Tüm alıştırma soruları

Soru 7301Soru →

A physical quantity $C$ is calculated using the formula $C = \frac{x \cdot y}{z}$ , where $x$ , $y$ , and $z$ are positive variables. If $x$ is increased by $20\%$ , $y$ is decreased by $10\%$ , and $z$ is increased by $25\%$ , what is the net percentage change in $C$ ?

13.6% decrease

15.0% decrease

35.0% increase

15.7% decrease

Cevabı ve açıklamayı göster

Cevap: 13.6% decrease

Cevap

13.6% decrease

The new value

C'

is related to the original value

C

by the scale factor

\frac{1.20 \times 0.90}{1.25} = \frac{1.08}{1.25} = 0.864

. Subtracting 1 gives

-0.136

, which corresponds to a 13.6% decrease.

Adım Adım Çözüm

1

Express the modified variables in terms of their original values using decimal multipliers.

x' = 1.20x

,

y' = 0.90y

, and

z' = 1.25z

.

An increase of 20% corresponds to a factor of 1.20, a decrease of 10% corresponds to 0.90, and an increase of 25% corresponds to 1.25.

2

Substitute the modified variables into the formula for

C

.

C' = \frac{(1.20x)(0.90y)}{1.25z} = \left(\frac{1.20 \times 0.90}{1.25}\right) \frac{xy}{z} = \left(\frac{1.08}{1.25}\right) C

.

Separate the numerical scale factor from the original expression

C = \frac{xy}{z}

.

3

Evaluate the combined scale factor as a decimal.

\frac{1.08}{1.25} = \frac{108}{125} = \frac{864}{1000} = 0.864

.

Multiplying the numerator and denominator by 8 simplifies the fraction to a thousandths decimal.

4

Calculate the net percentage change in

C

.

\text{Percentage Change} = (0.864 - 1) \times 100\% = -0.136 \times 100\% = -13.6\%

.

A factor less than 1 indicates a decrease of

(1 - 0.864) \times 100\% = 13.6\%

.

Anahtar Kavram

Compound Percentage Change and Variable Multipliers

Soru 7302Soru →

The test scores of six students in a mathematics quiz are $12$ , $18$ , $25$ , $31$ , $47$ , and $50$ . What is the range of these scores?

38

50

30.5

28

Cevabı ve açıklamayı göster

Cevap: $38$

Cevap

The range of the scores is

38

.

The range is a measure of dispersion defined as the difference between the maximum value (

50

) and the minimum value (

12

). Subtracting

12

from

50

gives

38

.

Adım Adım Çözüm

1

Identify the maximum and minimum values in the dataset.

Maximum score =

50

, Minimum score =

12

.

Range measures the spread between the highest and lowest values in a distribution.

2

Calculate the range using the formula

\text{Range} = \text{Maximum} - \text{Minimum}

.

\text{Range} = 50 - 12 = 38

.

Subtracting the minimum value from the maximum value yields the measure of dispersion known as range.

Anahtar Kavram

Range of a Dataset

Soru 7303Soru →

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 40\}$ . Subsets $P$ , $Q$ , and $R$ of $\mathcal{U}$ are defined as follows:
$P = \{x \in \mathcal{U} : x \text{ is a multiple of } 3\}$ ,
$Q = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\}$ , and
$R = \{x \in \mathcal{U} : x \text{ is a perfect square\}}$ .
What is the cardinality of the set $(P \cup Q)' \cap R'$ ?

18

22

20

17

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

By De Morgan's Law, the expression

(P \cup Q)' \cap R'

is equivalent to

(P \cup Q \cup R)'

. The universal set contains 40 integers. The set

P \cup Q \cup R

consists of all integers from 1 to 40 that are multiples of 3, multiples of 4, or perfect squares, giving a total of 22 distinct elements. Subtracting this from 40 leaves 18 elements that belong to neither

P

,

Q

, nor

R

.

