All practice questions

Question 6881Question →

Meridian Logistics Plc was registered with an authorized share capital of $5,000,000$ ordinary shares of $\text{₦}1.50$ each. The directors issued $3,000,000$ shares to the public and called up $\text{₦}1.00$ per share. If all called-up funds were received except a call of $\text{₦}0.20$ per share on $150,000$ shares, calculate the total paid-up share capital of the company in Naira.

Show answer & explanation

Answer: 2970000

Answer

The paid-up capital of Meridian Logistics Plc is ₦2,970,000.

Paid-up capital represents the actual cash received from shareholders for called-up shares. The called-up capital is

3,000,000 \text{ shares} \times \text{₦}1.00 = \text{₦}3,000,000

. Calls in arrears are

150,000 \text{ shares} \times \text{₦}0.20 = \text{₦}30,000

. Deducting calls in arrears from called-up capital gives

\text{₦}3,000,000 - \text{₦}30,000 = \text{₦}2,970,000

.

Step-by-Step Solution

1

Calculate Total Called-up Capital

₦3,000,000

Multiply issued shares by called-up value per share (3,000,000 shares × ₦1.00).

2

Calculate Calls in Arrears

₦30,000

Multiply defaulting shares by unpaid call per share (150,000 shares × ₦0.20).

3

Compute Paid-up Capital

₦2,970,000

Subtract Calls in Arrears from Total Called-up Capital (₦3,000,000 - ₦30,000).

Key Concept

Classification of Share Capital - Paid-up Capital Calculation

Estimated Time:1m 30s

Question 6882Question →

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 6883Question →

When the polynomial $P(x) = 3x^3 + ax^2 + bx - 10$ is divided by $(x - 2)$ , the remainder is $14$ , and when it is divided by $(x + 1)$ , the remainder is $-16$ . What is the value of $a + b$ ?

Show answer & explanation

Answer: 1

Answer

The value of

a + b

is

1

.

According to the Remainder Theorem, dividing

P(x)

by

(x - 2)

and

(x + 1)

gives

P(2) = 14

and

P(-1) = -16

respectively. Expanding these expressions forms two linear equations:

2a + b = 0

and

a - b = -3

. Solving these simultaneously gives

a = -1

and

b = 2

, so

a + b = 1

.

Step-by-Step Solution

1

Apply the Remainder Theorem for the first divisor

(x - 2)

2a + b = 0

By the Remainder Theorem,

P(2) = 14

. Substituting

x = 2

into

P(x)

yields

3(8) + 4a + 2b - 10 = 14

, which simplifies to

2a + b = 0

.

2

Apply the Remainder Theorem for the second divisor

(x + 1)

a - b = -3

By the Remainder Theorem,

P(-1) = -16

. Substituting

x = -1

into

P(x)

yields

3(-1) + a - b - 10 = -16

, which simplifies to

a - b = -3

.

3

Solve the simultaneous equations for

a

and

b

a = -1

and

b = 2

Adding

2a + b = 0

and

a - b = -3

yields

3a = -3

, giving

a = -1

. Substituting

a = -1

into

2a + b = 0

gives

b = 2

.

4

Calculate the value of

a + b

1

Adding the computed values yields

a + b = -1 + 2 = 1

.

Key Concept

Polynomial Remainder Theorem and Systems of Linear Equations

Estimated Time:2m 0s

Question 6884Question →

If $\log_4 x + \log_2 y = \frac{5}{2}$ and $3^x \cdot 9^{-y} = 81$ , what is the value of $x + y$ ?

6

8

10

12

Show answer & explanation

Answer: 10

Answer

The value of

x + y

is

10

.

Converting

\log_4 x

to base

2

gives

\frac{1}{2}\log_2 x

, leadings to

x y^2 = 32

. Simplifying

3^x \cdot 9^{-y} = 81

gives

x - 2y = 4

. Solving these simultaneously gives

x = 8

and

y = 2

, whose sum is

10

.

