Number and Numeration Alıştırma Soruları — JAMB UTME — Sayfa 9

Soru 161Soru →

Pumps $P_1$ and $P_2$ operating together can fill a water storage tank in $12\text{ hours}$ , while pumps $P_2$ and $P_3$ operating together can fill the same tank in $20\text{ hours}$ . When all three pumps operate simultaneously for $5\text{ hours}$ , they fill exactly $\frac{1}{2}$ of the tank. If pump $P_2$ is then shut off, how many additional hours will it take pumps $P_1$ and $P_3$ working together to fill the remainder of the tank?

7.5\text{ hours}

15.0\text{ hours}

6.0\text{ hours}

12.5\text{ hours}

Cevabı ve açıklamayı göster

Cevap: $7.5\text{ hours}$

Cevap

7.5\text{ hours}

The three pumps combined have a work rate of

\frac{1}{10}\text{ tank/hr}

. Subtracting the rate of

P_1 + P_2

(which is

\frac{1}{12}

) gives

P_3

's rate as

\frac{1}{60}\text{ tank/hr}

. Subtracting the rate of

P_2 + P_3

(which is

\frac{1}{20}

) gives

P_1

's rate as

\frac{1}{20}\text{ tank/hr}

. Together,

P_1

and

P_3

have a combined rate of

\frac{1}{20} + \frac{1}{60} = \frac{1}{15}\text{ tank/hr}

. To fill the remaining

\frac{1}{2}

of the tank, it will take

\frac{1/2}{1/15} = 7.5\text{ hours}

.

Adım Adım Çözüm

1

Determine the combined work rate of all three pumps working together.

Since all three pumps fill

\frac{1}{2}

of the tank in

5\text{ hours}

, their combined rate is

r_1 + r_2 + r_3 = \frac{1/2}{5} = \frac{1}{10}\text{ tank per hour}

.

Work rate is defined as the fraction of work completed per unit time.

2

Find the individual work rates of pumps

P_1

and

P_3

.

r_3 = (r_1 + r_2 + r_3) - (r_1 + r_2) = \frac{1}{10} - \frac{1}{12} = \frac{1}{60}\text{ tank per hour}

. Also,

r_1 = (r_1 + r_2 + r_3) - (r_2 + r_3) = \frac{1}{10} - \frac{1}{20} = \frac{1}{20}\text{ tank per hour}

.

Subtracting known pair rates from the total three-pump rate yields individual rates.

3

Calculate the combined work rate of

P_1

and

P_3

.

r_1 + r_3 = \frac{1}{20} + \frac{1}{60} = \frac{3 + 1}{60} = \frac{4}{60} = \frac{1}{15}\text{ tank per hour}

.

The rate of two pumps operating together is the sum of their individual rates.

4

Compute the time needed to fill the remaining portion of the tank.

Remaining fraction to fill is

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

. Time required =

\frac{1/2}{1/15} = \frac{15}{2} = 7.5\text{ hours}

.

Dividing the remaining workload by the combined rate gives the additional operating time.

Anahtar Kavram

Work-rate problems involving simultaneous rates and partial work completion

Tahmini Süre:2m 0s

Soru 162Soru →

If $\log_3 x - 2\log_x 27 = 1$ , what is the sum of all possible real values of $x$ ?

\frac{244}{9}

18

1

27

Cevabı ve açıklamayı göster

Cevap: $\frac{244}{9}$

Cevap

\frac{244}{9}

By applying the change of base rule

\log_x 27 = \frac{\log_3 27}{\log_3 x} = \frac{3}{\log_3 x}

, the given equation simplifies to

\log_3 x - \frac{6}{\log_3 x} = 1

. Setting

u = \log_3 x

yields

u^2 - u - 6 = 0

, which factors as

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

. Thus,

u = 3

or

u = -2

, giving solutions

x = 3^3 = 27

and

x = 3^{-2} = \frac{1}{9}

. Adding these valid real solutions gives

27 + \frac{1}{9} = \frac{244}{9}

.

Adım Adım Çözüm

1

Apply the change of base formula to

\log_x 27

.

\log_x 27 = \frac{\log_3 27}{\log_3 x} = \frac{3}{\log_3 x}

Converting all logarithmic terms to a common base (base 3) allows substitution.

2

Substitute

\log_x 27

back into the original equation and let

u = \log_3 x

.

u - 2\left(\frac{3}{u}\right) = 1 \implies u - \frac{6}{u} = 1

This transforms the logarithmic equation into an algebraic equation.

