

Question 512Question →

Find the product of all real solutions to the exponential equation $9^{x+1} - 28 \cdot 3^x + 3 = 0$ .

Show answer & explanation

Answer: -2

Answer

The product of all real solutions is -2.

Using the index laws

9^{x+1} = 9 \cdot (3^x)^2

, let

u = 3^x

. The equation reduces to the quadratic

9u^2 - 28u + 3 = 0

, which factors as

(9u - 1)(u - 3) = 0

. This gives

u = 1/9

or

u = 3

. Solving

3^x = 1/9

yields

x = -2

, and solving

3^x = 3

yields

x = 1

. The product of these solutions is

(-2) \times 1 = -2

.

Step-by-Step Solution

1

Express

9^{x+1}

in terms of

3^x

9^{x+1} = 9^1 \cdot 9^x = 9 \cdot (3^2)^x = 9 \cdot (3^x)^2

Apply index laws

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

and

(a^m)^n = (a^n)^m

to establish a common base of 3.

2

Substitute

u = 3^x

to form a quadratic equation

9u^2 - 28u + 3 = 0

Transform the exponential equation into a standard quadratic algebraic equation.

3

Solve the quadratic equation for

u

(9u - 1)(u - 3) = 0 \implies u = \frac{1}{9}

or

u = 3

Factorize the quadratic expression to determine its roots.

4

Substitute back

u = 3^x

to solve for

x

3^x = 3^{-2} \implies x = -2

, and

3^x = 3^1 \implies x = 1

Equate exponents with matching bases to find all valid real solutions for

x

.

5

Find the product of the two solutions

(-2) \times 1 = -2

Calculate the required mathematical product of the solutions.

Key Concept

Solving exponential equations reducible to quadratic form using laws of indices

Question 513Question →

Calcium hydride ( $\text{CaH}_2$ ) reacts vigorously with water to produce calcium hydroxide and hydrogen gas. What volume of dry hydrogen gas, in $\text{dm}^3$ , measured at standard temperature and pressure (s.t.p.), is liberated when $10.5\text{ g}$ of pure calcium hydride reacts completely with excess water?

[Relative atomic masses: $\text{Ca} = 40$ , $\text{H} = 1$ ; Molar volume of gas at s.t.p. = $22.4\text{ dm}^3\text{ mol}^{-1}$ ]

Show answer & explanation

Answer: 11.2

Answer

The volume of dry hydrogen gas liberated at s.t.p. is 11.2 dm³.

Calcium hydride reacts with water according to the reaction CaH₂ + 2H₂O → Ca(OH)₂ + 2H₂. Given 10.5 g of CaH₂ (molar mass 42 g/mol), there are 0.25 moles of CaH₂. Based on the 1:2 stoichiometric ratio, 0.50 moles of H₂ gas are generated. Multiplying by the molar volume at s.t.p. (22.4 dm³/mol) yields 11.2 dm³.

Step-by-Step Solution

1

Write the balanced chemical equation for the reaction of calcium hydride with water

CaH₂ + 2H₂O → Ca(OH)₂ + 2H₂

Establishing the stoichiometric mole ratio between the reactant CaH₂ and the product H₂ gas.

2

Calculate the molar mass of CaH₂

42 g/mol

Molar mass is required to convert the given mass of CaH₂ into moles.

3

Determine the amount of CaH₂ in moles

0.25 mol

Moles = Mass / Molar mass = 10.5 g / 42 g/mol.

4

Calculate the moles of H₂ gas liberated using the 1:2 stoichiometric ratio

0.50 mol

1 mole of CaH₂ produces 2 moles of H₂ gas.

5

Calculate the volume of H₂ gas produced at standard temperature and pressure (s.t.p.)

11.2 dm³

Volume at s.t.p. = Moles × Molar volume at s.t.p. = 0.50 mol × 22.4 dm³/mol.

Key Concept

Laboratory and industrial preparation of hydrogen using metal hydrides and mole-volume stoichiometric calculations at s.t.p.

