All practice questions

Question 4181Question →

A biased four-sector spinner has sectors labeled 1, 2, 3, and 4. The theoretical probabilities of landing on sectors 1, 2, and 3 are in the ratio $1 : 2 : 2$ , respectively, while the theoretical probability of landing on sector 4 is $0.20$ . In an experiment consisting of $200$ spins, sector 2 was recorded $80$ times and sector 4 was recorded $60$ times. What is the positive difference between the expected number of even outcomes in $500$ future spins based on the experimental probability and that based on the theoretical probability?

90

50

318

60

Show answer & explanation

Answer: $90$

Answer

90

The theoretical probability of getting an even outcome (sector 2 or 4) is

P(2) + P(4) = 0.32 + 0.20 = 0.52

, which predicts

500 \times 0.52 = 260

even outcomes in

500

spins. Empirically, even outcomes occurred

80 + 60 = 140

times in

200

spins, giving an experimental probability of

140 / 200 = 0.70

and an estimated frequency of

500 \times 0.70 = 350

. The positive difference between these two expected values is

350 - 260 = 90

.

Step-by-Step Solution

1

Calculate the theoretical probabilities for each sector

P(1) = 0.16

,

P(2) = 0.32

,

P(3) = 0.32

,

P(4) = 0.20

Since

P(4) = 0.20

, the remaining probability

1 - 0.20 = 0.80

is shared among sectors 1, 2, and 3 in the ratio

1 : 2 : 2

(total 5 parts). Each part equals

0.80 / 5 = 0.16

.

2

Determine the theoretical probability and expected frequency of an even outcome

P_{\text{theo}}(\text{Even}) = 0.52

; Expected frequency in 500 spins

= 260

The even outcomes are sectors 2 and 4.

P_{\text{theo}}(\text{Even}) = P(2) + P(4) = 0.32 + 0.20 = 0.52

. In

500

spins,

500 \times 0.52 = 260

.

3

Calculate the experimental probability and expected frequency of an even outcome

P_{\text{exp}}(\text{Even}) = 0.70

; Expected frequency in 500 spins

= 350

Out of

200

spins, even outcomes occurred

80 + 60 = 140

times.

P_{\text{exp}}(\text{Even}) = 140 / 200 = 0.70

. In

500

spins,

500 \times 0.70 = 350

.

4

Find the positive difference between the two expected values

350 - 260 = 90

Subtracting the theoretical expected frequency from the experimental expected frequency gives

|350 - 260| = 90

.

Key Concept

Experimental vs. Theoretical Probability and Expected Frequency

Question 4182Question →

Find the smallest positive integer $n$ that simultaneously satisfies the linear modular congruences $3n \equiv 5 \pmod{13}$ and $4n \equiv 2 \pmod{9}$ .

Show answer & explanation

Answer: 32

Answer

The smallest positive integer satisfying both congruences is 32.

Solving the first congruence

3n \equiv 5 \pmod{13}

gives

n \equiv 6 \pmod{13}

(since

3 \times 9 \equiv 1 \pmod{13}

and

5 \times 9 = 45 \equiv 6 \pmod{13}

), which means

n

can be written as

13k + 6

. Substituting this into the second congruence

4n \equiv 2 \pmod{9}

yields

4(13k + 6) \equiv 2 \pmod{9} \implies 52k + 24 \equiv 2 \pmod{9}

. Reducing the coefficients modulo 9 gives

7k + 6 \equiv 2 \pmod{9} \implies 7k \equiv -4 \equiv 5 \pmod{9}

. Multiplying by 4 (the modular inverse of 7 modulo 9) yields

k \equiv 20 \equiv 2 \pmod{9}

. Setting

k = 2

yields the smallest positive integer

n = 13(2) + 6 = 32

.

Step-by-Step Solution

1

Solve the first modular congruence

3n \equiv 5 \pmod{13}

for

n

n \equiv 6 \pmod{13}

, which implies

n = 13k + 6

Multiplying both sides by the modular inverse of 3 modulo 13 (which is 9) isolates

n

.

