Tüm alıştırma soruları

Soru 781Soru →

Two capacitors with capacitances of $10\text{ }\mu\text{F}$ and $15\text{ }\mu\text{F}$ are connected in parallel. What is the equivalent capacitance of the combination in microfarads ( $\mu\text{F}$ )?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The equivalent capacitance of the parallel combination is

25\text{ }\mu\text{F}

.

When capacitors are connected in parallel, the total equivalent capacitance is equal to the direct sum of the individual capacitances:

C_{\text{eq}} = C_1 + C_2 = 10\text{ }\mu\text{F} + 15\text{ }\mu\text{F} = 25\text{ }\mu\text{F}

.

Adım Adım Çözüm

1

Identify the relationship for parallel capacitors

C_{\text{eq}} = C_1 + C_2

Capacitors connected in parallel store charge independently across the same potential difference, so their capacitances add directly.

2

Substitute the given values into the formula

C_{\text{eq}} = 10\text{ }\mu\text{F} + 15\text{ }\mu\text{F}

The circuit contains two capacitors of

10\text{ }\mu\text{F}

and

15\text{ }\mu\text{F}

in parallel.

3

Calculate the total capacitance

25\text{ }\mu\text{F}

Simple addition of the two values yields

25\text{ }\mu\text{F}

.

Anahtar Kavram

Equivalent Capacitance of Parallel Connected Capacitors

Soru 782Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 30000

Cevap

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\%

.

Adım Adım Çözüm

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.

Anahtar Kavram

Simple Interest and Allocation of Principal across Different Interest Rates

Tahmini Süre:1m 30s

Soru 783Soru →

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?

Cevabı ve açıklamayı göster

Cevap: 280000

Cevap

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

.

Adım Adım Çözüm

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.

Anahtar Kavram

Percentage Profit and Selling Price

Soru 784Soru →

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 $k$ .

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

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

.

Adım Adım Çözüm

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

.

Anahtar Kavram

Rationalization of Denominators and Difference of Conjugate Surd Fractions

Tahmini Süre:1m 30s

Soru 785Soru →

If $y = e^{3x}\cos(2x) + \ln(x + 1)$ , calculate the value of $\frac{dy}{dx}$ at $x = 0$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The derivative evaluated at

x = 0

is 4.

Differentiating

y = e^{3x}\cos(2x) + \ln(x + 1)

with respect to

x

yields

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

. Evaluating this derivative at

x = 0

gives

3(1)(1) - 2(1)(0) + 1 = 4

.

Adım Adım Çözüm

1

Differentiate the product

u(x) = e^{3x}\cos(2x)

using the product rule and chain rule.

\frac{du}{dx} = 3e^{3x}\cos(2x) - 2e^{3x}\sin(2x)

By the product rule

\frac{d}{dx}(uv) = u'v + uv'

, where

\frac{d}{dx}(e^{3x}) = 3e^{3x}

and

\frac{d}{dx}(\cos(2x)) = -2\sin(2x)

.

2

Differentiate the logarithmic term

v(x) = \ln(x + 1)

.

\frac{dv}{dx} = \frac{1}{x + 1}

The derivative of

\ln(g(x))

is

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

.

3

Combine the terms to write the complete derivative

\frac{dy}{dx}

.

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

The derivative of a sum of functions is the sum of their individual derivatives.

4

Evaluate

\frac{dy}{dx}

at

x = 0

.

\frac{dy}{dx}\Big|_{x=0} = 3e^0\cos(0) - 2e^0\sin(0) + \frac{1}{0 + 1} = 3(1)(1) - 2(1)(0) + 1 = 4

Substitute

x = 0

using

e^0 = 1

,

\cos(0) = 1

, and

\sin(0) = 0

.

Anahtar Kavram

Differentiation of Trigonometric, Exponential, and Logarithmic Functions

Tahmini Süre:1m 30s

Soru 786Soru →

A convex polygon has $44$ diagonals. What is the total number of distinct triangles that can be formed by joining any three vertices of this polygon?

