Number and Numeration Practice Questions — JAMB UTME — Page 4

Question 61Question →

An entrepreneur bought a commercial printing machine for $\text{₦}300,000$ . If the machine depreciates in value at a compound rate of $15\%$ per annum, what is its value in Naira ( $\text{₦}$ ) at the end of $2$ years?

Show answer & explanation

Answer: 216750

Answer

The value of the machine at the end of 2 years is ₦216,750.

The value of the asset after 2 years is calculated using the reducing balance formula:

V_2 = V_0(1 - r)^2 = 300,000(0.85)^2 = \text{₦}216,750

.

Step-by-Step Solution

1

Identify given parameters

Initial principal value

V_0 = 300,000

, annual depreciation rate

r = 0.15

, duration

n = 2

years.

Establishing known variables simplifies formula substitution.

2

Apply compound depreciation formula

V_2 = V_0(1 - r)^2 = 300,000(1 - 0.15)^2

Asset values decrease multiplicatively per period under compound depreciation.

3

Compute the final depreciated value

V_2 = 300,000 \times (0.85)^2 = 300,000 \times 0.7225 = 216,750

Multiplying the initial amount by the combined depreciation factor yields the residual asset value.

Key Concept

Compound Depreciation

Question 62Question →

In an agricultural survey of $100$ local farmers in a community, $55$ grow Cassava ( $C$ ), $45$ grow Yam ( $Y$ ), and $40$ grow Maize ( $M$ ). It is observed that $20$ farmers grow both Cassava and Yam, $15$ grow both Yam and Maize, and $16$ grow both Cassava and Maize. If $7$ farmers grow all three crops, how many farmers grow none of these three crops?

4

11

16

24

Show answer & explanation

Answer: $4$

Answer

The number of farmers who grow none of the three crops is 4.

The number of farmers who grow at least one crop is found using

n(C \cup Y \cup M) = 55 + 45 + 40 - 20 - 15 - 16 + 7 = 96

. Subtracting this union from the total number of farmers (

100

) gives

100 - 96 = 4

farmers who grow none of the three crops.

Step-by-Step Solution

1

Apply the Principle of Inclusion-Exclusion for three sets to find the total number of farmers growing at least one crop

n(C \cup Y \cup M)

.

n(C \cup Y \cup M) = n(C) + n(Y) + n(M) - n(C \cap Y) - n(Y \cap M) - n(C \cap M) + n(C \cap Y \cap M)

Elements in pairwise intersections are double-counted and elements in all three sets are triple-counted, requiring correction.

2

Substitute the given cardinalities into the formula.

n(C \cup Y \cup M) = 55 + 45 + 40 - 20 - 15 - 16 + 7 = 96

Calculating the total number of elements inside the union of the three sets.

3

Subtract the number of farmers in the union from the universal set size

n(U) = 100

.

n((C \cup Y \cup M)') = 100 - 96 = 4

The complement of the union gives the number of elements belonging to none of the sets.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Universal Complement

Question 63Question →

A motorist travels from Town A to Town B at a constant speed of $60\text{ km/h}$ for $2\text{ hours}$ , and then continues from Town B to Town C at a constant speed of $90\text{ km/h}$ for $3\text{ hours}$ . What is the average speed of the motorist for the entire journey in $\text{km/h}$ ?

Show answer & explanation

Answer: 78

Answer

The average speed for the entire journey is

78\text{ km/h}

.

The correct average speed is

78\text{ km/h}

, obtained by dividing the total distance of

390\text{ km}

(

120\text{ km} + 270\text{ km}

) by the total duration of

5\text{ hours}

(

2\text{ h} + 3\text{ h}

).

Step-by-Step Solution

1

Find the distance traveled during each segment of the journey

First segment distance =

120\text{ km}

, second segment distance =

270\text{ km}

Distance equals speed multiplied by time (

D = v \times t

).

