All practice questions

Question 4161Question →

If $x$ , $y$ , and $z$ are non-zero real numbers satisfying the exponential equation $2^x = 5^y = 100^z$ , what is the numerical value of the expression $z \left( \frac{2}{x} + \frac{2}{y} \right)$ ?

Show answer & explanation

Answer: 1

Answer

The numerical value of

z \left( \frac{2}{x} + \frac{2}{y} \right)

is

1

.

By setting

2^x = 5^y = 100^z = k

, we can write

2 = k^{1/x}

,

5 = k^{1/y}

, and

100 = k^{1/z}

. Factoring

100 = 2^2 \times 5^2

gives

k^{1/z} = (k^{1/x})^2 \times (k^{1/y})^2 = k^{2/x + 2/y}

. Equating exponents gives

\frac{1}{z} = \frac{2}{x} + \frac{2}{y}

, which upon multiplying by

z

yields

1

.

Step-by-Step Solution

1

Equate the given exponential expressions to a common constant

k

.

2^x = 5^y = 100^z = k

Introducing a common variable allows isolating each base exponent combination.

2

Express bases

2

,

5

, and

100

in terms of

k

using fractional indices.

2 = k^{\frac{1}{x}}

,

5 = k^{\frac{1}{y}}

,

100 = k^{\frac{1}{z}}

Applying the power law

(a^m)^{\frac{1}{m}} = a

isolates each base.

3

Express

100

using prime factorization of the other bases.

100 = 2^2 \times 5^2

Establishing a numerical relationship between

100

,

2

, and

5

links the exponential variables.

4

Substitute the

k

-expressions into

100 = 2^2 \times 5^2

and apply index multiplication laws.

k^{\frac{1}{z}} = \left(k^{\frac{1}{x}}\right)^2 \times \left(k^{\frac{1}{y}}\right)^2 = k^{\frac{2}{x}} \times k^{\frac{2}{y}} = k^{\frac{2}{x} + \frac{2}{y}}

Multiplying powers with the same base requires adding the exponents:

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

.

5

Equate exponents of identical bases.

\frac{1}{z} = \frac{2}{x} + \frac{2}{y}

If

k^a = k^b

for

k > 1

, then

a = b

.

6

Multiply both sides of the equation by

z

.

z \left( \frac{2}{x} + \frac{2}{y} \right) = 1

Rearranging the equation yields the exact numerical value of the requested expression.

Key Concept

Equating Exponents of Common Bases and Fractional Indices

Question 4162Question →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b - 2ab$ . If $(x \ast 3) \ast 2 = 38$ , find the value of $x$ .

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

Applying the binary operation definition sequentially yields

x \ast 3 = 3 - 5x

for the inner expression, and

(3 - 5x) \ast 2 = 15x - 7

for the composite expression. Equating

15x - 7 = 38

leads to

15x = 45

, giving

x = 3

.

Step-by-Step Solution

1

Evaluate the inner binary operation expression

x \ast 3

x \ast 3 = 3 - 5x

Apply the definition

a \ast b = a + b - 2ab

with

a = x

and

b = 3

.

2

Evaluate the outer binary operation

(3 - 5x) \ast 2

(3 - 5x) \ast 2 = 15x - 7

Substitute the result from step 1 into the outer operation definition with

a = 3 - 5x

and

b = 2

.

3

Set the resulting expression equal to 38 and solve the linear equation

x = 3

Solve

15x - 7 = 38

by adding 7 to both sides to get

15x = 45

, then dividing by 15.

Key Concept

Nested composition of defined binary operations

Question 4163Question →

In a survey of $150$ subscribers of a digital media platform, $75$ prefer High-Definition Audio ( $H$ ), $70$ prefer Offline Downloads ( $D$ ), and $65$ prefer Ad-free Listening ( $A$ ). It is observed that $35$ subscribers prefer both $H$ and $D$ , $30$ prefer both $D$ and $A$ , and $25$ prefer both $H$ and $A$ . If $15$ subscribers prefer none of these three features, how many subscribers prefer exactly two of these features?

