All practice questions

What is the simplified form of the expression $\frac{a^3 \times a^4}{a^2}$ ?

a^5

a^{10}

a^6

a^9

Show answer & explanation

Answer: $a^5$

Answer

The simplified expression is

a^5

.

Combining terms with the same base using the product law gives

a^{3+4} = a^7

in the numerator. Next, dividing by

a^2

using the quotient law yields

a^{7-2} = a^5

.

Step-by-Step Solution

1

Simplify the numerator using the product law of indices

a^3 \times a^4 = a^{3+4} = a^7

When multiplying terms with the same base, add their exponents.

2

Divide by the denominator using the quotient law of indices

\frac{a^7}{a^2} = a^{7-2} = a^5

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Key Concept

Laws of Indices (Product and Quotient Rules)

Estimated Time:45s

Question 4122Question →

Find the total number of distinct solutions to the trigonometric equation $2\cos^2(2x) + \sin(2x) - 1 = 0$ in the interval $0^\circ \le x \le 360^\circ$ .

Show answer & explanation

Answer: 6

Answer

The total number of distinct solutions is 6.

Substituting

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

gives the quadratic

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

, which factors into

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

. For

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

, the angle argument

2x

covers

0^\circ \le 2x \le 720^\circ

. The equation

\sin(2x) = 1

provides 2 values for

x

(

45^\circ, 225^\circ

), while

\sin(2x) = -\frac{1}{2}

provides 4 values for

x

(

105^\circ, 165^\circ, 285^\circ, 345^\circ

). Summing these gives 6 distinct solutions in total.

Step-by-Step Solution

1

Use the Pythagorean trigonometric identity

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

to express the entire equation in terms of

\sin(2x)

.

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

Converting all trigonometric terms to a single function allows the equation to be solved as a polynomial.

2

Rearrange and factorize the resulting quadratic equation in terms of

\sin(2x)

.

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

Factorization splits the quadratic trigonometric equation into two simple linear trigonometric equations.

3

Determine the expanded domain for

2x

given

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

.

0^\circ \le 2x \le 720^\circ

Multiplying the bounds of

x

by 2 accounts for two full rotations in the unit circle.

4

Solve the first linear equation

\sin(2x) = 1

within

0^\circ \le 2x \le 720^\circ

.

2x = 90^\circ, 450^\circ \implies x = 45^\circ, 225^\circ

(2 distinct solutions)

The sine function equals 1 at

90^\circ

in the first revolution and at

90^\circ + 360^\circ = 450^\circ

in the second revolution.

5

Solve the second linear equation

\sin(2x) = -\frac{1}{2}

within

0^\circ \le 2x \le 720^\circ

.

2x = 210^\circ, 330^\circ, 570^\circ, 690^\circ \implies x = 105^\circ, 165^\circ, 285^\circ, 345^\circ

(4 distinct solutions)

The sine function is negative in the 3rd and 4th quadrants of both revolutions.

6

Combine the solution counts from both cases.

Total number of solutions =

2 + 4 = 6

.

Adding the valid solutions from both factor equations gives the complete set of roots.

Key Concept

Solving quadratic trigonometric equations across multiple revolutions

Estimated Time:3m 0s

Question 4123Question →

Which of the following ranges of $x$ satisfies the quadratic inequality $2x^2 - 7x + 3 < 0$ ?

\frac{1}{2} < x < 3

x < \frac{1}{2} \text{ or } x > 3

-3 < x < -\frac{1}{2}

x < -3 \text{ or } x > -\frac{1}{2}

Show answer & explanation

Answer: $\frac{1}{2} < x < 3$

Answer

\frac{1}{2} < x < 3

Factorizing

2x^2 - 7x + 3 < 0

gives

(2x - 1)(x - 3) < 0

. The roots of the quadratic equation

(2x - 1)(x - 3) = 0

are

x = \frac{1}{2}

and

x = 3

. Because the coefficient of

x^2

is positive, the quadratic curve opens upwards and is strictly negative between the two roots. Therefore, the inequality is satisfied when

\frac{1}{2} < x < 3

.

Step-by-Step Solution

1

Factorize the quadratic expression

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

Factorization allows us to find the critical values of the inequality.

2

Determine the critical values by setting the factored expression to zero

x = \frac{1}{2}

and

x = 3

These points mark the boundary values where the quadratic expression changes sign.

3

Test intervals or apply sign analysis for strict inequality

< 0

The product

(2x - 1)(x - 3)

is negative for

\frac{1}{2} < x < 3

Since the leading coefficient is positive (

2 > 0

), the parabola opens upwards and takes negative values strictly between its real roots.

Key Concept

Solving Quadratic Inequalities by Factorization and Interval Sign Analysis

Estimated Time:1m 15s

Question 4124Question →

The table below shows the distribution of masses (in kg) of cocoa bags harvested on a farm:

Mass (kg)	Frequency ( $f$ )
$10 - 19$	$6$
$20 - 29$	$k$
$30 - 39$	$15$
$40 - 49$	$12$
$50 - 59$	$7$

If the estimated mean mass of the distribution is $35.3\text{ kg}$ , calculate the value of the missing frequency $k$ .

Show answer & explanation

Answer: 10

Answer

The value of the missing frequency is 10.

To find the missing frequency, compute the midpoints (

x

) of each mass class interval: 14.5, 24.5, 34.5, 44.5, and 54.5. Next, express the sum of frequencies as

\sum f = 40 + k

and the sum of products of frequency and midpoint as

\sum fx = 6(14.5) + k(24.5) + 15(34.5) + 12(44.5) + 7(54.5) = 1520 + 24.5k

. Equating the mean expression

\frac{\sum fx}{\sum f}

to

35.3

gives

\frac{1520 + 24.5k}{40 + k} = 35.3

. Cross-multiplying and solving yields

1520 + 24.5k = 1412 + 35.3k

, which simplifies to

10.8k = 108

, giving

k = 10

.

Step-by-Step Solution

1

Determine the class midpoints (

x

) for all intervals.

Class midpoints are 14.5, 24.5, 34.5, 44.5, and 54.5.

Grouped mean calculations require representative midpoint values for each class interval.

2

Formulate expressions for total frequency

\sum f

and total weighted sum

\sum fx

.

\sum f = 40 + k and \sum fx = 1520 + 24.5k.

These algebraic expressions are necessary to substitute into the mean formula.

3

Set up and solve the linear equation using the given mean of 35.3.

\frac{1520 + 24.5k}{40 + k} = 35.3 \implies 10.8k = 108 \implies k = 10.

Equating the algebraic mean expression to the numerical mean allows solving for the unknown frequency k.

Key Concept

Measures of Central Tendency for Grouped Data

Question 4125Question →

In triangle $LMN$ , the side lengths are given as $l = 7\text{ cm}$ , $m = 8\text{ cm}$ , and $n = 13\text{ cm}$ . What is the measure of angle $N$ in degrees?

Show answer & explanation

Answer: 120

Answer

The measure of angle

N

is

120^\circ

.

Using the Cosine Rule for angle

N

,

\cos N = \frac{l^2 + m^2 - n^2}{2lm} = \frac{7^2 + 8^2 - 13^2}{2(7)(8)} = \frac{-56}{112} = -0.5

. The angle whose cosine is

-0.5

within the interior angles of a triangle (

0^\circ < N < 180^\circ

) is

120^\circ

.

Step-by-Step Solution

1

Set up the Cosine Rule formula for angle

N

\cos N = \frac{l^2 + m^2 - n^2}{2lm}

The Cosine Rule relates the three sides of any triangle to the cosine of one of its interior angles.

2

Substitute the known side lengths into the formula

\cos N = \frac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8} = \frac{49 + 64 - 169}{112}

Side

n = 13\text{ cm}

is opposite to angle

N

and must be subtracted in the numerator.

3

Simplify the fraction

\cos N = \frac{-56}{112} = -0.5

Evaluating the numerical expression gives a negative value, indicating that angle

N

is obtuse.

4

Calculate the principal inverse cosine angle for the triangle

N = 120^\circ

Since

\cos 60^\circ = 0.5

,

\cos(180^\circ - 60^\circ) = -0.5

, giving

N = 120^\circ

.

Key Concept

Applying the Cosine Rule to find an obtuse angle given three side lengths (SSS)

Question 4126Question →

The fourth term of an arithmetic progression (A.P.) is $15$ and the ninth term is $35$ . What is the common difference of the progression?

Show answer & explanation

Answer: 4

Answer

The common difference of the arithmetic progression is

4

.

Using the A.P. term formula

T_n = a + (n-1)d

, the fourth term gives

a + 3d = 15

and the ninth term gives

a + 8d = 35

. Subtracting the two equations yields

5d = 20

, which simplifies directly to

d = 4

.

Step-by-Step Solution

1

Express the given terms using the n-th term formula

T_n = a + (n-1)d

a + 3d = 15

and

a + 8d = 35

The

n^{\text{th}}

term formula relates any term to the first term (

a

) and common difference (

d

).

2

Subtract the equation for the fourth term from the ninth term

5d = 20

Subtracting eliminates the first term

a

and leaves a simple equation in terms of

d

.

3

Divide by

5

to solve for

d

d = 4

Dividing both sides of

5d = 20

by

5

gives the common difference.

Key Concept

Finding the common difference of an Arithmetic Progression given two non-consecutive terms

Question 4127Question →

What is the mean deviation of the data set $4, 7, 8, 11, 15$ ?

Show answer & explanation

Answer: 3.2

Answer

The mean deviation of the data set is 3.2.

To find the mean deviation, first calculate the mean of the dataset, which is 9. Then, compute the absolute difference of each number from 9, obtaining values of 5, 2, 1, 2, and 6. Finally, divide the sum of these absolute values (16) by the total number of items (5) to get 3.2.

Step-by-Step Solution

1

Calculate the arithmetic mean of the given data set.

The mean

\bar{x} = 9

.

The mean is needed to evaluate how far each data point deviates from the central value.

2

Find the absolute difference between each value and the mean.

The absolute deviations are 5, 2, 1, 2, and 6.

Mean deviation measures dispersion using absolute distances, ignoring negative signs.

3

Calculate the mean of the absolute deviations.

Mean deviation = 3.2.

Dividing the total sum of absolute deviations (16) by the number of observations (5) gives the mean deviation.

Key Concept

Mean Deviation for Ungrouped Data

Question 4128Question →

A surveyor stands at a point on level ground $50\text{ m}$ away from the base of a vertical transmission tower. If the angle of elevation from the observer's position on the ground to the top of the tower is $45^\circ$ , what is the height of the tower in meters?

Show answer & explanation

Answer: 50

Answer

The height of the transmission tower is

50\text{ meters}

.

In a right-angled triangle, the tangent of the angle of elevation equals the ratio of the height (opposite side) to the horizontal distance (adjacent side). Since

\tan(45^\circ) = 1

, the height of the tower must be equal to the horizontal distance of

50\text{ m}

.

Step-by-Step Solution

1

Formulate the trigonometric relationship using the right triangle formed by the observer, the base of the tower, and the top of the tower.

\tan(45^\circ) = \frac{h}{50}

, where

h

is the height of the tower.

The tangent ratio relates the opposite side (height of tower) to the adjacent side (distance along level ground).

2

Substitute the value of

\tan(45^\circ) = 1

and solve for

h

.

h = 50 \times 1 = 50\text{ m}

.

Multiplying the adjacent side length by

\tan(45^\circ)

yields the exact height.

Key Concept

Angle of elevation using basic right-triangle trigonometry

Question 4129Question →

A vertical tower $TC$ of height $h$ metres stands on a horizontal plane with its base at $C$ . Point $A$ on the plane is due West of $C$ , and point $B$ is due South of $A$ , such that the distance $AB = 40\text{ m}$ . If the angles of elevation of the top $T$ of the tower from $A$ and $B$ are $60^\circ$ and $30^\circ$ respectively, what is the height of the tower?

10\sqrt{6}\text{ m}

4\sqrt{30}\text{ m}

20\sqrt{2}\text{ m}

20\sqrt{3}\text{ m}

Show answer & explanation

Answer: $10\sqrt{6}\text{ m}$

Answer

The height of the tower is

10\sqrt{6}\text{ m}

.

Using the trigonometric tangent ratios, the horizontal distances from the base of the tower are

AC = \frac{h}{\sqrt{3}}

and

BC = h\sqrt{3}

. Because point

A

is West of

C

and point

B

is South of

A

, the angle

\angle CAB = 90^\circ

. Applying Pythagoras' theorem

BC^2 = AC^2 + AB^2

yields

3h^2 = \frac{h^2}{3} + 1600

, which simplifies to

h = 10\sqrt{6}\text{ m}

.

Step-by-Step Solution

1

Express the horizontal distances

AC

and

BC

in terms of height

h

.

In vertical right-angled triangle

TAC

,

\tan(60^\circ) = \frac{h}{AC} \implies AC = \frac{h}{\sqrt{3}}

. In vertical right-angled triangle

TBC

, \tan(30^\circ) = \frac{h}{BC} \implies BC = h\sqrt{3}$.

Relate the vertical height to the horizontal ground distances using tangent ratios.

2

Identify the geometry of the ground plane triangle

\triangle CAB

.

Since

A

is due West of

C

and

B

is due South of

A

, the line segments

CA

(East-West) and

AB

(North-South) are perpendicular. Thus,

\triangle CAB

is right-angled at

A

.

Cardinal directions (West and South from

A

) are perpendicular to each other.

3

Apply Pythagoras' theorem to

\triangle CAB

.

BC^2 = AC^2 + AB^2 \implies (h\sqrt{3})^2 = \left(\frac{h}{\sqrt{3}}\right)^2 + 40^2 \implies 3h^2 = \frac{h^2}{3} + 1600

.

Hypotenuse

BC

relates the two legs

AC

and

AB

on the horizontal plane.

4

Solve the equation for

h

.

3h^2 - \frac{h^2}{3} = 1600 \implies \frac{8h^2}{3} = 1600 \implies 8h^2 = 4800 \implies h^2 = 600 \implies h = \sqrt{600} = 10\sqrt{6}\text{ m}

.

Simplify algebraic terms to determine

h

in surd form.

Key Concept

Combining angles of elevation in 3D vertical planes with horizontal plane coordinate geometry and Pythagoras' theorem.

Question 4130Question →

Evaluate the value of the logarithmic expression $\log_3 7 \times \log_7 81$ .

Show answer & explanation

Answer: 4

Answer

The value of the expression is 4.

By applying the change of base formula

\log_7 81 = \frac{\log_3 81}{\log_3 7}

, the expression becomes

\log_3 7 \times \frac{\log_3 81}{\log_3 7} = \log_3 81

. Since

3^4 = 81

, the result is 4.

Step-by-Step Solution

1

Apply the change of base chain rule

\log_a b \cdot \log_b c = \log_a c

\log_3 7 \times \log_7 81 = \log_3 81

By change of base,

\log_7 81 = \frac{\log_3 81}{\log_3 7}

, so multiplying by

\log_3 7

cancels out the common factor.

2

Evaluate

\log_3 81

4

Since

3^4 = 81

, the logarithm base 3 of 81 is equal to 4.

Key Concept

Change of Base Property of Logarithms

Question 4131Question →

A student spent $\frac{2}{5}$ of his monthly allowance on books and $\frac{1}{3}$ of the remaining amount on food. If he was left with $\text{N}4,000$ , what was his total monthly allowance in Naira ( $\text{N}$ )?

Show answer & explanation

Answer: 10000

Answer

The total monthly allowance was 10,000 Naira.

After spending

\frac{2}{5}

on books,

\frac{3}{5}

of the allowance remains. Spending

\frac{1}{3}

of this remainder on food accounts for

\frac{1}{5}

of the original allowance. Subtracting

\frac{1}{5}

from

\frac{3}{5}

leaves

\frac{2}{5}

of the total allowance, which is equal to

\text{N}4,000

. Solving

\frac{2}{5} \times \text{Total} = 4,000

yields

10,000

Naira.

Step-by-Step Solution

1

Determine the remaining fraction after the first expenditure

Fraction left =

\frac{3}{5}

The student spent

\frac{2}{5}

on books, leaving

1 - \frac{2}{5} = \frac{3}{5}

of the total allowance.

2

Calculate the fraction of the total allowance spent on food

Fraction spent on food =

\frac{1}{5}

He spent

\frac{1}{3}

of the remaining

\frac{3}{5}

, which equals

\frac{1}{3} \times \frac{3}{5} = \frac{1}{5}

of the whole allowance.

3

Calculate the final remaining fraction of the allowance

Final fraction left =

\frac{2}{5}

The remaining fraction is

\frac{3}{5} - \frac{1}{5} = \frac{2}{5}

.

4

Solve for the total allowance

Total allowance = 10,000 Naira

Since

\frac{2}{5}

of the total allowance equals

\text{N}4,000

, the total allowance is

\frac{4,000 \times 5}{2} = 10,000

Naira.

Key Concept

Sequential Fraction of Remainder Problems

Question 4132Question →

Identify the word nearest in meaning to 'industrious' that appropriately fills the blank in the sentence to complete its meaning.

Fill in the blanks below

The manager commended the young secretary for being exceptionally in organizing all official documents.

Show answer & explanation

Answer

diligent

The word 'diligent' means showing steady, earnest, and energetic effort, making it the exact synonym for 'industrious' in a workplace context.

Step-by-Step Solution

1

Determine the contextual meaning required for the blank based on the clue 'industrious'.

The word must convey active dedication, thoroughness, and steady effort in performing tasks.

The sentence highlights praise from a manager for careful and hardworking document organization.

2

Select a direct synonym that fits grammatically and semantically.

The adjective 'diligent' (or 'hardworking' / 'assiduous') fits the context precisely.

'Diligent' means showing care and conscientiousness in one's work, which is synonymous with 'industrious'.

Key Concept

Synonyms and Words Nearest in Meaning

Question 4133Question →

Find the number of non-negative integer values of $x$ that satisfy the quadratic inequality $x^2 - 5x - 14 \le 0$ .

Show answer & explanation

Answer: 8

Answer

8

Factoring the quadratic expression gives

(x - 7)(x + 2) \le 0

, which evaluates to the solution interval

-2 \le x \le 7

. Restricting this interval to non-negative integers (

x \ge 0

) yields the set

\{0, 1, 2, 3, 4, 5, 6, 7\}

, which contains exactly 8 values.

Step-by-Step Solution

1

Factor the quadratic equation

x^2 - 5x - 14 = 0

(x - 7)(x + 2) = 0

, yielding critical roots at

x = 7

and

x = -2

Finding the roots determines the boundary points for the quadratic inequality.

2

Determine the solution set for the inequality

x^2 - 5x - 14 \le 0

-2 \le x \le 7

The quadratic expression is negative or zero between its two real roots.

3

Identify and count the non-negative integers in the interval

[-2, 7]

The non-negative integers are

0, 1, 2, 3, 4, 5, 6, 7

, giving a total of 8 values.

Non-negative integers consist of zero and all positive whole numbers within the solution range.

Key Concept

Solving quadratic inequalities and identifying discrete non-negative integer solution sets

Question 4134Question →

If $y = \ln\left(\frac{1 + \sin x}{1 - \sin x}\right)$ , what is $\frac{dy}{dx}$ ?

2\sec x

-2\sec x

2\tan x

\sec x

Show answer & explanation

Answer: $2\sec x$

Answer

The derivative

\frac{dy}{dx}

is equal to

2\sec x

.

Rewriting the function as

y = \ln(1 + \sin x) - \ln(1 - \sin x)

and differentiating both terms yields

\frac{\cos x}{1 + \sin x} + \frac{\cos x}{1 - \sin x}

. Combining these over the common denominator

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

simplifies to

\frac{2\cos x}{\cos^2 x} = 2\sec x

.

Step-by-Step Solution

1

Apply the logarithmic law

\ln\left(\frac{u}{v}\right) = \ln u - \ln v

y = \ln(1 + \sin x) - \ln(1 - \sin x)

Simplifies the quotient into separate terms prior to differentiation.

2

Differentiate each logarithmic term with respect to

x

using the chain rule

\frac{dy}{dx} = \frac{1}{1 + \sin x} \cdot \cos x - \frac{1}{1 - \sin x} \cdot (-\cos x) = \frac{\cos x}{1 + \sin x} + \frac{\cos x}{1 - \sin x}

The derivative of

\ln(f(x))

is

\frac{f'(x)}{f(x)}

.

3

Combine the fractions over a common denominator

\frac{dy}{dx} = \frac{\cos x (1 - \sin x) + \cos x (1 + \sin x)}{(1 + \sin x)(1 - \sin x)} = \frac{2\cos x}{1 - \sin^2 x}

Adding two fractions with denominators

(1 + \sin x)

and

(1 - \sin x)

.

4

Use the Pythagorean trigonometric identity

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

and simplify

\frac{dy}{dx} = \frac{2\cos x}{\cos^2 x} = \frac{2}{\cos x} = 2\sec x

Simplifying

\frac{2\cos x}{\cos^2 x}

yields

2\sec x

.

Key Concept

Differentiation of Logarithmic and Trigonometric Functions via Chain Rule and Log Properties

Alternative Method

Alternatively, express

y = \ln(\sec x + \tan x)^2 = 2 \ln(\sec x + \tan x)

. The derivative of

\ln(\sec x + \tan x)

is

\frac{\sec x \tan x + \sec^2 x}{\sec x + \tan x} = \sec x

. Multiplying by 2 gives

2\sec x

.

Estimated Time:2m 0s

Question 4135Question →

Given the function $y = x^2 \sqrt{3x^2 + 4}$ , find the numerical value of $\frac{dy}{dx}$ at $x = 2$ .

Show answer & explanation

Answer: 22

Answer

The numerical value of the derivative at

x = 2

is 22.

Applying both the Product Rule and Chain Rule correctly yields

\frac{dy}{dx} = 2x\sqrt{3x^2 + 4} + \frac{3x^3}{\sqrt{3x^2 + 4}}

. Evaluating this expression at

x = 2

gives

16 + 6 = 22

.

Step-by-Step Solution

1

Decompose the function into a product of two functions

Let

u(x) = x^2

and

v(x) = \sqrt{3x^2 + 4} = (3x^2 + 4)^{1/2}

.

The function

y

is expressed as the product of

u(x)

and

v(x)

, requiring the Product Rule for differentiation.

2

Find the derivative of the inner square root function using the Chain Rule

v'(x) = \frac{1}{2}(3x^2 + 4)^{-1/2} \cdot 6x = \frac{3x}{\sqrt{3x^2 + 4}}

.

The Chain Rule states that

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

.

3

Combine derivatives using the Product Rule formula

\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x) = 2x\sqrt{3x^2 + 4} + \frac{3x^3}{\sqrt{3x^2 + 4}}

.

The Product Rule formula is

\frac{d}{dx}[u \cdot v] = u'v + uv'

.

4

Evaluate the derivative at

x = 2

\frac{dy}{dx}\Big|_{x=2} = 2(2)\sqrt{3(2)^2 + 4} + \frac{3(2)^3}{\sqrt{3(2)^2 + 4}} = 4(4) + \frac{24}{4} = 16 + 6 = 22

.

Substituting

x = 2

gives the specific slope of the tangent line to the curve at that point.

Key Concept

Combining the Product Rule and Chain Rule to evaluate derivatives of composite product functions

Question 4136Question →

Let $A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} x & 2 \\ 1 & y \end{pmatrix}$ be $2 \times 2$ matrices with integer entries $x$ and $y$ . If the matrix product $AB$ is singular and $\det(A + B) = 0$ , what is the value of $x + y$ ?

-3

3

-5

7

Show answer & explanation

Answer: $-3$

Answer

The value of

x + y

is

-3

.

Since matrix

AB

is singular,

\det(AB) = \det(A)\det(B) = 0

. Evaluating

\det(A) = 7 \neq 0

, we find

\det(B) = xy - 2 = 0

, giving

xy = 2

. Computing

A+B = \begin{pmatrix} 2+x & 3 \\ 0 & 3+y \end{pmatrix}

, its determinant is

(2+x)(3+y) = 6 + 3x + 2y + xy = 0

. Substituting

xy = 2

yields

3x + 2y = -8

. The integer pair satisfying both

xy = 2

and

3x + 2y = -8

is

x = -2

and

y = -1

. Therefore,

x + y = -3

.

Step-by-Step Solution

1

Calculate the determinant of matrix

A

.

\det(A) = (2)(3) - (1)(-1) = 6 + 1 = 7

.

Knowing

\det(A)

helps simplify the condition

\det(AB) = 0

using determinant properties.

2

Use the singularity of

AB

to find a relationship between

x

and

y

.

\det(AB) = \det(A)\det(B) = 7(xy - 2) = 0 \implies xy = 2

.

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

3

Form the matrix

A + B

and compute its determinant.

A + B = \begin{pmatrix} 2+x & 3 \\ 0 & 3+y \end{pmatrix} \implies \det(A+B) = (2+x)(3+y) - (3)(0) = (2+x)(3+y) = 6 + 3x + 2y + xy

.

Setting this determinant to zero gives a second equation involving

x

and

y

.

4

Substitute

xy = 2

into

\det(A+B) = 0

and solve for integer values of

x

and

y

.

6 + 3x + 2y + 2 = 0 \implies 3x + 2y = -8

. Testing integer pairs

(x,y)

for

xy = 2

gives

x = -2

and

y = -1

.

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

is satisfied only by

x = -2, y = -1

among integer pairs.

5

Compute

x + y

.

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

.

This answers the question directly.

Key Concept

Determinants of matrix products and sums, matrix singularity, and integer solutions to matrix equations.

Question 4137Question →

If the matrix $P = \begin{pmatrix} x & 3 \\ 8 & 6 \end{pmatrix}$ is a singular matrix, what is the value of $x$ ?

-4

4

5

16

Show answer & explanation

Answer: 4

Answer

The value of

x

is

4

.

For a

2 \times 2

matrix to be singular, its determinant must be zero. For matrix

P

, the determinant is

(x \cdot 6) - (3 \cdot 8) = 6x - 24

. Setting

6x - 24 = 0

gives

6x = 24

, which simplifies to

x = 4

.

Step-by-Step Solution

1

State the condition for a matrix to be singular.

A matrix is singular if its determinant is equal to zero:

\det(P) = 0

.

By definition, a singular matrix has a determinant equal to zero.

2

Calculate the determinant of matrix

P

.

\det(P) = (x \times 6) - (3 \times 8) = 6x - 24.

The determinant of a

2 \times 2

matrix

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

is calculated using the formula

ad - bc

.

3

Solve the linear equation

6x - 24 = 0

for

x

.

6x = 24 \implies x = 4.

Adding

24

to both sides and dividing by

6

isolates

x

.

Key Concept

Singular matrix condition and

2 \times 2

determinant calculation

Question 4138Question →

In a survey of $100$ agricultural market traders in Lagos, $52$ sell cassava, $45$ sell yam, and $60$ sell plantain. Furthermore, $25$ sell both cassava and yam, $22$ sell both yam and plantain, and $28$ sell both cassava and plantain. If the number of traders who sell none of these three crops is twice the number of traders who sell all three crops, how many traders sell exactly two of these crops?

Show answer & explanation

Answer: 57

Answer

57 traders sell exactly two of these crops.

Using inclusion-exclusion, the total number of traders selling at least one crop is

82 + x

. Adding the

2x

traders selling none gives

82 + 3x = 100

, so

x = 6

. The number of traders selling exactly two crops is

(25 - 6) + (22 - 6) + (28 - 6) = 19 + 16 + 22 = 57

.

Step-by-Step Solution

1

Apply the Principle of Inclusion-Exclusion for three set unions.

n(C \cup Y \cup P) = 82 + x

, where

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

.

Summing single set cardinalities, subtracting pairwise intersections, and adding back the triple intersection accounts for all region overlaps.

2

Set up and solve the universal set cardinality equation.

x = 6

Since total traders

|U| = 100

and non-sellers equal

2x

, the equation

100 = (82 + x) + 2x

simplifies to

3x = 18

, giving

x = 6

.

3

Compute the sum of elements in regions representing exactly two sets.

57

Subtracting

x = 6

from each pairwise intersection isolates traders who sell only cassava & yam (19), only yam & plantain (16), and only cassava & plantain (22). Summing these gives

19 + 16 + 22 = 57

.

Key Concept

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

Question 4139Question →

What is the gradient of a line that is perpendicular to the line passing through the points $(2, 5)$ and $(4, 11)$ ?

3

-\frac{1}{3}

\frac{1}{3}

-3

Show answer & explanation

Answer: $-\frac{1}{3}$

Answer

The gradient of the perpendicular line is

-\frac{1}{3}

.

The gradient of the line connecting

(2, 5)

and

(4, 11)

is

m = \frac{11-5}{4-2} = 3

. The gradient of a line perpendicular to it must satisfy

m_1 \cdot m_2 = -1

, yielding

-\frac{1}{3}

.

Step-by-Step Solution

1

Calculate the gradient

m_1

of the line passing through

(2, 5)

and

(4, 11)

.

m_1 = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

The gradient formula between two points

(x_1, y_1)

and

(x_2, y_2)

is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Apply the condition for perpendicular lines to find the required gradient

m_2

.

m_2 = -\frac{1}{m_1} = -\frac{1}{3}

Two non-vertical lines are perpendicular if and only if the product of their gradients is

-1

(

m_1 \cdot m_2 = -1

).

Key Concept

Perpendicular Line Gradients

Question 4140Question →

A straight line $L_1$ passes through the points $(k, 2)$ and $(3, 8)$ . If $L_1$ is perpendicular to the line $L_2$ defined by the equation $2x - 3y + 5 = 0$ , what is the value of $k$ ?

7

-6

-1

12

Show answer & explanation

Answer: $7$

Answer

The value of

k

is

7

.

Rearranging the line equation

2x - 3y + 5 = 0

gives

y = \frac{2}{3}x + \frac{5}{3}

, so its gradient is

\frac{2}{3}

. The line perpendicular to it must have a gradient equal to the negative reciprocal, which is

-\frac{3}{2}

. Equating this to the slope formula

\frac{8 - 2}{3 - k}

gives

\frac{6}{3 - k} = -\frac{3}{2}

. Cross-multiplying yields

12 = -9 + 3k

, which simplifies to

3k = 21

, giving the correct answer

7

.

Step-by-Step Solution

1

Find the gradient (

m_2

) of the line

L_2

Rearrange

2x - 3y + 5 = 0

into

y = mx + c

form:

3y = 2x + 5 \implies y = \frac{2}{3}x + \frac{5}{3}

. Thus,

m_2 = \frac{2}{3}

.

The coefficient of

x

in slope-intercept form gives the gradient of

L_2

.

2

Determine the gradient (

m_1

) of the perpendicular line

L_1

Using the perpendicular condition

m_1 \cdot m_2 = -1

:

m_1 = -\frac{1}{m_2} = -\frac{3}{2}

.

Perpendicular lines have gradients that are negative reciprocals of each other.

3

Express the gradient of

L_1

using the two given points

(k, 2)

and

(3, 8)

m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{3 - k} = \frac{6}{3 - k}

.

The gradient of a line through

(x_1, y_1)

and

(x_2, y_2)

is given by

\frac{y_2 - y_1}{x_2 - x_1}

.

4

Equate the two gradient expressions and solve for

k

\frac{6}{3 - k} = -\frac{3}{2} \implies 6 \cdot 2 = -3(3 - k) \implies 12 = -9 + 3k \implies 3k = 21 \implies k = 7

.

Solving the linear algebraic equation yields the required value of the unknown coordinate

k

.

Key Concept

Two non-vertical lines are perpendicular if and only if the product of their gradients is

-1

(

m_1 \cdot m_2 = -1

).

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