Number and Numeration Practice Questions — JAMB UTME — Page 10

Question 181Question →

A delivery van travels from Town A to Town B, a distance of $120\text{ km}$ , at an average speed of $60\text{ km/h}$ . On the return journey along the same route, traffic congestion reduces its average speed by one-third. What is the average speed of the van for the entire round trip?

48\text{ km/h}

50\text{ km/h}

30\text{ km/h}

24\text{ km/h}

Show answer & explanation

Answer: $48\text{ km/h}$

Answer

The average speed for the entire round trip is

48\text{ km/h}

.

The correct answer is found by dividing the total round-trip distance of

240\text{ km}

by the total journey time of

5\text{ hours}

(

2\text{ hours}

outbound and

3\text{ hours}

return), yielding

48\text{ km/h}

.

Step-by-Step Solution

1

Calculate the time taken for the outbound journey.

\text{Time}_{\text{out}} = \frac{120\text{ km}}{60\text{ km/h}} = 2\text{ hours}

.

Time equals distance divided by speed.

2

Calculate the return speed and return time.

\text{Return speed} = 60 - \left(\frac{1}{3} \times 60\right) = 40\text{ km/h}. \text{Time}_{\text{return}} = \frac{120\text{ km}}{40\text{ km/h}} = 3\text{ hours}.

Speed reduction of one-third leaves two-thirds of the initial speed.

3

Compute total distance and total time for the round trip.

\text{Total distance} = 120 + 120 = 240\text{ km}; \text{Total time} = 2 + 3 = 5\text{ hours}.

A round trip covers the distance twice.

4

Divide total distance by total time to find average speed.

\text{Average speed} = \frac{240\text{ km}}{5\text{ hours}} = 48\text{ km/h}.

Average rate over unequal sub-intervals requires total quantity over total time.

Key Concept

Average Rate for Round-Trip Journeys

Question 182Question →

Given that $x$ is a positive real number and the determinant of the matrix $A = \begin{pmatrix} x & 1 & 0 \\ 2 & x & 3 \\ 0 & 1 & 4 \end{pmatrix}$ is $2$ , find the value of $x$ .

Show answer & explanation

Answer: 2

Answer

The positive value of

x

is

2

.

Expanding the determinant along the first row yields

\det(A) = x(4x - 3) - 1(8 - 0) = 4x^2 - 3x - 8

. Equating this to

2

gives

4x^2 - 3x - 10 = 0

. Factoring the quadratic gives

(4x + 5)(x - 2) = 0

, which yields

x = 2

or

x = -1.25

. Because

x

must be positive, the correct value is

2

.

Step-by-Step Solution

1

Expand the

3 \times 3

determinant along the first row

\det(A) = x(4x - 3) - 1(8 - 0) + 0 = 4x^2 - 3x - 8

Expanding along the first row leverages the zero entry to simplify calculation of the determinant.

2

Equate the determinant expression to the given determinant value

4x^2 - 3x - 10 = 0

Setting the calculated determinant equal to

2

creates a quadratic equation in terms of

x

.

3

Factor the quadratic equation to find the candidate values for

x

(4x + 5)(x - 2) = 0 \implies x = 2 \text{ or } x = -1.25

Factoring determines all algebraic solutions that satisfy the determinant equation.

4

Apply the positivity constraint given in the problem statement

x = 2

The question restricts

x

to positive real numbers, discarding the negative root.

Key Concept

Determinant of a 3x3 matrix and quadratic equations

Estimated Time:2m 0s

Question 183Question →

A fruit juice mixture is prepared by combining orange extract and water in the ratio $3 : 5$ . If a bottle contains $40\text{ litres}$ of this mixture, how many litres of water does it contain?

Show answer & explanation

Answer: 25

Answer

The volume of water in the mixture is 25 litres.

The mixture ratio of orange extract to water is

3 : 5

, giving a total of

3 + 5 = 8

equal parts. Dividing the total volume of

40\text{ litres}

by

8

parts gives

5\text{ litres}

per part. Since water makes up

5

parts of the ratio, its volume is

5 \times 5 = 25\text{ litres}

.

