All practice questions

Question 4221Question →

A trapezium has parallel sides of lengths $8\text{ cm}$ and $12\text{ cm}$ . If the perpendicular distance between these parallel sides is $6\text{ cm}$ , what is the area of the trapezium?

60\text{ cm}^2

120\text{ cm}^2

10\text{ cm}^2

288\text{ cm}^2

Show answer & explanation

Answer: $60\text{ cm}^2$

Answer

60\text{ cm}^2

The area of a trapezium is given by

A = \frac{1}{2}(a + b)h

, where

a

and

b

are the parallel sides and

h

is the perpendicular distance between them. Substituting

a = 8

,

b = 12

, and

h = 6

gives

A = \frac{1}{2}(8 + 12)(6) = \frac{1}{2}(20)(6) = 60\text{ cm}^2

.

Step-by-Step Solution

1

Identify the given dimensions and the formula for the area of a trapezium.

Parallel sides

a = 8\text{ cm}

,

b = 12\text{ cm}

, height

h = 6\text{ cm}

. Formula:

A = \frac{1}{2}(a + b)h

.

The area of a trapezium depends on the sum of its parallel sides and its perpendicular height.

2

Sum the parallel sides.

a + b = 8 + 12 = 20\text{ cm}

.

The average of the parallel sides forms the effective width of an equivalent rectangle.

3

Multiply by half of the height to find the total area.

A = \frac{1}{2} \times 20 \times 6 = 60\text{ cm}^2

.

Completing the formula gives the exact plane surface measure in square centimeters.

Key Concept

Area of a Trapezium

Question 4222Question →

Complete the sentence below by supplying the correct prepositions in the blanks.

Fill in the blanks below

The tribunal declared that the submitted documentation was completely bereft credible financial records, rendering the director's defence tantamount an outright admission of corruption.

Show answer & explanation

Answer

The first blank requires 'of' to form the adjectival collocation 'bereft of', and the second blank requires 'to' to form the dependent prepositional structure 'tantamount to'.

The correct prepositions are 'of' for the first blank and 'to' for the second blank. In standard formal English, the adjective 'bereft' takes the dependent preposition 'of' to signify complete lack or deprivation. Similarly, the adjective 'tantamount' takes the fixed preposition 'to' when describing something that is equal in seriousness or significance to another thing.

Step-by-Step Solution

1

Determine the required dependent preposition for the adjective 'bereft'.

The adjective 'bereft' strictly requires the preposition 'of' when indicating a state of being deprived of or lacking something.

Prepositions such as 'with', 'from', or 'in' are ungrammatical collocations when paired with 'bereft'.

2

Determine the required preposition for the adjective 'tantamount'.

The adjective 'tantamount' obligatorily pairs with 'to' when denoting equivalence in meaning, force, or effect.

Using 'with', 'for', or 'of' after 'tantamount' violates standard English prepositional rules.

Key Concept

Dependent Prepositions and Adjectival Collocations

Question 4223Question →

A micrometer screw gauge has a pitch of $0.5\text{ mm}$ and $50$ divisions on its circular thimble scale. When the anvil and spindle are brought into contact without an object between them, the $47\text{th}$ division on the thimble scale aligns exactly with the datum line of the main scale. When used to measure the diameter of a uniform metal rod, the main scale reads $1.5\text{ mm}$ and the $18\text{th}$ division on the thimble scale aligns with the datum line. What is the true diameter of the metal rod?

1.65\text{ mm}

1.68\text{ mm}

1.71\text{ mm}

2.15\text{ mm}

Show answer & explanation

Answer: $1.71\text{ mm}$

Answer

The true diameter of the metal rod is

1.71\text{ mm}

.

The least count of the micrometer is

0.01\text{ mm}

. Because the zero position when closed shows the

47\text{th}

division aligned, it has a negative zero error of

-0.03\text{ mm}

. Subtracting this negative error from the observed reading of

1.68\text{ mm}

gives

1.68 - (-0.03) = 1.71\text{ mm}

.