2

Determine the total distance traveled and total time taken

Total distance =

390\text{ km}

, Total time =

5\text{ hours}

Average rate requires the sum of all distances and the sum of all durations.

3

Calculate the average speed

Average speed =

78\text{ km/h}

Average speed is defined as total distance divided by total time.

Key Concept

Average Rate and Speed

Question 64Question →

A boarding school store has sufficient provisions to feed $120$ students for $25$ days. After $5$ days of regular consumption, $30$ additional students join the school. Assuming the daily consumption rate per student remains unchanged, for how many more days will the remaining provisions last?

16\text{ days}

20\text{ days}

25\text{ days}

24\text{ days}

Show answer & explanation

Answer: $16\text{ days}$

Answer

The remaining provisions will last for

16\text{ days}

.

After 5 days, the remaining food is sufficient to feed 120 students for 20 more days, giving a total of

120 \times 20 = 2,400

student-days of provisions. When 30 new students join, the group increases to 150 students. Dividing 2,400 student-days by 150 students gives 16 days.

Step-by-Step Solution

1

Calculate the remaining days of food for the original number of students

\text{Remaining days} = 25 - 5 = 20\text{ days}

Food has already been consumed for 5 days out of the original 25 days.

2

Determine the total remaining food capacity in student-days

\text{Remaining supply} = 120 \text{ students} \times 20 \text{ days} = 2,400 \text{ student-days}

One student-day represents the amount of food consumed by one student in one day.

3

Calculate the new total number of students

\text{Total students} = 120 + 30 = 150 \text{ students}

30 additional students joined the original 120 students.

4

Find the number of days the remaining food will last the new total number of students

\text{Days} = \frac{2,400 \text{ student-days}}{150 \text{ students}} = 16 \text{ days}

Since consumption rate is inversely proportional to the number of consumers, divide total student-days by the total student count.

Key Concept

Inverse Proportion and Consumption Rate

Question 65Question →

A cooperative society invested $\text{₦}80,000$ in a fixed deposit fund that pays compound interest at a rate of $10\%$ per annum compounded annually. What is the total compound interest earned by the cooperative society at the end of $2$ years?

\text{₦}16,000

\text{₦}16,800

\text{₦}96,000

\text{₦}96,800

Show answer & explanation

Answer: $\text{₦}16,800$

Answer

\text{₦}16,800

Using the compound interest formula for total amount,

A = P\left(1 + \frac{r}{100}\right)^n

, with principal

P = \text{₦}80,000

, rate

r = 10\%

, and time

n = 2

years:

A = 80000 \times (1.1)^2 = \text{₦}96,800

. The interest earned is the difference between the total amount and the principal:

I = 96800 - 80000 = \text{₦}16,800

.

Step-by-Step Solution

1

Calculate the total accumulated amount

A

using the compound interest formula

A = P\left(1 + \frac{r}{100}\right)^n

A = 80000 \times \left(1 + \frac{10}{100}\right)^2 = 80000 \times (1.1)^2 = 80000 \times 1.21 = \text{₦}96,800

The formula yields the full final balance after compound growth over 2 years.

2

Subtract the initial principal

P

from the total amount

A

to find the compound interest earned

I

I = A - P = 96800 - 80000 = \text{₦}16,800

Interest earned represents only the financial growth beyond the starting capital.

Key Concept

Compound Interest and Total Accumulated Amount

Question 66Question →

The sum of the first three terms of an increasing geometric progression of positive real numbers is $21$ , and the sum of their squares is $189$ . What is the common ratio of this progression?

Show answer & explanation

Answer: 2

Answer

The common ratio of the geometric progression is 2.