Adım Adım Çözüm

1

Apply De Morgan's Laws to simplify the required set expression

(P \cup Q)' \cap R' = (P \cup Q \cup R)'

The intersection of the complements of two sets is equal to the complement of their union.

2

List and count the elements belonging to sets

P

,

Q

, and

R

within

\mathcal{U} = \{1, 2, \dots, 40\}

P

has 13 elements (multiples of 3);

Q

adds 7 elements not in

P

(multiples of 4 excluding 12, 24, 36);

R

adds 2 elements not in

P \cup Q

(

1

and

25

)

Counting unique elements avoids double-counting shared elements across sets.

3

Calculate the cardinality of

P \cup Q \cup R

n(P \cup Q \cup R) = 13 + 7 + 2 = 22

Summing the non-overlapping subsets gives the total size of the union.

4

Subtract the union cardinality from the universal set cardinality

n(\mathcal{U}) = 40

n((P \cup Q \cup R)') = 40 - 22 = 18

The cardinality of the complement is the size of the universal set minus the size of the set itself.

Anahtar Kavram

De Morgan's Laws and Set Complements

Soru 7304Soru →

What is the gradient of the normal to the curve $y = x^2 - 4x + 5$ at the point where $x = 3$ ?

-\frac{1}{2}

2

\frac{1}{2}

-2

Cevabı ve açıklamayı göster

Cevap: $-\frac{1}{2}$

Cevap

The gradient of the normal to the curve is

-\frac{1}{2}

.

Differentiating

y = x^2 - 4x + 5

gives

\frac{dy}{dx} = 2x - 4

. Substituting

x = 3

yields a tangent gradient of

2

. Since the normal line is perpendicular to the tangent line, its gradient is the negative reciprocal,

-\frac{1}{2}

.

Adım Adım Çözüm

1

Differentiate the equation of the curve to determine the gradient function.

\frac{dy}{dx} = 2x - 4

The first derivative represents the gradient of the tangent to the curve at any point

x

.

2

Evaluate the derivative at

x = 3

to find the tangent gradient

m_t

.

m_t = 2(3) - 4 = 2

Substituting

x = 3

gives the slope of the tangent line at the given point.

3

Calculate the gradient of the normal

m_n

using

m_n = -\frac{1}{m_t}

.

m_n = -\frac{1}{2}

The normal line is perpendicular to the tangent line, so its gradient is the negative reciprocal of the tangent's gradient.

Anahtar Kavram

Gradient of a Normal Line

Soru 7305Soru →

A solid right circular cylinder has a height of $10\text{ cm}$ and a total surface area of $192\pi\text{ cm}^2$ . What is the volume of the cylinder in $\text{cm}^3$ expressed in terms of $\pi$ ?

Cevabı ve açıklamayı göster

Cevap: 360

Cevap

360

The total surface area of a closed cylinder is

A = 2\pi r^2 + 2\pi r h

. Given

A = 192\pi

and

h = 10\text{ cm}

, we have

2\pi r^2 + 20\pi r = 192\pi

. Dividing by

2\pi

yields

r^2 + 10r - 96 = 0

. Factoring the quadratic gives

(r + 16)(r - 6) = 0

, so

r = 6\text{ cm}

(since radius must be positive). Substituting

r = 6\text{ cm}

and

h = 10\text{ cm}

into the volume formula

V = \pi r^2 h

yields

V = \pi (6^2)(10) = 360\pi\text{ cm}^3

. The numerical coefficient of

\pi

is

360

.

Adım Adım Çözüm

1

Set up the equation for the total surface area of the cylinder

2\pi r^2 + 2\pi r(10) = 192\pi

The total surface area of a solid cylinder comprises the areas of the top and bottom circular bases (

2\pi r^2

) plus the curved surface area (

2\pi r h

).