Step-by-Step Solution

1

Calculate the least count (precision) of the micrometer screw gauge.

\text{Least count} = \frac{\text{Pitch}}{\text{Number of thimble divisions}} = \frac{0.5\text{ mm}}{50} = 0.01\text{ mm}

.

The least count determines the value of each thimble division.

2

Determine the zero error of the instrument.

Since the

47\text{th}

division coincides with the datum line when closed, the zero mark lies above the reference line.

\text{Zero error} = -(50 - 47) \times 0.01\text{ mm} = -0.03\text{ mm}

.

When the thimble zero mark has passed the datum line in the reverse direction, the instrument exhibits a negative zero error.

3

Calculate the observed reading from the main scale and thimble scale.

\text{Observed reading} = 1.5\text{ mm} + (18 \times 0.01\text{ mm}) = 1.5\text{ mm} + 0.18\text{ mm} = 1.68\text{ mm}

.

The observed reading is the sum of the main scale reading and the thimble scale reading.

4

Compute the true reading by correcting for zero error.

\text{True reading} = \text{Observed reading} - \text{Zero error} = 1.68\text{ mm} - (-0.03\text{ mm}) = 1.71\text{ mm}

.

True value is found by subtracting the zero error (including its sign) from the observed value.

Key Concept

Zero Error Correction in Micrometer Screw Gauge

Question 4224Question →

The perpendicular bisector of the line segment joining the points $P(2, -1)$ and $Q(6, 7)$ intersects the $y$ -axis at $(0, c)$ . Find the value of $c$ .

Show answer & explanation

Answer: 5

Answer

The value of

c

is 5.

The midpoint of

PQ

is

(4, 3)

and the gradient of

PQ

is

2

. The perpendicular bisector has a gradient of

-\frac{1}{2}

and passes through

(4, 3)

. Substituting these into the line equation gives

y - 3 = -\frac{1}{2}(x - 4)

, which simplifies to

y = -\frac{1}{2}x + 5

. The line intersects the

y

-axis at

(0, 5)

, so

c = 5

.

Step-by-Step Solution

1

Find the midpoint of the line segment

PQ

Midpoint

M = (4, 3)

The perpendicular bisector must pass through the midpoint of the segment.

2

Calculate the gradient of

PQ

Gradient

m_{PQ} = 2

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

for points

(2, -1)

and

(6, 7)

.

3

Determine the gradient of the perpendicular line

Perpendicular gradient

m_{\perp} = -\frac{1}{2}

Perpendicular lines have gradients that are negative reciprocals (

m_1 m_2 = -1

).

4

Formulate the equation of the perpendicular bisector and solve for the

y

-intercept

y = -\frac{1}{2}x + 5

, hence

c = 5

Using point-slope form

y - y_1 = m(x - x_1)

with point

(4, 3)

and

m = -\frac{1}{2}

, setting

x = 0

gives the

y

-intercept.

Key Concept

Perpendicular Bisector and Line Equations

Question 4225Question →

The physical quantity impulse is defined as the product of force and time. Which of the following physical quantities has the same dimensions as impulse?

Linear momentum

Work done

Power

Force

Show answer & explanation

Answer: Linear momentum

Answer

Linear momentum

Linear momentum is calculated as mass times velocity, which yields the base dimensions

[M L T^{-1}]

. This is identical to impulse, which is force times time (

[M L T^{-2}] \times [T] = [M L T^{-1}]

).

Step-by-Step Solution

1

Determine the dimensions of impulse.

\text{Impulse} = \text{Force} \times \text{Time} = [M L T^{-2}][T] = [M L T^{-1}]

Force has base dimensions

[M L T^{-2}]

and time has dimension

[T]

.

2

Determine the dimensions of linear momentum.

\text{Linear Momentum} = \text{Mass} \times \text{Velocity} = [M][L T^{-1}] = [M L T^{-1}]

Mass has base dimension

[M]

and velocity has dimensions

[L T^{-1}]

.

3

Compare the dimensions of impulse and linear momentum.