2

Solve the second modular congruence

4n \equiv 2 \pmod{9}

for

n

n \equiv 5 \pmod{9}

Multiplying both sides by the modular inverse of 4 modulo 9 (which is 7) isolates

n

.

3

Substitute

n = 13k + 6

into

n \equiv 5 \pmod{9}

and simplify modulo 9

4k \equiv 8 \pmod{9}

Reducing 13 modulo 9 yields

4k

, and subtracting 6 from 5 yields

-1 \equiv 8 \pmod{9}

.

4

Solve for

k

and calculate the smallest positive integer

n

k \equiv 2 \pmod{9}

, giving

n = 13(2) + 6 = 32

Setting the integer parameter

k

to its minimum non-negative value

2

provides the smallest positive integer solution.

Key Concept

System of Linear Modular Congruences and Modular Inverses

Question 4183Question →

What is the $10^{\text{th}}$ term of the arithmetic progression $3, 7, 11, 15, \dots$ ?

Show answer & explanation

Answer: 39

Answer

The

10^{\text{th}}

term of the arithmetic progression is

39

.

By applying the

n^{\text{th}}

term formula for an arithmetic progression

T_n = a + (n - 1)d

with first term

a = 3

, common difference

d = 4

, and term index

n = 10

, the calculation yields

T_{10} = 3 + (10 - 1) \times 4 = 3 + 36 = 39

.

Step-by-Step Solution

1

Identify the key parameters of the arithmetic progression from the given sequence.

First term

a = 3

, common difference

d = 7 - 3 = 4

, and number of terms

n = 10

.

These parameters are required to use the

n^{\text{th}}

term formula of an A.P.

2

Substitute the values into the formula

T_n = a + (n - 1)d

.

T_{10} = 3 + (10 - 1) \times 4

The formula relates the

n^{\text{th}}

term to the first term, common difference, and term position.

3

Evaluate the mathematical expression.

T_{10} = 3 + 9 \times 4 = 3 + 36 = 39

Perform multiplication before addition according to standard order of operations.

Key Concept

nth term of an Arithmetic Progression

Question 4184Question →

The newly appointed committee chairman promised that all members would strictly abide _____ the regulations established by the board. Which preposition correctly completes the sentence?

by

with

to

from

Show answer & explanation

Answer: by

Answer

The preposition 'by' correctly completes the phrase 'abide by', meaning to accept and obey a decision or rule.

The verb 'abide' conventionally pairs with the preposition 'by' to form the fixed phrasal verb meaning to follow or obey rules, laws, or agreements.

Step-by-Step Solution

1

Identify the main verb requiring a prepositional complement in the sentence

The main verb is 'abide'.

The sentence tests the fixed prepositional collocation associated with the verb 'abide'.

2

Determine the correct fixed preposition for 'abide' when referring to rules or regulations

'Abide' takes the preposition 'by'.

In standard English grammar, 'abide by' is a fixed prepositional expression meaning to observe or conform to rules.

Key Concept

Dependent Prepositions and Fixed Verb-Preposition Collocations

Estimated Time:45s

Question 4185Question →

Read the sentence carefully: 'Despite her strong personal reservations regarding the executive board's new corporate policy, the senior director chose to toe the line during the press briefing.' Which of the following options best conveys the meaning of the idiomatic expression 'toe the line' as used in the sentence?

strictly conform to the rules and expectations of authority

openly voice defiance and challenge official decisions

physically stand on a designated floor boundary line

tender an immediate letter of resignation to the board

Show answer & explanation

Answer: strictly conform to the rules and expectations of authority

Answer

strictly conform to the rules and expectations of authority

The idiom 'toe the line' means to follow established rules, standards, or orders strictly, particularly when expected by an organization or authority. In the context provided, the senior director put aside her personal doubts to publicly adhere to the official corporate stance.

Step-by-Step Solution

1

Analyze the context of the sentence

The subject has 'strong personal reservations' but acts in a particular way during an official press briefing.

Understanding the contrast between private reservations and public conduct is essential to deciphering the intended meaning.

2

Determine the idiomatic meaning of 'toe the line'

'Toe the line' is an established English idiom meaning to conform strictly to established rules or authority.