2

Simplify and solve the resulting quadratic equation for radius

r

r^2 + 10r - 96 = 0 \implies (r + 16)(r - 6) = 0 \implies r = 6\text{ cm}

Dividing the equation by

2\pi

yields

r^2 + 10r - 96 = 0

. Since radius must be positive,

r = 6\text{ cm}

.

3

Calculate the volume of the cylinder using the radius and height

V = \pi r^2 h = \pi (6)^2 (10) = 360\pi\text{ cm}^3

The volume of a cylinder is calculated using the formula

V = \pi r^2 h

.

Anahtar Kavram

Surface Area and Volume of 3D Solids

Soru 7306Soru →

A piece of wire of length $88\text{ cm}$ is bent to form the perimeter of a sector of a circle of radius $21\text{ cm}$ . What is the area of the sector in $\text{cm}^2$ ?

Cevabı ve açıklamayı göster

Cevap: 483

Cevap

The area of the sector is

483\text{ cm}^2

.

The total length of the wire forms the perimeter of the sector, which consists of two radii and the arc length (

P = 2r + l

). With

P = 88\text{ cm}

and

r = 21\text{ cm}

, the arc length is

l = 88 - 2(21) = 46\text{ cm}

. Using the sector area formula

A = \frac{1}{2} r l

, the area is

\frac{1}{2} \times 21 \times 46 = 483\text{ cm}^2

.

Adım Adım Çözüm

1

Determine the arc length of the sector from the given perimeter

l = 46\text{ cm}

The total perimeter of a sector includes two straight radii and the curved arc length:

P = 2r + l

. Subtracting twice the radius (

2 \times 21 = 42\text{ cm}

) from the total wire length of

88\text{ cm}

gives an arc length of

46\text{ cm}

.

2

Calculate the area of the sector

A = 483\text{ cm}^2

The area of a sector can be computed using the formula

A = \frac{1}{2} r l

. Substituting

r = 21\text{ cm}

and

l = 46\text{ cm}

yields

A = \frac{1}{2} \times 21 \times 46 = 483\text{ cm}^2

.

Anahtar Kavram

Perimeter and Area of a Sector of a Circle

Soru 7307Soru →

Evaluate the limit:

\lim_{x \to 2} \frac{x^3 - 8}{\sqrt{x + 2} - 2}

What is the numerical value of this limit?

Cevabı ve açıklamayı göster

Cevap: 48

Cevap

The numerical value of the limit is 48.

Evaluating the limit of

\frac{x^3 - 8}{\sqrt{x + 2} - 2}

as

x \to 2

gives an indeterminate form

\frac{0}{0}

. Factorizing the numerator gives

(x - 2)(x^2 + 2x + 4)

, and rationalizing the denominator by multiplying numerator and denominator by

(\sqrt{x + 2} + 2)

converts the denominator to

x - 2

. Canceling

(x - 2)

leaves

(x^2 + 2x + 4)(\sqrt{x + 2} + 2)

. Evaluating at

x = 2

gives

(4 + 4 + 4)(\sqrt{4} + 2) = 12 \times 4 = 48

.

Adım Adım Çözüm

1

Identify the limit form via direct substitution

Substituting

x = 2

yields

\frac{0}{0}

.

Direct evaluation results in an indeterminate form, requiring algebraic manipulation to eliminate the zero factor.

2

Factorize the numerator using the difference of cubes formula

x^3 - 8 = (x - 2)(x^2 + 2x + 4)

Exposing the factor

(x - 2)

is essential to resolving the zero denominator.

3

Rationalize the denominator using its algebraic conjugate

Multiply top and bottom by

(\sqrt{x + 2} + 2)

to get denominator

(x + 2) - 4 = x - 2

.

Applying

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

eliminates the square root from the denominator.

4

Cancel the common factor and compute the final value

\lim_{x \to 2} (x^2 + 2x + 4)(\sqrt{x + 2} + 2) = (12)(4) = 48.

With

(x - 2)

cancelled for

x \neq 2

, direct substitution now yields a defined real number.