Let the terms be

a

,

ar

, and

ar^2

. The given conditions yield

a(1+r+r^2) = 21

and

a^2(1+r^2+r^4) = 189

. Squaring the first equation gives

a^2(1+r+r^2)^2 = 441

. Dividing the sum of squares equation by this squared equation gives

\frac{1-r+r^2}{1+r+r^2} = \frac{189}{441} = \frac{3}{7}

. Simplifying

7(1-r+r^2) = 3(1+r+r^2)

results in

2r^2 - 5r + 2 = 0

, which factors into

(2r-1)(r-2) = 0

. Because the progression is increasing,

r > 1

, making

r = 2

the correct common ratio.

Step-by-Step Solution

1

Formulate algebraic expressions for the sum of terms and sum of squares.

a(1 + r + r^2) = 21

and

a^2(1 + r^2 + r^4) = 189

The first three terms of any geometric progression can be expressed as

a

,

ar

, and

ar^2

.

2

Eliminate the first term

a

by squaring the first equation and dividing.

\frac{a^2(1 + r^2 + r^4)}{a^2(1 + r + r^2)^2} = \frac{189}{441} \implies \frac{(1 + r + r^2)(1 - r + r^2)}{(1 + r + r^2)^2} = \frac{3}{7}

Using the algebraic factorization

1 + r^2 + r^4 = (1 + r + r^2)(1 - r + r^2)

allows cancellation of

a^2

and

(1 + r + r^2)

.

3

Solve the resulting equation for the common ratio

r

.

7(1 - r + r^2) = 3(1 + r + r^2) \implies 4r^2 - 10r + 4 = 0 \implies 2r^2 - 5r + 2 = 0

Cross-multiplying reduces the ratio to a standard quadratic equation.

4

Factor the quadratic equation and select the correct root.

(2r - 1)(r - 2) = 0 \implies r = 2 \text{ or } r = 0.5

Since the geometric progression is specified as increasing, the common ratio must be greater than 1 (

r = 2

).

Key Concept

Geometric progression term representations, sum formulas, and algebraic identity factorization.

Alternative Method

Find

a

and

r

by testing factors of

21

:

21 = 3 \times 7

, so the terms could be

3, 6, 12

(

a=3, r=2

). Check squares:

3^2 + 6^2 + 12^2 = 9 + 36 + 144 = 189

, which confirms

r = 2

.

Estimated Time:1m 30s

Question 67Question →

A merchant bought a power generator for $\text{₦}120,000$ and sold it to a retailer at a profit of $20\%$ . The retailer later sold the generator to a customer at a loss of $15\%$ . How much did the customer pay for the generator?

\text{₦}122,400

\text{₦}126,000

\text{₦}110,400

\text{₦}144,000

Show answer & explanation

Answer: $\text{₦}122,400$

Answer

The customer paid

\text{₦}122,400

for the generator.

The merchant sells the generator for

120\%

of

\text{₦}120,000

, which equals

\text{₦}144,000

. The retailer then sells it at

85\%

of

\text{₦}144,000

, yielding

\text{₦}122,400

.

Step-by-Step Solution

1

Calculate the selling price of the generator from the merchant to the retailer.

\text{Selling Price}_1 = \text{₦}120,000 \times \left(1 + \frac{20}{100}\right) = \text{₦}120,000 \times 1.20 = \text{₦}144,000

The merchant makes a

20\%

profit on the initial cost price of

\text{₦}120,000

.

2

Calculate the selling price from the retailer to the final customer.

\text{Selling Price}_2 = \text{₦}144,000 \times \left(1 - \frac{15}{100}\right) = \text{₦}144,000 \times 0.85 = \text{₦}122,400

The retailer incurs a

15\%

loss calculated relative to their purchase price of

\text{₦}144,000

.

Key Concept

Successive Percentage Profit and Loss

Question 68Question →

The $3^{\text{rd}}$ term of an arithmetic progression (AP) is $10$ and the $7^{\text{th}}$ term is $22$ . What is the sum of the first $12$ terms of the progression?

228

246

264

444

Show answer & explanation

Answer: $246$

Answer

The sum of the first

12

terms of the arithmetic progression is

246

.