3

Clear the denominator by multiplying through by

u

and rearrange into standard quadratic form.

u^2 - u - 6 = 0

Multiplying by

u

(where

u \neq 0

) forms a standard quadratic equation.

4

Factor the quadratic equation to solve for

u

.

(u - 3)(u + 2) = 0 \implies u = 3 \text{ or } u = -2

Factoring determines the values of

\log_3 x

.

5

Convert back to

x

using

x = 3^u

.

For

u = 3

,

x = 3^3 = 27

. For

u = -2

,

x = 3^{-2} = \frac{1}{9}

. Both

x = 27

and

x = \frac{1}{9}

are valid bases (

x > 0, x \neq 1

).

Exponentiation resolves the logarithmic variable.

6

Find the sum of all valid values of

x

.

\text{Sum} = 27 + \frac{1}{9} = \frac{243 + 1}{9} = \frac{244}{9}

Combines the two real solutions into the final requested sum.

Anahtar Kavram

Change of base formula

\log_a b = \frac{\log_c b}{\log_c a}

and solving equations reducible to quadratics.

Tahmini Süre:2m 0s

Soru 163Soru →

An arithmetic progression has a first term of $5$ and a common difference of $4$ . What is the $12^{\text{th}}$ term of this progression?

49

53

57

45

Cevabı ve açıklamayı göster

Cevap: $49$

Cevap

49

The

n^{\text{th}}

term of an arithmetic progression is determined using the formula

T_n = a + (n - 1)d

. Substituting

a = 5

,

d = 4

, and

n = 12

yields

T_{12} = 5 + 11 \times 4 = 49

, making

49

the correct value.

Adım Adım Çözüm

1

Identify the given values from the problem statement

First term

a = 5

, common difference

d = 4

, and position

n = 12

These are the standard variables required for calculating terms in an arithmetic progression.

2

Apply the

n^{\text{th}}

term formula for an arithmetic progression

T_{12} = 5 + (12 - 1) \times 4

The standard formula for the

n^{\text{th}}

term of an AP is

T_n = a + (n - 1)d

.

3

Simplify the expression to find the final value

T_{12} = 5 + 11 \times 4 = 5 + 44 = 49

Perform multiplication before addition according to standard order of operations.

Anahtar Kavram

Arithmetic Progression

n^{\text{th}}

Term Formula

Tahmini Süre:45s

Soru 164Soru →

If $\left( \frac{16}{81} \right)^{x-1} \times \left( \frac{27}{8} \right)^{x+2} = \frac{9}{4}$ , find the value of $x$ .

12

8

-4

10

Cevabı ve açıklamayı göster

Cevap: $8$

Cevap

The value of

x

is

8

.

By writing

\frac{16}{81}

as

\left(\frac{2}{3}\right)^4

,

\frac{27}{8}

as

\left(\frac{2}{3}\right)^{-3}

, and

\frac{9}{4}

as

\left(\frac{2}{3}\right)^{-2}

, the equation simplifies via exponent addition to

\left(\frac{2}{3}\right)^{4x - 4 - 3x - 6} = \left(\frac{2}{3}\right)^{-2}

. Equating indices gives

x - 10 = -2

, which yields the correct solution

x = 8

.

Adım Adım Çözüm

1

Express all fractional terms with a common base of

\frac{2}{3}

\frac{16}{81} = \left(\frac{2}{3}\right)^4

,

\frac{27}{8} = \left(\frac{3}{2}\right)^3 = \left(\frac{2}{3}\right)^{-3}

, and

\frac{9}{4} = \left(\frac{3}{2}\right)^2 = \left(\frac{2}{3}\right)^{-2}

Converting all terms to a single common base allows exponents to be combined using the laws of indices.

2

Substitute the common base expressions back into the original equation

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

Applying the power of a power law

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

to simplify each term.

3

Apply the power law and multiplication law of indices

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

When multiplying exponential terms with identical bases, add their exponents:

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

.

4

Equate the exponents and solve for

x

4(x-1) - 3(x+2) = -2 \implies 4x - 4 - 3x - 6 = -2 \implies x - 10 = -2 \implies x = 8

Since the bases on both sides are equal and non-zero, their indices must be equal.