Step-by-Step Solution

1

Determine the total number of parts in the ratio.

3 + 5 = 8 parts

The full volume of the mixture is divided into equal portions represented by the sum of the ratio parts.

2

Calculate the volume per ratio part.

40 / 8 = 5 litres per part

Dividing the total volume by the total number of parts gives the size of a single part.

3

Find the quantity of water.

5 * 5 = 25 litres

Water accounts for 5 parts of the ratio, so multiply 5 parts by 5 litres per part.

Key Concept

Ratio Sharing and Proportional Division

Question 184Question →

A trader bought a quantity of palm oil. He sold $\frac{1}{3}$ of the total quantity at a profit of $20\%$ , $\frac{1}{2}$ of the remaining palm oil at a loss of $10\%$ , and the rest of the palm oil at a profit of $x\%$ . If his overall profit on the entire transaction was $15\%$ , what is the value of $x$ ?

Show answer & explanation

Answer: 35

Answer

The value of

x

is

35

.

The palm oil is divided into three equal portions of

\frac{1}{3}

each. The first portion contributes

\frac{20}{3}\%

profit, the second portion contributes a loss of

-\frac{10}{3}\%

, and the final portion contributes

\frac{x}{3}\%

profit. Summing these contributions yields

\frac{10 + x}{3}\%

. Equating this to the target overall profit of

15\%

gives

10 + x = 45

, which solves to

x = 35

.

Step-by-Step Solution

1

Determine the fractional portion of palm oil sold at each stage.

First portion =

\frac{1}{3}

. Remaining after first sale =

1 - \frac{1}{3} = \frac{2}{3}

. Second portion =

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

. Final portion =

1 - \left(\frac{1}{3} + \frac{1}{3}\right) = \frac{1}{3}

.

Finding the exact fraction of the total quantity sold at each step is necessary to weight their respective profit/loss rates correctly.

2

Formulate the net percentage profit equation.

\text{Net Profit } \% = \left(\frac{1}{3} \times 20\%\right) + \left(\frac{1}{3} \times (-10\%)\right) + \left(\frac{1}{3} \times x\%\right) = \frac{20 - 10 + x}{3}\% = \frac{10 + x}{3}\%

The overall profit percentage is the sum of individual fractional contributions, treating loss as a negative percentage.

3

Set the net profit equal to

15\%

and solve for

x

.

\frac{10 + x}{3} = 15 \implies 10 + x = 45 \implies x = 35

Solving this linear equation gives the required profit percentage for the final portion.

Key Concept

Weighted Average Profit and Loss across fractional components of an asset

Estimated Time:2m 0s

Question 185Question →

What is the canonical non-negative remainder when $3^{45} - 11$ is divided by $7$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Using modular exponentiation,

3^3 = 27 \equiv -1 \pmod{7}

, so

3^{45} = (3^3)^{15} \equiv (-1)^{15} = -1 \pmod{7}

. Subtracting 11 yields

-1 - 11 = -12 \pmod{7}

. Adding 14 (a multiple of 7) gives

-12 + 14 = 2

, which is the canonical non-negative remainder.

Step-by-Step Solution

1

Evaluate

3^{45} \pmod{7}

using power rules

3^{45} \equiv -1 \pmod{7}

Since

3^3 = 27 \equiv -1 \pmod{7}

, raising both sides to the power of 15 gives

(3^3)^{15} \equiv (-1)^{15} = -1 \pmod{7}

.

2

Subtract 11 from the modular result

-12 \pmod{7}

Replacing

3^{45}

with

-1

gives

-1 - 11 = -12

.

3

Reduce -12 to its canonical non-negative remainder in

[0, 6]

2

Adding the smallest multiple of 7 that makes the result non-negative gives

-12 + 14 = 2

.

Key Concept

Modular Exponentiation and Negative Remainder Reduction

Question 186Question →

In a survey of $120$ tech entrepreneurs at an innovation hub, $65$ secured Angel Investment ( $A$ ), $55$ received Venture Capital ( $V$ ), and $50$ obtained Government Grants ( $G$ ). Furthermore, $25$ received both $A$ and $V$ , $20$ received both $V$ and $G$ , $22$ received both $A$ and $G$ , while $12$ received no funding from any of these three sources. How many entrepreneurs secured funding from exactly two of these sources?