Using the

n^{\text{th}}

term formula

T_n = a + (n-1)d

, the equations

a + 2d = 10

and

a + 6d = 22

yield

d = 3

and

a = 4

. Substituting these values into

S_{12} = \frac{12}{2}[2(4) + 11(3)]

gives

6 \times 41 = 246

.

Step-by-Step Solution

1

Set up simultaneous equations using the

n^{\text{th}}

term formula

T_n = a + (n-1)d

.

a + 2d = 10

and

a + 6d = 22

.

The

3^{\text{rd}}

term corresponds to

n=3

and the

7^{\text{th}}

term corresponds to

n=7

.

2

Subtract the first equation from the second to find the common difference

d

.

4d = 12 \implies d = 3

.

Subtracting eliminates the first term

a

.

3

Substitute

d = 3

back into the first equation to find the first term

a

.

a + 2(3) = 10 \implies a = 4

.

Determining the first term is necessary to calculate the sum.

4

Calculate the sum of the first

12

terms using

S_n = \frac{n}{2}[2a + (n-1)d]

.

S_{12} = \frac{12}{2}[2(4) + (12-1)(3)] = 6[8 + 33] = 6(41) = 246

.

Applying the AP sum formula with

n = 12

,

a = 4

, and

d = 3

.

Key Concept

Arithmetic Progression: Finding common difference, first term, and sum of terms

Estimated Time:1m 30s

Question 69Question →

An arithmetic progression (AP) has a first term of $2$ and a common difference of $3$ . A geometric progression (GP) has a first term of $1$ and a common ratio of $2$ . Arrange the following quantities in ascending order of their numerical values:

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is the 4th term of the GP (8), followed by the 4th term of the AP (11), the sum of the first 3 terms of the AP (15), and the 5th term of the GP (16).

Evaluating each term individually gives: the 4th term of the GP equals 8, the 4th term of the AP equals 11, the sum of the first 3 terms of the AP equals 15, and the 5th term of the GP equals 16. Arranging these calculated values from smallest to largest gives the order 8, 11, 15, 16.

Step-by-Step Solution

1

Calculate the 4th term of the AP

T_4 = 2 + (4 - 1) \times 3 = 2 + 9 = 11

Using the AP

n^{\text{th}}

term formula

T_n = a + (n-1)d

.

2

Calculate the 5th term of the GP

T_5 = 1 \times 2^{5-1} = 2^4 = 16

Using the GP

n^{\text{th}}

term formula

T_n = a r^{n-1}

.

3

Calculate the sum of the first 3 terms of the AP

S_3 = \frac{3}{2}[2(2) + (3-1)3] = \frac{3}{2}[4 + 6] = 15

Using the AP sum formula

S_n = \frac{n}{2}[2a + (n-1)d]

.

4

Calculate the 4th term of the GP

T_4 = 1 \times 2^{4-1} = 2^3 = 8

Using the GP

n^{\text{th}}

term formula

T_n = a r^{n-1}

.

5

Compare and arrange the values in ascending order

8 < 11 < 15 < 16

Arranging from smallest to largest numerical value.

Key Concept

Nth term and sum formulas of Arithmetic and Geometric Progressions

Question 70Question →

The $2^{\text{nd}}$ term of a geometric progression (GP) is $6$ and the $5^{\text{th}}$ term is $48$ . What is the sum of the first $6$ terms of the progression?

189

93

162

192

Show answer & explanation

Answer: $189$

Answer

The sum of the first 6 terms of the geometric progression is 189.

By using the GP term formula

T_n = a r^{n-1}

, we establish

a r = 6

and

a r^4 = 48

. Dividing the fifth term by the second term yields

r^3 = 8

, giving

r = 2

. Substituting

r = 2

into

a r = 6

gives

a = 3

. Finally, using the sum formula

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

, we evaluate

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

.

Step-by-Step Solution

1

Set up equations for the given terms using the nth term formula

T_n = a r^{n-1}

.