Both quantities have identical dimensions of

[M L T^{-1}]

.

By the impulse-momentum theorem, impulse equals change in momentum.

Key Concept

Dimensional analysis of physical quantities

Question 4226Question →

A body of mass $0.2\text{ kg}$ undergoes simple harmonic motion with an angular frequency of $10\text{ rad/s}$ and an amplitude of $0.04\text{ m}$ . What is the magnitude of the maximum restoring force acting on the body in newtons?

Show answer & explanation

Answer: 0.8

Answer

The magnitude of the maximum restoring force acting on the body is

0.8\text{ N}

.

The maximum restoring force in simple harmonic motion occurs at maximum displacement (

y = A

) and is given by

F_{\text{max}} = m \omega^2 A

. Substituting

m = 0.2\text{ kg}

,

\omega = 10\text{ rad/s}

, and

A = 0.04\text{ m}

yields

F_{\text{max}} = 0.2 \times (10)^2 \times 0.04 = 0.8\text{ N}

.

Step-by-Step Solution

1

Identify the given physical quantities from the problem statement.

m = 0.2\text{ kg}

,

\omega = 10\text{ rad/s}

, and

A = 0.04\text{ m}

.

These parameters are required to calculate acceleration and restoring force in simple harmonic motion.

2

Calculate the maximum acceleration of the oscillating body.

a_{\text{max}} = \omega^2 A = (10)^2 \times 0.04 = 4.0\text{ m/s}^2

.

In simple harmonic motion, maximum acceleration occurs at maximum displacement (the amplitude).

3

Determine the maximum restoring force.

F_{\text{max}} = m a_{\text{max}} = 0.2 \times 4.0 = 0.8\text{ N}

.

According to Newton's second law, force is the product of mass and acceleration.

Key Concept

Maximum Restoring Force in Simple Harmonic Motion

Question 4227Question →

If $y = e^{2x} \cos 3x$ , what is $\frac{dy}{dx}$ ?

e^{2x}(2\cos 3x - 3\sin 3x)

e^{2x}(2\cos 3x + 3\sin 3x)

e^{2x}(2\cos 3x - \sin 3x)

e^{2x}(\cos 3x - 3\sin 3x)

Show answer & explanation

Answer: $e^{2x}(2\cos 3x - 3\sin 3x)$

Answer

\frac{dy}{dx} = e^{2x}(2\cos 3x - 3\sin 3x)

Applying the product rule to

u = e^{2x}

and

v = \cos 3x

yields

\frac{du}{dx} = 2e^{2x}

and

\frac{dv}{dx} = -3\sin 3x

. Substituting these into

u\frac{dv}{dx} + v\frac{du}{dx}

gives

e^{2x}(-3\sin 3x) + \cos 3x(2e^{2x}) = e^{2x}(2\cos 3x - 3\sin 3x)

.

Step-by-Step Solution

1

Identify the components for the product rule

Let

u = e^{2x}

and

v = \cos 3x

.

The function

y = e^{2x} \cos 3x

is a product of two differentiable functions.

2

Differentiate

u = e^{2x}

with respect to

x

\frac{du}{dx} = 2e^{2x}

By the chain rule,

\frac{d}{dx}(e^{kx}) = k e^{kx}

.

3

Differentiate

v = \cos 3x

with respect to

x

\frac{dv}{dx} = -3\sin 3x

By the chain rule,

\frac{d}{dx}(\cos kx) = -k \sin kx

.

4

Apply the product rule formula

\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}

and factor out

e^{2x}

\frac{dy}{dx} = e^{2x}(-3\sin 3x) + (\cos 3x)(2e^{2x}) = e^{2x}(2\cos 3x - 3\sin 3x)

Combining the products and factoring out the common exponential factor simplifies the expression.

Key Concept

Product Rule and Chain Rule for Exponential and Trigonometric Functions

Question 4228Question →

If $\log_{10}(x) + \log_{10}(5) = 2$ , what is the value of $x$ ?