Step-by-Step Solution

1

Simplify the logarithmic equation using the change of base formula.

\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{1}{2}\log_2 x

. Substituting this gives

\frac{1}{2}\log_2 x + \log_2 y = \frac{5}{2} \implies \log_2 x + 2\log_2 y = 5

.

Converting logarithms to a common base of

2

allows combining terms.

2

Combine logarithmic terms and express as an algebraic relation.

\log_2(x y^2) = 5 \implies x y^2 = 2^5 = 32

.

Applying the logarithmic laws

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

and

\log_b M + \log_b N = \log_b(MN)

.

3

Simplify the exponential equation.

3^x \cdot (3^2)^{-y} = 3^4 \implies 3^{x - 2y} = 3^4 \implies x - 2y = 4

.

Expressing both sides with base

3

allows equating exponents.

4

Solve the system of equations for

x

and

y

.

From

x = 4 + 2y

, substitute into

x y^2 = 32

:

(4 + 2y)y^2 = 32 \implies 2y^3 + 4y^2 - 32 = 0 \implies y^3 + 2y^2 - 16 = 0

. Testing positive factors yields

y = 2

. Then

x = 4 + 2(2) = 8

.

Simultaneous substitution yields the unique real positive solutions

x = 8

and

y = 2

.

5

Calculate

x + y

.

x + y = 8 + 2 = 10

.

Adding the computed values of

x

and

y

gives the required sum.

Key Concept

Simultaneous Exponential and Logarithmic Systems

Estimated Time:2m 0s

Question 6885Question →

How many integer values of $x$ satisfy both the linear inequality $\frac{3 - 2x}{5} \ge \frac{x + 4}{2}$ and the quadratic inequality $x^2 + 4x - 5 \le 0$ ?

4

3

5

2

Show answer & explanation

Answer: 4

Answer

4

Solving the linear inequality yields

x \le -\frac{14}{9} \approx -1.56

. Solving the quadratic inequality gives

-5 \le x \le 1

. The overlap between both sets is

-5 \le x \le -\frac{14}{9}

. The integers falling within this interval are

-5, -4, -3

, and

-2

, making 4 valid integer solutions in total.

Step-by-Step Solution

1

Solve the linear inequality

\frac{3 - 2x}{5} \ge \frac{x + 4}{2}

.

2(3 - 2x) \ge 5(x + 4) \implies 6 - 4x \ge 5x + 20 \implies -9x \ge 14 \implies x \le -\frac{14}{9} \approx -1.56

.

Clear denominators by multiplying by 10 and reverse the inequality sign when dividing both sides by

-9

.

2

Solve the quadratic inequality

x^2 + 4x - 5 \le 0

.

(x + 5)(x - 1) \le 0 \implies -5 \le x \le 1

.

Factorize the quadratic expression to find critical points at

x = -5

and

x = 1

. The region where the product is non-positive is between the roots.

3

Determine the intersection of the two solution sets.

-5 \le x \le -\frac{14}{9}

.

Combine the conditions

x \le -1.56

and

-5 \le x \le 1

to find the set of values satisfying both inequalities simultaneously.

4

Count the integer values within the intersection set

[-5, -1.56]

.

The integers are

-5, -4, -3, -2

, giving a total of 4 integers.

Identify all whole numbers within the combined solution interval.

Key Concept

Simultaneous Linear and Quadratic Inequalities

Question 6886Question →

When the polynomial $P(x) = x^3 - ax^2 + bx - 6$ is divided by $(x - 1)$ , the remainder is $-4$ . If $(x - 2)$ is a factor of $P(x)$ , what is the value of $a + b$ ?

5

1

-1

-3

Show answer & explanation

Answer: $5$

Answer

The value of

a + b

is

5

.

Using the Remainder Theorem with

P(1) = -4

gives the equation

-a + b = 1

. Using the Factor Theorem with

P(2) = 0

gives

2a - b = 1

. Solving this simultaneous system gives

a = 2

and

b = 3

, leading to

a + b = 5

.