Anahtar Kavram

Laws of Indices: Base Conversion and Exponential Equations

Tahmini Süre:2m 0s

Soru 165Soru →

Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and a subset $P = \{x \in U : x \text{ is a prime number}\}$ , which of the following sets represents the complement of $P$ , denoted as $P'$ ?

\{1, 4, 6, 8, 9, 10\}

\{4, 6, 8, 9, 10\}

\{1, 3, 5, 7, 9\}

\{2, 4, 6, 8, 10\}

Cevabı ve açıklamayı göster

Cevap: $\{1, 4, 6, 8, 9, 10\}$

Cevap

The set

\{1, 4, 6, 8, 9, 10\}

The prime numbers in

U

are

2, 3, 5,

and

7

, making

P = \{2, 3, 5, 7\}

. The complement

P'

consists of all elements in

U

that do not belong to

P

, which yields

\{1, 4, 6, 8, 9, 10\}

.

Adım Adım Çözüm

1

Identify the elements of the universal set

U

U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

The universal set defines the boundary of all possible elements under consideration.

2

Identify the elements belonging to set

P

(prime numbers in

U

)

P = \{2, 3, 5, 7\}

Prime numbers have exactly two distinct positive divisors: 1 and themselves. Note that 1 is not a prime number.

3

Compute the complement

P' = U \setminus P

P' = \{1, 4, 6, 8, 9, 10\}

The complement of set

P

consists of all elements present in

U

that are not present in

P

.

Anahtar Kavram

Set Complement and Universal Set Boundaries

Tahmini Süre:45s

Soru 166Soru →

In a survey of $200$ agricultural exporters regarding three major commodities—Cocoa ( $C$ ), Palm Oil ( $P$ ), and Rubber ( $R$ )—it was found that $110$ export Cocoa, $90$ export Palm Oil, and $75$ export Rubber. Exactly $20$ exporters export none of the three commodities, and $45$ export Cocoa only. Furthermore, the number of exporters who export Cocoa and Palm Oil only is twice the number of exporters who export all three commodities; the number who export Palm Oil and Rubber only is equal to the number who export all three; and the number who export Cocoa and Rubber only is $5$ more than the number who export all three. Find the total number of exporters who export at least two of the three commodities.

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

The total number of exporters who export at least two of the three commodities is 80.

The correct answer of 80 is obtained by solving for the number of exporters trading in all three commodities (

x = 15

) using the Cocoa set equation

45 + 2x + (x + 5) + x = 110

, and then evaluating the region sum for at least two commodities (

2x + x + (x + 5) + x = 5x + 5 = 80

).

Adım Adım Çözüm

1

Assign a variable to the triple intersection

Let

x = n(C \cap P \cap R)

be the number of exporters of all three commodities.

The intersection of all three sets serves as the common parameter for all double-intersection regions.

2

Write algebraic expressions for the three pairwise-only intersections

n(C \cap P \text{ only}) = 2x

,

n(P \cap R \text{ only}) = x

, and

n(C \cap R \text{ only}) = x + 5

.

These expressions are derived directly from the relationships given in the problem statement.

3

Formulate and solve an equation using the set of Cocoa exporters

45 + 2x + (x + 5) + x = 110 \implies 50 + 4x = 110 \implies x = 15

.

The set of Cocoa exporters consists of four mutually exclusive regions whose cardinalities sum to 110.

4

Sum the regions corresponding to 'at least two commodities'

(2x) + (x) + (x + 5) + x = 5x + 5 = 5(15) + 5 = 80

.

'At least two' encompasses everyone who exports exactly two commodities plus those who export all three.

Anahtar Kavram

Three-set principle of inclusion-exclusion and cardinal region decomposition

Soru 167Soru →

What is the determinant of the matrix $P = \begin{pmatrix} 5 & 3 \\ 2 & 7 \end{pmatrix}$ ?

Cevabı ve açıklamayı göster

Cevap: 29

Cevap

29

For a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

, the determinant is given by

ad - bc

. Substituting the values

a = 5

,

b = 3

,

c = 2

, and

d = 7

yields

(5 \times 7) - (3 \times 2) = 35 - 6 = 29

.

Adım Adım Çözüm

1

Identify the values of

a, b, c,

and

d

from the given

2 \times 2

matrix

P = \begin{pmatrix} 5 & 3 \\ 2 & 7 \end{pmatrix}

.

a = 5, b = 3, c = 2, d = 7

To set up the values for the

2 \times 2

determinant formula.

