All practice questions

Simplify the trigonometric expression $\frac{\tan 60^\circ + \sin 45^\circ}{\cos 45^\circ}$ to its exact surd form.

\sqrt{6} + 1

\frac{\sqrt{10}}{2}

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

\sqrt{3} + 1

Show answer & explanation

Answer: $\sqrt{6} + 1$

Answer

\sqrt{6} + 1

Substituting the exact values gives

\tan 60^\circ = \sqrt{3}

,

\sin 45^\circ = \frac{1}{\sqrt{2}}

, and

\cos 45^\circ = \frac{1}{\sqrt{2}}

. Simplifying

\frac{\sqrt{3} + 1/\sqrt{2}}{1/\sqrt{2}}

gives

\sqrt{3} \cdot \sqrt{2} + 1 = \sqrt{6} + 1

.

Step-by-Step Solution

1

Substitute the exact trigonometric values for the special angles

\tan 60^\circ = \sqrt{3}

,

\sin 45^\circ = \frac{1}{\sqrt{2}}

, and

\cos 45^\circ = \frac{1}{\sqrt{2}}

Special angle values must be expressed in exact surd form.

2

Set up the fractional expression

\frac{\sqrt{3} + \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}

Replace each ratio with its exact surd equivalent.

3

Divide numerator terms by the denominator

\frac{\sqrt{3}}{\frac{1}{\sqrt{2}}} + \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = \sqrt{3} \cdot \sqrt{2} + 1 = \sqrt{6} + 1

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Key Concept

Evaluation of Special Angle Trigonometric Ratios and Simplification of Surds

Question 4022Question →

Consider the following list of numbers representing test scores of seven students: $15$ , $4$ , $22$ , $9$ , $19$ , $2$ , and $13$ . What is the median of this data set?

9

12

13

15

Show answer & explanation

Answer: $13$

Answer

The median of the data set is

13

.

First arrange the numbers in ascending order:

2, 4, 9, 13, 15, 19, 22

. Since there are

7

numbers, the median is the middle (

4\text{th}

) number, which is

13

.

Step-by-Step Solution

1

Arrange the given numbers in ascending order.

The ordered data set is

2, 4, 9, 13, 15, 19, 22

.

To find the median of ungrouped data, the values must first be placed in numerical order.

2

Determine the position of the median for

n = 7

items.

Position =

\frac{n + 1}{2} = \frac{7 + 1}{2} = 4\text{th}

value.

For an odd number of observations

n

, the median is the

\frac{n+1}{2}\text{th}

term.

3

Identify the 4th value from the ordered list.

The 4th value is

13

.

Counting from the smallest value,

2

is 1st,

4

is 2nd,

9

is 3rd, and

13

is 4th.

Key Concept

Median of Ungrouped Data

Question 4023Question →

A container holds a liquid mixture where water constitutes $\frac{3}{8}$ of the total volume. When $15\text{ liters}$ of pure water is added to the mixture, water then accounts for $50\%$ of the new total volume. What was the initial total volume of the mixture in liters?

Show answer & explanation

Answer: 60

Answer

The initial total volume of the mixture was 60 liters.

The correct answer is 60 liters. By modeling the initial volume of water as

\frac{3}{8}V

, adding 15 liters yields a new water volume of

\frac{3}{8}V + 15

out of a total volume of

V + 15

. Setting

\frac{3}{8}V + 15 = 0.5(V + 15)

and solving gives

V = 60

.

Step-by-Step Solution

1

Define the variable and write an algebraic expression for the initial amount of water.

If

V

is the initial total volume in liters, initial water volume =

\frac{3}{8}V

.

Water makes up

\frac{3}{8}

of the total initial volume.

2

Account for the addition of 15 liters of pure water to both water volume and total volume.

New water volume =

\frac{3}{8}V + 15

; New total volume =

V + 15

.

Adding pure water increases both the specific water volume and the total mixture volume by 15 liters.

3

Formulate an equation relating new water volume to 50% of the new total volume.