Step-by-Step Solution

1

Apply the Remainder Theorem for division by

(x - 1)

.

-a + b = 1

According to the Remainder Theorem,

P(1) = -4

. Substituting

x = 1

into

P(x)

yields

1^3 - a(1)^2 + b(1) - 6 = -4 \implies -a + b = 1

.

2

Apply the Factor Theorem for the factor

(x - 2)

.

2a - b = 1

Since

(x - 2)

is a factor,

P(2) = 0

. Substituting

x = 2

into

P(x)

yields

2^3 - a(2)^2 + b(2) - 6 = 0 \implies 8 - 4a + 2b - 6 = 0 \implies 2a - b = 1

.

3

Solve the system of linear equations simultaneously.

a = 2, b = 3

Adding the two equations

(-a + b) + (2a - b) = 1 + 1

yields

a = 2

. Substituting

a = 2

back into

-a + b = 1

gives

b = 3

.

4

Calculate the required value

a + b

.

5

a + b = 2 + 3 = 5

.

Key Concept

Polynomial Factor and Remainder Theorems

Estimated Time:1m 30s

Question 6887Question →

In how many different ways can $5$ boys and $3$ girls be seated in a straight row such that all $3$ girls must sit together?

Show answer & explanation

Answer: 4320

Answer

The total number of ways to seat

5

boys and

3

girls in a row such that all

3

girls sit together is

4320

.

To arrange

5

boys and

3

girls so that the girls are always together, treat the

3

girls as

1

single unit. Combined with the

5

boys, there are

6

units to arrange in a straight line, which can be done in

6! = 720

ways. Within their group, the

3

girls can be arranged in

3! = 6

ways. By the multiplication principle, the total number of seating arrangements is

720 \times 6 = 4320

.

Step-by-Step Solution

1

Group the restricted items into a single block

The

3

girls form

1

unit. Combined with the

5

boys, there are

5 + 1 = 6

units to arrange.

Since all

3

girls must sit together, treating them as a single block ensures they are not separated.

2

Calculate the linear arrangements of the combined units

The

6

units can be arranged in

6! = 720

ways.

The number of distinct ways to arrange

n

items in a line is

n!

.

3

Calculate internal arrangements of the girls' block

The

3

girls can be arranged among themselves in

3! = 6

ways.

The

3

girls inside the block are distinct individuals and can swap positions.

4

Apply the fundamental counting principle

Total arrangements =

6! \times 3! = 720 \times 6 = 4320

.

The total number of arrangements is the product of external block arrangements and internal block arrangements.

Key Concept

Permutations with grouping constraints (string method)

Question 6888Question →

When the polynomial $P(x) = 2x^4 + ax^3 + bx^2 - 5x + 6$ is divided by $(x - 2)(x + 1)$ , the remainder is $6x + 8$ . What is the value of $a - b$ ?

11

-11

1

-1

Show answer & explanation

Answer: 11

Answer

11

According to the Remainder Theorem, dividing

P(x)

by

(x - 2)(x + 1)

leaves a remainder

R(x) = 6x + 8

. Substituting

x = -1

gives

R(-1) = 2

and

P(-1) = 2 - a + b + 5 + 6 = 13 - a + b

. Equating

13 - a + b = 2

gives

-a + b = -11

, which directly leads to

a - b = 11

.

Step-by-Step Solution

1

Apply the Remainder Theorem for linear factors of the divisor

(x - 2)(x + 1)

.

Since the divisor is

(x - 2)(x + 1)

, the roots of the divisor are

x = 2

and

x = -1

. The remainder function is

R(x) = 6x + 8

, so

P(2) = R(2)

and

P(-1) = R(-1)

.

By the Polynomial Division Algorithm,

P(x) = (x - 2)(x + 1)Q(x) + R(x)

.

2

Evaluate

R(x)

and

P(x)

at

x = -1

.

R(-1) = 6(-1) + 8 = 2

.