Idioms cannot be interpreted literally; their accepted figurative meanings dictate sentence comprehension.

Key Concept

Idioms and Idiomatic Expressions

Question 4186Question →

Which of the following groups consists exclusively of fundamental physical quantities?

Mass, thermodynamic temperature, and luminous intensity

Mass, length, and electric charge

Electric current, force, and time

Amount of substance, weight, and temperature

Show answer & explanation

Answer: Mass, thermodynamic temperature, and luminous intensity

Answer

The group containing mass, thermodynamic temperature, and luminous intensity consists exclusively of fundamental physical quantities.

Mass, thermodynamic temperature, and luminous intensity are three of the seven internationally recognized SI base (fundamental) physical quantities.

Step-by-Step Solution

1

Identify the seven fundamental SI physical quantities

The seven base quantities are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

Fundamental quantities are independent quantities that cannot be defined in terms of other physical quantities.

2

Evaluate each provided option against the list of base quantities

Electric charge, force, and weight are derived quantities. Only mass, thermodynamic temperature, and luminous intensity are all fundamental.

Any quantity derived by multiplying or dividing base quantities is derived.

Key Concept

Fundamental quantities are basic physical quantities that are independent of one another and form the basis from which derived quantities are obtained.

Question 4187Question →

What is the measure, in degrees, of each interior angle of a regular octagon (an 8-sided regular polygon)?

Show answer & explanation

Answer: 135

Answer

Each interior angle of a regular octagon measures 135 degrees.

The sum of the interior angles of a polygon with

n

sides is

(n - 2) \times 180^\circ

. For a regular octagon (

n = 8

), the total interior angle sum is

(8 - 2) \times 180^\circ = 1080^\circ

. Since all 8 interior angles of a regular octagon are congruent, dividing the total sum by 8 yields

135^\circ

for each interior angle.

Step-by-Step Solution

1

Find the sum of all interior angles of the regular octagon.

Sum of interior angles =

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

.

The sum of interior angles for any

n

-sided polygon is

(n - 2) \times 180^\circ

.

2

Calculate the measure of a single interior angle.

Interior angle =

\frac{1080^\circ}{8} = 135^\circ

.

In a regular polygon, all interior angles are equal in measure.

Key Concept

Interior Angle of a Regular Polygon

Question 4188Question →

What is the value of

x

that satisfies the exponential equation

\frac{4^{2x + 1} \times 8^{x - 1}}{16^x} = 32

?

2

1

\frac{5}{3}

11

Show answer & explanation

Answer: $2$

Answer

The value of

x

is

2

.

Converting all terms to base

2

gives

\frac{2^{4x+2} \times 2^{3x-3}}{2^{4x}} = 2^5

. Combining the powers on the left side yields

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

. Equating exponents gives

3x - 1 = 5

, which solves to

x = 2

.

Step-by-Step Solution

1

Express all base numbers in terms of a common prime base (base 2).

4 = 2^2

,

8 = 2^3

,

16 = 2^4

, and

32 = 2^5

.

Converting all terms to powers of 2 enables the application of index laws.

2

Substitute these prime base powers into the original equation and expand exponents.

\frac{(2^2)^{2x + 1} \times (2^3)^{x - 1}}{(2^4)^x} = 2^5 \implies \frac{2^{4x + 2} \times 2^{3x - 3}}{2^{4x}} = 2^5

Applying

(a^m)^n = a^{m \cdot n}

requires multiplying the outer exponent by every term in the inner exponent.

3

Apply index laws for multiplication (

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

) and division (

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

) on the left-hand side.

2^{(4x + 2) + (3x - 3) - 4x} = 2^5 \implies 2^{3x - 1} = 2^5

Powers with the same base are combined by adding numerator exponents and subtracting denominator exponents.

4

Equate the exponents since the bases are identical, and solve for

x

.

3x - 1 = 5 \implies 3x = 6 \implies x = 2

If

a^x = a^y

for

a > 0

and

a \neq 1

, then

x = y

.