\frac{3}{8}V + 15 = 0.5(V + 15)

Water constitutes 50% (or

\frac{1}{2}

) of the updated mixture.

4

Solve the linear equation for

V

.

7.5 = \frac{1}{8}V \implies V = 60

Subtracting

\frac{3}{8}V

and

7.5

from both sides isolates

\frac{1}{8}V

on one side.

Key Concept

Solving multi-step fraction and percentage mixture problems

Question 4024Question →

The table below shows the distribution of weights (in kg) of 10 packages in a warehouse:

Weight Interval (kg)	Frequency ( $f$ )
$10 - 14$	$2$
$15 - 19$	$3$
$20 - 24$	$5$

What is the mean weight of the packages?

16.5\text{ kg}

17.0\text{ kg}

18.5\text{ kg}

20.5\text{ kg}

Show answer & explanation

Answer: $18.5\text{ kg}$

Answer

The mean weight of the packages is

18.5\text{ kg}

.

The correct mean is calculated by finding the midpoint of each class interval (

12, 17, 22

), multiplying each by its frequency to obtain

fx

(

24, 51, 110

), and dividing the sum of

fx

(

185

) by the total frequency (

10

), giving

18.5\text{ kg}

.

Step-by-Step Solution

1

Calculate the class midpoint (

x

) for each interval

Midpoints are:

\frac{10+14}{2} = 12

,

\frac{15+19}{2} = 17

, and

\frac{20+24}{2} = 22

.

For grouped data, each class interval is represented by its midpoint.

2

Multiply each midpoint (

x

) by its corresponding frequency (

f

) to find

fx

2 \times 12 = 24

,

3 \times 17 = 51

, and

5 \times 22 = 110

.

This yields the total contribution of values from each class interval.

3

Sum the frequencies (

\sum f

) and the products (

\sum fx

)

\sum f = 2 + 3 + 5 = 10

and

\sum fx = 24 + 51 + 110 = 185

.

These sums are needed to compute the weighted mean.

4

Divide

\sum fx

by

\sum f

Mean

\bar{x} = \frac{185}{10} = 18.5\text{ kg}

.

The mean formula for grouped data is

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

.

Key Concept

Mean of Grouped Data using Class Marks

Question 4025Question →

In a geometric progression of positive terms, the sum of the first two terms is $12$ and the sum of the third and fourth terms is $48$ . What is the $6^{\text{th}}$ term of the progression?

Show answer & explanation

Answer: 128

Answer

The 6th term of the geometric progression is 128.

Dividing

ar^2(1+r) = 48

by

a(1+r) = 12

yields

r^2 = 4

, so

r = 2

for positive terms. Substituting

r = 2

into

a(1+r) = 12

gives

a = 4

. Using

T_n = a r^{n-1}

for

n=6

, we get

T_6 = 4 \times 2^5 = 128

.

Step-by-Step Solution

1

Set up algebraic equations for the given sums using first term

a

and common ratio

r

.

a(1+r) = 12

and

ar^2(1+r) = 48

The terms of a geometric progression are given by

T_n = a r^{n-1}

.

2

Divide the equation for the third and fourth terms by the equation for the first and second terms.

r^2 = 4 \implies r = 2

Dividing eliminates

a

and

(1+r)

, giving

r^2 = 4

. Since terms are positive,

r > 0

.

3

Substitute

r = 2

into

a(1+r) = 12

to solve for

a

.

a = 4

3a = 12

leads directly to

a = 4

.

4

Evaluate the 6th term using the formula

T_6 = a r^{5}

.

T_6 = 4 \times 2^5 = 128

Applying the general term formula

T_n = a r^{n-1}

with

n=6

.

Key Concept

Geometric Progression term relations and finding the common ratio from consecutive term pairs

Question 4026Question →

In how many different ways can a chairperson and a secretary be selected from a committee of $5$ members?

10

20

25

120

Show answer & explanation

Answer: $20$

Answer

The number of ways to select a chairperson and a secretary from

5

members is

20

.

Selecting

2

individuals for distinct positions (chairperson and secretary) from

5

candidates is an ordered selection problem. The number of ways is given by the permutation formula

_5P_2 = 5 \times 4 = 20

.