20

95

10

40

Show answer & explanation

Answer: 20

Answer

20

Using the logarithmic product identity

\log_b(A) + \log_b(B) = \log_b(AB)

, the left side simplifies to

\log_{10}(5x) = 2

. Converting to exponential form gives

5x = 10^2 = 100

, which simplifies directly to

x = 20

.

Step-by-Step Solution

1

Apply the product law of logarithms

\log_{10}(5x) = 2

The sum of logarithms with the same base equals the logarithm of their product:

\log_b(A) + \log_b(B) = \log_b(AB)

.

2

Convert the logarithmic equation to exponential form

5x = 10^2 = 100

By definition of logarithms, if

\log_b(y) = c

, then

y = b^c

.

3

Solve for

x

x = \frac{100}{5} = 20

Divide both sides of the linear equation by

5

.

Key Concept

Logarithmic Product Law and Exponential Conversion

Estimated Time:45s

Question 4229Question →

If $\sqrt{75} + \sqrt{12} - \sqrt{27} = k\sqrt{3}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is 4.

Simplifying each surd into its basic form yields

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

,

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

, and

\sqrt{27} = 3\sqrt{3}

. Combining the coefficients gives

(5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

. Equating

4\sqrt{3}

to

k\sqrt{3}

shows that

k = 4

.

Step-by-Step Solution

1

Simplify each individual surd by factoring out perfect squares

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

,

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

,

\sqrt{27} = 3\sqrt{3}

To combine surds through addition or subtraction, they must be converted to similar surds.

2

Combine the coefficients of the like surds

5\sqrt{3} + 2\sqrt{3} - 3\sqrt{3} = (5 + 2 - 3)\sqrt{3} = 4\sqrt{3}

Like terms with the same radical factor

\sqrt{3}

can be added and subtracted directly.

3

Compare the resulting coefficient with

k\sqrt{3}

k = 4

By direct comparison of coefficients of

\sqrt{3}

,

k

equals 4.

Key Concept

Simplification and combining of similar surds

Question 4230Question →

Two $2 \times 2$ matrices are given as $A = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 2 & k \end{pmatrix}$ . If the determinant of the product matrix $AB$ is equal to $30$ , what is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of k is 3.

The determinant of a product of square matrices equals the product of their individual determinants:

\det(AB) = \det(A) \cdot \det(B)

. Evaluating

\det(A)

yields

(2)(3) - (-1)(4) = 10

, and

\det(B)

yields

(1)(k) - (0)(2) = k

. Substituting these into

\det(AB) = 30

gives

10k = 30

, leading directly to

k = 3

.

Step-by-Step Solution

1

Evaluate the determinant of matrix A

\det(A) = 10

Using the

2 \times 2

determinant formula

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

, we calculate

\det(A) = (2)(3) - (-1)(4) = 6 + 4 = 10

.

2

Evaluate the determinant of matrix B

\det(B) = k

Calculating the determinant of matrix

B

yields

\det(B) = (1)(k) - (0)(2) = k

.

3

Use the product property of determinants to solve for k

k = 3

Since

\det(AB) = \det(A) \cdot \det(B)

, we have

10k = 30

. Dividing both sides by 10 yields

k = 3

.

Key Concept

Determinant of Matrix Product

Estimated Time:1m 30s

Question 4231Question →

The sum of the first $n$ terms of an arithmetic progression (AP) is given by $S_n = 2n^2 + 3n$ . The $3^{\text{rd}}$ term of this AP is equal to the $2^{\text{nd}}$ term of a geometric progression (GP). If the common ratio of the GP is $2$ , what is the sum of the first $4$ terms of the GP?

97.5

127.5

48.75

45.5

Show answer & explanation

Answer: $97.5$

Answer

97.5

Evaluating

S_3 - S_2

gives the 3rd AP term as

27 - 14 = 13

. Setting the 2nd GP term

a(2) = 13

yields

a = 6.5

. The sum of the first 4 terms of the GP is

6.5 \times (2^4 - 1) = 6.5 \times 15 = 97.5

.