Show answer & explanation

Answer: 52

Answer

The number of entrepreneurs who secured funding from exactly two sources is 52.

Subtracting the

12

unfunded entrepreneurs from the total

120

yields

108

funded entrepreneurs. Applying inclusion-exclusion gives

108 = 65 + 55 + 50 - (25 + 20 + 22) + n(A \cap V \cap G)

, which gives

n(A \cap V \cap G) = 5

. Subtracting

5

from each pairwise intersection yields

20

,

15

, and

17

for entrepreneurs receiving funding from exactly two sources. Summing these gives

52

.

Step-by-Step Solution

1

Determine the cardinality of the union of all three funding sets

n(A \cup V \cup G) = 108

Subtracting the 12 unfunded entrepreneurs from the universal set of 120 gives the total number of entrepreneurs who received at least one form of funding.

2

Solve for the number of entrepreneurs who received funding from all three sources using inclusion-exclusion

n(A \cap V \cap G) = 5

Using

n(A \cup V \cup G) = n(A) + n(V) + n(G) - n(A \cap V) - n(V \cap G) - n(A \cap G) + n(A \cap V \cap G)

, we get

108 = 65 + 55 + 50 - 25 - 20 - 22 + x

, which simplifies to

108 = 103 + x

, giving

x = 5

.

3

Calculate the count for each region representing exactly two funding sources

Only

A \cap V = 20

, Only

V \cap G = 15

, Only

A \cap G = 17

Subtracting the 3-set intersection (

x = 5

) from each pairwise intersection isolates the elements belonging to strictly two sets.

4

Sum the three strictly two-set regions

20 + 15 + 17 = 52

Adding the individual counts for the three disjoint regions gives the total number of entrepreneurs who received funding from exactly two sources.

Key Concept

Principle of Inclusion-Exclusion for three sets and cardinal partitioning of Venn diagrams

Question 187Question →

What is the determinant of the matrix $A = \begin{pmatrix} 6 & -3 \\ 2 & 1 \end{pmatrix}$ ?

12

0

-12

-36

Show answer & explanation

Answer: 12

Answer

12

For any

2 \times 2

matrix

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

, the determinant is given by

ad - bc

. Substituting

a = 6

,

b = -3

,

c = 2

, and

d = 1

yields

(6)(1) - (-3)(2) = 6 - (-6) = 12

.

Step-by-Step Solution

1

Identify the elements of the 2x2 matrix

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

a = 6

,

b = -3

,

c = 2

,

d = 1

Standard entries for determinant formula

\det(A) = ad - bc

.

2

Compute the product of the main diagonal elements (

ad

) and off-diagonal elements (

bc

)

ad = (6)(1) = 6

and

bc = (-3)(2) = -6

These form the components of the 2x2 determinant formula.

3

Subtract

bc

from

ad

\det(A) = 6 - (-6) = 6 + 6 = 12

Subtracting a negative quantity is equivalent to addition.

Key Concept

Determinant of a 2x2 Matrix

Estimated Time:45s

Question 188Question →

In a survey of $100$ book club members regarding their reading preferences among Science Fiction ( $S$ ), Mystery ( $M$ ), and Historical Fiction ( $H$ ), it was found that $48$ read Science Fiction, $42$ read Mystery, and $38$ read Historical Fiction. Furthermore, $18$ read both Science Fiction and Mystery, $15$ read both Science Fiction and Historical Fiction, $14$ read both Mystery and Historical Fiction, and $8$ read all three genres. How many of the members read exactly one of these three genres?

Show answer & explanation

Answer: 58

Answer

The number of members who read exactly one of the three genres is

58

.

By isolating the disjoint regions of the 3-set Venn diagram, the number of members reading only Science Fiction is

23

, only Mystery is

18

, and only Historical Fiction is

17

. Adding these disjoint sets yields

58

.