Step-by-Step Solution

1

Identify whether order matters

Order matters because the roles of chairperson and secretary are distinct.

When distinct roles are assigned, the arrangement is a permutation rather than a combination.

2

Apply the permutation formula

_nP_r = \frac{n!}{(n-r)!}

for

n=5

and

r=2

_5P_2 = \frac{5!}{(5-2)!} = \frac{5!}{3!} = 5 \times 4 = 20

There are

5

choices for chairperson and

4

remaining choices for secretary.

Key Concept

Permutation of

n

distinct items taken

r

at a time

Estimated Time:45s

Question 4027Question →

Find the set of real values of $x$ that satisfies both inequalities $x^2 - 3x - 10 < 0$ and $3 - 2x \le 1$ .

1 \le x < 5

-2 < x \le 1

x < -2 \text{ or } x > 5

-2 < x < 5

Show answer & explanation

Answer: $1 \le x < 5$

Answer

The set of real values satisfying both inequalities is

1 \le x < 5

.

Solving the quadratic inequality

x^2 - 3x - 10 < 0

gives the open interval

-2 < x < 5

. Solving the linear inequality

3 - 2x \le 1

gives

-2x \le -2

, which upon dividing by

-2

and reversing the inequality sign becomes

x \ge 1

. Finding the overlapping values that satisfy both inequalities gives

1 \le x < 5

.

Step-by-Step Solution

1

Solve the quadratic inequality

x^2 - 3x - 10 < 0

.

Factor into

(x - 5)(x + 2) < 0

. Critical values are

x = -2

and

x = 5

. Since the inequality is strictly less than zero, the solution region is

-2 < x < 5

.

The quadratic expression is negative between its real roots.

2

Solve the linear inequality

3 - 2x \le 1

.

Subtract 3 from both sides:

-2x \le -2

. Divide by

-2

and flip the inequality sign:

x \ge 1

.

Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.

3

Find the intersection of the two solution sets.

Combine

-2 < x < 5

and

x \ge 1

to get

1 \le x < 5

.

Values of

x

must satisfy both conditions simultaneously.

Key Concept

Solving simultaneous linear and quadratic inequalities

Question 4028Question →

What is the value of the expression $\frac{0.00048 \times 0.025}{0.0016}$ expressed in standard form?

7.5 \times 10^{-3}

7.5 \times 10^{-2}

7.5 \times 10^{-4}

1.2 \times 10^{-2}

Show answer & explanation

Answer: $7.5 \times 10^{-3}$

Answer

7.5 \times 10^{-3}

The expression evaluates step-by-step to

0.75 \times 10^{-2}

, which when written in standard form

A \times 10^n

(where

1 \le A < 10

) gives

7.5 \times 10^{-3}

.

Step-by-Step Solution

1

Convert each decimal in the expression to standard form (powers of 10).

0.00048 = 4.8 \times 10^{-4}

,

0.025 = 2.5 \times 10^{-2}

, and

0.0016 = 1.6 \times 10^{-3}

.

Converting decimals into standard form simplifies multiplication and division using laws of indices.

2

Simplify the numerator.

(4.8 \times 10^{-4}) \times (2.5 \times 10^{-2}) = (4.8 \times 2.5) \times 10^{-4 + (-2)} = 12 \times 10^{-6} = 1.2 \times 10^{-5}

.

Multiply the numerical coefficients together and add the exponents of 10.

3

Divide the numerator by the denominator.

\frac{1.2 \times 10^{-5}}{1.6 \times 10^{-3}} = \left(\frac{1.2}{1.6}\right) \times 10^{-5 - (-3)} = 0.75 \times 10^{-2}

.

Divide the coefficients and subtract the exponent of the denominator from that of the numerator.

4

Convert the result into standard form (

A \times 10^n

where

1 \le A < 10

).

0.75 \times 10^{-2} = (7.5 \times 10^{-1}) \times 10^{-2} = 7.5 \times 10^{-3}

.

Standard form requires the leading coefficient to be between 1 and 10.