P(-1) = 2(-1)^4 + a(-1)^3 + b(-1)^2 - 5(-1) + 6 = 2 - a + b + 5 + 6 = 13 - a + b

.
Setting

P(-1) = R(-1)

gives

13 - a + b = 2

.

Substituting

x = -1

eliminates the quotient term since

(-1 + 1) = 0

.

3

Rearrange the equation to solve for

a - b

.

13 - a + b = 2 \implies -a + b = -11 \implies a - b = 11

.

Multiplying both sides of

-a + b = -11

by

-1

gives

a - b = 11

.

Key Concept

Remainder Theorem for Composite Linear Divisors

Question 6889Question →

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 6890Question →

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 6891Question →

The mean of five numbers arranged in ascending order is $28$ . The mean of the first three numbers is $22$ , while the mean of the last three numbers is $36$ . Find the median of these five numbers.

Show answer & explanation

Answer: 34

Answer

The median of the five numbers is

34

.

For five numbers ordered from smallest to largest (

x_1, x_2, x_3, x_4, x_5

), the median is the middle value

x_3

. The sum of all five numbers is

5 \times 28 = 140

. The sum of the first three numbers is

x_1 + x_2 + x_3 = 3 \times 22 = 66

, and the sum of the last three numbers is

x_3 + x_4 + x_5 = 3 \times 36 = 108

. Adding these two partial sums gives

(x_1 + x_2 + x_3 + x_4 + x_5) + x_3 = 66 + 108 = 174

. Substituting the overall sum of

140

into the equation yields

140 + x_3 = 174

, which simplifies to

x_3 = 34

.

Step-by-Step Solution

1

Calculate the sum of all five numbers.

Sum of all 5 numbers is

5 \times 28 = 140

.

The total sum of a set of data equals the number of items multiplied by the mean.

2

Calculate the partial sums of the first three and last three numbers.

First three numbers sum to

3 \times 22 = 66

; last three numbers sum to

3 \times 36 = 108

.

Multiplying each sub-group mean by the count of numbers in that sub-group yields the sub-group sum.

3

Set up an equation relating the partial sums to the total sum and the median.

Adding the partial sums counts the third number (median) twice:

66 + 108 = 140 + \text{median}

.

In an ordered set of 5 numbers, the 3rd term is the median and is shared by both the first three and last three elements.

4

Solve for the median.

Median

= 174 - 140 = 34

.

Subtracting the total sum from the combined partial sums isolates the overlapping median value.

Key Concept

Measures of Central Tendency for Ungrouped Data (Relationship between sub-group means, total sum, and median in ordered data)

Question 6892Question →

The masses, in kilograms, of five wooden crates stored in a warehouse are $3$ , $7$ , $8$ , $10$ , and $12$ . What is the variance of the masses of these crates?

9.2

2.4

11.5

46.0

Show answer & explanation

Answer: $9.2$

Answer

The variance of the masses is

9.2

.

The mean of the numbers is

8

. The squared deviations are

25, 1, 0, 4,

and

16

. The sum of these squared deviations is

46

. Dividing by the count of numbers (

5

) gives

9.2

.

Step-by-Step Solution

1

Calculate the mean (

\bar{x}

) of the dataset.

\bar{x} = \frac{3 + 7 + 8 + 10 + 12}{5} = \frac{40}{5} = 8

The mean is required to find individual deviations.

2

Compute the deviation of each observation from the mean,

(x - \bar{x})

.

3 - 8 = -5

,

7 - 8 = -1

,

8 - 8 = 0

,

10 - 8 = 2

,

12 - 8 = 4

Deviations quantify how far each data point lies from the central mean.

3

Square each deviation,

(x - \bar{x})^2

.

(-5)^2 = 25

,

(-1)^2 = 1

,

0^2 = 0

,

2^2 = 4

,

4^2 = 16

Squaring eliminates negative signs and weights larger deviations.

4

Sum the squared deviations and divide by the total number of observations

N = 5

.

Variance

\sigma^2 = \frac{25 + 1 + 0 + 4 + 16}{5} = \frac{46}{5} = 9.2

Variance is defined as the mean of the squared deviations.