Step-by-Step Solution

1

Determine exclusive two-set intersection regions

n(S \cap M \text{ only}) = 10

,

n(S \cap H \text{ only}) = 7

,

n(M \cap H \text{ only}) = 6

The given pairwise totals include members who read all three genres, so subtracting

n(S \cap M \cap H) = 8

isolates those in exactly two sets.

2

Determine exclusive single-set regions

n(S \text{ only}) = 23

,

n(M \text{ only}) = 18

,

n(H \text{ only}) = 17

Subtracting all overlapping regions within each set's boundary gives the number of members reading only that specific genre.

3

Sum the single-set regions

23 + 18 + 17 = 58

The set of members reading exactly one genre is the disjoint union of the three exclusive single-set regions.

Key Concept

3-Set Venn Diagram Cardinality and Disjoint Region Analysis

Question 189Question →

Given the $3 \times 3$ matrix $M = \begin{pmatrix} k & 2 & 1 \\ 3 & k-1 & 2 \\ 1 & 4 & 1 \end{pmatrix}$ , where $k > 5$ . If $\det(M^2) = 100$ and $\det(M) < 0$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

7

Using the matrix determinant power identity

\det(M^2) = (\det(M))^2 = 100

and the condition

\det(M) < 0

, we find

\det(M) = -10

. Expanding

\det(M)

along the first row yields

\det(M) = k(k-9) - 2(1) + 1(13-k) = k^2 - 10k + 11

. Setting this equal to

-10

gives the quadratic equation

k^2 - 10k + 21 = 0

, which factors as

(k-3)(k-7) = 0

. Given

k > 5

, the unique solution is

k = 7

.

Step-by-Step Solution

1

Apply determinant properties for matrix powers

\det(M) = -10

Because

\det(M^2) = (\det(M))^2 = 100

and it is given that

\det(M) < 0

, taking the negative square root gives

\det(M) = -10

.

2

Evaluate the determinant of matrix

M

using first-row expansion

\det(M) = k^2 - 10k + 11

Expanding along row 1 gives

k((k-1)(1) - 8) - 2(3(1) - 2(1)) + 1(3(4) - (k-1)(1)) = k^2 - 9k - 2 + 13 - k = k^2 - 10k + 11

.

3

Set up and simplify the quadratic equation for

k

k^2 - 10k + 21 = 0

Equating

k^2 - 10k + 11

to

-10

yields

k^2 - 10k + 21 = 0

.

4

Solve the quadratic equation and enforce the inequality condition

k > 5

k = 7

Factoring

(k-3)(k-7) = 0

gives

k = 3

or

k = 7

. Applying the restriction

k > 5

selects

k = 7

.

Key Concept

Determinant Properties of Matrix Powers and 3x3 Matrix Expansion

Question 190Question →

Given the matrices $P = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 0 \\ -2 & 1 \end{pmatrix}$ , what is the product matrix $PQ$ ?

\begin{pmatrix} 10 & 0 \\ -2 & 4 \end{pmatrix}

\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

\begin{pmatrix} 10 & 15 \\ -3 & -2 \end{pmatrix}

\begin{pmatrix} 16 & 3 \\ 13 & 4 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}$

Answer

The product matrix

PQ

is

\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

.

To find

PQ

, multiply each row of matrix

P

by each column of matrix

Q

. The entry in row 1, column 1 is

2(5) + 3(-2) = 4

. The entry in row 1, column 2 is

2(0) + 3(1) = 3

. The entry in row 2, column 1 is

1(5) + 4(-2) = -3

. The entry in row 2, column 2 is

1(0) + 4(1) = 4

. Putting these together yields

\begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

.

Step-by-Step Solution

1

Compute the entries of the top row of

PQ

using the first row of

P

and both columns of

Q

.

First entry:

(2)(5) + (3)(-2) = 10 - 6 = 4

. Second entry:

(2)(0) + (3)(1) = 0 + 3 = 3

.

Matrix entry

(1,1)

is the dot product of Row 1 of

P

and Column 1 of

Q

; entry

(1,2)

is the dot product of Row 1 of

P

and Column 2 of

Q

.