Cevabı ve açıklamayı göster

Cevap: 165

Cevap

165

The number of diagonals of an

n

-sided convex polygon is given by

\binom{n}{2} - n = \frac{n(n-3)}{2}

. Setting this equal to

44

yields

n(n-3) = 88

, which simplifies to

n^2 - 3n - 88 = 0

. Factoring gives

(n-11)(n+8) = 0

, so

n = 11

. The total number of distinct triangles formed by choosing any 3 vertices from an 11-sided polygon is

\binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165

.

Adım Adım Çözüm

1

Set up the equation for the number of diagonals in terms of the number of vertices

n

\frac{n(n-3)}{2} = 44

Choosing any 2 vertices from

n

vertices gives

\binom{n}{2}

total connecting line segments. Subtracting the

n

boundary sides leaves the diagonals.

2

Solve the quadratic equation for

n

n^2 - 3n - 88 = 0 \implies (n-11)(n+8) = 0 \implies n = 11

A polygon must have a positive integer number of vertices, so

n = 11

.

3

Compute the number of distinct triangles using combinations

\binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165

In a convex polygon, no three vertices are collinear, so every unique combination of 3 vertices forms a distinct triangle.

Anahtar Kavram

Application of combinations to geometry (polygon diagonals and triangle selection)

Tahmini Süre:2m 0s

Soru 787Soru →

What is the magnitude of the electric field intensity, in $\text{N C}^{-1}$ , at a point $2.0\text{ m}$ away from a isolated point charge of $+4.0 \times 10^{-6}\text{ C}$ in a vacuum? (Take Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ )

Cevabı ve açıklamayı göster

Cevap: 9000

Cevap

The magnitude of the electric field intensity is

9000\text{ N C}^{-1}

.

The electric field intensity

E

produced by a point charge

q

at distance

r

is given by

E = \frac{kq}{r^2}

. Substituting

k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}

,

q = 4.0 \times 10^{-6}\text{ C}

, and

r = 2.0\text{ m}

yields

E = \frac{9.0 \times 10^9 \times 4.0 \times 10^{-6}}{4.0} = 9000\text{ N C}^{-1}

.

Adım Adım Çözüm

1

Identify the given physical quantities and formula

q = 4.0 \times 10^{-6}\text{ C}

,

r = 2.0\text{ m}

,

k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}

. Formula:

E = \frac{kq}{r^2}

The magnitude of electric field intensity due to a single point charge is given by Coulomb's field law.

2

Substitute the values and calculate the electric field strength

E = \frac{9.0 \times 10^9 \times 4.0 \times 10^{-6}}{2.0^2} = \frac{36000}{4} = 9000\text{ N C}^{-1}

Perform basic arithmetic simplification to determine the numerical result.

Anahtar Kavram

Electric Field Intensity due to a Point Charge

Soru 788Soru →

If $\frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}} + \frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}} = p + q\sqrt{21}$ , where $p$ and $q$ are integers, what is the value of $p + q$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

p + q

is 6.

Rationalising each fraction yields

(\sqrt{21} - 3)

and

(7 + \sqrt{21})

. Summing these expressions gives

4 + 2\sqrt{21}

. Comparing this to

p + q\sqrt{21}

yields

p = 4

and

q = 2

, so

p + q = 6

.

Adım Adım Çözüm

1

Rationalise the first term

\frac{4\sqrt{3}}{\sqrt{7} + \sqrt{3}}

by multiplying the numerator and denominator by the conjugate

(\sqrt{7} - \sqrt{3})

.

\frac{4\sqrt{3}(\sqrt{7} - \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{4\sqrt{21} - 12}{7 - 3} = \frac{4\sqrt{21} - 12}{4} = \sqrt{21} - 3

Multiplying by the conjugate eliminates the surd from the denominator using the difference of two squares.

2

Rationalise the second term

\frac{4\sqrt{7}}{\sqrt{7} - \sqrt{3}}

by multiplying the numerator and denominator by the conjugate

(\sqrt{7} + \sqrt{3})

.