Anahtar Kavram

Limits of Indeterminate Forms using Difference of Cubes and Surd Rationalization

Tahmini Süre:2m 30s

Soru 7308Soru →

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

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of the derivative of

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

at

x = 0

is 6.

Differentiating each transcendental term individually using the chain rule yields

\frac{dy}{dx} = 6e^{2x} - \tan x

. Substituting

x = 0

gives

6e^0 - \tan 0 = 6 - 0 = 6

.

Adım Adım Çözüm

1

Differentiate the exponential term

3e^{2x}

6e^{2x}

Applying the derivative rule for exponential functions

\frac{d}{dx}[a e^{kx}] = a k e^{kx}

.

2

Differentiate the logarithmic term

\ln(\cos x)

-\tan x

Applying the chain rule

\frac{d}{dx}[\ln(u)] = \frac{1}{u}\frac{du}{dx}

where

u = \cos x

gives

\frac{-\sin x}{\cos x} = -\tan x

.

3

Combine terms and evaluate at

x = 0

6

Substituting

x = 0

into

\frac{dy}{dx} = 6e^{2x} - \tan x

yields

6e^0 - \tan 0 = 6(1) - 0 = 6

.

Anahtar Kavram

Differentiation of Trigonometric, Exponential, and Logarithmic Functions

Soru 7309Soru →

Determine the number of integer values of $x$ that satisfy the quadratic inequality $3x^2 - 14x - 5 < 0$ .

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The total number of integer values satisfying the inequality is 5.

Factoring

3x^2 - 14x - 5 < 0

gives

(3x + 1)(x - 5) < 0

. The critical roots are

x = -\frac{1}{3}

and

x = 5

. Since the parabola opens upward, the expression is negative between the roots, yielding the interval

-\frac{1}{3} < x < 5

. The integer values contained in this interval are

0, 1, 2, 3,

and

4

, which total 5 values.

Adım Adım Çözüm

1

Factor the quadratic expression

(3x + 1)(x - 5) < 0

Factoring identifies the critical values where the quadratic expression changes sign.

2

Determine the critical points

x = -\frac{1}{3}

and

x = 5

Setting each factor to zero gives the boundary roots of the equation.

3

Formulate the solution interval

-\frac{1}{3} < x < 5

Since the coefficient of

x^2

is positive, the quadratic curve is below the x-axis strictly between the two roots.

4

List and count the integer solutions

The integers are

0, 1, 2, 3, 4

, yielding 5 integer solutions.

Counting integers strictly greater than

-\frac{1}{3}

and strictly less than

5

.

Anahtar Kavram

Solving quadratic inequalities and finding integer solution counts

Tahmini Süre:1m 30s

Soru 7310Soru →

A solid wooden block is shaped as a frustum of a right circular cone with a top base radius of $4\text{ cm}$ , a bottom base radius of $12\text{ cm}$ , and a vertical height of $15\text{ cm}$ . A cylindrical hole of radius $4\text{ cm}$ is drilled vertically through the center of the frustum from the top base straight down to the bottom base. What is the volume of the remaining wooden solid in $\text{cm}^3$ ?

800\pi\text{ cm}^3

2880\pi\text{ cm}^3

560\pi\text{ cm}^3

80\pi\text{ cm}^3

Cevabı ve açıklamayı göster

Cevap: $800\pi\text{ cm}^3$

Cevap

The volume of the remaining wooden solid is

800\pi\text{ cm}^3

.

The volume of a frustum of a cone with radii

R=12\text{ cm}

,

r=4\text{ cm}

, and height

h=15\text{ cm}

is

\frac{1}{3}\pi(15)(12^2 + 4^2 + 12 \times 4) = 1040\pi\text{ cm}^3

. Subtracting the volume of the central cylindrical hole of radius

4\text{ cm}

and height

15\text{ cm}

, which is

\pi(4^2)(15) = 240\pi\text{ cm}^3

, yields

1040\pi - 240\pi = 800\pi\text{ cm}^3

.