Key Concept

Fractions, Decimals, and Standard Form Conversions

Question 4029Question →

Given the function $y = \frac{(x^2 + 1)^3}{3x - 5}$ , calculate the value of $\frac{dy}{dx}$ at $x = 2$ .

Show answer & explanation

Answer: -75

Answer

The value of

\frac{dy}{dx}

at

x = 2

is

-75

.

Applying the Quotient Rule

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

along with the Chain Rule to differentiate

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

yields

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

. Evaluating at

x=2

gives

u(2)=125

,

u'(2)=300

,

v(2)=1

, and

v'(2)=3

, leading to

\frac{300(1) - 125(3)}{1^2} = -75

.

Step-by-Step Solution

1

Set up the Quotient Rule framework

Let

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

and

v(x) = 3x - 5

, so that

y = \frac{u(x)}{v(x)}

and

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

.

The function is expressed as a quotient of two differentiable terms.

2

Differentiate the numerator using the Chain Rule

u'(x) = 3(x^2 + 1)^2 \cdot \frac{d}{dx}(x^2 + 1) = 6x(x^2 + 1)^2

.

The numerator is a composite function requiring the inner derivative derivative of

x^2+1

to be multiplied.

3

Differentiate the denominator

v'(x) = 3

.

The derivative of a linear function

3x - 5

with respect to

x

is its coefficient 3.

4

Evaluate

u(x)

,

u'(x)

,

v(x)

, and

v'(x)

at

x = 2

u(2) = 125

,

u'(2) = 300

,

v(2) = 1

,

v'(2) = 3

.

Substituting

x = 2

into each evaluated component simplifies the numerical calculation.

5

Substitute numerical values into the Quotient Rule formula

\left.\frac{dy}{dx}\right|_{x=2} = \frac{(300)(1) - (125)(3)}{1^2} = \frac{300 - 375}{1} = -75

.

Completing the arithmetic calculation yields the final numerical derivative value.

Key Concept

Combining the Quotient Rule and Chain Rule for composite fractional functions

Estimated Time:1m 30s

Question 4030Question →

The chairman criticized the committee's ephemeral interest in the reform policy, noting that their enthusiasm faded as soon as challenges arose. Which word is nearest in meaning to the underlined word as used in the sentence?

fleeting

enduring

superficial

profound

Show answer & explanation

Answer: fleeting

Answer

fleeting

The word 'ephemeral' describes something that lasts for a very short time. In the given sentence, the phrase 'their enthusiasm faded as soon as challenges arose' provides a clear contextual clue that the interest was brief. Therefore, 'fleeting' is the word nearest in meaning.

Step-by-Step Solution

1

Analyze the context of the sentence

The sentence indicates that the committee's enthusiasm 'faded as soon as challenges arose', showing their interest was short-lived.

Contextual clues clarify the specific sense in which the target word is being used.

2

Determine the meaning of the underlined word 'ephemeral'

'Ephemeral' refers to something transitory, momentary, or lasting for a very short duration.

Matching the definition with contextual clues narrows down the exact synonym.

3

Evaluate the options against the target meaning

'Fleeting' directly conveys the meaning of lasting a very short time.

'Fleeting' is the accurate word nearest in meaning to 'ephemeral' in this scenario.

Key Concept

Contextual Synonyms in Lexis and Vocabulary

Question 4031Question →

A trader buys a quantity of goods. He sells $\frac{1}{4}$ of the total goods at a profit of $20\%$ , and $\frac{1}{2}$ of the remaining goods at a loss of $10\%$ . If the rest of the goods are sold at cost price, resulting in an overall net profit of $\text{N}1,500$ , what was the total cost price of the goods in Naira?

Show answer & explanation

Answer: 120000

Answer

The total cost price of the goods is 120,000 Naira.