T_2 = a r = 6

and

T_5 = a r^4 = 48

.

The nth term of a GP is defined by

T_n = a r^{n-1}

.

2

Solve for the common ratio

r

by dividing

T_5

by

T_2

.

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

.

Dividing the terms eliminates the first term

a

.

3

Find the first term

a

.

a(2) = 6 \implies a = 3

.

Substitute

r = 2

back into the equation for

T_2

.

4

Calculate the sum of the first 6 terms using

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

.

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

.

Apply the sum formula for a GP with

r > 1

.

Key Concept

Geometric Progression nth term and sum formulas

Question 71Question →

Given the matrices $A = \begin{pmatrix} 4 & x \\ -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , if the determinant of the product matrix $AB$ is $95$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x

is

7

.

Using the property

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

, we find

\det(B) = (2)(4) - (1)(3) = 5

and

\det(A) = (4)(3) - (x)(-1) = 12 + x

. Setting

5(12 + x) = 95

yields

60 + 5x = 95

, which solves to

x = 7

.

Step-by-Step Solution

1

Calculate the determinant of matrix

B

\det(B) = (2)(4) - (1)(3) = 5

The determinant of a

2 \times 2

matrix

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

is

ad - bc

.

2

Express the determinant of matrix

A

in terms of

x

\det(A) = (4)(3) - (x)(-1) = 12 + x

Applying the determinant formula to matrix

A

gives

12 - (-x) = 12 + x

.

3

Apply the determinant product property

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

5(12 + x) = 95

The determinant of the product of two square matrices is equal to the product of their individual determinants.

4

Solve the linear equation for

x

60 + 5x = 95 \implies 5x = 35 \implies x = 7

Subtract

60

from both sides and divide by

5

to isolate

x

.

Key Concept

Determinant of Matrix Product Property

Question 72Question →

What is the simplified form of the expression $\sqrt{50} - \sqrt{18} + \sqrt{8}$ ?

4\sqrt{2}

2\sqrt{10}

6\sqrt{2}

2\sqrt{2}

Show answer & explanation

Answer: $4\sqrt{2}$

Answer

The simplified form of the expression is

4\sqrt{2}

.

Simplifying each radical into basic surd form yields

\sqrt{50} = 5\sqrt{2}

,

\sqrt{18} = 3\sqrt{2}

, and

\sqrt{8} = 2\sqrt{2}

. Combining these like terms gives

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

.

Step-by-Step Solution

1

Simplify each surd term into basic surd form

\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}

,

\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}

, and

\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

Converting each surd to have a common radicand allows for addition and subtraction of like terms.

2

Substitute the simplified surds back into the expression and combine like terms

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

Surds with identical radicands can be combined by operating on their coefficients.

Key Concept

Simplification and Addition/Subtraction of Like Surds

Question 73Question →

Simplify the expression $\frac{3\frac{1}{3} - 1\frac{1}{4}}{\frac{2}{3} \text{ of } 2\frac{1}{2} + 1\frac{1}{6}}$ and express the answer correct to three significant figures. What is the resulting value?

0.735

0.852

1.07

0.736

Show answer & explanation

Answer: $0.735$

Answer

The simplified value correct to three significant figures is

0.735

.

Simplifying the numerator yields

\frac{25}{12}

. For the denominator, applying BODMAS gives

\frac{2}{3} \times \frac{5}{2} = \frac{5}{3}

, and adding

\frac{7}{6}

results in

\frac{17}{6}

. Dividing the numerator by the denominator gives

\frac{25}{34} = 0.73529...

. Rounding to three significant figures gives

0.735

.

Step-by-Step Solution

1

Convert mixed fractions to improper fractions and simplify the numerator.

3\frac{1}{3} - 1\frac{1}{4} = \frac{10}{3} - \frac{5}{4} = \frac{40 - 15}{12} = \frac{25}{12}

Find a common denominator (

12

) to subtract the fractions.