\frac{4\sqrt{7}(\sqrt{7} + \sqrt{3})}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{28 + 4\sqrt{21}}{7 - 3} = \frac{28 + 4\sqrt{21}}{4} = 7 + \sqrt{21}

Conjugate rationalisation simplifies the second fraction into linear surd terms.

3

Add the two simplified expressions together and equate to

p + q\sqrt{21}

.

(\sqrt{21} - 3) + (7 + \sqrt{21}) = 4 + 2\sqrt{21}

Combining like surd terms yields the simplified form

p + q\sqrt{21}

.

4

Identify the values of

p

and

q

and evaluate

p + q

.

p = 4, q = 2 \implies p + q = 4 + 2 = 6

Equating the rational parts gives

p = 4

and the irrational coefficients gives

q = 2

.

Anahtar Kavram

Rationalisation of surds with binomial denominators

Alternatif Yöntem

Combine the two fractions directly over the common denominator

(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3}) = 4

: \frac{4\sqrt{3}(\sqrt{7} - \sqrt{3}) + 4\sqrt{7}(\sqrt{7} + \sqrt{3})}{4} = \frac{4\sqrt{21} - 12 + 28 + 4\sqrt{21}}{4} = \frac{16 + 8\sqrt{21}}{4} = 4 + 2\sqrt{21}.

Tahmini Süre:1m 30s

Soru 789Soru →

The frequency distribution table below shows the daily profits (in thousands of Naira, $\text{₦}$ ) recorded by a sample of $40$ small-scale market traders:

Daily Profit ( $\text{₦'000}$ )	Frequency ( $f$ )
$10 - 14$	$6$
$15 - 19$	$10$
$20 - 24$	$12$
$25 - 29$	$8$
$30 - 34$	$4$

What is the mean daily profit of the traders, in thousands of Naira?

Cevabı ve açıklamayı göster

Cevap: 21.25

Cevap

The mean daily profit of the traders is

21.25

thousand Naira.

The mean daily profit is found by dividing the sum of the product of each class midpoint and its frequency (

\sum fx = 850

) by the total frequency (

\sum f = 40

), yielding

21.25

.

Adım Adım Çözüm

1

Find the class midpoints (

x

) for each interval

Midpoints are

12

,

17

,

22

,

27

, and

32

.

For grouped data, each class interval is represented by its midpoint.

2

Calculate the product of frequency and midpoint (

f \cdot x

) for each interval

Products are

72

,

170

,

264

,

216

, and

128

.

Multiplying class midpoint by class frequency estimates the sum of values within that class.

3

Calculate total frequency (

\sum f

) and total sum of products (

\sum fx

)

\sum f = 40

and

\sum fx = 850

.

The sum of frequencies gives the total number of observations, and the sum of products gives the estimated grand total.

4

Apply the grouped mean formula

\bar{x} = \frac{\sum fx}{\sum f}

\bar{x} = \frac{850}{40} = 21.25.

Dividing total sum by total frequency gives the mean value.

Anahtar Kavram

Measures of Central Tendency for Grouped Data - Mean

Soru 790Soru →

A mechanical longitudinal wave of frequency $250\text{ Hz}$ propagates from Medium 1 into Medium 2. In Medium 1, the distance between two consecutive compressions is $1.40\text{ m}$ . Upon entering Medium 2, the wave speed increases by $40\%$ . What is the distance, in meters, between a compression and the immediately adjacent rarefaction in Medium 2?

Cevabı ve açıklamayı göster

Cevap: 0.98

Cevap

The distance between a compression and the adjacent rarefaction in Medium 2 is

0.98\text{ m}

.

When a mechanical wave travels between media, its frequency remains unchanged. The wave speed equation

v = f\lambda

indicates that wavelength is directly proportional to wave speed. A

40\%

increase in speed increases the wavelength in Medium 2 from

1.40\text{ m}

to

1.40 \times 1.40\text{ m} = 1.96\text{ m}

. In a longitudinal wave, the distance between a compression and an adjacent rarefaction is half a wavelength, yielding

\frac{1.96\text{ m}}{2} = 0.98\text{ m}

.