Let

C

be the total cost price of the goods. The first portion sold is

\frac{1}{4}C

at a profit of

20\%

, giving a gain of

0.20 \times \frac{1}{4}C = 0.05C

. The remaining portion is

1 - \frac{1}{4} = \frac{3}{4}C

. Half of this remainder is

\frac{1}{2} \times \frac{3}{4}C = \frac{3}{8}C

, which is sold at a

10\%

loss, causing a loss of

0.10 \times \frac{3}{8}C = 0.0375C

. The final remaining

\frac{3}{8}C

is sold at cost price (0 profit). Thus, the net profit is

0.05C - 0.0375C = 0.0125C

. Setting

0.0125C = 1,500

and solving for

C

gives

C = \frac{1,500}{0.0125} = 120,000

Naira.

Step-by-Step Solution

1

Define the unknown total cost price

Let

C

represent the total cost price of the goods in Naira.

Establishing a variable allows for algebraic formulation of profits and losses.

2

Calculate profit from the first portion

\text{Profit}_1 = 20\% \text{ of } \frac{1}{4}C = 0.20 \times 0.25C = +0.05C

The trader sells a quarter of the total value at a 20% gain.

3

Determine the remaining quantity and calculate loss from the second portion

\text{Remaining} = C - \frac{1}{4}C = \frac{3}{4}C; \quad \text{Second Portion} = \frac{1}{2} \times \frac{3}{4}C = \frac{3}{8}C = 0.375C; \quad \text{Loss}_2 = 10\% \text{ of } 0.375C = -0.0375C

The second sale applies to half of what was left after the first sale, incurred as a loss.

4

Equate net profit to given numerical value and solve for total cost price

\text{Net Profit} = 0.05C - 0.0375C = 0.0125C = 1,500 \implies C = \frac{1,500}{0.0125} = 120,000

The rest of the goods were sold at cost price (zero profit/loss), so net profit equals gain from portion 1 minus loss from portion 2.

Key Concept

Fractions and Percentages of Quantities

Estimated Time:1m 30s

Question 4032Question →

A cubic curve is defined by the equation $y = ax^3 + bx^2 + cx + d$ . The curve has a point of inflexion at $(0, 1)$ and a stationary point at $(1, 5)$ . What is the local minimum value of the function?

-3

-7

5

-1

Show answer & explanation

Answer: $-3$

Answer

The local minimum value of the function is

-3

.

By applying the conditions for a point of inflexion (

y''=0

) and a stationary point (

y'=0

), the curve equation is identified as

y = -2x^3 + 6x + 1

. Solving

y' = 0

gives

x = -1

for the local minimum, resulting in a minimum value of

-3

.

Step-by-Step Solution

1

Use the point of inflexion and given point

(0, 1)

to find

b

and

d

.

d = 1

and

b = 0

.

Since

(0,1)

lies on the curve, substituting

x=0, y=1

gives

d = 1

. The second derivative is

y'' = 6ax + 2b

. At a point of inflexion

x = 0

,

y'' = 0

, giving

2b = 0 \implies b = 0

.

2

Use the stationary point condition at

(1, 5)

to determine

a

and

c

.

a = -2

and

c = 6

.

The equation reduces to

y = ax^3 + cx + 1

. Since

(1,5)

is on the curve,

a + c + 1 = 5 \implies a + c = 4

. Also,

y' = 3ax^2 + c = 0

at

x = 1

, so

3a + c = 0

. Solving

3a + c = 0

and

a + c = 4

simultaneously yields

a = -2

and

c = 6

.

3

Find all stationary points of

y = -2x^3 + 6x + 1

and test their nature using the second derivative.

Stationary points are at

x = 1

(maximum) and

x = -1

(minimum).

Setting

y' = -6x^2 + 6 = 0

yields

x^2 = 1 \implies x = \pm 1

. The second derivative is

y'' = -12x

. For

x = -1

,

y'' = 12 > 0

, confirming a local minimum.

4

Evaluate the function at

x = -1

to find the local minimum value.

y(-1) = -3

.

Substituting

x = -1

into

y = -2x^3 + 6x + 1

yields

y = -2(-1)^3 + 6(-1) + 1 = 2 - 6 + 1 = -3

.

Key Concept

Determining curve constants using stationary points and points of inflexion, followed by identifying local extrema.