Key Concept

Variance of Ungrouped Data

Estimated Time:1m 30s

Question 6893Question →

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 6894Question →

What is the $y$ -intercept of the normal line to the curve $y = \frac{2x - 1}{x + 1}$ at the point where the curve crosses the $x$ -axis?

\frac{3}{8}

-\frac{2}{3}

\frac{9}{8}

\frac{2}{3}

Show answer & explanation

Answer: $\frac{3}{8}$

Answer

The

y

-intercept of the normal line is

\frac{3}{8}

.

To find the

y

-intercept of the normal line, set

y = 0

to find the point of contact on the

x

-axis, which gives

\left(\frac{1}{2}, 0\right)

. Differentiating

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

gives

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

. At

x = \frac{1}{2}

, the tangent slope is

\frac{4}{3}

, making the normal slope

-\frac{3}{4}

. The equation of the normal line is

y = -\frac{3}{4}x + \frac{3}{8}

, so its

y

-intercept is

\frac{3}{8}

.

Step-by-Step Solution

1

Find the point of intersection of the curve with the

x

-axis

Set

y = 0 \implies \frac{2x - 1}{x + 1} = 0 \implies 2x - 1 = 0 \implies x = \frac{1}{2}

. The point is

\left(\frac{1}{2}, 0\right)

.

The normal line is drawn at the point where the curve crosses the

x

-axis.

2

Differentiate

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

using the quotient rule

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

The derivative provides the gradient function of the tangent line to the curve.

3

Evaluate the gradient of the tangent and normal lines at

x = \frac{1}{2}

Tangent gradient

m_t = \frac{3}{\left(\frac{1}{2} + 1\right)^2} = \frac{3}{\frac{9}{4}} = \frac{4}{3}

. Normal gradient

m_n = -\frac{1}{m_t} = -\frac{3}{4}

.

The normal line is perpendicular to the tangent line at the point of contact.

4

Determine the equation of the normal line and evaluate its

y

-intercept

Using

y - 0 = -\frac{3}{4}\left(x - \frac{1}{2}\right) \implies y = -\frac{3}{4}x + \frac{3}{8}

. Setting

x = 0

gives

y = \frac{3}{8}

.

The

y

-intercept is the value of

y

when

x = 0

on the line.

Key Concept

Tangents and Normals to Curves

Question 6895Question →

Given that the matrix $A = \begin{pmatrix} x & 2 \\ 5 & x - 3 \end{pmatrix}$ is a singular matrix, what is the positive value of $x$ ?

2

3

5

10

Show answer & explanation

Answer: $5$

Answer

The positive value of

x

is

5

.

For a matrix to be singular, its determinant must be zero. Calculating the determinant of

A

yields

\det(A) = x(x - 3) - (2)(5) = x^2 - 3x - 10

. Setting this to zero and factoring gives

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

, leading to

x = 5

or

x = -2

. The positive solution is

5

.

Step-by-Step Solution

1

Set the determinant of the matrix equal to zero.

\det(A) = (x)(x - 3) - (2)(5) = 0

By definition, a matrix is singular if and only if its determinant is zero.

2

Expand and simplify the algebraic equation.

x^2 - 3x - 10 = 0

Expanding

x(x - 3)

gives

x^2 - 3x

, and subtracting

10

forms a standard quadratic equation.

3

Factor the quadratic equation to solve for

x

.

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

The factors of

-10

that sum to

-3

are

-5

and

+2

.

4

Select the positive value requested by the question.

x = 5

The question specifically asks for the positive value of

x

.

Key Concept

Singular Matrices and Determinants

Question 6896Question →

The second term of a geometric progression (G.P.) is $6$ and its fifth term is $48$ . What is the sum of the first $6$ terms of the progression?