Adım Adım Çözüm

1

Identify the wavelength in Medium 1 from the compression spacing.

\lambda_1 = 1.40\text{ m}

The distance between two successive compressions in a longitudinal wave corresponds to one complete wavelength.

2

Calculate the wavelength in Medium 2 using the constant frequency principle across media boundaries.

\lambda_2 = 1.40 \times 1.40\text{ m} = 1.96\text{ m}

The frequency of a wave is determined by the source and does not change upon entering a new medium. Since

v = f\lambda

, a

40\%

increase in wave speed results in a proportional

40\%

increase in wavelength.

3

Find the distance between a compression and the adjacent rarefaction in Medium 2.

d = \frac{\lambda_2}{2} = \frac{1.96\text{ m}}{2} = 0.98\text{ m}

In any longitudinal wave, a compression and its adjacent rarefaction are out of phase by half a cycle, corresponding to half a wavelength.

Anahtar Kavram

Wave propagation across boundaries and spatial separation of compressions and rarefactions in longitudinal waves.

Soru 791Soru →

The derived SI unit of dynamic viscosity, the pascal-second ( $\text{Pa}\cdot\text{s}$ ), can be expressed in fundamental SI base units as $\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c$ . Determine the numerical value of the exponent of length, $b$ .

Cevabı ve açıklamayı göster

Cevap: -1

Cevap

The numerical value of the exponent of length

b

is

-1

.

Dynamic viscosity is measured in pascal-seconds (

\text{Pa}\cdot\text{s}

). Substituting

1\text{ Pa} = 1\text{ N/m}^2 = 1\text{ kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

into the formula gives

1\text{ Pa}\cdot\text{s} = 1\text{ kg}^1 \cdot \text{m}^{-1} \cdot \text{s}^{-1}

. Comparing this with

\text{kg}^a \cdot \text{m}^b \cdot \text{s}^c

, the exponent of length (meters) is

b = -1

.

Adım Adım Çözüm

1

Express the newton in base SI units using

F = ma

.

\text{N} = \text{kg}\cdot\text{m}\cdot\text{s}^{-2}

.

Force is the product of mass and acceleration.

2

Derive the SI base unit expression for pressure (pascal,

\text{Pa}

).

\text{Pa} = \frac{\text{N}}{\text{m}^2} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

.

Pressure is defined as force per unit area.

3

Multiply the base unit expression of pressure by seconds.

\text{Pa}\cdot\text{s} = (\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}) \cdot \text{s}^1 = \text{kg}^1 \cdot \text{m}^{-1} \cdot \text{s}^{-1}

.

Dynamic viscosity is measured in pascal-seconds.

4

Extract the exponent of length (

b

) corresponding to the meter unit.

b = -1

.

The power of meters (

\text{m}

) in

\text{kg}^1 \cdot \text{m}^{-1} \cdot \text{s}^{-1}

is

-1

.

Anahtar Kavram

Deriving base SI units for derived physical quantities

Tahmini Süre:1m 30s

Soru 792Soru →

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.

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

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}

.

Adım Adım Çözüm

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}

.

Anahtar Kavram

Average Rate and Speed Calculations

Soru 793Soru →

A coastal monitoring station at point $O$ tracks two vessels on horizontal water. Vessel $A$ is located $15\text{ km}$ from $O$ on a bearing of $070^\circ$ , while Vessel $B$ is located $20\text{ km}$ from $O$ on a bearing of $160^\circ$ . What is the direct distance between Vessel $A$ and Vessel $B$ in kilometers?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The direct distance between Vessel A and Vessel B is 25 km.

The difference between the two bearings (

160^\circ - 070^\circ = 90^\circ

) establishes that triangle

AOB

is a right-angled triangle at station

O

. Applying Pythagoras' theorem yields

AB = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25\text{ km}

.