Show answer & explanation

Answer: 45

Answer

The number of subscribers who prefer exactly two of the features is 45.

To find the number of subscribers who prefer exactly two features, we first calculate the cardinality of the union of all three sets as

150 - 15 = 135

. Applying the 3-set inclusion-exclusion formula gives the number of subscribers preferring all three features as

15

. Subtracting

15

from each pairwise intersection gives the exclusive regions:

20

for

H

and

D

only,

15

for

D

and

A

only, and

10

for

H

and

A

only. Summing these three exclusive regions gives

20 + 15 + 10 = 45

.

Step-by-Step Solution

1

Determine the total number of subscribers who prefer at least one feature.

|H \cup D \cup A| = 135

Subtracting the number of subscribers who prefer none of the features (

15

) from the universal set size (

150

) gives

|H \cup D \cup A| = 150 - 15 = 135

.

2

Apply the Principle of Inclusion-Exclusion for three sets to find the number of subscribers who prefer all three features.

|H \cap D \cap A| = 15

Substitute the known cardinalities into

|H \cup D \cup A| = |H| + |D| + |A| - |H \cap D| - |D \cap A| - |H \cap A| + |H \cap D \cap A|

to get

135 = 75 + 70 + 65 - (35 + 30 + 25) + |H \cap D \cap A|

, which simplifies to

135 = 120 + |H \cap D \cap A|

, giving

|H \cap D \cap A| = 15

.

3

Calculate the number of subscribers preferring exactly two features by subtracting the triple intersection from each pairwise intersection.

45 subscribers

Subscribers preferring only

H

and

D = 35 - 15 = 20

, only

D

and

A = 30 - 15 = 15

, and only

H

and

A = 25 - 15 = 10

. Summing these exclusive regions yields

20 + 15 + 10 = 45

.

Key Concept

Cardinality of set operations and 3-set inclusion-exclusion principle

Question 4164Question →

Although the regional manager was initially hesitant, he eventually became reconciled _____ the sweeping corporate reforms, especially after the board prevailed _____ him to lead the implementation committee.

Which of the following pairs of prepositions correctly completes the sentence above?

to / upon

with / over

with / against

at / for

Show answer & explanation

Answer: to / upon

Answer

The correct prepositions are 'to / upon'.

The preposition 'to' correctly completes the expression 'reconciled to', which signifies becoming resigned to or accepting an unwelcome circumstance. The preposition 'upon' correctly completes 'prevail upon', an established idiomatic construction meaning to persuade someone to take action.

Step-by-Step Solution

1

Analyze the first blank following 'reconciled' in the context of accepting an unwanted situation ('sweeping corporate reforms').

The correct preposition is 'to' because 'reconciled to' means coming to accept an unpleasant or undesirable state of affairs.

Using 'with' would incorrectly imply restoring personal harmony with an individual or group rather than resigning oneself to a situation.

2

Analyze the second blank following 'prevailed' in the context of influencing or persuading someone ('him to lead the implementation committee').

The correct preposition is 'upon' (or 'on') because 'prevail upon' is an idiomatic verb-preposition pair meaning to successfully persuade or induce someone to take action.

Using 'over' or 'against' would alter the meaning to defeating or gaining victory over an adversary.

Key Concept

Dependent prepositions and contextual prepositional collocations

Estimated Time:1m 0s

Question 4165Question →

Find the positive value of $x$ that satisfies the simultaneous equations $y = x + 2$ and $y = x^2 - 4$ .

Show answer & explanation

Answer: 3

Answer

The positive value of

x

is 3.

Equating the linear equation

y = x + 2

and the quadratic equation

y = x^2 - 4

yields

x^2 - x - 6 = 0

. Factorizing this quadratic equation gives

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

, which yields roots

x = 3

and

x = -2

. Selecting the positive value gives 3.