Key Concept

Solving Exponential Equations using Laws of Indices

Estimated Time:1m 30s

Question 4189Question →

Given the simultaneous equations $x + 2y = 7$ and $x^2 + 3xy + y^2 = 19$ , find the product of all possible values of $y$ that satisfy the system.

Show answer & explanation

Answer: -30

Answer

The product of all possible values of y that satisfy the system is -30.

Rearranging the linear equation gives x = 7 - 2y. Substituting this into the quadratic equation x^2 + 3xy + y^2 = 19 produces (7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19. Expanding and combining like terms yields y^2 + 7y - 30 = 0. Solving for y gives y = 3 and y = -10. Multiplying these values together gives a product of -30.

Step-by-Step Solution

1

Isolate x in the linear equation

x = 7 - 2y

Expressing one variable in terms of the other enables substitution into the quadratic equation.

2

Substitute x into the quadratic equation and expand

(7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19

This reduces the system to a single quadratic equation in terms of y.

3

Simplify the resulting expression into standard quadratic form

y^2 + 7y - 30 = 0

Expanding yields (49 - 28y + 4y^2) + (21y - 6y^2) + y^2 = 19, which reduces to y^2 + 7y - 30 = 0.

4

Calculate the product of the roots of y

y_1 * y_2 = -30

By Vieta's formulas, the product of roots for y^2 + ay + b = 0 is b/1 = -30 (or factoring gives y = 3 and y = -10, with product 3 * (-10) = -30).

Key Concept

Solving simultaneous linear and quadratic equations via substitution and applying quadratic root properties

Estimated Time:2m 0s

Question 4190Question →

The grouped frequency distribution table below shows the marks scored by $100$ candidates in a Mathematics screening test:

Class Interval	Frequency
$10 - 19$	15
$20 - 29$	25
$30 - 39$	40
$40 - 49$	20

Match each data representation component on the left with its correct calculated numerical value on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The sector angle matches

144^\circ

, frequency density matches

4.0

, preceding upper class boundary matches

29.5

, and median class cumulative frequency matches

80

.

The items match based on direct statistical computations: the modal sector angle is

144^\circ

, frequency density is

4.0

, upper class boundary of the preceding interval is

29.5

, and cumulative frequency at the upper boundary of the median class is

80

.

Step-by-Step Solution

1

Determine the modal class and total frequency

N

.

The highest frequency is

40

, so the modal class is

30 - 39

. Total frequency

N = 15 + 25 + 40 + 20 = 100

.

Modal class identification is essential for pie chart sector, frequency density, and class boundary calculations.

2

Calculate the sector angle for the modal class in a pie chart.

Sector Angle

= \frac{40}{100} \times 360^\circ = 144^\circ

.

The sector angle represents the class frequency as a fraction of total frequency multiplied by

360^\circ

.

3

Calculate the frequency density of the modal class.

Class boundary range for

30 - 39

is

29.5 - 39.5

, so width

= 10

. Frequency density

= \frac{40}{10} = 4.0

.

Frequency density is defined as class frequency divided by class interval width.

4

Determine the preceding upper class boundary and the cumulative frequency for the median class.

Preceding interval is

20 - 29

, so its upper boundary is

29.5

. Median is at position

50

(in interval

30 - 39

). Cumulative frequency up to boundary

39.5

is

15 + 25 + 40 = 80

.

Class boundaries are midpoints between adjacent class limits, and cumulative frequency sums all preceding frequencies up to the upper boundary.

Key Concept

Interpretation and calculation of pie chart sector angles, histogram frequency densities, real class boundaries, and cumulative frequencies from grouped data.

Question 4191Question →

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

Show answer & explanation

Answer: 54

Answer

54

Applying the chain rule gives

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

. Evaluating at

x = 1

gives

6(3)^2 = 54

.

Step-by-Step Solution

1

Differentiate

y = (2x + 1)^3

using the chain rule.

\frac{dy}{dx} = 6(2x + 1)^2

According to the chain rule,

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

, where

u = 2x + 1

and

\frac{du}{dx} = 2

.

2

Evaluate the derivative at

x = 1

.