189

93

96

378

Show answer & explanation

Answer: 189

Answer

189

Using the geometric progression terms

T_2 = a r = 6

and

T_5 = a r^4 = 48

, dividing

T_5

by

T_2

gives

r^3 = 8

, so

r = 2

. Substituting

r = 2

into

a r = 6

yields

a = 3

. The sum of the first 6 terms is calculated as

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

.

Step-by-Step Solution

1

Set up equations using the

n^{\text{th}}

term formula for a G.P.,

T_n = a r^{n-1}

.

T_2 = a r = 6

and

T_5 = a r^4 = 48

Relate given terms to the first term

a

and common ratio

r

.

2

Divide the equation for

T_5

by the equation for

T_2

to find the common ratio

r

.

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

Eliminate the variable

a

to solve for

r

.

3

Substitute

r = 2

back into

a r = 6

to find the first term

a

.

a(2) = 6 \implies a = 3

Determine the first term of the progression.

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(63) = 189

Apply the sum formula for a finite geometric progression.

Key Concept

Geometric Progression (G.P.)

n^{\text{th}}

term and sum of finite terms

Question 6897Question →

If the matrix $M = \begin{pmatrix} 2x & 4 \\ 3 & 6 \end{pmatrix}$ is a singular matrix, what is the value of $x$ ?

1

-1

2

0

Show answer & explanation

Answer: $1$

Answer

The value of

x

is

1

.

A matrix is singular when its determinant equals zero. Evaluating the determinant of

\begin{pmatrix} 2x & 4 \\ 3 & 6 \end{pmatrix}

gives

(2x)(6) - (4)(3) = 12x - 12

. Setting

12x - 12 = 0

leads directly to

x = 1

.

Step-by-Step Solution

1

Recall the condition for a matrix to be singular.

A matrix is singular if and only if its determinant is equal to zero, so

\det(M) = 0

.

Singular matrices have no inverse because their determinant is zero.

2

Calculate the determinant of matrix

M = \begin{pmatrix} 2x & 4 \\ 3 & 6 \end{pmatrix}

.

\det(M) = (2x)(6) - (4)(3) = 12x - 12.

The determinant of a

2 \times 2

matrix

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

is

ad - bc

.

3

Set the determinant equal to zero and solve for

x

.

12x - 12 = 0 \implies 12x = 12 \implies x = 1

.

Solving the linear equation yields the required parameter value.

Key Concept

Condition for Singular Matrix and 2x2 Determinant Evaluation

Estimated Time:45s

Question 6898Question →

For the matrix $M = \begin{pmatrix} k & 3 & 1 \\ 2 & k & 0 \\ 1 & 5 & 2 \end{pmatrix}$ , the determinant of $M$ is equal to $4$ . What is the positive value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The positive value of

k

is

2

.

Expanding the determinant along the second row yields

\det(M) = -2(6-5) + k(2k-1) = 2k^2 - k - 2

. Setting this equal to

4

yields

2k^2 - k - 6 = 0

, which factors into

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

. The positive value is

2

.

Step-by-Step Solution

1

Evaluate the determinant of

M

in terms of

k

using row 2 cofactor expansion

\det(M) = 2k^2 - k - 2

Expanding along the second row gives

-2(6 - 5) + k(2k - 1) - 0 = -2 + 2k^2 - k

.

2

Set the determinant equal to the given value 4 and rearrange into standard quadratic form

2k^2 - k - 6 = 0

Subtracting 4 from both sides gives

2k^2 - k - 6 = 0

.

3

Solve the quadratic equation by factorization

k = -1.5 \text{ or } k = 2

Factoring

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

gives roots

k = -1.5

and

k = 2

.

4

Select the positive root

k = 2

The question specifically asks for the positive value of

k

.

Key Concept

Evaluating a 3x3 matrix determinant using cofactor expansion and solving the resulting quadratic equation for an unknown parameter.

Question 6899Question →

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

-3

10

-1

1

Show answer & explanation

Answer: $-3$

Answer

The value of

x + y

is

-3

.