2

Compute the entries of the bottom row of

PQ

using the second row of

P

and both columns of

Q

.

Third entry:

(1)(5) + (4)(-2) = 5 - 8 = -3

. Fourth entry:

(1)(0) + (4)(1) = 0 + 4 = 4

.

Matrix entry

(2,1)

is the dot product of Row 2 of

P

and Column 1 of

Q

; entry

(2,2)

is the dot product of Row 2 of

P

and Column 2 of

Q

.

3

Assemble the computed entries into a

2 \times 2

matrix.

PQ = \begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix}

.

Combining row results gives the final product matrix.

Key Concept

Matrix Multiplication (Row-by-Column Dot Product)

Estimated Time:1m 15s

Question 191Question →

A businesswoman invested $\text{₦}50,000$ in a financial fund. Part of the money was invested at a simple interest rate of $6\%$ per annum, and the remaining part was invested at $8\%$ simple interest per annum. If the total interest earned at the end of $1$ year was $\text{₦}3,600$ , what was the amount, in Naira, invested at the $8\%$ interest rate?

Show answer & explanation

Answer: 30000

Answer

The amount invested at the

8\%

interest rate is

\text{₦}30,000

.

Setting up the linear equation for annual simple interest gives

0.06(50,000 - x) + 0.08x = 3,600

. Simplifying this yields

3,000 + 0.02x = 3,600

, which solves to

x = 30,000

. Thus,

\text{₦}30,000

was invested at

8\%

.

Step-by-Step Solution

1

Define variables for the two investment amounts.

Let

x

be the amount in Naira invested at

8\%

, so

(50,000 - x)

is the amount invested at

6\%

.

The total capital of

\text{₦}50,000

is split into two distinct portions.

2

Formulate the total interest expression using the simple interest formula

I = \frac{P \times R \times T}{100}

.

\frac{6}{100}(50,000 - x) + \frac{8}{100}x = 3,600

The sum of annual interests from both parts equals the total interest earned of

\text{₦}3,600

.

3

Expand the terms and solve the linear equation for

x

.

3,000 + 0.02x = 3,600 \implies 0.02x = 600 \implies x = 30,000

Isolating

x

yields the exact principal amount allocated to the

8\%

interest rate.

Key Concept

Simple Interest and Allocation of Principal across Different Interest Rates

Estimated Time:1m 30s

Question 192Question →

A farmer bought a motorcycle for $\text{₦}250,000$ and later sold it at a profit of $12\%$ . What is the selling price of the motorcycle in Naira?

Show answer & explanation

Answer: 280000

Answer

The selling price of the motorcycle is ₦280,000.

The cost price of the motorcycle is

\text{₦}250,000

. A profit of

12\%

means an additional

\frac{12}{100} \times 250,000 = \text{₦}30,000

. Adding this profit to the cost price gives a selling price of

\text{₦}250,000 + \text{₦}30,000 = \text{₦}280,000

.

Step-by-Step Solution

1

Calculate the profit amount in Naira

Profit = ₦30,000

Profit is calculated as 12% of the original cost price of ₦250,000.

2

Determine the final selling price

Selling Price = ₦280,000

Selling price equals cost price plus the profit made.

Key Concept

Percentage Profit and Selling Price

Question 193Question →

If $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} - \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = k\sqrt{15}$ , find the value of $k$ .

Show answer & explanation

Answer: 2

Answer

The value of

k

is

2

.

Combining the fractions gives a common denominator of

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

. Expanding the numerator gives

(8+2\sqrt{15}) - (8-2\sqrt{15}) = 4\sqrt{15}

. Dividing by 2 yields

2\sqrt{15}

, giving

k = 2

.

Step-by-Step Solution

1

Combine the fractions on the left-hand side over a common denominator.

\frac{(\sqrt{5} + \sqrt{3})^2 - (\sqrt{5} - \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}

Combining two fractions with conjugate denominators simplifies the expression.

2

Evaluate the denominator using the difference of two squares formula

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

.