Step-by-Step Solution

1

Equate the linear and quadratic equations

x + 2 = x^2 - 4

Since both expressions are equal to

y

, set them equal to each other to solve for

x

.

2

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - x - 6 = 0

Subtract

x

and

2

from both sides of the equation.

3

Factorize the quadratic expression

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

Find two factors of

-6

that add up to

-1

, which are

-3

and

2

.

4

Determine the roots and select the positive value

x = 3

Setting each factor to zero gives

x = 3

or

x = -2

. Selecting the positive root yields

3

.

Key Concept

Solving simultaneous linear and quadratic equations by substitution

Question 4166Question →

What is the smallest non-negative integer $k$ that satisfies the modular congruence $7^{99} + k \equiv -15 \pmod{11}$ ?

10

1

7

8

Show answer & explanation

Answer: 10

Answer

The correct answer is 10.

Using Fermat's Little Theorem,

7^{10} \equiv 1 \pmod{11}

, which simplifies

7^{99} \pmod{11}

to

7^9 \equiv 8 \pmod{11}

. Reducing the right-hand side gives

-15 \equiv 7 \pmod{11}

. The modular equation

8 + k \equiv 7 \pmod{11}

yields

k \equiv -1 \pmod{11}

. Adding the modulus

11

gives the canonical positive remainder

10

.

Step-by-Step Solution

1

Simplify

7^{99} \pmod{11}

using Fermat's Little Theorem.

7^{10} \equiv 1 \pmod{11}

, so

7^{99} = (7^{10})^9 \times 7^9 \equiv 1^9 \times 7^9 \equiv 7^9 \pmod{11}

.

Since

11

is prime and

\gcd(7, 11) = 1

, Fermat's Little Theorem allows exponent reduction modulo

10

.

2

Compute

7^9 \pmod{11}

.

7^1 \equiv 7

,

7^2 = 49 \equiv 5

,

7^4 \equiv 5^2 = 25 \equiv 3

,

7^5 \equiv 7 \times 3 = 21 \equiv 10

,

7^9 = 7^5 \times 7^4 \equiv 10 \times 3 = 30 \equiv 8 \pmod{11}

.

Repeated squaring and modular multiplication efficiently reduces

7^9

modulo

11

.

3

Reduce the right-hand side

-15 \pmod{11}

.

-15 = -2(11) + 7 \equiv 7 \pmod{11}

.

Converting negative numbers into the standard non-negative remainder range

[0, 10]

.

4

Substitute remainders into the congruence and solve for

k

.

8 + k \equiv 7 \implies k \equiv 7 - 8 = -1 \pmod{11}

.

Linear algebraic rearrangement in modular arithmetic.

5

Convert the negative remainder

-1

to canonical non-negative form.

k = -1 + 11 = 10

.

The standard remainder must satisfy

0 \leq k < 11

.

Key Concept

Modular Exponentiation & Negative Remainder Reduction

Question 4167Question →

Which of the following sets contains all the values of $x$ in the interval $0^\circ \le x \le 360^\circ$ that satisfy the trigonometric equation $2\cos^2 x + 3\sin x - 3 = 0$ ?

30^\circ, 90^\circ, 150^\circ

60^\circ, 90^\circ, 120^\circ

30^\circ, 150^\circ

60^\circ, 300^\circ

Show answer & explanation

Answer: $30^\circ, 90^\circ, 150^\circ$

Answer

The values of

x

in the interval

0^\circ \le x \le 360^\circ

satisfying the equation are

30^\circ, 90^\circ,

and

150^\circ

.

By substituting

\cos^2 x = 1 - \sin^2 x

, the equation reduces to

2\sin^2 x - 3\sin x + 1 = 0

, which factors into

(2\sin x - 1)(\sin x - 1) = 0

. Solving

\sin x = 1/2

gives

x = 30^\circ

and

x = 150^\circ

within the specified domain. Solving

\sin x = 1

gives

x = 90^\circ

. Combining these yields the set of solutions

30^\circ, 90^\circ, 150^\circ

.