Evaluating the matrix products

AB

and

BA

using standard matrix multiplication rules yields

AB = \begin{pmatrix} 2+2x & 8+xy \\ 5 & -4+3y \end{pmatrix}

and

BA = \begin{pmatrix} -2 & x+12 \\ 4-y & 2x+3y \end{pmatrix}

. Equating entry-by-entry gives

2+2x = -2 \implies x = -2

and

5 = 4-y \implies y = -1

. Adding these together gives

x + y = -3

.

Step-by-Step Solution

1

Compute the product matrix

AB

AB = \begin{pmatrix} 2(1) + x(2) & 2(4) + x(y) \\ -1(1) + 3(2) & -1(4) + 3(y) \end{pmatrix} = \begin{pmatrix} 2 + 2x & 8 + xy \\ 5 & -4 + 3y \end{pmatrix}

Matrix multiplication requires taking the dot product of rows from the first matrix and columns from the second matrix.

2

Compute the product matrix

BA

BA = \begin{pmatrix} 1(2) + 4(-1) & 1(x) + 4(3) \\ 2(2) + y(-1) & 2(x) + y(3) \end{pmatrix} = \begin{pmatrix} -2 & x + 12 \\ 4 - y & 2x + 3y \end{pmatrix}

Evaluate

BA

by multiplying rows of

B

by columns of

A

.

3

Equate corresponding entries of

AB

and

BA

since

AB = BA

From row 1, col 1:

2 + 2x = -2 \implies 2x = -4 \implies x = -2

.
From row 2, col 1:

5 = 4 - y \implies y = -1

.

Two matrices are equal if and only if all corresponding entries are equal.

4

Verify consistency on remaining entries and calculate

x + y

Row 1, col 2 check:

8 + (-2)(-1) = 10

and

-2 + 12 = 10

.
Row 2, col 2 check:

-4 + 3(-1) = -7

and

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

.
Sum:

x + y = -2 + (-1) = -3

.

Verifying consistency ensures the system of equations has a valid unique solution.

Key Concept

Matrix Multiplication Commutativity and Matrix Equality

Estimated Time:2m 30s

Question 6900Question →

A thermometer is calibrated on a custom scale, $X$ , where the ice point ( $0^\circ\text{C}$ ) is marked as $-10^\circ\text{X}$ and the steam point ( $100^\circ\text{C}$ ) is marked as $110^\circ\text{X}$ . What is the reading on this custom scale when a standard Celsius thermometer reads $35^\circ\text{C}$ ?

Show answer & explanation

Answer: 32

Answer

32 °X

Using the linear temperature interpolation formula

\frac{X - \text{LFP}_X}{\text{UFP}_X - \text{LFP}_X} = \frac{C - \text{LFP}_C}{\text{UFP}_C - \text{LFP}_C}

, substituting

\text{LFP}_X = -10

,

\text{UFP}_X = 110

,

C = 35

,

\text{LFP}_C = 0

, and

\text{UFP}_C = 100

gives

\frac{X - (-10)}{110 - (-10)} = \frac{35 - 0}{100 - 0}

. Simplifying gives

\frac{X + 10}{120} = 0.35

, leading to

X + 10 = 42

, so

X = 32^\circ\text{X}

.

Step-by-Step Solution

1

Determine fundamental intervals for both temperature scales

Celsius fundamental interval =

100 - 0 = 100^\circ\text{C}

; Custom scale fundamental interval =

110 - (-10) = 120^\circ\text{X}

Linear temperature scale interpolation requires calculating the total interval between the lower fixed point (LFP) and upper fixed point (UFP).

2

Set up the ratio equation between the two thermometric scales

\frac{X - (-10)}{120} = \frac{35 - 0}{100}

The fractional position of any given temperature relative to its fixed points must be equal on all linear scales.

3

Solve the algebraic equation for

X

X + 10 = 120 \times 0.35 = 42 \implies X = 32^\circ\text{X}

Isolating

X

gives the corresponding reading on the custom temperature scale.

Key Concept

Linear Temperature Scale Conversion and Interpolation

Answer

Step-by-Step Solution

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