Tüm alıştırma soruları

Soru 121Soru →

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?

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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 122Soru →

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

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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 123Soru →

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 ( $\text{N}$ )?

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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 124Soru →

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

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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 125Soru →

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

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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 126Soru →

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?

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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 127Soru →

The mean of eight consecutive odd numbers is $24$ . What is the median of the first four numbers?

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Cevap: 20

Cevap

The median of the first four numbers is

20

.

Letting the eight consecutive odd numbers be

x, x+2, \dots, x+14

, their sum is

8x + 56

. Dividing by

8

gives a mean of

x + 7 = 24

, so

x = 17

. The first four numbers are

17, 19, 21,

and

23

. The median of these four values is the average of the middle two values (

19

and

21

), which equals

20

.

Adım Adım Çözüm

1

Represent the eight consecutive odd numbers algebraically

Let the numbers be

x, x+2, x+4, x+6, x+8, x+10, x+12, x+14

.

Consecutive odd numbers increase by steps of

2

.

2

Set up and solve the mean equation

\frac{(x) + (x+2) + (x+4) + (x+6) + (x+8) + (x+10) + (x+12) + (x+14)}{8} = 24 \implies x + 7 = 24 \implies x = 17

.

The mean of ungrouped data is the sum of all data values divided by the total number of values.

3

Identify the first four numbers in the set

The first four numbers are

17, 19, 21, 23

.

Substitute

x = 17

into

x, x+2, x+4,

and

x+6

.

4

Find the median of the first four numbers

\text{Median} = \frac{19 + 21}{2} = 20

.

For an even count of ordered values (4 items), the median is the arithmetic mean of the two middle terms.

Anahtar Kavram

Measures of Central Tendency for Ungrouped Data

Soru 128Soru →

Find the maximum integer value of $m$ for which the quadratic inequality $x^2 - mx + 9 > 0$ holds for all real values of $x$ .

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Cevap: 5

Cevap

The maximum integer value of

m

is

5

.

For the quadratic expression

x^2 - mx + 9

to remain strictly positive for all real values of

x

, the quadratic curve must lie completely above the x-axis. Because the coefficient of

x^2

is positive (

1 > 0

), this requires the discriminant to be strictly negative (

D < 0

). Evaluating

b^2 - 4ac < 0

gives

m^2 - 36 < 0

, which simplifies to

-6 < m < 6

. The largest integer strictly less than

6

is

5

.

Adım Adım Çözüm

1

Determine the condition for the quadratic expression to be positive for all real values of

x

.

Since the leading coefficient is

1 > 0

, the condition is that the discriminant

D = b^2 - 4ac < 0

.

A parabola opening upward lies entirely above the horizontal axis when it has no real roots.

2

Calculate the discriminant using the coefficients of the quadratic expression.

D = (-m)^2 - 4(1)(9) = m^2 - 36 < 0

.

Here

a = 1

,

b = -m

, and

c = 9

.

3

Solve the quadratic inequality for

m

.

m^2 - 36 < 0 \implies -6 < m < 6

.

The roots of

m^2 - 36 = 0

are

m = -6

and

m = 6

, and the expression is negative strictly between these boundary values.

4

Determine the maximum integer value within the open interval

(-6, 6)

.

The maximum integer value is

5

.

The boundary value

6

is excluded by the strict inequality

m < 6

.

Anahtar Kavram

Quadratic Inequalities and Discriminant Conditions for Positive Definiteness

Soru 129Soru →

The sum of the first three terms of an arithmetic progression (AP) with a positive common difference $d$ is $21$ . If $2$ is added to the first term, $3$ is added to the second term, and $9$ is added to the third term, the resulting three numbers form consecutive terms of a geometric progression (GP). What is the sum of the first $10$ terms of this arithmetic progression?

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Cevap: 210

Cevap

The sum of the first 10 terms of the arithmetic progression is 210.

Representing the AP terms as

7-d, 7, 7+d

and adding the specified values produces GP terms

9-d, 10, 16+d

. Solving

10^2 = (9-d)(16+d)

yields

d=4

. Consequently, the first term of the AP is

3

. Using

S_{10} = \frac{10}{2}[2(3) + 9(4)]

gives the final answer 210.