Adım Adım Çözüm

1

Calculate the total volume of the conical frustum before drilling.

V_{\text{frustum}} = \frac{1}{3}\pi h (R^2 + r^2 + R r) = \frac{1}{3}\pi (15)(12^2 + 4^2 + 12 \times 4) = 5\pi(144 + 16 + 48) = 1040\pi\text{ cm}^3

.

The total volume of a frustum of a cone is determined by its vertical height and the radii of its top and bottom circular bases.

2

Calculate the volume of the cylindrical hole drilled through the solid.

V_{\text{cylinder}} = \pi r_{\text{hole}}^2 h = \pi (4^2)(15) = 240\pi\text{ cm}^3

.

The drilled hole forms a right circular cylinder of radius

4\text{ cm}

and height equal to the full height of the frustum (

15\text{ cm}

).

3

Subtract the cylinder's volume from the frustum's volume to find the remaining volume.

V_{\text{remaining}} = V_{\text{frustum}} - V_{\text{cylinder}} = 1040\pi - 240\pi = 800\pi\text{ cm}^3

.

Removing material by drilling decreases the overall volume of the original solid by the exact volume of the cylindrical bore.

Anahtar Kavram

Volume of Composite Solids and Conical Frustums

Tahmini Süre:2m 30s

Soru 7311Soru →

The monthly operating cost $C$ (in naira) of a commercial power generator is partly constant and partly varies directly as the square of its daily operating time $h$ (in hours). When the generator operates for $3$ hours per day, the monthly cost is ₦ $5,500$ . When it operates for $5$ hours per day, the monthly cost is ₦ $13,500$ . What is the daily operating time, in hours, when the monthly operating cost is ₦ $33,000$ ?

8 hours

7 hours

9 hours

10 hours

Cevabı ve açıklamayı göster

Cevap: 8 hours

Cevap

8 hours

The relationship is modeled by partial variation

C = k_1 + k_2 h^2

. Solving the simultaneous equations derived from

h = 3

(

C = 5500

) and

h = 5

(

C = 13500

) yields

k_2 = 500

and

k_1 = 1000

. Substituting

C = 33000

into

33000 = 1000 + 500 h^2

gives

500 h^2 = 32000

, so

h^2 = 64

, which yields

h = 8

hours.

Adım Adım Çözüm

1

Set up the general formula for partial variation.

C = k_1 + k_2 h^2

, where

k_1

and

k_2

are constants.

The cost consists of a constant part (

k_1

) and a part that varies directly as the square of daily hours (

k_2 h^2

).

2

Form simultaneous linear equations using the given data points.

Equation (1):

5500 = k_1 + 9 k_2

Equation (2):

13500 = k_1 + 25 k_2

Substitute

h = 3, C = 5500

and

h = 5, C = 13500

into the variation equation.

3

Solve for the variation constants

k_1

and

k_2

.

Subtract Equation (1) from Equation (2):

8000 = 16 k_2 \implies k_2 = 500

.
Substitute

k_2 = 500

into Equation (1):

5500 = k_1 + 9(500) \implies k_1 = 1000

.

Eliminating

k_1

gives

k_2

, which is then used to find the constant part

k_1

.

4

Calculate

h

when

C = 33000

.

33000 = 1000 + 500 h^2 \implies 32000 = 500 h^2 \implies h^2 = 64 \implies h = 8

hours.

Substitute the values of

k_1, k_2,

and target

C

into the relation equation to solve for

h

.

Anahtar Kavram

Partial Variation with Simultaneous Equations

Tahmini Süre:2m 30s

Soru 7312Soru →

Find the smallest integer $x$ that satisfies the linear inequality $5 - 3x \le -7$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The smallest integer value of

x

satisfying the inequality is

4

.