Step-by-Step Solution

1

Use the Pythagorean trigonometric identity

\cos^2 x = 1 - \sin^2 x

to rewrite the equation in terms of

\sin x

.

2(1 - \sin^2 x) + 3\sin x - 3 = 0 \implies 2 - 2\sin^2 x + 3\sin x - 3 = 0

Converting the equation to involve a single trigonometric function allows it to be solved as a quadratic equation.

2

Simplify and rearrange the equation into standard quadratic form.

-2\sin^2 x + 3\sin x - 1 = 0 \implies 2\sin^2 x - 3\sin x + 1 = 0

Multiplying by

-1

simplifies factoring.

3

Factor the quadratic equation

(2\sin x - 1)(\sin x - 1) = 0

to solve for

\sin x

.

\sin x = \frac{1}{2}

or

\sin x = 1

Setting each linear factor to zero yields the possible values for

\sin x

.

4

Determine all values of

x

in the domain

0^\circ \le x \le 360^\circ

for each case.

For

\sin x = \frac{1}{2}

,

x = 30^\circ

and

x = 180^\circ - 30^\circ = 150^\circ

. For

\sin x = 1

,

x = 90^\circ

.

Sine is positive in the first and second quadrants, and equals 1 at

90^\circ

.

Key Concept

Solving quadratic trigonometric equations by using fundamental identities to express the equation in terms of a single trigonometric function.

Estimated Time:2m 0s

Question 4168Question →

A binary operation $\circ$ defined on the set of real numbers $\mathbb{R} \setminus \{1\}$ is given by $a \circ b = a + b - ab$ . What is the inverse of $3$ under this operation?

\frac{3}{2}

-\frac{3}{2}

1

\frac{1}{3}

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

The inverse of

3

under the given operation is

\frac{3}{2}

.

To find the inverse of

3

under the operation

a \circ b = a + b - ab

, we must first determine the identity element

e

. Setting

a \circ e = a

yields

a + e - ae = a

, which simplifies to

e(1 - a) = 0

. For all

a \neq 1

, the identity element is

e = 0

. Next, using the definition of inverse

3 \circ x = 0

, we substitute into the operational formula to obtain

3 + x - 3x = 0 \implies 3 - 2x = 0 \implies x = \frac{3}{2}

. Thus, the option specifying

\frac{3}{2}

is correct.

Step-by-Step Solution

1

Find the identity element

e

of the operation

\circ

.

e = 0

By definition of identity,

a \circ e = a

. Substituting into the operational formula gives

a + e - ae = a \implies e(1 - a) = 0

. Since

a \neq 1

,

e = 0

.

2

Set up the inverse equation for

3

, letting

x

be the inverse of

3

.

3 \circ x = 0

By definition of an inverse element,

a \circ a^{-1} = e

.

3

Expand

3 \circ x

using the operation rule and solve for

x

.

x = \frac{3}{2}

3 + x - 3x = 0 \implies 3 - 2x = 0 \implies 2x = 3 \implies x = \frac{3}{2}

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 4169Question →

Using differentiation from first principles, what is the numerical value of the derivative of the function $f(x) = 3x^2 - 4x + 1$ at $x = 2$ ?

Show answer & explanation

Answer: 8

Answer

The numerical value of the derivative of

f(x) = 3x^2 - 4x + 1

at

x = 2

is 8.

Using the first-principles limit definition, the increment

f(2+h) - f(2)

simplifies to

8h + 3h^2

. Dividing by

h

yields

8 + 3h

, which evaluates to 8 as

h \to 0

.

Step-by-Step Solution

1

Calculate

f(2)

f(2) = 5

Substitute

x = 2

into

f(x) = 3x^2 - 4x + 1

.