2

Evaluate the denominator following BODMAS rules (performing 'of' before addition).

\frac{2}{3} \text{ of } 2\frac{1}{2} = \frac{2}{3} \times \frac{5}{2} = \frac{5}{3}

. Then add

1\frac{1}{6}

:

\frac{5}{3} + \frac{7}{6} = \frac{10 + 7}{6} = \frac{17}{6}

.

'of' signifies multiplication and takes precedence over addition.

3

Divide the numerator by the denominator.

\frac{\frac{25}{12}}{\frac{17}{6}} = \frac{25}{12} \times \frac{6}{17} = \frac{25}{2 \times 17} = \frac{25}{34}

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

4

Convert the resulting fraction to a decimal and round to three significant figures.

\frac{25}{34} = 0.735294... \approx 0.735

The first non-zero digit is

7

. The first three significant figures are

7, 3, 5

, followed by

2

, so we round down.

Key Concept

Compound Fraction Operations and Approximation

Question 74Question →

If the expression $\frac{\sqrt{48}}{\sqrt{3} + \sqrt{2}} + \frac{\sqrt{72}}{\sqrt{3} - \sqrt{2}}$ is simplified into the form $m + n\sqrt{6}$ , where $m$ and $n$ are integers, find the value of $m + n$ .

Show answer & explanation

Answer: 26

Answer

The simplified expression is

24 + 2\sqrt{6}

, giving

m = 24

and

n = 2

, so

m + n = 26

.

Simplifying

\sqrt{48}

to

4\sqrt{3}

and

\sqrt{72}

to

6\sqrt{2}

allows rationalization of each fraction by its conjugate. The first fraction becomes

12 - 4\sqrt{6}

and the second becomes

12 + 6\sqrt{6}

. Adding these expressions results in

24 + 2\sqrt{6}

, so

m = 24

and

n = 2

, giving

m + n = 26

.

Step-by-Step Solution

1

Simplify the radical numerators

\sqrt{48} = 4\sqrt{3}

and

\sqrt{72} = 6\sqrt{2}

Decomposing surds into perfect square factors simplifies subsequent algebraic expansion.

2

Rationalize the first term

\frac{4\sqrt{3}}{\sqrt{3} + \sqrt{2}}

12 - 4\sqrt{6}

Multiplying the numerator and denominator by the conjugate

(\sqrt{3} - \sqrt{2})

removes the surd from the denominator using the difference of squares

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

.

3

Rationalize the second term

\frac{6\sqrt{2}}{\sqrt{3} - \sqrt{2}}

12 + 6\sqrt{6}

Multiplying the numerator and denominator by the conjugate

(\sqrt{3} + \sqrt{2})

yields a rational denominator of

1

.

4

Combine like surd terms

24 + 2\sqrt{6}

Summing the rational components

(12 + 12 = 24)

and combining similar surd terms

(-4\sqrt{6} + 6\sqrt{6} = 2\sqrt{6})

.

5

Calculate the target sum

m + n

26

Matching coefficients with

m + n\sqrt{6}

gives

m = 24

and

n = 2

, yielding

24 + 2 = 26

.

Key Concept

Rationalization of Binomial Denominators using Conjugates

Question 75Question →

During a seasonal sale, the price of a jacket was reduced from $\text{N}12,000$ to $\text{N}9,600$ . What is the percentage discount on the jacket?

16\%

20\%

25\%

80\%

Show answer & explanation

Answer: $20\%$

Answer

The percentage discount on the jacket is

20\%

.

The discount amount is

\text{N}12,000 - \text{N}9,600 = \text{N}2,400

. Dividing this discount by the original price of

\text{N}12,000

yields

\frac{2,400}{12,000} = 0.20

, which equals

20\%

.

Step-by-Step Solution

1

Calculate the amount of discount

Discount

= \text{N}12,000 - \text{N}9,600 = \text{N}2,400

The discount is the difference between the original price and the sale price.