\frac{dy}{dx}\Big|_{x=1} = 54

Substituting

x = 1

into

6(2x + 1)^2

yields

6(3)^2 = 54

.

Key Concept

Chain Rule for Differentiation

Question 4192Question →

A micrometer screw gauge with a positive zero error of $+0.04\text{ mm}$ gives an observed reading of $2.46\text{ mm}$ when measuring the diameter of a thin wire. What is the actual diameter of the wire?

2.42\text{ mm}

2.46\text{ mm}

2.50\text{ mm}

2.38\text{ mm}

Show answer & explanation

Answer: $2.42\text{ mm}$

Answer

The actual diameter of the wire is

2.42\text{ mm}

.

To calculate the actual reading on a measuring instrument, use the relationship

\text{Actual Reading} = \text{Observed Reading} - \text{Zero Error}

. Subtracting

+0.04\text{ mm}

from

2.46\text{ mm}

yields

2.42\text{ mm}

.

Step-by-Step Solution

1

Identify the given readings

Observed reading =

2.46\text{ mm}

, Zero error =

+0.04\text{ mm}

Extract values needed for zero error correction.

2

Apply zero error correction formula

\text{Actual Reading} = \text{Observed Reading} - \text{Zero Error} = 2.46\text{ mm} - (+0.04\text{ mm}) = 2.42\text{ mm}

A positive zero error means the instrument reads higher than true zero, so the zero error must be subtracted from the observed reading.

Key Concept

Instrument Zero Error Correction

Estimated Time:45s

Question 4193Question →

If $y = e^{4x} - \sin x$ , what is $\frac{dy}{dx}$ ?

4e^{4x} - \cos x

4e^{4x} + \cos x

e^{4x} - \cos x

e^{4x} + \cos x

Show answer & explanation

Answer: $4e^{4x} - \cos x$

Answer

4e^{4x} - \cos x

Differentiating

e^{4x}

gives

4e^{4x}

by applying the chain rule, and differentiating

-\sin x

yields

-\cos x

. Combining these terms gives the correct derivative

4e^{4x} - \cos x

.

Step-by-Step Solution

1

Apply the sum/difference rule of differentiation.

\frac{dy}{dx} = \frac{d}{dx}(e^{4x}) - \frac{d}{dx}(\sin x)

The derivative of a difference of two terms is the difference of their individual derivatives.

2

Differentiate the exponential term

e^{4x}

using the chain rule.

\frac{d}{dx}(e^{4x}) = 4e^{4x}

By the chain rule, \frac{d}{dx}(e^{k x}) = k e^{k x}.

3

Differentiate the trigonometric term

\sin x

.

\frac{d}{dx}(\sin x) = \cos x

The standard derivative of

\sin x

with respect to

x

is

\cos x

.

4

Combine the results.

\frac{dy}{dx} = 4e^{4x} - \cos x

Subtracting the derivative of

\sin x

from the derivative of

e^{4x}

gives the final answer.

Key Concept

Differentiation of Exponential and Trigonometric Functions

Question 4194Question →

Find the real value of

x

that satisfies the exponential equation

5^{2x - 1} \times 25^{x + 1} = 125^{x + 2}

Show answer & explanation

Answer: 5

Answer

The value of

x

is 5.

By converting all terms to base 5 (

25 = 5^2

and

125 = 5^3

), the equation becomes

5^{2x-1} \times 5^{2(x+1)} = 5^{3(x+2)}

. Simplifying exponents gives

5^{2x-1} \times 5^{2x+2} = 5^{3x+6}

. Adding the left-hand powers results in

5^{4x+1} = 5^{3x+6}

. Equating the exponents yields

4x + 1 = 3x + 6

, leading directly to

x = 5

.

Step-by-Step Solution

1

Express all terms with a common base of 5

5^{2x-1} \times 5^{2x+2} = 5^{3x+6}

Since

25 = 5^2

and

125 = 5^3

, using index laws

(a^m)^n = a^{mn}

allows all expressions to share base 5.

2

Apply the product rule of indices on the left side

5^{4x+1} = 5^{3x+6}

According to the product law

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

, the powers are added:

(2x-1) + (2x+2) = 4x+1

.