2

Expand

f(2+h)

f(2+h) = 5 + 8h + 3h^2

Substitute

x = 2+h

into

f(x)

and expand algebraically.

3

Simplify the difference quotient

\frac{f(2+h) - f(2)}{h}

\frac{8h + 3h^2}{h} = 8 + 3h

Subtract

f(2)

from

f(2+h)

and divide every term by

h

.

4

Evaluate the limit as

h \to 0

f'(2) = 8

As

h

approaches 0, the term

3h

vanishes, leaving 8.

Key Concept

Differentiation from First Principles

Estimated Time:1m 30s

Question 4170Question →

Evaluate the limit $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ . What is the numerical value of this limit?

Show answer & explanation

Answer: 4

Answer

The value of the limit is

4

.

Direct substitution of

x = 2

produces the indeterminate form

\frac{0}{0}

. Factoring the numerator gives

\frac{(x-2)(x+2)}{x-2}

. Canceling the non-zero factor

(x-2)

simplifies the expression to

x+2

. Evaluating the limit as

x

approaches

2

yields

2 + 2 = 4

.

Step-by-Step Solution

1

Check the form by direct substitution of

x = 2

Obtained the indeterminate form

\frac{0}{0}

Direct substitution results in division by zero, requiring algebraic simplification.

2

Factor the polynomial in the numerator

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

Difference of two squares factorization allows canceling common terms.

3

Cancel the common factor

(x - 2)

\frac{(x - 2)(x + 2)}{x - 2} = x + 2

For

x \neq 2

, division by

(x - 2)

is valid.

4

Evaluate the simplified limit as

x \to 2

2 + 2 = 4

Substitute

x = 2

directly into the continuous polynomial

x + 2

.

Key Concept

Evaluating indeterminate limits of the form

\frac{0}{0}

via factorization

Question 4171Question →

In the sentence below, what does the underlined idiomatic expression mean?

'Facing a severe revenue deficit, the chief executive officer decided to bite the bullet and implement the painful restructuring plan.'

face a painful or difficult situation with courage

physically bite down on metal objects to relieve stress

postpone making hard choices until conditions improve

blame external financial consultants for the revenue loss

Show answer & explanation

Answer: face a painful or difficult situation with courage

Answer

The expression 'bite the bullet' means to face a painful or difficult situation with courage.

The idiom 'bite the bullet' means to force oneself to undergo something difficult or unpleasant stoically. In the context of administrative decision-making during a financial crisis, it means courageously accepting the necessity of a hard decision.

Step-by-Step Solution

1

Analyze the context of the sentence

The chief executive officer must implement an uncomfortable restructuring plan due to financial difficulty.

Contextual clues point to accepting an unpleasant but necessary action.

2

Identify the figurative meaning of 'bite the bullet'

The idiom means accepting an unavoidable, tough situation bravely.

Idiomatic expressions convey non-literal meanings established through conventional usage.

Key Concept

Contextual interpretation of idiomatic expressions

Question 4172Question →

Three business partners share a total profit of ₦60,000 in the ratio $1 : 2 : 3$ . What is the share of the partner who receives the largest portion, in naira?

Show answer & explanation

Answer: 30000

Answer

The share of the partner receiving the largest portion is ₦30,000.

Sum the parts of the ratio (

1 + 2 + 3 = 6

). The largest share corresponds to 3 parts out of 6. Calculating

\frac{3}{6} \times 60,000

yields 30,000 naira.

Step-by-Step Solution

1

Calculate the sum of all ratio parts

1 + 2 + 3 = 6 parts

The total amount is divided into equal parts represented by the sum of the ratio numbers.

2

Determine the monetary value of a single part

₦60,000 / 6 = ₦10,000 per part

Dividing the total sum by the total number of parts gives the value of one unit part.

3

Calculate the largest share corresponding to 3 parts

3 × ₦10,000 = ₦30,000

The largest portion of the ratio is 3 parts.