2

Calculate the percentage discount

Percentage Discount

= \left(\frac{2,400}{12,000}\right) \times 100\% = 20\%

Percentage discount is the discount amount expressed as a fraction of the original price, multiplied by 100%.

Key Concept

Percentage Discount

Estimated Time:45s

Question 76Question →

If $x$ is the smallest positive integer satisfying the modular congruence $2^x \equiv 7 \pmod{11}$ , what is the value of $(3x^2 - 4x + 5) \pmod{11}$ expressed in standard non-negative remainder form?

Show answer & explanation

Answer: 3

Answer

The smallest positive integer exponent satisfying

2^x \equiv 7 \pmod{11}

is

x = 7

. Substituting

x = 7

into

3x^2 - 4x + 5

yields

124

, which simplifies to

3 \pmod{11}

.

Evaluating powers of 2 modulo 11 shows that

2^7 = 128 \equiv 7 \pmod{11}

, giving

x = 7

. Substituting

x = 7

into

3x^2 - 4x + 5

gives

124

, which leaves a remainder of

3

when divided by

11

.

Step-by-Step Solution

1

Find the smallest positive integer exponent

x

satisfying

2^x \equiv 7 \pmod{11}

x = 7

Evaluating consecutive powers of 2 modulo 11 shows

2^1 \equiv 2

,

2^2 \equiv 4

,

2^3 \equiv 8

,

2^4 \equiv 5

,

2^5 \equiv 10

,

2^6 \equiv 9

, and

2^7 \equiv 7

, making

x = 7

the smallest positive integer power.

2

Substitute

x = 7

into the expression

3x^2 - 4x + 5

124

Direct substitution gives

3(7)^2 - 4(7) + 5 = 3(49) - 28 + 5 = 147 - 28 + 5 = 124

.

3

Reduce 124 modulo 11 to standard non-negative remainder form

3

Dividing 124 by 11 yields a quotient of 11 with a remainder of 3 (

124 = 11 \times 11 + 3

).

Key Concept

Modular Exponentiation and Algebraic Evaluation in Modular Arithmetic

Question 77Question →

Convert the decimal number $18.625_{10}$ to a number in base 8. What is the equivalent value in base 8?

22.5_8

22.625_8

24.5_8

22.4_8

Show answer & explanation

Answer: $22.5_8$

Answer

The equivalent value in base 8 is

22.5_8

.

Converting the whole number part

18_{10}

to base 8 gives

22_8

because

18 = 2 \times 8^1 + 2 \times 8^0

. Converting the fractional part

0.625_{10}

to base 8 involves multiplying by 8:

0.625 \times 8 = 5.0

, giving

.5_8

. Thus,

18.625_{10} = 22.5_8

.

Step-by-Step Solution

1

Convert the integer part (

18

) from base 10 to base 8.

18 \div 8 = 2

with a remainder of

2

. Reading the digits upwards gives

22_8

.

Successive division by 8 extracts the octal place values for the whole number.

2

Convert the fractional part (

0.625

) from base 10 to base 8.

0.625 \times 8 = 5.0

. The whole number part of the product is

5

, so

0.625_{10} = 0.5_8

.

Successive multiplication of the fractional part by the target base isolates the negative powers of the base.

3

Combine the converted integer and fractional parts.

22_8 + 0.5_8 = 22.5_8

.

The full representation is the sum of the integer and fractional base 8 components.

Key Concept

Fractional Base Conversion

Question 78Question →

If $43_x + 56_x = 121_x$ , where $x$ represents a positive integer base, find the value of $x$ .

Show answer & explanation

Answer: 8

Answer

The value of the base

x

is 8.