2

Multiply the elements of the main diagonal and the secondary diagonal.

Main diagonal product:

5 \times 7 = 35

; Secondary diagonal product:

3 \times 2 = 6

.

The determinant of a

2 \times 2

matrix is the difference between the main diagonal product and the secondary diagonal product.

3

Subtract the product of the secondary diagonal from the product of the main diagonal.

Determinant =

35 - 6 = 29

.

Completes the formula

\det(P) = ad - bc

.

Anahtar Kavram

Determinant of a 2x2 Matrix

Soru 168Soru →

What is the result of the subtraction $524_8 - 267_8$ expressed in base 8?

235_8

257_8

345_8

343_8

Cevabı ve açıklamayı göster

Cevap: $235_8$

Cevap

The correct answer is

235_8

.

Subtracting

267_8

from

524_8

using direct base-8 subtraction requires borrowing 8 when a digit is smaller than the digit subtracted from it. In the units column, borrowing 1 (which equals 8) gives

(4 + 8) - 7 = 5

. In the eights column, the borrowed-from digit 2 becomes 1; borrowing 1 (which equals 8) gives

(1 + 8) - 6 = 3

. In the sixty-fours column, the borrowed-from digit 5 becomes 4;

4 - 2 = 2

. Thus, the difference is

235_8

.

Adım Adım Çözüm

1

Convert both octal numbers to base 10.

524_8 = (5 \times 8^2) + (2 \times 8^1) + (4 \times 8^0) = 320 + 16 + 4 = 340_{10}

, and

267_8 = (2 \times 8^2) + (6 \times 8^1) + (7 \times 8^0) = 128 + 48 + 7 = 183_{10}

.

Converting to decimal simplifies standard arithmetic operations.

2

Subtract the decimal values.

340_{10} - 183_{10} = 157_{10}

.

Obtain the difference in base 10.

3

Convert the decimal difference

157_{10}

back to base 8.

157 \div 8 = 19

remainder

5

;

19 \div 8 = 2

remainder

3

;

2 \div 8 = 0

remainder

2

. Reading remainders from bottom to top gives

235_8

.

Successive division by 8 yields the octal representation.

Anahtar Kavram

Non-decimal base arithmetic and borrowing rules

Soru 169Soru →

The $5^{\text{th}}$ term of an arithmetic progression (AP) is $17$ and its common difference is $3$ . What is the first term of the progression?

5

2

29

-1

Cevabı ve açıklamayı göster

Cevap: $5$

Cevap

The first term of the progression is

5

.

The

n^{\text{th}}

term formula of an arithmetic progression is

T_n = a + (n - 1)d

. Substituting

T_5 = 17

,

n = 5

, and

d = 3

gives

17 = a + 4(3) = a + 12

. Isolating

a

yields

a = 17 - 12 = 5

.

Adım Adım Çözüm

1

Identify the given values and formula for the nth term of an AP.

The formula is

T_n = a + (n - 1)d

, with

T_5 = 17

,

n = 5

, and

d = 3

.

This formula connects the nth term, the first term, the number of terms, and the common difference.

2

Substitute the known values into the equation.

17 = a + (5 - 1) \times 3 \implies 17 = a + 12

.

Subtracting 1 from the term index 5 gives 4, and multiplying by 3 gives 12.

3

Solve for the first term

a

.

a = 17 - 12 = 5

.

Subtracting 12 from both sides isolates

a

.

Anahtar Kavram

n-th term of an Arithmetic Progression

Tahmini Süre:1m 0s

Soru 170Soru →

What is the simplified numerical value of $\frac{\sqrt{12}}{\sqrt{3} + 1} + \frac{\sqrt{12}}{\sqrt{3} - 1}$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The simplified numerical value of the given expression is 6.

Combining the fractions using their common conjugate denominator

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

leads to a numerator of

2\sqrt{3}[(\sqrt{3}-1)+(\sqrt{3}+1)] = 2\sqrt{3}(2\sqrt{3}) = 12

. Dividing 12 by 2 yields the final answer of 6.

Adım Adım Çözüm

1

Simplify the surd in the numerator

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

Simplifying surds into basic form makes subsequent calculations simpler.

2

Combine the fractions by finding a common denominator

\frac{2\sqrt{3}(\sqrt{3}-1) + 2\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}+1)(\sqrt{3}-1)}

Multiplying denominators forms a conjugate pair, which rationalizes the combined denominator.