Estimated Time:2m 0s

Question 514Question →

Calculate the solubility in $\text{mol/dm}^3$ of sodium nitrate ( $\text{NaNO}_3$ ) at $25^\circ\text{C}$ , if $17.0\text{ g}$ of the salt dissolves in $100.0\text{ g}$ of water to form a saturated solution. [Relative atomic masses: $\text{Na} = 23, \text{N} = 14, \text{O} = 16$ ; density of water $= 1.0\text{ g/cm}^3$ ]

Show answer & explanation

Answer: 2

Answer

The solubility of sodium nitrate at

25^\circ\text{C}

is

2.0\text{ mol/dm}^3

.

To find solubility in

\text{mol/dm}^3

, first convert

17.0\text{ g}

of

\text{NaNO}_3

into moles by dividing by its molar mass (

85.0\text{ g/mol}

), obtaining

0.20\text{ mol}

. Next, convert

100.0\text{ g}

of water into volume, which equals

0.100\text{ dm}^3

. Dividing

0.20\text{ mol}

by

0.100\text{ dm}^3

yields

2.0\text{ mol/dm}^3

.

Step-by-Step Solution

1

Determine the molar mass of the solute (

\text{NaNO}_3

)

Molar mass of

\text{NaNO}_3 = 23 + 14 + (3 \times 16) = 85.0\text{ g/mol}

Molar mass is required to convert mass of solute to amount in moles.

2

Calculate the amount of

\text{NaNO}_3

in moles

\text{Moles} = \frac{17.0\text{ g}}{85.0\text{ g/mol}} = 0.20\text{ mol}

Solubility in

\text{mol/dm}^3

measures the amount of solute in moles per cubic decimeter of solvent.

3

Convert the mass of solvent (water) to volume in

\text{dm}^3

100.0\text{ g of water} = 100.0\text{ cm}^3 = 0.100\text{ dm}^3

Density of water is

1.0\text{ g/cm}^3

, and

1000\text{ cm}^3 = 1\text{ dm}^3

.

4

Calculate the solubility in

\text{mol/dm}^3

\text{Solubility} = \frac{0.20\text{ mol}}{0.100\text{ dm}^3} = 2.0\text{ mol/dm}^3

Dividing the moles of solute by the volume of solvent in

\text{dm}^3

gives the concentration in

\text{mol/dm}^3

.

Key Concept

Solubility expressed in molar concentration (mol/dm³)

Question 515Question →

In a nuclear fusion reaction, two light nuclei fuse together to form a heavier nucleus. The total mass of the reactants before fusion is $4.028\text{ u}$ , and the total mass of the products after fusion is $4.003\text{ u}$ . Given that $1\text{ u} = 931.5\text{ MeV}$ , what is the total energy released in this reaction in $\text{MeV}$ ?

Show answer & explanation

Answer: 23.2875

Answer

The total energy released in the reaction is 23.2875 MeV.

The energy released in nuclear fusion is calculated using the mass defect $\Delta m = m_{\text{reactants}} - m_{\text{products}}$. Subtracting

4.003\text{ u}

from

4.028\text{ u}

yields a mass defect of

0.025\text{ u}

. Multiplying this mass defect by

931.5\text{ MeV/u}

gives the total energy released as

23.2875\text{ MeV}

.

Step-by-Step Solution

1

Calculate the mass defect (\Delta m)

\Delta m = 4.028\text{ u} - 4.003\text{ u} = 0.025\text{ u}

Mass defect is the loss of mass during nuclear fusion that gets converted into energy.

2

Convert mass defect into energy released

E = 0.025\text{ u} \times 931.5\text{ MeV/u} = 23.2875\text{ MeV}

According to mass-energy equivalence, each atomic mass unit (u) of missing mass yields 931.5 MeV of energy.

Key Concept

Mass Defect and Energy Release in Nuclear Reactions

Question 516Question →

A radioactive sample has a half-life of $12\text{ minutes}$ . If its initial activity is $96\text{ Bq}$ , what is the remaining activity of the sample, in $\text{Bq}$ , after an elapsed time of $36\text{ minutes}$ ?