Estimated Time:3m 0s

Question 4033Question →

A standard six-sided die was rolled $100$ times during a probability experiment, and the outcome $4$ was recorded $25$ times. What is the experimental probability of rolling a $4$ ?

\frac{1}{4}

\frac{1}{6}

\frac{3}{4}

\frac{1}{25}

Show answer & explanation

Answer: $\frac{1}{4}$

Answer

The experimental probability of rolling a

4

is

\frac{1}{4}

.

The experimental probability of an event is calculated by taking the ratio of the number of times the event occurs to the total number of trials performed. Since the outcome

4

appeared

25

times out of

100

rolls, the experimental probability is

\frac{25}{100}

, which simplifies directly to

\frac{1}{4}

.

Step-by-Step Solution

1

Identify the number of favorable trials and total trials

Favorable trials (rolling a

4

) =

25

; Total trials =

100

.

Experimental probability depends on empirical data collected during the experiment.

2

Apply the experimental probability formula:

P(E) = \frac{\text{Frequency of Event}}{\text{Total Number of Trials}}

P(\text{rolling a } 4) = \frac{25}{100}

.

The experimental probability is defined as the relative frequency of the outcome.

3

Simplify the fraction to its lowest terms

\frac{25}{100} = \frac{1}{4}

.

Dividing both the numerator and denominator by their greatest common divisor,

25

, gives the simplified fraction.

Key Concept

Experimental Probability

Question 4034Question →

The second term of a geometric progression (G.P.) with positive terms is $6$ and the fifth term is $48$ . What is the sum of the first $6$ terms of the progression?

93

96

189

192

Show answer & explanation

Answer: $189$

Answer

The sum of the first

6

terms is

189

.

Using the

n

-th term formula

T_n = a r^{n-1}

, we form two equations:

a r = 6

and

a r^4 = 48

. Dividing the fifth term equation by the second term equation gives

r^3 = 8

, so the common ratio

r = 2

. Substituting

r = 2

back gives the first term

a = 3

. Finally, applying the sum formula

S_6 = \frac{3(2^6 - 1)}{2 - 1}

gives

3 \times 63 = 189

.

Step-by-Step Solution

1

Set up equations for the given terms using the

n

-th term formula

T_n = a r^{n-1}

.

T_2 = a r = 6

and

T_5 = a r^4 = 48

.

The

n

-th term of a G.P. is defined by

T_n = a r^{n-1}

.

2

Divide the equation for

T_5

by the equation for

T_2

to find the common ratio

r

.

\frac{a r^4}{a r} = \frac{48}{6} \implies r^3 = 8 \implies r = 2.

Dividing eliminates the first term

a

and allows solving for

r

directly.

3

Substitute

r = 2

into

T_2 = 6

to find the first term

a

.

a(2) = 6 \implies a = 3.

Knowing

r

allows calculating

a

from any known term.

4

Calculate the sum of the first

6

terms using

S_n = \frac{a(r^n - 1)}{r - 1}

.

S_6 = \frac{3(2^6 - 1)}{2 - 1} = \frac{3(64 - 1)}{1} = 3 \times 63 = 189.

Applying the G.P. sum formula for

n = 6

,

a = 3

, and

r = 2

gives the total sum.

Key Concept

Geometric Progression term formula (

T_n = a r^{n-1}

) and sum of

n

terms formula (

S_n = \frac{a(r^n - 1)}{r - 1}

).

Estimated Time:1m 30s

Question 4035Question →

In a mathematics test, the mean score of a group of $12$ boys is $60$ , and the mean score of a group of $18$ girls is $70$ . What is the mean score of all $30$ students combined?

66

65

64

67

Show answer & explanation

Answer: 66

Answer

The combined mean score of all 30 students is 66.

To find the overall mean score for the entire class, determine the total sum of all test scores and divide by the total number of students. The total score contributed by the boys is

12 \times 60 = 720

, and the total score contributed by the girls is

18 \times 70 = 1260

. The combined total score is

720 + 1260 = 1980

. Dividing this total by

30

students yields a combined mean of

66

.