Subtracting 5 from both sides of

5 - 3x \le -7

gives

-3x \le -12

. Dividing both sides by

-3

requires reversing the inequality sign to obtain

x \ge 4

. Therefore, the smallest integer value in the solution set is 4.

Adım Adım Çözüm

1

Subtract 5 from both sides of the inequality to isolate the variable term

-3x \le -12

Subtracting a constant from both sides maintains the inequality direction.

2

Divide both sides by

-3

and flip the inequality sign

x \ge 4

Dividing an inequality by a negative number reverses the direction of the inequality symbol.

3

Identify the smallest integer satisfying the condition

4

The solution set contains all real numbers greater than or equal to 4, making 4 the minimum integer value.

Anahtar Kavram

Solving Linear Inequalities with Negative Coefficients

Tahmini Süre:45s

Soru 7313Soru →

If $y = 2\sin(3x) + e^{4x}$ , what is the value of $\frac{dy}{dx}$ at $x = 0$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

Differentiating

y = 2\sin(3x) + e^{4x}

yields

\frac{dy}{dx} = 6\cos(3x) + 4e^{4x}

. Evaluating this expression at

x = 0

gives

6\cos(0) + 4e^{0} = 6(1) + 4(1) = 10

.

Adım Adım Çözüm

1

Differentiate each term of

y = 2\sin(3x) + e^{4x}

with respect to

x

.

\frac{dy}{dx} = 6\cos(3x) + 4e^{4x}

Applying the chain rule gives

\frac{d}{dx}[2\sin(3x)] = 2 \times 3\cos(3x) = 6\cos(3x)

and

\frac{d}{dx}[e^{4x}] = 4e^{4x}

.

2

Evaluate the derivative at

x = 0

.

6\cos(0) + 4e^{0} = 6(1) + 4(1) = 10

Substituting

x = 0

gives

\cos(0) = 1

and

e^{0} = 1

.

Anahtar Kavram

Differentiation of trigonometric and exponential functions

Soru 7314Soru →

At what point on the curve $y = 2x^2 - 8x + 5$ is the normal line parallel to the straight line $x + 4y - 7 = 0$ ?

(3, -1)

(1, -1)

(3, 11)

(4, 5)

Cevabı ve açıklamayı göster

Cevap: $(3, -1)$

Cevap

The point of contact on the curve is

(3, -1)

.

The line

x + 4y - 7 = 0

has a slope of

-\frac{1}{4}

. Since the normal line is parallel to this line, the normal gradient is

m_n = -\frac{1}{4}

. Consequently, the tangent gradient must be

m_t = 4

because

m_t \cdot m_n = -1

. Equating the derivative

\frac{dy}{dx} = 4x - 8

to

4

yields

x = 3

. Substituting

x = 3

into the curve equation

y = 2x^2 - 8x + 5

gives

y = -1

, yielding the point

(3, -1)

.

Adım Adım Çözüm

1

Find the gradient of the given straight line.

Rearranging

x + 4y - 7 = 0

into slope-intercept form gives

y = -\frac{1}{4}x + \frac{7}{4}

, so the line's gradient is

m = -\frac{1}{4}

.

Parallel lines have equal gradients, so the normal line to the curve must have gradient

m_n = -\frac{1}{4}

.

2

Determine the required gradient of the tangent line.

Since

m_t \cdot m_n = -1

, we have

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

.

The tangent and normal lines are perpendicular to each other.

3

Differentiate the curve's equation to find the

x

-coordinate.

\frac{dy}{dx} = 4x - 8

. Setting

\frac{dy}{dx} = 4

gives

4x - 8 = 4 \implies 4x = 12 \implies x = 3

.

The derivative represents the slope of the tangent line at any point

x

.

4

Substitute

x = 3

back into the curve's equation to find

y

.

y = 2(3)^2 - 8(3) + 5 = 2(9) - 24 + 5 = 18 - 24 + 5 = -1

.

The point of contact lies on the original curve.