Expanding each base

x

number into polynomial form gives

(4x + 3) + (5x + 6) = x^2 + 2x + 1

. Simplifying yields the quadratic equation

x^2 - 7x - 8 = 0

, which factors as

(x - 8)(x + 1) = 0

. Since a number base must be a positive integer greater than any digit present in the problem (

x > 6

),

x = 8

.

Step-by-Step Solution

1

Convert all base

x

numbers into base 10 algebraic expressions.

43_x = 4x + 3

,

56_x = 5x + 6

, and

121_x = x^2 + 2x + 1

.

Place-value expansion expresses numbers in base

x

as polynomials in

x

.

2

Set up the algebraic equation corresponding to the addition.

(4x + 3) + (5x + 6) = x^2 + 2x + 1 \implies 9x + 9 = x^2 + 2x + 1

.

The sum of the left-hand terms equals the right-hand term.

3

Rearrange into standard quadratic form and factor.

x^2 - 7x - 8 = 0 \implies (x - 8)(x + 1) = 0

.

Moving all terms to one side allows solving for the roots of the quadratic equation.

4

Determine the valid base value.

x = 8

.

Number bases must be positive integers greater than all individual digits present in the expression (

x > 6

).

Key Concept

Unknown base equations and expansion

Question 79Question →

If the surd expression $\sqrt{75} - \sqrt{12} + \frac{6}{\sqrt{3}}$ is simplified to the form $k\sqrt{3}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is 5.

Simplifying

\sqrt{75}

yields

5\sqrt{3}

, simplifying

\sqrt{12}

yields

2\sqrt{3}

, and rationalizing

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

gives

2\sqrt{3}

. Adding these together gives

5\sqrt{3} - 2\sqrt{3} + 2\sqrt{3} = 5\sqrt{3}

. Equating

5\sqrt{3}

to

k\sqrt{3}

yields

k = 5

.

Step-by-Step Solution

1

Simplify the individual square roots.

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}

and

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

.

Factoring out perfect square numbers allows surds to be written in basic radical form.

2

Rationalize the fractional surd term

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

.

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

.

Multiplying numerator and denominator by

\sqrt{3}

eliminates the radical from the denominator.

3

Combine like surd terms.

5\sqrt{3} - 2\sqrt{3} + 2\sqrt{3} = 5\sqrt{3}

.

Like surds share the same radical factor and can be added or subtracted algebraically.

4

Equate the simplified expression to

k\sqrt{3}

.

k = 5

.

Comparing coefficients of

\sqrt{3}

reveals the value of

k

.

Key Concept

Surd Simplification and Rationalization of Denominators

Estimated Time:1m 30s

Question 80Question →

What is the value of $\log_3 16 \times \log_2 27$ ?

12

7

432

6

Show answer & explanation

Answer: $12$

Answer

The value of

\log_3 16 \times \log_2 27

is

12

.

By applying the power law of logarithms,

\log_3 16 = 4\log_3 2

and

\log_2 27 = 3\log_2 3

. Multiplying these gives

(4 \times 3)(\log_3 2 \times \log_2 3)

. Using the change of base identity

\log_3 2 \times \log_2 3 = 1

, the product simplifies to

12

.

Step-by-Step Solution

1

Rewrite

16

and

27

as prime powers

\log_3 16 = \log_3 (2^4)

and

\log_2 27 = \log_2 (3^3)

Express numbers in terms of base prime factors to simplify logarithmic powers.

2

Apply the power law of logarithms

\log_b (a^k) = k \log_b a

\log_3 (2^4) = 4 \log_3 2

and

\log_2 (3^3) = 3 \log_2 3

Bring the exponents out as multipliers.

3

Multiply the expressions and apply the change of base reciprocal property

\log_a b \times \log_b a = 1

(4 \log_3 2) \times (3 \log_2 3) = 12 \times (\log_3 2 \times \log_2 3) = 12 \times 1 = 12

The logarithmic terms are reciprocals of each other, simplifying their product to 1.

Key Concept

Change of Base and Power Laws of Logarithms

Estimated Time:45s

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