3

Simplify the numerator and the denominator

\text{Numerator} = 2\sqrt{3}(\sqrt{3}-1 + \sqrt{3}+1) = 2\sqrt{3}(2\sqrt{3}) = 12; \quad \text{Denominator} = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2

Expanding the numerator combines like surd terms, and using the difference of squares simplifies the denominator to a rational integer.

4

Perform the final division

\frac{12}{2} = 6

Dividing the simplified numerator by the rationalized denominator gives the final integer value.

Anahtar Kavram

Rationalization of binomial denominators using conjugate surds

Soru 171Soru →

The sum of the first $n$ terms of a sequence is given by $S_n = 2n^2 + 3n$ . Find the $5^{\text{th}}$ term of a geometric progression whose first term is the $3^{\text{rd}}$ term of this sequence, and whose common ratio is equal to the common difference of this sequence.

3328

13312

6912

1280

Cevabı ve açıklamayı göster

Cevap: $3328$

Cevap

The 5th term of the geometric progression is 3328.

Evaluating

S_1

,

S_2

, and

S_3

gives sequence terms

T_1 = 5

,

T_2 = 9

, and

T_3 = 13

. The common difference is

d = 9 - 5 = 4

. Using

T_3 = 13

as the first term of the GP and

r = 4

as the common ratio, the 5th term of the GP is

13 \times 4^{5-1} = 13 \times 256 = 3328

.

Adım Adım Çözüm

1

Calculate the first few terms of the sequence using the sum formula

S_n = 2n^2 + 3n

.

S_1 = 2(1)^2 + 3(1) = 5

,

S_2 = 2(2)^2 + 3(2) = 14

,

S_3 = 2(3)^2 + 3(3) = 27

.

The sum formula gives cumulative sums, from which individual terms can be derived.

2

Find the 3rd term (

T_3

) and the common difference (

d

) of the arithmetic progression.

T_1 = 5

,

T_2 = S_2 - S_1 = 9

,

T_3 = S_3 - S_2 = 13

. Common difference

d = T_2 - T_1 = 4

.

The difference between consecutive cumulative sums gives the individual sequence terms, and their constant difference gives the common difference.

3

Define the parameters of the geometric progression (GP).

First term of GP

a = T_3 = 13

, common ratio

r = d = 4

.

The problem specifies that the first term of the GP is the 3rd term of the sequence and the common ratio equals the common difference.

4

Compute the 5th term of the geometric progression using

G_n = a \cdot r^{n-1}

.

G_5 = 13 \cdot 4^{5-1} = 13 \cdot 4^4 = 13 \cdot 256 = 3328

.

Applying the standard nth term formula for a GP with

n = 5

.

Anahtar Kavram

Arithmetic Progression sum formula to term conversion and Geometric Progression nth term evaluation

Tahmini Süre:2m 0s

Soru 172Soru →

Three business partners, $X$ , $Y$ , and $Z$ , invest in a venture. $X$ contributes $\frac{1}{3}$ of the total capital for $\frac{1}{4}$ of the total duration of the business. $Y$ contributes $\frac{1}{4}$ of the capital for $\frac{1}{2}$ of the duration, while $Z$ contributes the remaining capital for the entire duration. If the total profit realized from the business is $\mathcal{N}78,000$ , what is $Z$ 's share of the profit?

\mathcal{N}32,500

\mathcal{N}26,000

\mathcal{N}52,000

\mathcal{N}60,000

Cevabı ve açıklamayı göster

Cevap: $\mathcal{N}52,000$

Cevap

\mathcal{N}52,000

Profit distribution in compound partnerships depends on both capital invested and investment time duration.

X

's product is

\frac{1}{12}

,

Y

's product is

\frac{1}{8}

, and

Z

's product is

\frac{5}{12}

. Scaling these by 24 gives a ratio of

2 : 3 : 10

. With 15 total parts,

Z

receives

\frac{10}{15}

of

\mathcal{N}78,000

, which equals

\mathcal{N}52,000

.

Adım Adım Çözüm

1

Determine

Z

's fraction of the total capital

Z

's capital fraction

= 1 - \left(\frac{1}{3} + \frac{1}{4}\right) = 1 - \frac{7}{12} = \frac{5}{12}

The sum of all capital contributions must equal 1.

2

Calculate the compound product of (Capital Share

\times

Time Duration) for each partner

X = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}

,

Y = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}

,

Z = \frac{5}{12} \times 1 = \frac{5}{12}

Profit sharing in compound partnerships is proportional to the product of capital invested and the time period.