Step-by-Step Solution

1

Calculate the total score for the boys

Total boys' score = 12 × 60 = 720

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

2

Calculate the total score for the girls

Total girls' score = 18 × 70 = 1260

Multiply the number of girls by their mean score.

3

Find the combined total score and total student count

Combined total score = 720 + 1260 = 1980; Total students = 12 + 18 = 30

Sum the total scores and the total counts for both groups.

4

Calculate the combined mean score

Combined mean = 1980 / 30 = 66

Divide the combined total score by the total number of students.

Key Concept

Combined Mean of Two Ungrouped Datasets

Question 4036Question →

Despite the intricate technical challenges and severe time constraints, the engineering team managed to __________ the complex structural renovation successfully. Which phrasal verb correctly completes the sentence?

bring about

bring off

bring out

bring up

Show answer & explanation

Answer: bring off

Answer

The phrasal verb 'bring off' correctly completes the sentence because it means to successfully perform or achieve something difficult.

The phrasal verb 'bring off' means to succeed in doing something difficult or unexpected. Given that the team faced technical challenges and tight deadlines, executing the renovation successfully is precisely expressed by 'bring off'.

Step-by-Step Solution

1

Analyze the contextual clues in the sentence stem.

The sentence describes overcoming 'intricate technical challenges' and 'severe time constraints' to complete a renovation 'successfully'.

The blank requires a phrasal verb that signifies achieving or pulling off a challenging task.

2

Evaluate the particle meanings associated with the verb root 'bring'.

'Bring off' specifically denotes achieving success in an arduous undertaking, whereas 'bring about' implies causation, 'bring out' implies publication/emphasis, and 'bring up' implies mention/rearing.

Matching the precise idiomatic sense of the particle prevents misinterpretation of the verb's contextual role.

Key Concept

Distinguishing contextual meanings of phrasal verbs sharing the same root verb

Estimated Time:1m 0s

Question 4037Question →

In a probability experiment, a card is drawn at random with replacement from a bag containing red, green, and blue cards. After conducting $250$ trials, a green card was drawn $85$ times. Given that the theoretical probability of drawing a green card is $0.30$ , calculate the positive difference between the experimental probability and the theoretical probability of drawing a green card.

Show answer & explanation

Answer: 0.04

Answer

The positive difference between the experimental probability and the theoretical probability is 0.04.

The experimental probability is calculated as the ratio of observed favorable outcomes to total trials: $\frac{85}{250} = 0.34$. Subtracting the given theoretical probability of $0.30$ yields a positive difference of $|0.34 - 0.30| = 0.04$.

Step-by-Step Solution

1

Determine experimental probability from trial data

Experimental probability = 85 / 250 = 0.34

Experimental probability is calculated as the ratio of observed favorable trials to the total number of trials executed.

2

Subtract theoretical probability from experimental probability

|0.34 - 0.30| = 0.04

Finding the positive difference requires subtracting the theoretical probability (0.30) from the experimental relative frequency (0.34).

Key Concept

Experimental probability is determined empirically by dividing the number of times an event occurs by the total number of trials, whereas theoretical probability is based on expected outcomes under ideal conditions.

Question 4038Question →

A rectangular coil of $100$ turns with dimensions $0.10\text{ m}$ by $0.20\text{ m}$ is positioned perpendicular to a uniform magnetic field of $0.50\text{ T}$ . The coil is rotated through $90^\circ$ about an axis perpendicular to the field lines in a time interval of $0.040\text{ s}$ , bringing its plane parallel to the magnetic field. What is the magnitude of the average electromotive force (e.m.f.) induced in the coil in volts?

Show answer & explanation

Answer: 25

Answer

The magnitude of the average induced electromotive force in the coil is

25\text{ V}

.