Step-by-Step Solution

1

Find the 3rd term (

T_3

) of the AP using the given sum formula

S_n = 2n^2 + 3n

T_3 = S_3 - S_2 = [2(3)^2 + 3(3)] - [2(2)^2 + 3(2)] = [18 + 9] - [8 + 6] = 27 - 14 = 13

The

n

-th term of a sequence is equal to

S_n - S_{n-1}

.

2

Determine the first term (

a

) of the GP

Since

G_2 = 13

and common ratio

r = 2

,

a \cdot r^{2-1} = 13 \implies 2a = 13 \implies a = 6.5

The

n

-th term of a GP is given by

G_n = a r^{n-1}

.

3

Calculate the sum of the first 4 terms of the GP

S_4 = \frac{a(r^4 - 1)}{r - 1} = \frac{6.5(2^4 - 1)}{2 - 1} = 6.5 \times 15 = 97.5

The sum of the first

n

terms of a GP with

r > 1

is

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

.

Key Concept

Combining Arithmetic Progression sum formula with Geometric Progression term and sum formulas

Estimated Time:2m 0s

Question 4232Question →

The table below shows the frequency distribution of marks obtained by a group of students in a mathematics test:

Mark ( $x$ )	2	4	6	8	10
Frequency ( $f$ )	2	1	4	1	2

Find the mean deviation of the distribution.

Show answer & explanation

Answer: 2

Answer

The mean deviation of the distribution is 2.

To find the mean deviation, first calculate the mean

\bar{x} = \frac{\sum fx}{\sum f} = \frac{60}{10} = 6

. Next, sum the absolute deviations multiplied by their frequencies:

\sum f|x - \bar{x}| = 2(4) + 1(2) + 4(0) + 1(2) + 2(4) = 20

. Dividing this total by the sum of frequencies

10

yields a mean deviation of

2

.

Step-by-Step Solution

1

Calculate the arithmetic mean of the distribution

\bar{x} = \frac{\sum f x}{\sum f} = \frac{(2 \times 2) + (1 \times 4) + (4 \times 6) + (1 \times 8) + (2 \times 10)}{2 + 1 + 4 + 1 + 2} = \frac{60}{10} = 6

The mean is required as the central benchmark from which individual deviations are measured.

2

Calculate the sum of absolute deviations weighted by frequency

\sum f |x - \bar{x}| = 2|2 - 6| + 1|4 - 6| + 4|6 - 6| + 1|8 - 6| + 2|10 - 6| = 8 + 2 + 0 + 2 + 8 = 20

Each absolute difference from the mean must be multiplied by its frequency to account for the total deviation.

3

Divide the total absolute deviation by the total frequency

\text{Mean Deviation} = \frac{\sum f |x - \bar{x}|}{\sum f} = \frac{20}{10} = 2

The mean deviation represents the average distance of all observations from the arithmetic mean.

Key Concept

Mean Deviation of a Frequency Distribution

Estimated Time:1m 30s

Question 4233Question →

The mechanical power $P$ dissipated by a circular disc of radius $R$ rotating at an angular velocity $\omega$ in a fluid of density $\rho$ is given by the relation $P = k \cdot \rho^a \cdot \omega^b \cdot R^c$ , where $k$ is a dimensionless constant. Using dimensional analysis, which of the following represents the correct values of the exponents $a$ , $b$ , and $c$ ?

a = 1, b = 3, c = 5

a = 1, b = 2, c = 4

a = 1, b = 3, c = 2

a = 1, b = 1, c = 5

Show answer & explanation

Answer: $a = 1, b = 3, c = 5$

Answer

a = 1, b = 3, c = 5

By writing the dimensions of power as

[M L^2 T^{-3}]

, density as

[M L^{-3}]

, angular velocity as

[T^{-1}]

, and radius as

[L]

, dimensional homogeneity requires that

M L^2 T^{-3} = M^a L^{-3a + c} T^{-b}

. Equating powers gives

a = 1

,

b = 3

, and

c = 5

.