Key Concept

Direct Ratio Sharing

Question 4173Question →

Two independent events, $A$ and $B$ , have probabilities $P(A) = 0.4$ and $P(B) = 0.5$ . What is the probability that both event $A$ and event $B$ occur, $P(A \cap B)$ ?

Show answer & explanation

Answer: 0.2

Answer

The probability that both events occur is 0.2.

For independent events, the joint probability of both events occurring simultaneously is found by multiplying their individual probabilities:

P(A \cap B) = P(A) \times P(B) = 0.4 \times 0.5 = 0.2

.

Step-by-Step Solution

1

Identify event independence and the required probability operation

Events

A

and

B

are independent, and the question requires calculating their intersection

P(A \cap B)

.

The problem explicitly states that the events are independent.

2

Apply the multiplication law for independent events

P(A \cap B) = P(A) \times P(B)

For any two independent events, the probability of both occurring together is the product of their individual probabilities.

3

Perform the multiplication

P(A \cap B) = 0.4 \times 0.5 = 0.2

Multiplying 0.4 by 0.5 gives 0.2.

Key Concept

Multiplication Law of Probability for Independent Events

Question 4174Question →

The sum of the first $n$ terms of an arithmetic progression is given by $S_n = 2n^2 + 3n$ . What is the value of the $7^{\text{th}}$ term of the progression?

Show answer & explanation

Answer: 29

Answer

The

7^{\text{th}}

term of the arithmetic progression is

29

.

For any sequence, the

n

-th term is found using

T_n = S_n - S_{n-1}

. Substituting

n = 7

gives

S_7 = 2(7)^2 + 3(7) = 119

and

S_6 = 2(6)^2 + 3(6) = 90

. Thus,

T_7 = 119 - 90 = 29

.

Step-by-Step Solution

1

State the relationship between the

n

-th term

T_n

and the sum of first

n

terms

S_n

T_n = S_n - S_{n-1}

The sum of the first

n

terms minus the sum of the first

n-1

terms equals the

n

-th term.

2

Calculate the sum of the first 7 terms (

S_7

)

S_7 = 2(7)^2 + 3(7) = 119

Substitute

n = 7

into the sum formula

S_n = 2n^2 + 3n

.

3

Calculate the sum of the first 6 terms (

S_6

)

S_6 = 2(6)^2 + 3(6) = 90

Substitute

n = 6

into the sum formula

S_n = 2n^2 + 3n

.

4

Compute the

7^{\text{th}}

term (

T_7

)

T_7 = 119 - 90 = 29

Subtract

S_6

from

S_7

.

Key Concept

Relationship between the nth term and the sum of first n terms of an AP

Question 4175Question →

A curve is defined by the equation $y = x^3 + px^2 + qx + 5$ , where $p$ and $q$ are constants. If the curve has a stationary point with a local maximum at $x = -1$ and a local minimum at $x = 3$ , what is the value of $p + q$ ?

-12

12

-15

-6

Show answer & explanation

Answer: $-12$

Answer

The value of

p + q

is

-12

.

The derivative of

y = x^3 + px^2 + qx + 5

is

\frac{dy}{dx} = 3x^2 + 2px + q

. Setting

\frac{dy}{dx} = 0

at the stationary points

x = -1

and

x = 3

means

\frac{dy}{dx} = 3(x + 1)(x - 3) = 3x^2 - 6x - 9

. Comparing coefficients gives

2p = -6 \implies p = -3

and

q = -9

. Summing these constants gives

p + q = -12

. Evaluating the second derivative

\frac{d^2y}{dx^2} = 6x + 2p = 6x - 6

confirms a maximum at

x = -1

(

\frac{d^2y}{dx^2} = -12 < 0

) and a minimum at

x = 3

(

\frac{d^2y}{dx^2} = 12 > 0

).

Step-by-Step Solution

1

Find the derivative of the given function.