Adım Adım Çözüm

1

Express AP terms symmetrically and solve for the middle term.

The middle term is

a = 7

, making the terms

7-d

,

7

, and

7+d

.

Choosing terms

a-d, a, a+d

allows the sum equation

3a = 21

to directly isolate the middle term.

2

Set up the geometric progression relation to determine common difference

d

.

The GP terms are

9-d

,

10

, and

16+d

. Solving

10^2 = (9-d)(16+d)

gives

d^2 + 7d - 44 = 0

, yielding

d = 4

.

In any geometric progression, the square of the middle term equals the product of the first and third terms.

3

Determine the first term

a_1

and calculate

S_{10}

.

The first term is

a_1 = 7 - 4 = 3

, and the sum

S_{10} = \frac{10}{2}[2(3) + (10-1)(4)] = 210

.

Applying the AP sum formula

S_n = \frac{n}{2}[2a_1 + (n-1)d]

with

n=10

,

a_1=3

, and

d=4

.

Anahtar Kavram

Integrating AP and GP structural relationships to solve for sequence parameters and evaluate finite sums

Soru 130Soru →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b + 5$ . What is the identity element of this operation?

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Cevap: -5

Cevap

The identity element of the binary operation is

-5

.

By the definition of an identity element

e

, the relation

a \ast e = a

must hold for all real numbers

a

. Applying the rule

a \ast b = a + b + 5

gives

a + e + 5 = a

. Subtracting

a

from both sides leads to

e + 5 = 0

, which gives

e = -5

.

Adım Adım Çözüm

1

Set up the identity element equation

a \ast e = a \implies a + e + 5 = a

By definition of an identity element, operating any element

a

with

e

yields

a

.

2

Solve the equation for

e

e = -5

Subtracting

a

from both sides gives

e + 5 = 0

, which yields

e = -5

.

Anahtar Kavram

Identity Element of a Binary Operation

Soru 131Soru →

A straight line passing through the points $(1, 3)$ and $(5, k)$ has a gradient of $2$ . What is the value of $k$ ?

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Cevap: 11

Cevap

The value of

k

is

11

.

The gradient

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Substituting

(1, 3)

,

(5, k)

, and

m = 2

yields

2 = \frac{k - 3}{5 - 1} = \frac{k - 3}{4}

. Multiplying by

4

gives

8 = k - 3

, so

k = 11

.

Adım Adım Çözüm

1

Recall the slope/gradient formula for a straight line.

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

The gradient of a straight line passing through two points is the ratio of vertical change to horizontal change.

2

Substitute the known point coordinates and gradient into the formula.

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

We are given

(x_1, y_1) = (1, 3)

,

(x_2, y_2) = (5, k)

, and gradient

m = 2

.

3

Simplify and solve for

k

.

2 = \frac{k - 3}{4} \implies 8 = k - 3 \implies k = 11

Multiplying both sides by

4

clears the fraction, and adding

3

isolates

k

.

Anahtar Kavram

Gradient of a straight line passing through two points

Tahmini Süre:45s

Soru 132Soru →

A committee of $3$ members is to be selected from a group of $7$ people. In how many different ways can this committee be formed?

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Cevap: 35

Cevap

35

Selecting a committee of

3

members from

7

people requires calculating the number of combinations, given by

^7C_3 = \frac{7!}{3!4!} = 35

.

Adım Adım Çözüm

1

Identify total elements and group size

n = 7

and

r = 3

Since the arrangement or order of members in the committee does not matter, this is a selection problem (combinations).

2

Apply the combinations formula

^nC_r = \frac{n!}{r!(n-r)!}

^7C_3 = \frac{7!}{3!4!}

This formula counts the distinct subsets of size

r

that can be chosen from

n

items.

3

Evaluate the factorial expression

^7C_3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35

Expanding

7!

as

7 \times 6 \times 5 \times 4!

allows cancelling

4!

, leaving

\frac{210}{6} = 35

.

Anahtar Kavram

Combinations formula

^nC_r = \frac{n!}{r!(n-r)!}

Tahmini Süre:45s

Soru 133Soru →

In a class of $40$ students, $25$ study Mathematics and $18$ study Physics. If $3$ students study neither of the two subjects, how many students study both Mathematics and Physics?