Show answer & explanation

Answer: 12

Answer

The remaining activity of the radioactive sample after

36\text{ minutes}

is

12\text{ Bq}

.

In

36\text{ minutes}

, exactly

3

half-lives elapse (

36 / 12 = 3

). The remaining activity reduces to

(1/2)^3 = 1/8

of the initial value, giving

96\text{ Bq} / 8 = 12\text{ Bq}

.

Step-by-Step Solution

1

Calculate the number of half-lives that have elapsed

n = 3

Divide the total elapsed time (

36\text{ minutes}

) by the half-life duration (

12\text{ minutes}

).

2

Calculate the remaining activity of the isotope

A = 12\text{ Bq}

After

n = 3

half-lives, the remaining activity fraction is

\left(\frac{1}{2}\right)^3 = \frac{1}{8}

. Multiplying initial activity

96\text{ Bq}

by

\frac{1}{8}

yields

12\text{ Bq}

.

Key Concept

Radioactive Decay Law and Half-life

Question 517Question →

A weak monobasic acid, $\text{HA}$ , has an acid dissociation constant ( $K_a$ ) of $4.5 \times 10^{-5}\text{ mol dm}^{-3}$ at $25^\circ\text{C}$ . Calculate the percentage ionization of a $0.05\text{ mol dm}^{-3}$ solution of this acid.

Show answer & explanation

Answer: 3

Answer

The percentage ionization of the weak monobasic acid solution is 3.0%.

According to Ostwald's dilution law for weak monobasic acids,

K_a = \alpha^2 C

. Rearranging to solve for the degree of ionization gives

\alpha = \sqrt{K_a / C} = \sqrt{(4.5 \times 10^{-5}) / 0.05} = \sqrt{9.0 \times 10^{-4}} = 0.03

. Expressed as a percentage,

0.03 \times 100\% = 3.0\%

.

Step-by-Step Solution

1

Relate acid dissociation constant (

K_a

), initial molar concentration (

C

), and degree of ionization (

\alpha

)

K_a = \alpha^2 C

For a weak monobasic acid undergoing partial ionization (

\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-

), Ostwald's dilution law simplifies to

K_a = \frac{\alpha^2 C}{1 - \alpha} \approx \alpha^2 C

because

\alpha \ll 1

.

2

Substitute the given values into the simplified expression and calculate

\alpha

\alpha = \sqrt{\frac{4.5 \times 10^{-5}}{0.05}} = \sqrt{9.0 \times 10^{-4}} = 0.03

Rearranging the equation yields

\alpha = \sqrt{K_a / C}

. Dividing the acid dissociation constant by the molar concentration gives

9.0 \times 10^{-4}

, and taking the square root yields

0.03

.

3

Convert the fractional degree of ionization to percentage ionization

3.0%

Multiplying the fractional degree of ionization by 100% gives the percentage of acid molecules ionized in solution.

Key Concept

Ostwald's Dilution Law and Degree of Ionization of Weak Acids

Estimated Time:1m 30s

Question 518Question →

At a specific temperature and pressure, $2.0\text{ moles}$ of a real gas occupy a volume of $18.0\text{ dm}^3$ . Under the exact same conditions, $2.0\text{ moles}$ of an ideal gas occupy a volume of $22.5\text{ dm}^3$ . What is the compressibility factor ( $Z$ ) of the real gas under these conditions?

Show answer & explanation

Answer: 0.8

Answer

0.8

The compressibility factor

Z

measures the deviation of a real gas from ideal gas behavior and is calculated as

Z = \frac{V_{\text{real}}}{V_{\text{ideal}}}

. Substituting the given values yields

Z = \frac{18.0}{22.5} = 0.80

. A value of

Z < 1

indicates that intermolecular attractive forces dominate, causing the real gas to occupy less volume than predicted by the ideal gas equation.

Step-by-Step Solution

1

Identify the formula for compressibility factor

Z = \frac{V_{\text{real}}}{V_{\text{ideal}}}

The compressibility factor

Z

quantifies deviation from ideality as the ratio of molar volume of a real gas to that of an ideal gas at the same temperature and pressure.