3

Equate powers of equal bases to solve for x

x = 5

Since bases are equal, exponents must be equal:

4x + 1 = 3x + 6 \implies 4x - 3x = 6 - 1 \implies x = 5

.

Key Concept

Indices and Laws of Indices

Question 4195Question →

In a probability experiment, two fair six-sided dice were rolled $180$ times, yielding an experimental probability of $\frac{5}{18}$ for obtaining a sum divisible by $3$ . If $m$ additional consecutive rolls were conducted and every single one resulted in a sum divisible by $3$ , the updated overall experimental probability equaled the theoretical probability that the absolute difference between the numbers shown on two fair six-sided dice is at most $1$ . Calculate the value of $m$ .

Show answer & explanation

Answer: 54

Answer

54

The initial number of successful trials is

180 \times \frac{5}{18} = 50

. The theoretical probability of rolling two dice with an absolute difference of at most

1

is calculated by counting

6

outcomes with difference

0

and

10

outcomes with difference

1

, giving

\frac{16}{36} = \frac{4}{9}

. Equating the updated experimental probability

\frac{50 + m}{180 + m}

to

\frac{4}{9}

yields

9(50 + m) = 4(180 + m)

, which simplifies to

5m = 270

or

m = 54

.

Step-by-Step Solution

1

Calculate the initial number of successful trials from the given experimental probability.

Initial successful outcomes =

180 \times \frac{5}{18} = 50

.

Experimental probability is defined as the number of successful trials divided by the total number of trials.

2

Calculate the theoretical probability that the absolute difference between two rolled six-sided dice is at most 1.

Favorable outcomes = 16, so

P(\text{theoretical}) = \frac{16}{36} = \frac{4}{9}

.

Outcomes with difference 0:

(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

(6 outcomes). Outcomes with difference 1:

(1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5)

(10 outcomes). Total sample space

= 6 \times 6 = 36

.

3

Formulate the algebraic equation relating the updated experimental probability to the theoretical probability.

\frac{50 + m}{180 + m} = \frac{4}{9}

.

Adding

m

consecutive successful rolls increases both the number of successful outcomes (to

50 + m

) and the total number of trials (to

180 + m

).

4

Solve the equation for

m

.

9(50 + m) = 4(180 + m) \implies 450 + 9m = 720 + 4m \implies 5m = 270 \implies m = 54

.

Cross-multiplication converts the rational expression into a linear equation.

Key Concept

Experimental and Theoretical Probability Synthesis

Estimated Time:3m 0s

Question 4196Question →

A box contains $50$ marbles of three different colors: red, blue, and green. The theoretical probability of selecting a red marble at random is $\frac{3}{10}$ . In a probability experiment, a marble was drawn with replacement $250$ times, resulting in a red marble being drawn $85$ times. What is the positive difference between the experimental probability and the theoretical probability of selecting a red marble?

\frac{1}{25}

\frac{7}{25}

\frac{1}{5}

\frac{17}{50}

Show answer & explanation

Answer: $\frac{1}{25}$

Answer

The positive difference between the experimental probability and the theoretical probability of selecting a red marble is

\frac{1}{25}

.

The experimental probability is the ratio of observed successful trials to total trials, which is

\frac{85}{250} = \frac{17}{50}

. The theoretical probability is given as

\frac{3}{10} = \frac{15}{50}

. Subtracting the theoretical probability from the experimental probability gives

\frac{17}{50} - \frac{15}{50} = \frac{2}{50} = \frac{1}{25}

.

Step-by-Step Solution

1

Calculate the experimental probability (relative frequency) of drawing a red marble.

\text{Experimental Probability} = \frac{\text{Number of successful outcomes}}{\text{Total number of trials}} = \frac{85}{250} = \frac{17}{50} = 0.34

Experimental probability is calculated using empirical trial outcomes.

2

Identify the given theoretical probability of drawing a red marble.

\text{Theoretical Probability} = \frac{3}{10} = \frac{15}{50} = 0.30

The theoretical probability is provided directly as

\frac{3}{10}

.