3

Convert the fractions to an equivalent integer ratio

Multiplying

\frac{1}{12} : \frac{1}{8} : \frac{5}{12}

by 24 gives the ratio

2 : 3 : 10

Finding the least common multiple (24) simplifies the fractions into integer ratio parts.

4

Compute

Z

's profit share from the total profit

Total ratio parts

= 2 + 3 + 10 = 15

.

Z

's share

= \frac{10}{15} \times \mathcal{N}78,000 = \mathcal{N}52,000

Multiply total profit by

Z

's ratio fraction.

Anahtar Kavram

Compound Ratio and Partnership Profit Sharing

Soru 173Soru →

Given that the matrix $A = \begin{pmatrix} 1 & 2 & k \\ 0 & k & 3 \\ 2 & -1 & 1 \end{pmatrix}$ is singular, what is the positive value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The positive value of

k

is

3

.

A square matrix is singular when its determinant is zero. Expanding

\det(A)

along the first column yields

1 \cdot (k + 3) + 2 \cdot (6 - k^2) = -2k^2 + k + 15

. Equating this to zero gives the quadratic equation

2k^2 - k - 15 = 0

, which factors as

(2k + 5)(k - 3) = 0

. The roots are

k = -2.5

and

k = 3

. The positive value is

3

.

Adım Adım Çözüm

1

Calculate the determinant of matrix A

\det(A) = -2k^2 + k + 15

Expanding along the first column simplifies the calculation because of the zero entry.

2

Set the determinant to zero for singularity

2k^2 - k - 15 = 0

A matrix is singular if and only if its determinant equals zero.

3

Solve the quadratic equation for k

k = 3 or k = -2.5

Factoring

2k^2 - k - 15 = (2k + 5)(k - 3) = 0

yields two roots.

4

Choose the positive solution

k = 3

The question explicitly requires the positive value of

k

.

Anahtar Kavram

Singular matrix definition and 3x3 determinant evaluation

Soru 174Soru →

In a survey of $180$ cloud computing engineers regarding their proficiency in three major platforms—AWS ( $A$ ), Azure ( $B$ ), and Google Cloud ( $C$ )—it was found that $95$ are proficient in AWS, $80$ in Azure, and $75$ in Google Cloud. Furthermore, $40$ are proficient in both AWS and Azure, $35$ in both Azure and Google Cloud, $30$ in both AWS and Google Cloud, and $15$ are not proficient in any of the three platforms. How many engineers are proficient in exactly one of these platforms?

Cevabı ve açıklamayı göster

Cevap: 100

Cevap

100 engineers are proficient in exactly one platform.

Using the 3-set inclusion-exclusion formula, the number of engineers proficient in all three platforms is solved as 20. Subtracting the relevant overlap regions from each set gives 45 for AWS only, 25 for Azure only, and 30 for Google Cloud only. Adding these single-set values yields a total of 100 engineers proficient in exactly one platform.

Adım Adım Çözüm

1

Determine the cardinality of the union of all three sets.

n(A \cup B \cup C) = 180 - 15 = 165

.

Subtracting the engineers who are not proficient in any of the three platforms from the universal set.

2

Apply the Principle of Inclusion-Exclusion to calculate the triple intersection

n(A \cap B \cap C)

.

n(A \cap B \cap C) = 20

.

Substituting known values gives

165 = 250 - 105 + n(A \cap B \cap C)

, which simplifies to

n(A \cap B \cap C) = 20

.

3

Calculate the counts for regions representing exactly two platforms.

AWS & Azure only = 20, Azure & GCP only = 15, AWS & GCP only = 10.

Subtracting the triple intersection count (20) from each pairwise intersection.

4

Calculate the single-set exclusive regions.

AWS only = 45, Azure only = 25, GCP only = 30.

Subtracting all multi-platform overlap regions from each total set size.

5

Sum the single-set exclusive regions.

Total =

45 + 25 + 30 = 100

.

Combining the counts of engineers proficient in AWS only, Azure only, and Google Cloud only.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets and Venn Diagram Region Partitioning

Soru 175Soru →

A business owner borrowed $\text{₦}150,000$ to expand his store at a compound interest rate of $8\%$ per annum, compounded annually. What is the total compound interest he will pay at the end of 2 years?