Adım Adım Çözüm

1

Find the angle between the lines of sight to the two vessels

\angle AOB = 160^\circ - 070^\circ = 90^\circ

Subtracting the smaller bearing angle from the larger bearing angle from the same origin point gives the included angle.

2

Set up the equation for distance AB using Pythagoras' theorem

AB^2 = 15^2 + 20^2 = 225 + 400 = 625

Since the included angle is 90 degrees, the three points form a right-angled triangle where AB is the hypotenuse.

3

Calculate the principal square root of 625

AB = \sqrt{625} = 25\text{ km}

Taking the square root converts the squared distance into the direct linear distance between the vessels.

Anahtar Kavram

Calculating the distance between two points using bearings and right-angled triangle properties (Pythagoras' theorem).

Tahmini Süre:1m 30s

Soru 794Soru →

A binary operation $\ast$ defined on the set of real numbers $\mathbb{R}$ is given by $a \ast b = a + b + 7$ . What is the identity element under this operation?

Cevabı ve açıklamayı göster

Cevap: -7

Cevap

The identity element under the operation is

-7

.

The identity element

e

satisfies

a \ast e = a

for any real number

a

. Substituting into the definition gives

a + e + 7 = a

, which simplifies to

e = -7

.

Adım Adım Çözüm

1

Set up the identity element equation using the definition

a \ast e = a

.

a + e + 7 = a

By definition, operating any element

a

with the identity element

e

yields

a

.

2

Subtract

a

from both sides of the equation.

e + 7 = 0

Isolating terms involving

e

.

3

Subtract

7

from both sides to solve for

e

.

e = -7

Determining the numerical value of the identity element.

Anahtar Kavram

Identity Element in Binary Operations

Soru 795Soru →

Calculate the area of the finite region bounded by the parabola $y = 3x^2 - 12x + 9$ and the $x$ -axis.

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The area of the bounded region is 4 square units.

Finding the x-intercepts of

y = 3x^2 - 12x + 9

gives

x = 1

and

x = 3

. Integrating

-y

from

1

to

3

gives

\left[-x^3 + 6x^2 - 9x\right]_{1}^{3} = 0 - (-4) = 4

square units.

Adım Adım Çözüm

1

Determine the limits of integration by finding the x-intercepts of the curve.

x = 1

and

x = 3

The bounded region lies between the points where the curve intersects the x-axis (

y = 0

).

2

Set up the definite integral with the correct integrand sign.

A = \int_{1}^{3} (9 - 12x + 3x^2) \, dx = \int_{1}^{3} (-3x^2 + 12x - 9) \, dx

Since

y \le 0

on

[1, 3]

, negating the function ensures the calculated area is positive.

3

Integrate term-by-term and evaluate between upper limit 3 and lower limit 1.

\left[-x^3 + 6x^2 - 9x\right]_{1}^{3} = (0) - (-4) = 4

Applying the Fundamental Theorem of Calculus yields the exact value of 4.

Anahtar Kavram

Area bounded by a curve and the x-axis lying below the x-axis

Soru 796Soru →

A line segment joins the points $A(1, 4)$ and $B(7, 10)$ . Point $P$ divides the line segment $AB$ internally in the ratio $1:2$ . A second line $L_2$ passes through $P$ and is perpendicular to $AB$ . If line $L_2$ intersects the y-axis at the point $(0, k)$ , find the value of $k$ .

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Cevap: 9

Cevap

The value of k is 9.

Using the section formula for internal division in a 1:2 ratio, the coordinates of point P are found to be (3, 6). The gradient of the segment AB is 1, which means the perpendicular line L_2 has a gradient of -1. Writing the equation of line L_2 passing through (3, 6) yields y = -x + 9. Evaluating at x = 0 gives the y-intercept k = 9.

Adım Adım Çözüm

1

Calculate the coordinates of point P dividing segment AB internally in the ratio 1:2.