Anahtar Kavram

Relationship between gradients of parallel lines, tangent lines, and normal lines to a curve

Soru 7315Soru →

Find the value of $r$ such that $-29 \equiv r \pmod{8}$ , where $0 \le r < 8$ .

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

r

in standard non-negative remainder form is

3

.

Dividing

-29

by

8

yields a quotient of

-4

and a remainder of

3

, because

8 \times (-4) + 3 = -29

. Since

0 \le 3 < 8

,

3

is the standard non-negative remainder.

Adım Adım Çözüm

1

Express

-29

in the form

8q + r

, where

q

is an integer and

0 \le r < 8

.

-29 = 8(-4) + 3

To find the standard remainder modulo

8

, the remainder

r

must satisfy

0 \le r < 8

.

2

Identify the remainder value

r

.

r = 3

Since

3

lies within the required range

[0, 7]

, it is the canonical remainder.

Anahtar Kavram

Modular Arithmetic and Non-Negative Remainders

Soru 7316Soru →

The mean score of $9$ students in a mathematics test was calculated as $12$ . It was later discovered that a score of $5$ was incorrectly recorded as $14$ , and an additional student's score of $31$ was omitted entirely. What is the correct mean score of all $10$ students?

13

13.9

14.4

14.8

Cevabı ve açıklamayı göster

Cevap: $13$

Cevap

The correct mean score of all 10 students is

13

.

The original sum of

9

scores is

108

. Subtracting the incorrect value (

14

) and adding the true value (

5

) reduces the sum of the

9

scores to

99

. Adding the

10^{\text{th}}

score of

31

gives a total sum of

130

. Dividing

130

by

10

gives the correct mean of

13

.

Adım Adım Çözüm

1

Calculate the initial total sum of the original 9 scores.

Initial sum

= 9 \times 12 = 108

.

Mean is defined as total sum divided by number of items, so total sum equals mean times count.

2

Adjust the total sum for the misread score.

Corrected sum of 9 scores

= 108 - 14 + 5 = 99

.

Subtract the incorrect value (

14

) and add the actual value (

5

).

3

Add the omitted 10th score to the sum.

New total sum

= 99 + 31 = 130

.

Including the omitted score increases the total score sum.

4

Divide the new total sum by the updated total number of students.

Correct mean

= \frac{130}{10} = 13

.

The total number of students increased from 9 to 10.

Anahtar Kavram

Correcting the mean of ungrouped data after data entry errors or additions

Tahmini Süre:1m 30s

Soru 7317Soru →

If $32_x = 23_{10}$ , find the value of the base $x$ .

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Expanding the number

32_x

into base 10 gives

3 \times x + 2 = 3x + 2

. Setting

3x + 2 = 23

leads to

3x = 21

, giving

x = 7

.

Adım Adım Çözüm

1

Expand

32_x

in terms of powers of

x

3x + 2

In base

x

, the rightmost digit has a weight of

x^0 = 1

and the next digit to the left has a weight of

x^1 = x

.

2

Equate the expanded form to the given base 10 value and solve for

x

x = 7

Setting

3x + 2 = 23

gives

3x = 21

, which yields

x = 7

upon dividing by 3.

Anahtar Kavram

Expansion and conversion from Base x to Base 10

Tahmini Süre:45s

Soru 7318Soru →

Evaluate the limit:

\lim_{x \to \infty} (\sqrt{x^2 + 6x} - x)

What is the numerical value of this limit?

3

6

0

Undefined

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

To evaluate the limit of

\sqrt{x^2 + 6x} - x

as

x \to \infty

, multiply and divide by its conjugate

\sqrt{x^2 + 6x} + x

. The numerator simplifies to

(x^2 + 6x) - x^2 = 6x

. Dividing both the numerator and denominator by

x

yields

\frac{6}{\sqrt{1 + 6/x} + 1}

. Taking the limit as

x \to \infty

reduces

\frac{6}{x}

to

0

, resulting in

\frac{6}{\sqrt{1} + 1} = 3

.