3

Compute the positive difference between experimental and theoretical probabilities.

\text{Difference} = \frac{17}{50} - \frac{15}{50} = \frac{2}{50} = \frac{1}{25} = 0.04

Subtracting theoretical probability from experimental probability gives the required positive difference.

Key Concept

Experimental vs Theoretical Probability

Estimated Time:1m 30s

Question 4197Question →

The frequency table below shows the distribution of daily petrol consumption (in litres) recorded by $40$ commercial minibus drivers:

Daily Petrol Consumption (litres)	Frequency ( $f$ )
$10 - 14$	$6$
$15 - 19$	$10$
$20 - 24$	$12$
$25 - 29$	$8$
$30 - 34$	$4$

What is the mean daily petrol consumption of the minibus drivers?

19.25\text{ litres}

22.00\text{ litres}

21.25\text{ litres}

23.25\text{ litres}

Show answer & explanation

Answer: $21.25\text{ litres}$

Answer

The mean daily petrol consumption is

21.25\text{ litres}

.

The mean of a grouped frequency distribution is computed by multiplying the midpoint of each class interval by its frequency, summing these products, and dividing by the total frequency. For the given distribution,

\sum fx = 850

and

\sum f = 40

, giving a mean of

21.25\text{ litres}

.

Step-by-Step Solution

1

Calculate the class mark (midpoint)

x

for each class interval.

Midpoints are:

12

for

10 - 14

,

17

for

15 - 19

,

22

for

20 - 24

,

27

for

25 - 29

, and

32

for

30 - 34

.

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

2

Multiply each midpoint

x

by its corresponding frequency

f

to find

fx

.

6 \times 12 = 72

,

10 \times 17 = 170

,

12 \times 22 = 264

,

8 \times 27 = 216

,

4 \times 32 = 128

.

This yields the total sum contribution of each class.

3

Find total frequency

\sum f

and total sum of products

\sum fx

.

\sum f = 6 + 10 + 12 + 8 + 4 = 40

and

\sum fx = 72 + 170 + 264 + 216 + 128 = 850

.

Required components for the grouped mean formula.

4

Calculate the mean

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

.

\bar{x} = \frac{850}{40} = 21.25\text{ litres}.

Applying the formula for the mean of grouped frequency distribution.

Key Concept

Grouped Mean Calculation using Midpoints

Estimated Time:1m 30s

Question 4198Question →

Pump A can fill a water reservoir in $6$ hours, while a drain pipe can empty the full reservoir in $15$ hours. Pump A is switched on to fill an empty reservoir while the drain pipe is accidentally left open. After $3$ hours, an identical pump, Pump B, is also switched on to assist Pump A while the drain pipe remains open. How many total hours will it take for the reservoir to become completely full?

Show answer & explanation

Answer: 5.625

Answer

The total time required to fill the reservoir completely is

5.625

hours.

To solve multi-stage work and rate problems involving opposing forces (filling vs. draining), calculate the net rate of change per unit of time for each stage. In stage one, Pump A adds

\frac{1}{6}

while the drain removes

\frac{1}{15}

, giving a net rate of

\frac{1}{10}

per hour. In 3 hours,

\frac{3}{10}

of the reservoir is filled, leaving

\frac{7}{10}

. In stage two, adding identical Pump B increases the filling rate to

2 \times \frac{1}{6} = \frac{1}{3}

. Subtracting the drain rate

\frac{1}{15}

yields a net rate of

\frac{4}{15}

per hour. Dividing the remaining

\frac{7}{10}

by

\frac{4}{15}

gives

2.625

hours. Adding the initial 3 hours yields a total of

5.625

hours.

Step-by-Step Solution

1

Determine individual hourly rates

Pump A rate =

+\frac{1}{6}

reservoir/hr, Drain rate =

-\frac{1}{15}

reservoir/hr

Rate is the reciprocal of the time required to complete the full job.

2

Calculate net rate and progress for the first 3 hours

Net rate =

\frac{1}{6} - \frac{1}{15} = \frac{1}{10}

reservoir/hr. Progress in 3 hours =

3 \times \frac{1}{10} = \frac{3}{10}

of the reservoir.