(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2

Multiplying conjugate surds eliminates the radical signs in the denominator.

3

Expand both squared terms in the numerator and subtract them.

(8 + 2\sqrt{15}) - (8 - 2\sqrt{15}) = 4\sqrt{15}

Expanding

(a \pm b)^2 = a^2 \pm 2ab + b^2

gives

5 \pm 2\sqrt{15} + 3 = 8 \pm 2\sqrt{15}

.

4

Divide the resulting numerator by the denominator and solve for

k

.

\frac{4\sqrt{15}}{2} = 2\sqrt{15} \Rightarrow k = 2

Dividing

4\sqrt{15}

by

2

yields

2\sqrt{15}

, so matching the coefficients gives

k = 2

.

Key Concept

Rationalization of Denominators and Difference of Conjugate Surd Fractions

Estimated Time:1m 30s

Question 194Question →

A logistics firm purchased a commercial delivery van for $\text{₦}800,000$ . If the value of the van depreciates by $10\%$ in the first year and by $15\%$ in the second year, what is the value of the van at the end of the second year?

\text{₦}612,000

\text{₦}600,000

\text{₦}188,000

\text{₦}648,000

Show answer & explanation

Answer: $\text{₦}612,000$

Answer

\text{₦}612,000

To find the remaining value of an asset undergoing successive annual depreciation, multiply the initial cost by each annual retention factor

(1 - r)

. For Year 1 at

10\%

, the value retention factor is

0.90

, making the value

\text{₦}720,000

. For Year 2 at

15\%

, the retention factor is

0.85

, yielding a final value of

\text{₦}720,000 \times 0.85 = \text{₦}612,000

.

Step-by-Step Solution

1

Calculate the value of the van at the end of the first year after a 10% depreciation.

\text{Value after Year 1} = \text{₦}800,000 \times \left(1 - \frac{10}{100}\right) = \text{₦}800,000 \times 0.90 = \text{₦}720,000

Depreciation decreases the principal asset value by the specified percentage.

2

Calculate the value of the van at the end of the second year after a 15% depreciation on the year-one value.

\text{Value after Year 2} = \text{₦}720,000 \times \left(1 - \frac{15}{100}\right) = \text{₦}720,000 \times 0.85 = \text{₦}612,000

Successive depreciation is compounded on the reduced value at the start of each period.

Key Concept

Successive Asset Depreciation

Question 195Question →

A cyclist travels a total distance of $90\text{ km}$ . She completes the first $40\text{ km}$ of the journey at a constant speed of $20\text{ km/h}$ . If her average speed for the entire journey is $30\text{ km/h}$ , find her speed, in $\text{km/h}$ , over the remaining distance.

Show answer & explanation

Answer: 50

Answer

The speed over the remaining distance is

50\text{ km/h}

.

Average speed is total distance divided by total time. For a

90\text{ km}

trip with an average speed of

30\text{ km/h}

, the total trip time is

3\text{ hours}

. Covering the first

40\text{ km}

at

20\text{ km/h}

requires

2\text{ hours}

, leaving

1\text{ hour}

to cover the remaining

50\text{ km}

(

90 - 40

). Dividing

50\text{ km}

by

1\text{ hour}

results in a required speed of

50\text{ km/h}

.

Step-by-Step Solution

1

Calculate the total time for the complete trip

Total duration =

3\text{ hours}

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

T = \frac{D}{V_{\text{avg}}} = \frac{90}{30} = 3\text{ hours}

.

2

Calculate the time spent covering the first part of the journey

Time for first segment =

2\text{ hours}

Time taken for a segment is distance divided by speed:

t_1 = \frac{40}{20} = 2\text{ hours}

.

3

Find the remaining time and remaining distance

Remaining time =

1\text{ hour}

; Remaining distance =

50\text{ km}

Subtracting the first segment's time and distance from the totals gives

3 - 2 = 1\text{ hour}

and

90 - 40 = 50\text{ km}

.