Step-by-Step Solution

1

Express the base dimensions for each physical quantity

Power

[P] = M L^2 T^{-3}

, Density

[\rho] = M L^{-3}

, Angular velocity

[\omega] = T^{-1}

, and Radius

[R] = L

.

Dimensional analysis requires decomposing derived physical quantities into fundamental dimensions of mass (

M

), length (

L

), and time (

T

).

2

Substitute dimensions into the given equation

P = k \cdot \rho^a \cdot \omega^b \cdot R^c

M L^2 T^{-3} = (M L^{-3})^a \cdot (T^{-1})^b \cdot (L)^c = M^a \cdot L^{-3a + c} \cdot T^{-b}

.

The constant

k

is dimensionless, so its dimension is 1.

3

Equate the powers of

M

,

L

, and

T

on both sides of the equation

For mass

M

:

a = 1

. For time

T

:

-b = -3 \implies b = 3

. For length

L

:

-3a + c = 2 \implies -3(1) + c = 2 \implies c = 5

.

According to the principle of dimensional homogeneity, powers of fundamental dimensions must be equal on both sides of a physically correct equation.

Key Concept

Dimensional Homogeneity and Formula Derivation

Estimated Time:2m 0s

Question 4234Question →

Complete the sentence below by supplying a word nearest in meaning to the underlined word 'surreptitious'.

Fill in the blanks below

The auditor noted the treasurer's surreptitious transfer of company funds, observing that the transactions were executed in a completely manner to evade detection.

Show answer & explanation

Answer

A suitable word nearest in meaning to 'surreptitious' is 'stealthy' (or 'furtive', 'clandestine', 'covert', 'secret').

In this context, 'surreptitious' means done, made, or acquired by stealth or secret means. Words like 'stealthy', 'furtive', 'clandestine', and 'covert' directly capture this meaning of acting secretly to avoid observation.

Step-by-Step Solution

1

Analyze the context of the sentence

The word 'surreptitious' describes actions performed quietly, secretly, or evasively to avoid being noticed or detected.

Understanding the context of financial transactions designed to avoid detection clarifies the required meaning.

2

Identify synonymous terms

Terms such as 'stealthy', 'furtive', 'clandestine', or 'covert' accurately describe actions taken secretly.

The correct replacement must maintain the sense of concealment and stealth expressed by the target word.

Key Concept

Synonyms and Words Nearest in Meaning

Estimated Time:1m 0s

Question 4235Question →

In the dimensional equation for the period of oscillation of a simple pendulum, $T = k g^a l^b$ , where $T$ is the period, $g$ is the acceleration due to gravity, $l$ is the length of the pendulum, and $k$ is a dimensionless constant, what is the numerical value of the exponent $a$ ?

Show answer & explanation

Answer: -0.5

Answer

The numerical value of the exponent

a

is -0.5.

Applying dimensional analysis to

T = k g^a l^b

, the dimension of the left-hand side is

\text{T}^1

. The right-hand side has dimensions

(\text{L T}^{-2})^a (\text{L})^b = \text{L}^{a+b} \text{T}^{-2a}

. Equating the exponents of time

\text{T}

yields

1 = -2a

, which gives

a = -0.5

.

Step-by-Step Solution

1

Express the dimensions of all physical quantities involved in fundamental base dimensions (M, L, T).

The dimension of period

T

is

[\text{T}]

, length

l

is

[\text{L}]

, and gravitational acceleration

g

is

[\text{L T}^{-2}]

.

Dimensional analysis requires substituting each quantity with its fundamental dimensions.

2

Formulate the dimensional homogeneity equation.

\text{M}^0 \text{L}^0 \text{T}^1 = (\text{L T}^{-2})^a (\text{L})^b = \text{L}^{a+b} \text{T}^{-2a}

.

The principle of dimensional homogeneity states that the exponents of base dimensions on both sides of a physically correct equation must be equal.

3

Equate exponents of time

\text{T}

and solve for

a

.

1 = -2a \implies a = -\frac{1}{2} = -0.5

.