P = (3, 6)

Using the section formula x = (m x_2 + n x_1) / (m + n) and y = (m y_2 + n y_1) / (m + n) with ratio m:n = 1:2.

2

Calculate the gradient m_1 of the line segment AB.

m_1 = 1

Applying the gradient formula m = (y_2 - y_1) / (x_2 - x_1) gives (10 - 4) / (7 - 1) = 1.

3

Determine the gradient m_2 of the perpendicular line L_2.

m_2 = -1

Perpendicular lines satisfy m_1 * m_2 = -1, hence m_2 = -1 / 1 = -1.

4

Find the equation of line L_2 passing through P(3, 6) with gradient -1.

y = -x + 9

Using point-slope form y - y_1 = m(x - x_1) yields y - 6 = -1(x - 3).

5

Determine the y-intercept value k by setting x = 0.

k = 9

Substituting x = 0 into y = -x + 9 gives y = 9.

Anahtar Kavram

Section formula, perpendicular line gradients, and y-intercept determination

Tahmini Süre:3m 0s

Soru 797Soru →

A progressive wave traveling along a medium is described by the equation $y = 0.05 \sin(20\pi t - 4\pi x)$ , where $x$ and $y$ are measured in meters and $t$ is in seconds. What is the speed of the wave in $\text{m/s}$ ?

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Cevap: 5

Cevap

The speed of the wave is

5.0\text{ m/s}

.

Comparing

y = 0.05 \sin(20\pi t - 4\pi x)

to the standard wave equation

y = A \sin(\omega t - kx)

, we find

\omega = 20\pi\text{ rad/s}

and

k = 4\pi\text{ rad/m}

. The wave speed

v

is calculated as

v = \frac{\omega}{k} = \frac{20\pi}{4\pi} = 5.0\text{ m/s}

.

Adım Adım Çözüm

1

Identify the wave parameters from the standard equation form

\omega = 20\pi\text{ rad/s}

and

k = 4\pi\text{ rad/m}

Matching the given equation

y = 0.05 \sin(20\pi t - 4\pi x)

to

y = A \sin(\omega t - kx)

gives the values for angular frequency

\omega

and wave number

k

.

2

Compute wave speed using the relationship

v = \frac{\omega}{k}

v = \frac{20\pi}{4\pi} = 5.0\text{ m/s}

Wave speed is defined as the ratio of angular frequency to wave number.

Anahtar Kavram

Wave Speed from Wave Equation

Soru 798Soru →

Five of the interior angles of a convex polygon are each equal to $140^\circ$ , while the remaining interior angles are each equal to $160^\circ$ . Calculate the number of sides of the polygon.

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Cevap: 13

Cevap

The number of sides of the polygon is 13.

Each

140^\circ

interior angle has an exterior angle of

40^\circ

, contributing

5 \times 40^\circ = 200^\circ

to the exterior angle sum. Each

160^\circ

interior angle has an exterior angle of

20^\circ

, contributing

(n - 5) \times 20^\circ

. Since the sum of exterior angles of any convex polygon is

360^\circ

, setting

200 + 20(n - 5) = 360

yields

20n = 260

, giving

n = 13

.

Adım Adım Çözüm

1

Calculate the exterior angle measures

The exterior angles are

180^\circ - 140^\circ = 40^\circ

(for 5 vertices) and

180^\circ - 160^\circ = 20^\circ

(for the remaining

n - 5

vertices).

Interior and exterior angles at each vertex of a polygon form a linear pair and sum to

180^\circ

.

2

Apply the sum of exterior angles property

5(40^\circ) + (n - 5)(20^\circ) = 360^\circ

.

The sum of exterior angles of any convex polygon is always constant and equal to

360^\circ

.

3

Solve the linear equation for

n

200 + 20n - 100 = 360 \implies 20n = 260 \implies n = 13

.

Expanding terms and isolating

n

gives the exact number of sides.