According to Faraday's law, the induced electromotive force magnitude is

E = N \frac{\Delta \Phi}{\Delta t}

. The initial flux through each turn is

\Phi_1 = B A = 0.50\text{ T} \times 0.020\text{ m}^2 = 0.010\text{ Wb}

. When rotated parallel to the field, the final flux

\Phi_2

is

0\text{ Wb}

, so

\Delta \Phi = 0.010\text{ Wb}

. Substituting

N = 100

and

\Delta t = 0.040\text{ s}

yields an induced e.m.f. of

100 \times \frac{0.010}{0.040} = 25\text{ V}

.

Step-by-Step Solution

1

Calculate the cross-sectional area of the rectangular coil.

A = 0.10\text{ m} \times 0.20\text{ m} = 0.020\text{ m}^2

The surface area is required to find the magnetic flux passing through each turn.

2

Determine the change in magnetic flux through one turn of the coil.

Initial flux

\Phi_1 = B A = 0.50 \times 0.020 = 0.010\text{ Wb}

; Final flux

\Phi_2 = 0\text{ Wb}

; Change

\Delta \Phi = 0.010\text{ Wb}

When perpendicular to the magnetic field, maximum flux links the coil. Rotating it parallel reduces the flux linking the coil to zero.

3

Apply Faraday's law of electromagnetic induction to calculate induced e.m.f.

E = N \frac{\Delta \Phi}{\Delta t} = 100 \times \frac{0.010\text{ Wb}}{0.040\text{ s}} = 25\text{ V}

Faraday's law states that the induced e.m.f. magnitude equals the rate of change of total magnetic flux linkage.

Key Concept

Faraday's Law of Electromagnetic Induction

Question 4039Question →

What are the values of $y$ for the real solution pairs $(x, y)$ that satisfy the simultaneous equations $y - 3x = 2$ and $y = x^2 - x + 5$ ?

5

or

11

1

or

3

-1

or

-7

3

or

5

Show answer & explanation

Answer: $5$ or $11$

Answer

5

or

11

Substituting

y = 3x + 2

into the quadratic equation

y = x^2 - x + 5

gives

3x + 2 = x^2 - x + 5

, which simplifies to

x^2 - 4x + 3 = 0

. Solving this yields

x = 1

and

x = 3

. Substituting these into the linear equation gives

y = 3(1) + 2 = 5

and

y = 3(3) + 2 = 11

. Therefore, the possible values of

y

are

5

or

11

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation

y = 3x + 2

Linear equations can easily be substituted into quadratic equations.

2

Equate the linear expression for

y

to the quadratic equation

3x + 2 = x^2 - x + 5

Both expressions represent

y

.

3

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - 4x + 3 = 0

Subtract

3x

and

2

from both sides.

4

Factorize and solve for

x

(x - 1)(x - 3) = 0 \implies x = 1 \text{ or } x = 3

Finding the roots of the quadratic equation gives the

x

-coordinates of the solution pairs.

5

Substitute each

x

-value back into

y = 3x + 2

to find

y

When

x = 1

,

y = 3(1) + 2 = 5

. When

x = 3

,

y = 3(3) + 2 = 11

.

The question specifically asks for the values of

y

.

Key Concept

Simultaneous Linear and Quadratic Equations

Estimated Time:1m 30s

Question 4040Question →

A liquid will boil when its saturated vapour pressure becomes equal to the prevailing external atmospheric pressure.

True

False

Show answer & explanation

Answer: True

Answer

The statement is True. A liquid boils when its saturated vapour pressure equals the external atmospheric pressure.

The statement accurately expresses the fundamental thermodynamic condition for boiling: the temperature of the liquid must reach a point where its saturated vapour pressure equals the surrounding atmospheric pressure.

Step-by-Step Solution

1

Recall the definition of boiling point in thermal physics.

Boiling is the rapid conversion of liquid into gas occurring throughout the liquid body.

To determine the exact physical condition required for boiling to take place.

2

Relate saturated vapour pressure (SVP) to atmospheric pressure.

Bubbles of vapour can form within the liquid only when the pressure inside the bubbles (SVP) is equal to or greater than the pressure pushing down from the outside atmosphere.

If SVP is lower than atmospheric pressure, any vapour bubble attempting to form inside the liquid will immediately collapse.

Key Concept

Condition for Boiling of Liquids

Answer

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