Adım Adım Çözüm

1

Identify the indeterminate form

Direct evaluation gives

\infty - \infty

, which is an indeterminate form.

Substitution cannot be applied directly when subtracting infinite limits.

2

Multiply and divide by the algebraic conjugate

\lim_{x \to \infty} \frac{(\sqrt{x^2 + 6x} - x)(\sqrt{x^2 + 6x} + x)}{\sqrt{x^2 + 6x} + x} = \lim_{x \to \infty} \frac{(x^2 + 6x) - x^2}{\sqrt{x^2 + 6x} + x} = \lim_{x \to \infty} \frac{6x}{\sqrt{x^2 + 6x} + x}

The identity

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

eliminates the square root in the numerator.

3

Factor

x

out of the denominator

\lim_{x \to \infty} \frac{6x}{x \left(\sqrt{1 + \frac{6}{x}} + 1\right)} = \lim_{x \to \infty} \frac{6}{\sqrt{1 + \frac{6}{x}} + 1}

Dividing the numerator and denominator by

x

allows evaluation at infinity.

4

Compute the limit as

x \to \infty

Since

\lim_{x \to \infty} \frac{6}{x} = 0

, the expression becomes

\frac{6}{\sqrt{1 + 0} + 1} = \frac{6}{2} = 3

.

Terms with

x

in the denominator approach zero as

x

grows arbitrarily large.

Anahtar Kavram

Limits at infinity involving radical indeterminate forms of type

\infty - \infty

Tahmini Süre:2m 0s

Soru 7319Soru →

Find the sum of all integer values of $x$ that satisfy the system of inequalities $3x + 1 > 7$ and $x^2 - 4x - 21 \le 0$ .

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

Solving

3x + 1 > 7

gives

x > 2

. Solving

x^2 - 4x - 21 \le 0

gives

-3 \le x \le 7

. The combined condition is

2 < x \le 7

. The integer values satisfying this condition are 3, 4, 5, 6, and 7. The sum of these integers is

3 + 4 + 5 + 6 + 7 = 25

.

Adım Adım Çözüm

1

Solve the linear inequality

3x + 1 > 7

.

x > 2

Subtracting 1 from both sides gives

3x > 6

, and dividing by 3 yields

x > 2

.

2

Solve the quadratic inequality

x^2 - 4x - 21 \le 0

.

-3 \le x \le 7

Factoring the quadratic expression gives

(x - 7)(x + 3) \le 0

. The roots are

-3

and

7

, and the quadratic curve is non-positive between these roots.

3

Determine the intersection of the solution sets

x > 2

and

-3 \le x \le 7

.

2 < x \le 7

The lower bound is determined by the strict linear constraint

x > 2

, while the upper bound is determined by the inclusive quadratic constraint

x \le 7

.

4

List the integer values of

x

within

2 < x \le 7

and compute their sum.

Integers are

3, 4, 5, 6, 7

; their sum is

25

.

The strict inequality excludes

2

, while the inclusive inequality includes

7

.

Anahtar Kavram

System of linear and quadratic inequalities

Soru 7320Soru →

If $2^{3x - 1} = 32$ , find the value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

x

is 2.

Rewriting 32 as

2^5

allows equating the exponents

3x - 1 = 5

. Solving for

x

gives

3x = 6

, which simplifies to

x = 2

.

Adım Adım Çözüm

1

Express both sides of the equation using a common base of 2

2^{3x - 1} = 2^5

The number 32 can be rewritten in index form as

2^5

.

2

Equate the indices

3x - 1 = 5

If

a^m = a^n

for a non-zero base

a \neq 1

, then

m = n

.

3

Solve the linear equation for

x

x = 2

Adding 1 to both sides yields

3x = 6

, and dividing by 3 gives

x = 2

.

Anahtar Kavram

Solving exponential equations by expressing numbers with equal bases

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