\text{₦}24,960

\text{₦}174,960

\text{₦}24,000

\text{₦}26,960

Cevabı ve açıklamayı göster

Cevap: $\text{₦}24,960$

Cevap

The total compound interest paid at the end of 2 years is

\text{₦}24,960

.

The total compound interest is obtained by calculating the interest accrued in each period (

12,000

in Year 1 and

12,960

in Year 2) and summing them to get

\text{₦}24,960

. Alternatively, using the compound interest formula

I = P(1 + r)^n - P

,

I = 150,000(1.08)^2 - 150,000 = 174,960 - 150,000 = \text{₦}24,960

.

Adım Adım Çözüm

1

Calculate the interest for the first year.

\text{Interest}_1 = \frac{8}{100} \times 150,000 = \text{₦}12,000

Interest in the first year is calculated on the initial principal.

2

Find the principal for the second year.

\text{Principal}_2 = 150,000 + 12,000 = \text{₦}162,000

Under compound interest, the first year's interest is added to the initial principal.

3

Calculate the interest for the second year.

\text{Interest}_2 = \frac{8}{100} \times 162,000 = \text{₦}12,960

Interest in the second year is calculated on the updated principal.

4

Sum the interest from both years to find the total compound interest.

\text{Total Interest} = 12,000 + 12,960 = \text{₦}24,960

The total compound interest is the sum of interest accumulated in each compounding period.

Anahtar Kavram

Compound Interest

Tahmini Süre:1m 30s

Soru 176Soru →

A trader bought a set of solar panels for $\text{₦}160,000$ . He marked up the cost price by $25\%$ to fix the marked price. During a trade fair, he offered a $10\%$ discount on the marked price. What is his net profit in Naira?

Cevabı ve açıklamayı göster

Cevap: 20000

Cevap

The net profit made by the trader is ��20,000.

First, find the marked price by adding a 25% markup to the cost price: ₦160,000 × 1.25 = ₦200,000. Next, calculate the selling price after a 10% discount on the marked price: ₦200,000 × 0.90 = ₦180,000. Finally, subtract the cost price from the selling price to find the net profit: ₦180,000 - ₦160,000 = ₦20,000.

Adım Adım Çözüm

1

Calculate the marked price

Marked Price = ₦200,000

The marked price is 125% of the original cost price of ₦160,000.

2

Calculate the selling price after discount

Selling Price = ₦180,000

A 10% discount reduces the marked price by ₦20,000.

3

Determine the net profit

Net Profit = ₦20,000

Net profit is the difference between the selling price (₦180,000) and the cost price (₦160,000).

Anahtar Kavram

Markup, Discount, and Profit Calculations

Soru 177Soru →

If $5^{2x - 1} \times 25^{x + 1} = 125^{x + 2}$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is

5

.

Converting all terms to base

5

gives

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

. Simplifying the exponents yields

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

, which reduces to

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

. Setting the exponents equal to each other gives

4x + 1 = 3x + 6

, resulting in

x = 5

.

Adım Adım Çözüm

1

Express all bases in terms of prime base 5

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

All terms must share the same base to combine exponents using index laws.

2

Apply power of a power rule

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

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

Multiplication of inner and outer powers simplifies composite exponent expressions.

3

Apply the product rule

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

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

Adding the exponents on the left-hand side produces a single exponential term.

4

Equate exponents and solve for

x

x = 5

Equal bases imply equal exponents:

4x + 1 = 3x + 6

.

Anahtar Kavram

Solving exponential equations using common base conversion and laws of indices

Tahmini Süre:1m 30s

Soru 178Soru →

A rectangle has an area of $10\text{ cm}^2$ and a length of $(\sqrt{7} + \sqrt{2})\text{ cm}$ . What is the width of the rectangle in simplified surd form?

(2\sqrt{7} - 2\sqrt{2})\text{ cm}

(2\sqrt{7} + 2\sqrt{2})\text{ cm}

2\sqrt{5}\text{ cm}

\frac{10\sqrt{7} - 10\sqrt{2}}{9}\text{ cm}

Cevabı ve açıklamayı göster

Cevap: $(2\sqrt{7} - 2\sqrt{2})\text{ cm}$

Cevap

The width of the rectangle in simplified surd form is

(2\sqrt{7} - 2\sqrt{2})\text{ cm}

.