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Cevap: 6

Cevap

6 students study both Mathematics and Physics.

Subtracting the

3

students who study neither subject from the class total of

40

leaves

37

students studying at least one subject. Adding those studying Mathematics (

25

) and Physics (

18

) totals

43

. The excess of

43

over

37

represents the

6

students who study both subjects.

Adım Adım Çözüm

1

Subtract the number of students studying neither subject from the total number of students in the class.

n(M \cup P) = 40 - 3 = 37

This gives the number of students who belong to at least one of the two sets.

2

Set up the inclusion-exclusion formula

n(M \cup P) = n(M) + n(P) - n(M \cap P)

.

37 = 25 + 18 - n(M \cap P)

Summing

n(M)

and

n(P)

double-counts the students who study both subjects.

3

Solve for the intersection

n(M \cap P)

.

n(M \cap P) = 43 - 37 = 6

Subtracting the union from the sum of the individual sets isolates the intersection value.

Anahtar Kavram

Principle of Inclusion-Exclusion for Two Sets

Soru 134Soru →

In $\Delta ABC$ , the side lengths are given as $a = 3\text{ cm}$ , $b = 5\text{ cm}$ , and $c = 7\text{ cm}$ . What is the measure of angle $C$ in degrees?

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Cevap: 120

Cevap

The measure of angle

C

is

120^\circ

.

Using the Cosine Rule

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

, substituting

a=3

,

b=5

, and

c=7

yields

\cos C = \frac{9 + 25 - 49}{30} = -0.5

. Taking the inverse cosine of

-0.5

gives an angle of

120^\circ

.

Adım Adım Çözüm

1

Select the appropriate formula for calculating an interior angle given three sides (SSS).

Use the Cosine Rule rearranged for

\cos C

:

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

.

When all three side lengths of a non-right triangle are known, the Cosine Rule is required to find any of its angles.

2

Substitute side lengths

a=3

,

b=5

, and

c=7

into the Cosine Rule equation.

\cos C = \frac{3^2 + 5^2 - 7^2}{2(3)(5)} = \frac{9 + 25 - 49}{30} = -\frac{15}{30} = -0.5.

Simplifying the numerator and denominator determines the exact trigonometric ratio for angle

C

.

3

Find the inverse cosine of

-0.5

in degrees.

C = \arccos(-0.5) = 120^\circ.

A negative cosine value indicates an obtuse angle in the second quadrant (

90^\circ < C < 180^\circ

).

Anahtar Kavram

Using the Cosine Rule to determine obtuse angles in SSS triangles

Soru 135Soru →

The set of numbers $k - 2$ , $k$ , $k + 1$ , and $k + 5$ is given, where $k$ is any real constant. What is the variance of this set of numbers?

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Cevap: 6.5

Cevap

The variance of the given set of numbers is 6.5.

The mean of the set is

\bar{x} = k + 1

. Subtracting the mean from each data point gives deviations of

-3

,

-1

,

0

, and

4

. The squares of these deviations are

9

,

1

,

0

, and

16

, which sum to

26

. Dividing this sum by

4

gives a variance of

6.5

. A key statistical property illustrated here is that adding or subtracting a constant

k

from every value in a dataset shifts the mean by

k

but leaves measures of dispersion (such as variance and standard deviation) unchanged.

Adım Adım Çözüm

1

Find the mean (\bar{x}) of the given set {k - 2, k, k + 1, k + 5}.

\bar{x} = \frac{(k - 2) + k + (k + 1) + (k + 5)}{4} = \frac{4k + 4}{4} = k + 1

The mean is calculated by summing all values and dividing by the total count of numbers.

2

Determine the deviation of each value from the mean, (x_i - \bar{x}).

(k - 2) - (k + 1) = -3, k - (k + 1) = -1, (k + 1) - (k + 1) = 0, (k + 5) - (k + 1) = 4

Deviations measure how far each data value lies from the mean.

3

Square each individual deviation and sum the results.

(-3)^2 + (-1)^2 + 0^2 + 4^2 = 9 + 1 + 0 + 16 = 26

Squaring converts all deviations into non-negative values.

4

Divide the sum of squared deviations by the total number of observations (N = 4) to find the variance.