2

Substitute the given values into the equation

Z = \frac{18.0\text{ dm}^3}{22.5\text{ dm}^3}

Both volumes are measured under identical pressure, temperature, and mole count.

3

Calculate the numerical value

Z = 0.80

Dividing 18.0 by 22.5 gives 0.80, indicating

Z < 1

due to predominant intermolecular attraction.

Key Concept

Compressibility factor (Z = V_real / V_ideal) measuring real gas deviation

Question 519Question →

During a fiscal year, the Ministry of Transportation of a state government recorded the following financial disbursements:

- Construction of a modern bus terminal: $₦85,000,000$
- Routine servicing and minor repairs of fleet vehicles: $₦4,200,000$
- Payment of monthly salaries and allowances to ministry staff: $₦38,500,000$
- Major structural rehabilitation and capacity expansion of existing railway bridges: $₦52,000,000$
- Purchase of new traffic monitoring equipment: $₦19,800,000$
- Refueling of operational vehicles and office stationery supplies: $₦3,500,000$

What is the total recurrent expenditure (in $₦$ ) of the Ministry for the fiscal year?

Show answer & explanation

Answer: 46200000

Answer

The total recurrent expenditure of the Ministry for the fiscal year is ₦46,200,000.

Recurrent expenditures are ongoing operational expenses essential for daily administration, personnel payments, and regular asset maintenance. Adding routine servicing (

₦4,200,000

), salaries (

₦38,500,000

), and refueling/stationery (

₦3,500,000

) gives

₦46,200,000

. Capital expenditures such as building new terminals, acquiring equipment, and major bridge expansions are excluded because they yield long-term physical assets.

Step-by-Step Solution

1

Classify each public sector expenditure item as either recurrent expenditure or capital expenditure.

Recurrent expenditures: Routine servicing (

₦4,200,000

), staff salaries (

₦38,500,000

), and refueling/stationery (

₦3,500,000

). Capital expenditures: Bus terminal construction (

₦85,000,000

), major bridge expansion (

₦52,000,000

), and equipment acquisition (

₦19,800,000

).

Recurrent expenditures represent operational costs consumed within the financial year, whereas capital expenditures create non-current assets or increase their long-term capacity/lifespan.

2

Sum the classified recurrent expenditure amounts to obtain the total.

Total Recurrent Expenditure =

₦4,200,000 + ₦38,500,000 + ₦3,500,000 = ₦46,200,000

.

Aggregating all operational and recurring personnel expenses provides the true total of public recurrent spending.

Key Concept

Public sector recurrent expenditure comprises regular operational running costs, staff salaries, administrative expenses, and routine maintenance of existing public infrastructure, whereas capital expenditure involves creating or expanding long-term physical assets.

Question 520Question →

An agro-processing enterprise expands its production facility in the long run by increasing all of its input factors by $30\%$ . Consequently, its monthly output of processed cassava flour increases from $800\text{ bags}$ to $1,104\text{ bags}$ . What is the percentage increase in total output resulting from this scale expansion?

Show answer & explanation

Answer: 38

Answer

The percentage increase in total output is 38%.

The absolute increase in output is

1,104 - 800 = 304\text{ bags}

. Expressing this increase as a percentage of the base output gives

\frac{304}{800} \times 100 = 38\%

. Because the output increase (

38\%

) exceeds the input increase (

30\%

), the firm experiences increasing returns to scale.

Step-by-Step Solution

1

Find the absolute change in total output.

Change in output =

1,104 - 800 = 304\text{ bags}

.

To calculate a percentage change, the absolute difference between final and initial quantities must first be established.

2

Divide the change in output by the initial output and express as a percentage.

Percentage increase =

\left(\frac{304}{800}\right) \times 100 = 38\%

.

Percentage change measures the relative expansion of output relative to the original baseline quantity.

Key Concept

Calculating Percentage Change in Output for Returns to Scale Analysis

Answer

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