4

Calculate the required speed for the second segment

Speed for remaining distance =

50\text{ km/h}

Speed is calculated by dividing remaining distance by remaining time:

\frac{50\text{ km}}{1\text{ hour}} = 50\text{ km/h}

.

Key Concept

Average Rate and Speed Calculations

Question 196Question →

If $132_x - 54_x = 45_x$ , where $x$ is a positive integer base, what is the value of $x$ ?

6

7

8

9

Show answer & explanation

Answer: 7

Answer

The value of the base

x

is 7.

Expanding all numbers in terms of powers of

x

yields

132_x = x^2 + 3x + 2

,

54_x = 5x + 4

, and

45_x = 4x + 5

. Substituting these into

132_x - 54_x = 45_x

gives

(x^2 + 3x + 2) - (5x + 4) = 4x + 5

. Simplifying this equation results in

x^2 - 6x - 7 = 0

. Factoring

(x - 7)(x + 1) = 0

gives solutions

x = 7

and

x = -1

. Since a base must be a positive integer greater than 5, the base

x

is 7.

Step-by-Step Solution

1

Convert each term from base

x

to base 10 using place-value expansion.

132_x = 1 \cdot x^2 + 3 \cdot x^1 + 2 \cdot x^0 = x^2 + 3x + 2

,

54_x = 5x + 4

, and

45_x = 4x + 5

.

Converting all terms to a common base (base 10) allows standard algebraic solving.

2

Substitute the expanded expressions into the given equation.

(x^2 + 3x + 2) - (5x + 4) = 4x + 5

.

Set up the algebraic equation in terms of

x

.

3

Simplify the equation into standard quadratic form.

x^2 - 2x - 2 = 4x + 5 \implies x^2 - 6x - 7 = 0

.

Combine like terms and move all terms to one side.

4

Solve the quadratic equation for

x

.

(x - 7)(x + 1) = 0 \implies x = 7

or

x = -1

.

Factor the quadratic expression.

5

Select the valid base.

x = 7

.

A number base must be a positive integer greater than the largest digit present in the equation (which is 5).

Key Concept

Solving equations involving unknown number bases by expanding in powers of the base.

Question 197Question →

Given the matrices $A = \begin{pmatrix} x & 3 \\ 1 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ , what is the value of $x$ such that the determinant of the product matrix $AB$ is $25$ ?

4

2

-1

11.5

Show answer & explanation

Answer: 4

Answer

The value of

x

is 4.

The determinant of a product of matrices equals the product of their determinants,

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

. Since

\det(B) = 5

and

\det(A) = 2x - 3

, setting

5(2x - 3) = 25

yields

2x - 3 = 5

, giving

x = 4

.

Step-by-Step Solution

1

Calculate the determinant of matrix B

\det(B) = (2)(4) - (1)(3) = 8 - 3 = 5

The determinant of a 2x2 matrix

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

is given by

ad - bc

.

2

Express the determinant of matrix A in terms of x

\det(A) = (x)(2) - (3)(1) = 2x - 3

Apply the 2x2 determinant formula to matrix A.

3

Use the determinant product rule det(AB) = det(A) * det(B)

\det(AB) = (2x - 3) \cdot 5 = 10x - 15

The determinant of the product of two square matrices equals the product of their individual determinants.

4

Set det(AB) equal to 25 and solve for x

10x - 15 = 25 \implies 10x = 40 \implies x = 4

Equate the determinant expression to the given value 25 to isolate x.

Key Concept

Determinant of a Matrix Product

Question 198Question →

Given two matrices $A = \begin{pmatrix} x & 2 \\ 2 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} x & 1 \\ -1 & 1 \end{pmatrix}$ , where $x > 0$ . If the determinant of the product matrix $AB$ is $20$ , find the value of $x$ .

Show answer & explanation

Answer: 3

Answer

The positive value of

x

is 3.

Using the property

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

, we compute

\det(A) = 3x - 4

and

\det(B) = x + 1

. Equating their product to 20 gives

(3x - 4)(x + 1) = 20

, which simplifies to the quadratic

3x^2 - x - 24 = 0

. Factorizing yields

(3x + 8)(x - 3) = 0

. Since

x

must be a positive number, the correct value is

x = 3

.