Anahtar Kavram

Exterior angle sum property of convex polygons

Soru 799Soru →

Two identical positive point charges, each of magnitude $q = +2.5 \times 10^{-6}\text{ C}$ , are fixed in a vacuum at a distance of $0.60\text{ m}$ apart. A third point charge $q_0 = +1.0 \times 10^{-6}\text{ C}$ is placed on the perpendicular bisector of the line joining the two fixed charges, at a distance of $0.40\text{ m}$ from their midpoint. Taking Coulomb's constant $k = 9.0 \times 10^9\text{ N m}^2\text{ C}^{-2}$ , what is the magnitude of the net electrostatic force acting on the third charge, in newtons?

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Cevap: 0.144

Cevap

The magnitude of the net electrostatic force acting on the third charge is

0.144\text{ N}

.

Each fixed charge exerts an equal repulsive electrostatic force of

0.09\text{ N}

on the third charge. Due to the symmetrical arrangement, the force components perpendicular to the bisector cancel each other out, while the parallel components add together, giving a net force of

2 \times 0.09 \times 0.8 = 0.144\text{ N}

.

Adım Adım Çözüm

1

Determine the distance from each fixed charge to the third charge

r = 0.50\text{ m}

The charges form a right-angled triangle with base

0.30\text{ m}

(half of

0.60\text{ m}

) and height

0.40\text{ m}

, yielding a hypotenuse of

\sqrt{0.30^2 + 0.40^2} = 0.50\text{ m}

.

2

Calculate the magnitude of the individual repulsive force from one charge

F = 0.09\text{ N}

Applying Coulomb's law:

F = \frac{k q q_0}{r^2} = \frac{9.0 \times 10^9 \times 2.5 \times 10^{-6} \times 1.0 \times 10^{-6}}{0.25} = 0.09\text{ N}

.

3

Determine directional component of forces along the perpendicular bisector

\cos\theta = 0.8

The directional cosine along the axis of symmetry is the ratio of the adjacent side (

0.40\text{ m}

) to the hypotenuse (

0.50\text{ m}

).

4

Compute the net electrostatic force using vector addition

F_{\text{net}} = 0.144\text{ N}

Horizontal components cancel by symmetry, so

F_{\text{net}} = 2 F \cos\theta = 2 \times 0.09 \times 0.8 = 0.144\text{ N}

.

Anahtar Kavram

Vector superposition of Coulombic forces along an axis of symmetry

Soru 800Soru →

Find the sum of all integer values of $x$ that satisfy both the linear inequality $2x - 1 \ge 3$ and the quadratic inequality $x^2 - 5x - 14 \le 0$ .

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The sum of all integer values of

x

satisfying both inequalities is

27

.

Solving the linear inequality

2x - 1 \ge 3

yields

x \ge 2

. Solving the quadratic inequality

x^2 - 5x - 14 \le 0

by factoring gives

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

, which defines the interval

-2 \le x \le 7

. Taking the intersection of

x \ge 2

and

-2 \le x \le 7

gives

2 \le x \le 7

. The integer values satisfying this range are

2, 3, 4, 5, 6,

and

7

, and their sum is

27

.

Adım Adım Çözüm

1

Solve the linear inequality

2x \ge 4 \implies x \ge 2

Adding 1 to both sides and dividing by 2 isolates the variable

x

.

2

Factor and solve the quadratic inequality

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

The roots of the quadratic equation are

x = 7

and

x = -2

. The parabola opens upward, so the expression is non-positive between the roots.

3

Determine the intersection of both solution sets

2 \le x \le 7

The values of

x

must simultaneously satisfy

x \ge 2

and

-2 \le x \le 7

.

4

List all integer solutions within the valid interval

x \in \{2, 3, 4, 5, 6, 7\}

These are all the whole numbers contained in the closed interval

[2, 7]

.

5

Sum the integer solutions

2 + 3 + 4 + 5 + 6 + 7 = 27

Summing the identified integer values yields the final required numerical answer.

Anahtar Kavram

Linear and Quadratic Inequalities

Tahmini Süre:1m 30s

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