The area formula for a rectangle gives

\text{width} = \frac{10}{\sqrt{7} + \sqrt{2}}

. Multiplying both numerator and denominator by the conjugate

(\sqrt{7} - \sqrt{2})

produces

\frac{10(\sqrt{7} - \sqrt{2})}{7 - 2} = \frac{10(\sqrt{7} - \sqrt{2})}{5} = 2\sqrt{7} - 2\sqrt{2}

.

Adım Adım Çözüm

1

Set up the formula for the width of the rectangle

\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{10}{\sqrt{7} + \sqrt{2}}

The area of a rectangle is length multiplied by width.

2

Rationalize the denominator by multiplying top and bottom by the conjugate

(\sqrt{7} - \sqrt{2})

\text{Width} = \frac{10(\sqrt{7} - \sqrt{2})}{(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})}

Multiplying by the conjugate creates a difference of squares in the denominator, eliminating radicals.

3

Simplify the denominator using

(a+b)(a-b) = a^2 - b^2

(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5

Squaring a square root yields the underlying rational number.

4

Divide the numerator by the simplified denominator

\frac{10(\sqrt{7} - \sqrt{2})}{5} = 2(\sqrt{7} - \sqrt{2}) = 2\sqrt{7} - 2\sqrt{2}

Dividing

10

by

5

gives

2

, which is then distributed across the terms inside the parentheses.

Anahtar Kavram

Rationalization of Denominators with Binomial Surds

Soru 179Soru →

If $\log_2 x + \log_8 x = 4$ , what is the value of $x$ ?

4

6

8

64

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

By applying the change of base formula,

\log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{1}{3}\log_2 x

. Rewriting the equation gives

\log_2 x + \frac{1}{3}\log_2 x = \frac{4}{3}\log_2 x = 4

. Solving for

\log_2 x

yields

\log_2 x = 3

, which in exponential form gives

x = 2^3 = 8

.

Adım Adım Çözüm

1

Apply the change of base formula to express

\log_8 x

in base 2

\log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{\log_2 x}{3}

Logarithms must be converted to a common base before combining terms.

2

Substitute

\log_8 x = \frac{1}{3}\log_2 x

into the original equation

\log_2 x + \frac{1}{3}\log_2 x = 4 \implies \frac{4}{3}\log_2 x = 4

Combine like logarithmic terms.

3

Solve for

\log_2 x

\log_2 x = 4 \times \frac{3}{4} = 3

Isolate the logarithmic expression by multiplying both sides by

\frac{3}{4}

.

4

Convert the logarithmic equation to its exponential form to solve for

x

x = 2^3 = 8

Definition of logarithms states that

\log_b a = c \iff b^c = a

.

Anahtar Kavram

Logarithms and Change of Base

Soru 180Soru →

The $3^{\text{rd}}$ term of a geometric progression is $18$ and the $6^{\text{th}}$ term is $486$ . What is the sum of the first $5$ terms of the progression?

242

162

80

728

Cevabı ve açıklamayı göster

Cevap: $242$

Cevap

The sum of the first

5

terms is

242

.

Using

T_n = a r^{n-1}

, we set up

a r^2 = 18

and

a r^5 = 486

. Dividing these gives

r^3 = 27

, so

r = 3

, which leads to

a = 2

. Applying

S_5 = \frac{a(r^5 - 1)}{r - 1}

yields

\frac{2(243 - 1)}{2} = 242

.

Adım Adım Çözüm

1

Express the given terms using the geometric progression

n^{\text{th}}

term formula

T_n = a r^{n-1}

.

T_3 = a r^2 = 18

and

T_6 = a r^5 = 486

.

This establishes a system of equations in terms of the first term

a

and common ratio

r

.

2

Divide the expression for

T_6

by

T_3

to determine

r

.

\frac{a r^5}{a r^2} = \frac{486}{18} \implies r^3 = 27 \implies r = 3.

Dividing the equations eliminates

a

and allows direct solution for the common ratio

r

.

3

Substitute

r = 3

back into

T_3 = a r^2 = 18

to solve for

a

.

a (3)^2 = 18 \implies 9a = 18 \implies a = 2.

Determining the first term

a

is required to evaluate the sum.

4

Calculate the sum of the first

5

terms using

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

.

S_5 = \frac{2(3^5 - 1)}{3 - 1} = \frac{2(243 - 1)}{2} = 242.

Applying the GP sum formula yields the required value.

Anahtar Kavram

Geometric Progression nth term and sum formulas

Number and Numeration

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