Comparing the powers of

\text{T}

gives a linear equation in

a

.

Key Concept

Principle of Dimensional Homogeneity

Estimated Time:1m 15s

Question 4236Question →

The investigative panel noted that leaking confidential executive files prior to the hearing would be highly prejudicial _____ the proceedings and completely subversive _____ established board protocol. Which pair of prepositions correctly completes the sentence?

against / to

to / of

for / about

toward / for

Show answer & explanation

Answer: to / of

Answer

The pair 'to / of' correctly completes the sentence because 'prejudicial to' and 'subversive of' are the fixed, standard prepositional collocations.

The correct option provides the exact prepositional collocations required by the adjectives: 'prejudicial to' means harmful or damaging to an outcome, and 'subversive of' means seeking to undermine an established authority or protocol.

Step-by-Step Solution

1

Identify the dependent preposition required by the formal adjective 'prejudicial'.

In formal academic and legal contexts, 'prejudicial' requires the preposition 'to' when taking an object (e.g., prejudicial to the proceedings).

Adjectives denoting harm or detriment, such as 'prejudicial', 'inimical', and 'detrimental', grammatically govern 'to'.

2

Identify the dependent preposition required by the adjective 'subversive'.

The adjective 'subversive' requires the preposition 'of' when followed by a noun phrase complement (e.g., subversive of protocol).

Adjectives and nouns indicating destruction, undermining, or overthrowing typically collocate with 'of'.

Key Concept

Dependent prepositions following adjectives of impact and subversion

Question 4237Question →

A sector of a circle of radius $14\text{ cm}$ subtends an angle of $90^\circ$ at the centre of the circle. What is the area of the sector in $\text{cm}^2$ ? (Take $\pi = \frac{22}{7}$ )

Show answer & explanation

Answer: 154

Answer

The area of the sector is

154\text{ cm}^2

.

The area of a circular sector is given by

\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2

. Substituting

\theta = 90^\circ

,

r = 14\text{ cm}

, and

\pi = \frac{22}{7}

yields

\frac{90}{360} \times \frac{22}{7} \times 14^2 = \frac{1}{4} \times 616 = 154\text{ cm}^2

.

Step-by-Step Solution

1

Identify the formula for the area of a circular sector.

\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2

The area of a sector is proportional to the central angle it subtends relative to a full circle (

360^\circ

).

2

Substitute the given values into the formula.

\text{Area} = \frac{90^\circ}{360^\circ} \times \frac{22}{7} \times (14)^2

Given radius

r = 14\text{ cm}

, angle

\theta = 90^\circ

, and

\pi = \frac{22}{7}

.

3

Simplify the fraction and calculate the numerical value.

\text{Area} = \frac{1}{4} \times \frac{22}{7} \times 196 = \frac{1}{4} \times 22 \times 28 = 22 \times 7 = 154\text{ cm}^2

Simplifying

\frac{90}{360}

yields

\frac{1}{4}

and dividing

196

by

7

gives

28

.

Key Concept

Area of a sector of a circle

Estimated Time:45s

Question 4238Question →

A curve is defined by the equation $y = x^2 - 6x + 11$ . What is the minimum value of $y$ on this curve?

Show answer & explanation

Answer: 2

Answer

The minimum value of

y

on the curve is 2.

Differentiating

y = x^2 - 6x + 11

gives

\frac{dy}{dx} = 2x - 6

. Setting this derivative to zero yields

2x - 6 = 0

, so

x = 3

. Substituting

x = 3

into the original function gives

y = (3)^2 - 6(3) + 11 = 2

. Since

\frac{d^2y}{dx^2} = 2 > 0

, the point at

x = 3

is a local minimum, making 2 the minimum value of

y

.

Step-by-Step Solution

1

Find the first derivative of the curve function.

\frac{dy}{dx} = 2x - 6

Stationary points occur where the derivative is equal to zero.

2

Solve for the

x

-coordinate at the stationary point.

2x - 6 = 0 \implies x = 3

Setting the derivative to zero determines the input value where the slope is horizontal.