\text{Variance} = \frac{26}{4} = 6.5

Variance is defined as the arithmetic mean of the squared deviations from the mean.

Anahtar Kavram

Variance and Invariance under Constant Translation

Soru 136Soru →

If $x$ , $y$ , and $z$ are non-zero real numbers satisfying the exponential equation $2^x = 5^y = 100^z$ , what is the numerical value of the expression $z \left( \frac{2}{x} + \frac{2}{y} \right)$ ?

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Cevap: 1

Cevap

The numerical value of

z \left( \frac{2}{x} + \frac{2}{y} \right)

is

1

.

By setting

2^x = 5^y = 100^z = k

, we can write

2 = k^{1/x}

,

5 = k^{1/y}

, and

100 = k^{1/z}

. Factoring

100 = 2^2 \times 5^2

gives

k^{1/z} = (k^{1/x})^2 \times (k^{1/y})^2 = k^{2/x + 2/y}

. Equating exponents gives

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

, which upon multiplying by

z

yields

1

.

Adım Adım Çözüm

1

Equate the given exponential expressions to a common constant

k

.

2^x = 5^y = 100^z = k

Introducing a common variable allows isolating each base exponent combination.

2

Express bases

2

,

5

, and

100

in terms of

k

using fractional indices.

2 = k^{\frac{1}{x}}

,

5 = k^{\frac{1}{y}}

,

100 = k^{\frac{1}{z}}

Applying the power law

(a^m)^{\frac{1}{m}} = a

isolates each base.

3

Express

100

using prime factorization of the other bases.

100 = 2^2 \times 5^2

Establishing a numerical relationship between

100

,

2

, and

5

links the exponential variables.

4

Substitute the

k

-expressions into

100 = 2^2 \times 5^2

and apply index multiplication laws.

k^{\frac{1}{z}} = \left(k^{\frac{1}{x}}\right)^2 \times \left(k^{\frac{1}{y}}\right)^2 = k^{\frac{2}{x}} \times k^{\frac{2}{y}} = k^{\frac{2}{x} + \frac{2}{y}}

Multiplying powers with the same base requires adding the exponents:

a^m \cdot a^n = a^{m+n}

.

5

Equate exponents of identical bases.

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

If

k^a = k^b

for

k > 1

, then

a = b

.

6

Multiply both sides of the equation by

z

.

z \left( \frac{2}{x} + \frac{2}{y} \right) = 1

Rearranging the equation yields the exact numerical value of the requested expression.

Anahtar Kavram

Equating Exponents of Common Bases and Fractional Indices

Soru 137Soru →

A binary operation $\ast$ on the set of real numbers is defined by $a \ast b = a + b - 2ab$ . If $(x \ast 3) \ast 2 = 38$ , find the value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

x

is 3.

Applying the binary operation definition sequentially yields

x \ast 3 = 3 - 5x

for the inner expression, and

(3 - 5x) \ast 2 = 15x - 7

for the composite expression. Equating

15x - 7 = 38

leads to

15x = 45

, giving

x = 3

.

Adım Adım Çözüm

1

Evaluate the inner binary operation expression

x \ast 3

x \ast 3 = 3 - 5x

Apply the definition

a \ast b = a + b - 2ab

with

a = x

and

b = 3

.

2

Evaluate the outer binary operation

(3 - 5x) \ast 2

(3 - 5x) \ast 2 = 15x - 7

Substitute the result from step 1 into the outer operation definition with

a = 3 - 5x

and

b = 2

.

3

Set the resulting expression equal to 38 and solve the linear equation

x = 3

Solve

15x - 7 = 38

by adding 7 to both sides to get

15x = 45

, then dividing by 15.

Anahtar Kavram

Nested composition of defined binary operations

Soru 138Soru →

In a survey of $150$ subscribers of a digital media platform, $75$ prefer High-Definition Audio ( $H$ ), $70$ prefer Offline Downloads ( $D$ ), and $65$ prefer Ad-free Listening ( $A$ ). It is observed that $35$ subscribers prefer both $H$ and $D$ , $30$ prefer both $D$ and $A$ , and $25$ prefer both $H$ and $A$ . If $15$ subscribers prefer none of these three features, how many subscribers prefer exactly two of these features?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The number of subscribers who prefer exactly two of the features is 45.