\frac{dy}{dx} = 3x^2 + 2px + q

Stationary points occur where the first derivative

\frac{dy}{dx} = 0

.

2

Use the stationary points

x = -1

and

x = 3

to form a quadratic equation for the derivative.

\frac{dy}{dx} = 3(x + 1)(x - 3) = 3(x^2 - 2x - 3) = 3x^2 - 6x - 9

Since

x = -1

and

x = 3

are roots of

\frac{dy}{dx} = 0

, the derivative must factor as

3(x - (-1))(x - 3)

.

3

Equate coefficients of the two derivative expressions to solve for

p

and

q

.

2p = -6 \implies p = -3

and

q = -9

Matching corresponding terms gives

2p = -6

and

q = -9

.

4

Calculate the required sum

p + q

.

p + q = -3 + (-9) = -12

Summing the calculated constants yields

-12

.

Key Concept

Determining parameters of a polynomial function from given stationary points using differentiation and coefficient matching.

Question 4176Question →

The mean mark of a student in $6$ tests is $14$ . When the highest and lowest marks, which differ by $12$ , are excluded, the mean mark of the remaining $4$ tests becomes $13.5$ . What is the highest mark?

Show answer & explanation

Answer: 21

Answer

The highest mark is 21.

The total sum of all 6 tests is

6 \times 14 = 84

. When the highest (

H

) and lowest (

L

) marks are removed, the total sum of the remaining 4 tests is

4 \times 13.5 = 54

. The sum of the excluded marks is

H + L = 84 - 54 = 30

. Knowing that their difference is

H - L = 12

, we add the two equations to get

2H = 42

, which gives

H = 21

.

Step-by-Step Solution

1

Calculate the total sum of all 6 test marks

Sum of 6 marks =

6 \times 14 = 84

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

2

Calculate the sum of the remaining 4 test marks

Sum of 4 marks =

4 \times 13.5 = 54

Multiplying the new mean by 4 gives the sum of the test scores excluding the highest and lowest values.

3

Determine the combined sum of the highest (H) and lowest (L) marks

H + L = 84 - 54 = 30

Subtracting the sum of the 4 remaining marks from the total initial sum yields the sum of the two excluded marks.

4

Solve for H using the simultaneous linear equations

H = 21

Adding H + L = 30 and H - L = 12 gives 2H = 42, which solves to H = 21.

Key Concept

Mean of Ungrouped Data and Handling Excluded Values

Question 4177Question →

The sum of the first $n$ terms of an arithmetic progression (A.P.) is $210$ . If the first term is $3$ and the last term is $39$ , what is the value of $n$ ?

10

11

5

9

Show answer & explanation

Answer: 10

Answer

The number of terms

n

is

10

.

Using the standard sum formula

S_n = \frac{n}{2}(a + l)

for an A.P. with known first term

a = 3

and last term

l = 39

, we set

210 = \frac{n}{2}(3 + 39) = 21n

. Solving for

n

gives

n = 10

.

Step-by-Step Solution

1

Identify the given parameters of the arithmetic progression.

First term

a = 3

, last term

l = 39

, and sum

S_n = 210

.

These values are required to apply the sum formula for an A.P.

2

Apply the sum formula

S_n = \frac{n}{2}(a + l)

.

210 = \frac{n}{2}(3 + 39) = \frac{n}{2}(42) = 21n

.

The sum of

n

terms in an A.P. with a known first and last term is given by

\frac{n}{2}(a + l)

.

3

Solve for

n

.

n = \frac{210}{21} = 10

.

Dividing the total sum by

21

gives the exact number of terms.

Key Concept

Sum of an Arithmetic Progression using first and last terms

Estimated Time:1m 30s

Question 4178Question →

A convex polygon has $54$ diagonals. How many distinct triangles can be formed by joining any three of its vertices?