3

Calculate the corresponding

y

-value at

x = 3

.

y = (3)^2 - 6(3) + 11 = 2

Evaluating the original equation at

x = 3

yields the minimum value of

y

.

Key Concept

Finding the minimum value of a function using differentiation

Question 4239Question →

Given the matrices $A = \begin{pmatrix} 2 & 3 \\ 1 & m \end{pmatrix}$ and $B = \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ , if the matrix product $AB = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$ , what is the value of $m$ ?

-1

1

-3

-5

Show answer & explanation

Answer: $-1$

Answer

The value of

m

is

-1

.

Multiplying the

2 \times 2

matrix by the

2 \times 1

column vector yields a

2 \times 1

result where the second row entry is

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

. Setting this equal to the second row element

6

gives

4 - 2m = 6

, which simplifies to

m = -1

.

Step-by-Step Solution

1

Compute the matrix product

AB

AB = \begin{pmatrix} 2(4) + 3(-2) \\ 1(4) + m(-2) \end{pmatrix} = \begin{pmatrix} 8 - 6 \\ 4 - 2m \end{pmatrix} = \begin{pmatrix} 2 \\ 4 - 2m \end{pmatrix}

Matrix multiplication requires taking the dot product of each row of the first matrix with the column vector.

2

Equate the resulting vector to the given vector

\begin{pmatrix} 2 \\ 6 \end{pmatrix}

4 - 2m = 6

Corresponding elements of equal matrices must be equal.

3

Solve the linear equation for

m

-2m = 6 - 4 \Rightarrow -2m = 2 \Rightarrow m = -1

Isolating the variable

m

by subtracting

4

from both sides and dividing by

-2

.

Key Concept

Matrix Multiplication and Equality of Matrices

Question 4240Question →

The distribution of daily solar energy generation (in kWh) recorded at an agricultural research station over a period of days is summarized in the table below:

Daily Generation (kWh)	Frequency ( $f$ )
$10 - 19$	$3$
$20 - 29$	$x$
$30 - 39$	$8$
$40 - 49$	$5$
$50 - 59$	$x + 2$

If the mean daily solar energy generation for the recorded period is $37.5\text{ kWh}$ , what is the value of $x$ ?

1

3

6

11

Show answer & explanation

Answer: $6$

Answer

6

The exact class marks are

14.5, 24.5, 34.5, 44.5,

and

54.5

. The total frequency is

\sum f = 18 + 2x

and the sum of products is

\sum f x = 651 + 79x

. Substituting these into

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

with

\bar{x} = 37.5

gives

675 + 75x = 651 + 79x

, which solves to

4x = 24

, giving the value

6

.

Step-by-Step Solution

1

Determine the class mark (midpoint

x_i

) for each class interval

The midpoints are

14.5, 24.5, 34.5, 44.5,

and

54.5

.

The class mark is calculated as

\frac{\text{Lower Limit} + \text{Upper Limit}}{2}

.

2

Calculate total frequency

\sum f

and total product sum

\sum f x_i

\sum f = 3 + x + 8 + 5 + (x + 2) = 18 + 2x

.

\sum f x_i = 3(14.5) + x(24.5) + 8(34.5) + 5(44.5) + (x+2)(54.5) = 43.5 + 24.5x + 276 + 222.5 + 54.5x + 109 = 651 + 79x

To compute the mean of grouped data, sum the products of each frequency and its corresponding class mark.

3

Set up the mean formula equation and solve for

x

\bar{x} = \frac{\sum f x_i}{\sum f} \implies 37.5 = \frac{651 + 79x}{18 + 2x}

37.5(18 + 2x) = 651 + 79x

675 + 75x = 651 + 79x

79x - 75x = 675 - 651

4x = 24 \implies x = 6

Equating the expression for mean to the given mean value

37.5\text{ kWh}

yields a linear equation in

x

.

Key Concept

Measures of Central Tendency for Grouped Data - Finding Unknown Frequencies from Grouped Mean

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept