Tüm alıştırma soruları

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

a^5

a^{10}

a^6

a^9

Cevabı ve açıklamayı göster

Cevap: $a^5$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Laws of Indices (Product and Quotient Rules)

Tahmini Süre:45s

Soru 4122Soru →

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

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving quadratic trigonometric equations across multiple revolutions

Tahmini Süre:3m 0s

Soru 4123Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

\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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Solving Quadratic Inequalities by Factorization and Interval Sign Analysis

Tahmini Süre:1m 15s

Soru 4124Soru →

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 $k$ .

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Measures of Central Tendency for Grouped Data

Soru 4125Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Soru 4126Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 4127Soru →

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

Cevabı ve açıklamayı göster

Cevap: 3.2

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Mean Deviation for Ungrouped Data

Soru 4128Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

Angle of elevation using basic right-triangle trigonometry

Soru 4129Soru →

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}

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 4130Soru →

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

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Change of Base Property of Logarithms

Soru 4131Soru →

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 ( $\text{N}$ )?

Cevabı ve açıklamayı göster

Cevap: 10000

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Sequential Fraction of Remainder Problems

Soru 4132Soru →

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

Aşağıdaki boşlukları doldurun

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

Cevabı ve açıklamayı göster

Cevap

diligent

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

Adım Adım Çözüm

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'.

Anahtar Kavram

Synonyms and Words Nearest in Meaning

Soru 4133Soru →

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

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 4134Soru →

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

Cevabı ve açıklamayı göster

Cevap: $2\sec x$

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Alternatif Yöntem

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

.

Tahmini Süre:2m 0s

Soru 4135Soru →

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

Cevabı ve açıklamayı göster

Cevap: 22

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 4136Soru →

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

Cevabı ve açıklamayı göster

Cevap: $-3$

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru 4137Soru →

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

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Singular matrix condition and

2 \times 2

determinant calculation

Soru 4138Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 57

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

Soru 4139Soru →

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

Cevabı ve açıklamayı göster

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

Cevap

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}

.

Adım Adım Çözüm

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

).

Anahtar Kavram

Perpendicular Line Gradients

Soru 4140Soru →

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

Cevabı ve açıklamayı göster

Cevap: $7$

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

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

-1

(

m_1 \cdot m_2 = -1

).

Tahmini Süre:2m 0s

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