Only Pump A and the drain pipe are active during the initial 3-hour period.

3

Calculate remaining fraction of reservoir to be filled

Remaining portion =

1 - \frac{3}{10} = \frac{7}{10}

The total capacity of the reservoir is represented by

1

whole unit.

4

Calculate the combined rate after Pump B is added

New net rate =

\frac{1}{6} + \frac{1}{6} - \frac{1}{15} = \frac{1}{3} - \frac{1}{15} = \frac{4}{15}

reservoir/hr

Pump B is identical to Pump A, so its rate is also

\frac{1}{6}

reservoir/hr.

5

Find additional time needed and total time elapsed

Additional time =

\frac{7/10}{4/15} = \frac{7}{10} \times \frac{15}{4} = \frac{21}{8} = 2.625

hours. Total time =

3 + 2.625 = 5.625

hours.

Time equals remaining work divided by combined net rate, then added to elapsed time.

Key Concept

Work-Rate and Simultaneous Operations (Combined Filling and Emptying Rates)

Estimated Time:2m 30s

Question 4199Question →

After hours of heated debate, the parliamentary committee reached a tacit agreement regarding the new fiscal policies, avoiding any explicit formal endorsement. Which of the following words is nearest in meaning to the underlined word as used in the sentence?

implicit

explicit

tactful

rigid

Show answer & explanation

Answer: implicit

Answer

The word nearest in meaning to 'tacit' is 'implicit'.

The word 'tacit' describes an agreement or understanding that is expressed or carried out without words, implied rather than stated. In the given sentence, the phrase 'avoiding any explicit formal endorsement' confirms that the agreement was unwritten and unspoken. Therefore, 'implicit' is the word nearest in meaning.

Step-by-Step Solution

1

Analyze the contextual clue in the sentence.

The clause 'avoiding any explicit formal endorsement' indicates that the agreement was unstated or implied rather than openly spoken.

Contextual contrast helps determine the precise meaning of the target word.

2

Evaluate the options for the closest contextual synonym.

'Implicit' means suggested though not directly expressed, which perfectly fits 'tacit'.

Matching contextual definition with option meanings identifies the correct synonym.

Key Concept

Synonyms and Words Nearest in Meaning

Question 4200Question →

A physical quantity $Z$ is defined as the ratio of the product of impulse and linear velocity to the product of electric current and electric potential difference. When $Z$ is fully resolved into fundamental physical quantities, which fundamental quantity does $Z$ represent?

Time

Mass

Length

Electric current

Show answer & explanation

Answer: Time

Answer

Time

The correct answer is Time because reducing both the numerator (energy) and denominator (electrical power) to base fundamental dimensions yields

\frac{M L^2 T^{-2}}{M L^2 T^{-3}} = T

, which corresponds directly to the fundamental quantity of Time.

Step-by-Step Solution

1

Express impulse and velocity in terms of fundamental base quantities (Mass

M

, Length

L

, Time

T

)

\text{Impulse} = \text{Force} \times \text{Time} = M L T^{-1}

.

\text{Velocity} = L T^{-1}

. Product

= (M L T^{-1})(L T^{-1}) = M L^2 T^{-2}

.

To evaluate the numerator in fundamental base dimensions.

2

Express electric current and potential difference in terms of fundamental base quantities

\text{Current} = I

.

\text{Potential Difference} = \frac{\text{Work}}{\text{Charge}} = \frac{M L^2 T^{-2}}{I T} = M L^2 T^{-3} I^{-1}

. Product

= I \times (M L^2 T^{-3} I^{-1}) = M L^2 T^{-3}

.

To evaluate the denominator in fundamental base dimensions.

3

Divide the numerator by the denominator to simplify quantity

Z

Z = \frac{M L^2 T^{-2}}{M L^2 T^{-3}} = M^{1-1} L^{2-2} T^{-2 - (-3)} = T^1

.

Determining the net fundamental quantity after all derived units cancel out.

Key Concept

Fundamental and Derived Quantities

Answer

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