Show answer & explanation

Answer: 220

Answer

220 distinct triangles

Solving the equation for the number of diagonals

\frac{n(n-3)}{2} = 54

yields

n = 12

vertices. The number of triangles that can be formed by selecting any 3 of these 12 vertices is given by

\binom{12}{3} = \frac{12 \times 11 \times 10}{6} = 220

.

Step-by-Step Solution

1

Determine the number of vertices

n

of the polygon using the diagonals formula.

n = 12

The number of diagonals

D

in an

n

-sided convex polygon is given by

D = \binom{n}{2} - n = \frac{n(n-3)}{2}

. Setting

\frac{n(n-3)}{2} = 54

gives

n^2 - 3n - 108 = 0

. Factoring

(n - 12)(n + 9) = 0

yields

n = 12

since the number of vertices must be positive.

2

Calculate the number of distinct triangles formed by choosing 3 vertices from 12.

220

Each set of 3 distinct vertices forms one unique triangle. The order in which the vertices are chosen does not matter, so we use combinations:

\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220

.

Key Concept

Combinations applied to geometric figures and polygon properties

Estimated Time:2m 0s

Question 4179Question →

Complete the statement below by identifying the correct figure of speech. Which literary device is demonstrated in the sentence?

Fill in the blanks below

When the veteran described his traumatic combat experiences as merely 'a bit of an inconvenience', he was making use of .

Show answer & explanation

Answer

The correct figure of speech to complete the sentence is 'understatement' (or 'litotes').

Understatement is a figure of speech in which a situation is intentionally presented as less important, smaller, or less severe than it really is. Describing traumatic combat as merely 'a bit of an inconvenience' deliberately downplays the severity to achieve emphasis.

Step-by-Step Solution

1

Analyze the expression used in the sentence

The phrase 'a bit of an inconvenience' intentionally minimizes the severity of 'traumatic combat experiences'.

Recognizing the deliberate contrast between the magnitude of the situation and the mildness of the expression.

2

Identify the literary device

Representing something as much less intense or severe than it actually is constitutes an understatement.

Matching the definition of understatement to the contextual meaning of the excerpt.

Key Concept

Understatement (and Litotes) as a figure of speech used to intentionally minimize expression for emphasis or effect.

Estimated Time:45s

Question 4180Question →

Using differentiation from first principles, what is the derivative of the function $f(x) = x^2 + 3x$ with respect to $x$ ?

2x + 3

2x + 3 + h

x^2 + 3

2x

Show answer & explanation

Answer: $2x + 3$

Answer

2x + 3

Applying the first principles limit formula

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

gives

\frac{x^2 + 2xh + h^2 + 3x + 3h - (x^2 + 3x)}{h} = \frac{2xh + h^2 + 3h}{h} = 2x + h + 3

. As

h \to 0

, this expression evaluates to

2x + 3

.

Step-by-Step Solution

1

Set up the difference quotient using the definition of differentiation from first principles

\frac{f(x+h) - f(x)}{h} = \frac{[(x+h)^2 + 3(x+h)] - [x^2 + 3x]}{h}

The definition of derivative from first principles requires finding

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

.

2

Expand algebraic terms in the numerator

\frac{x^2 + 2xh + h^2 + 3x + 3h - x^2 - 3x}{h}

Expanding

(x+h)^2 = x^2 + 2xh + h^2

and

3(x+h) = 3x + 3h

allows terms to be combined.

3

Simplify the numerator by canceling like terms and dividing by

h

\frac{2xh + h^2 + 3h}{h} = 2x + h + 3

The terms

x^2 - x^2 = 0

and

3x - 3x = 0

cancel out, leaving terms containing

h

, which can be divided by

h

.

4

Evaluate the limit as

h \to 0

\lim_{h \to 0} (2x + h + 3) = 2x + 3

Setting

h = 0

in the simplified quotient yields the final derivative

f'(x) = 2x + 3

.

Key Concept

Differentiation from first principles

Answer

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