To find the number of subscribers who prefer exactly two features, we first calculate the cardinality of the union of all three sets as

150 - 15 = 135

. Applying the 3-set inclusion-exclusion formula gives the number of subscribers preferring all three features as

15

. Subtracting

15

from each pairwise intersection gives the exclusive regions:

20

for

H

and

D

only,

15

for

D

and

A

only, and

10

for

H

and

A

only. Summing these three exclusive regions gives

20 + 15 + 10 = 45

.

Adım Adım Çözüm

1

Determine the total number of subscribers who prefer at least one feature.

|H \cup D \cup A| = 135

Subtracting the number of subscribers who prefer none of the features (

15

) from the universal set size (

150

) gives

|H \cup D \cup A| = 150 - 15 = 135

.

2

Apply the Principle of Inclusion-Exclusion for three sets to find the number of subscribers who prefer all three features.

|H \cap D \cap A| = 15

Substitute the known cardinalities into

|H \cup D \cup A| = |H| + |D| + |A| - |H \cap D| - |D \cap A| - |H \cap A| + |H \cap D \cap A|

to get

135 = 75 + 70 + 65 - (35 + 30 + 25) + |H \cap D \cap A|

, which simplifies to

135 = 120 + |H \cap D \cap A|

, giving

|H \cap D \cap A| = 15

.

3

Calculate the number of subscribers preferring exactly two features by subtracting the triple intersection from each pairwise intersection.

45 subscribers

Subscribers preferring only

H

and

D = 35 - 15 = 20

, only

D

and

A = 30 - 15 = 15

, and only

H

and

A = 25 - 15 = 10

. Summing these exclusive regions yields

20 + 15 + 10 = 45

.

Anahtar Kavram

Cardinality of set operations and 3-set inclusion-exclusion principle

Soru 139Soru →

Find the positive value of $x$ that satisfies the simultaneous equations $y = x + 2$ and $y = x^2 - 4$ .

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The positive value of

x

is 3.

Equating the linear equation

y = x + 2

and the quadratic equation

y = x^2 - 4

yields

x^2 - x - 6 = 0

. Factorizing this quadratic equation gives

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

, which yields roots

x = 3

and

x = -2

. Selecting the positive value gives 3.

Adım Adım Çözüm

1

Equate the linear and quadratic equations

x + 2 = x^2 - 4

Since both expressions are equal to

y

, set them equal to each other to solve for

x

.

2

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - x - 6 = 0

Subtract

x

and

2

from both sides of the equation.

3

Factorize the quadratic expression

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

Find two factors of

-6

that add up to

-1

, which are

-3

and

2

.

4

Determine the roots and select the positive value

x = 3

Setting each factor to zero gives

x = 3

or

x = -2

. Selecting the positive root yields

3

.

Anahtar Kavram

Solving simultaneous linear and quadratic equations by substitution

Soru 140Soru →

Using differentiation from first principles, what is the numerical value of the derivative of the function $f(x) = 3x^2 - 4x + 1$ at $x = 2$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The numerical value of the derivative of

f(x) = 3x^2 - 4x + 1

at

x = 2

is 8.

Using the first-principles limit definition, the increment

f(2+h) - f(2)

simplifies to

8h + 3h^2

. Dividing by

h

yields

8 + 3h

, which evaluates to 8 as

h \to 0

.

Adım Adım Çözüm

1

Calculate

f(2)

f(2) = 5

Substitute

x = 2

into

f(x) = 3x^2 - 4x + 1

.

2

Expand

f(2+h)

f(2+h) = 5 + 8h + 3h^2

Substitute

x = 2+h

into

f(x)

and expand algebraically.

3

Simplify the difference quotient

\frac{f(2+h) - f(2)}{h}

\frac{8h + 3h^2}{h} = 8 + 3h

Subtract

f(2)

from

f(2+h)

and divide every term by

h

.

4

Evaluate the limit as

h \to 0

f'(2) = 8

As

h

approaches 0, the term

3h

vanishes, leaving 8.

Anahtar Kavram

Differentiation from First Principles

Tahmini Süre:1m 30s

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