Step-by-Step Solution

1

Calculate the determinant of matrix A

\det(A) = 3x - 4

The determinant of a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} is ad - bc.

2

Calculate the determinant of matrix B

\det(B) = x + 1

Applying ad - bc gives (x)(1) - (1)(-1) = x + 1.

3

Use the determinant product property det(AB) = det(A) * det(B)

(3x - 4)(x + 1) = 20

The determinant of the product of two square matrices equals the product of their individual determinants.

4

Form and solve the quadratic equation

3x^2 - x - 24 = 0, which factorizes into (3x + 8)(x - 3) = 0

Expanding (3x - 4)(x + 1) gives 3x^2 - x - 4. Subtracting 20 yields 3x^2 - x - 24 = 0.

5

Determine the positive solution for x

x = 3

Solving the factors gives x = -8/3 or x = 3. Since x must be positive (x > 0), x = 3.

Key Concept

Determinant of Matrix Product and 2x2 Determinants

Estimated Time:2m 0s

Question 199Question →

Evaluate the numerical expression $\frac{0.00054 \times 0.002}{0.00036}$ and state the final result as a decimal.

Show answer & explanation

Answer: 0.003

Answer

The correct value as a decimal is 0.003.

Converting all terms to scientific notation yields

\frac{(5.4 \times 10^{-4}) \times (2 \times 10^{-3})}{3.6 \times 10^{-4}} = \frac{10.8 \times 10^{-7}}{3.6 \times 10^{-4}} = 3 \times 10^{-3} = 0.003

.

Step-by-Step Solution

1

Convert each decimal in the expression to scientific notation (standard form)

0.00054 = 5.4 \times 10^{-4}

,

0.002 = 2 \times 10^{-3}

,

0.00036 = 3.6 \times 10^{-4}

Converting decimals with leading zeros to powers of 10 prevents errors in decimal point alignment during multiplication and division.

2

Simplify the numerator by multiplying coefficients and adding exponents

(5.4 \times 10^{-4}) \times (2 \times 10^{-3}) = 10.8 \times 10^{-7}

According to the laws of indices,

10^a \times 10^b = 10^{a+b}

, so

-4 + (-3) = -7

.

3

Divide the simplified numerator by the denominator

\frac{10.8 \times 10^{-7}}{3.6 \times 10^{-4}} = \left(\frac{10.8}{3.6}\right) \times 10^{-7 - (-4)} = 3.0 \times 10^{-3}

Dividing the coefficients gives

10.8 \div 3.6 = 3

, and subtracting the exponents gives

-7 - (-4) = -3

.

4

Express the result in standard decimal form

3.0 \times 10^{-3} = 0.003

Shifting the decimal point 3 positions to the left converts

10^{-3}

to standard decimal representation.

Key Concept

Simplifying Decimal Expressions using Standard Form and Laws of Indices

Question 200Question →

Given the matrices $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & k \end{pmatrix}$ , find the value of $k$ such that the determinant of the product matrix $AB$ is equal to 50.

Show answer & explanation

Answer: 3

Answer

The value of

k

is 3.

By applying the determinant product rule

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

, we find

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

. Given

\det(AB) = 50

, it follows that

\det(B) = 50 / 10 = 5

. Since

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

, setting

k + 2 = 5

gives

k = 3

.

Step-by-Step Solution

1

Calculate the determinant of matrix A

\det(A) = (4 \times 3) - (2 \times 1) = 12 - 2 = 10

The determinant of a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} is ad - bc.

2

Apply the product property of determinants

\det(B) = \frac{\det(AB)}{\det(A)} = \frac{50}{10} = 5

For any two square matrices of the same dimension, \det(AB) = \det(A) \cdot \det(B).

3

Express the determinant of matrix B in terms of k and solve

\det(B) = (1)(k) - (-1)(2) = k + 2 = 5 \implies k = 3

Equating the calculated determinant formula for B to its numerical value of 5.

Key Concept

Determinant Product Property and 2x2 Matrix Determinant

Estimated Time:1m 30s

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