Algebra Alıştırma Soruları — JAMB UTME — Sayfa 8

Soru 141Soru →

Find the product matrix $AB$ given that $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix}$ .

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

\begin{pmatrix} 4 & 10 \\ 0 & 6 \end{pmatrix}

\begin{pmatrix} 4 & 23 \\ 1 & 8 \end{pmatrix}

\begin{pmatrix} 5 & 7 \\ 1 & 5 \end{pmatrix}

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Cevap: $\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}$

Cevap

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

The matrix product

AB

is found by performing row-by-column multiplication. Computing row 1 gives

(1)(4)+(2)(1) = 6

and

(1)(5)+(2)(2) = 9

. Computing row 2 gives

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

and

(0)(5)+(3)(2) = 6

. Thus, the resulting matrix is

\begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

.

Adım Adım Çözüm

1

Set up row-by-column matrix multiplication formula for

2 \times 2

matrices.

The entry at row

i

, column

j

of product

AB

is the dot product of row

i

of

A

and column

j

of

B

.

Matrix multiplication requires combining rows of the left matrix with columns of the right matrix.

2

Calculate row 1 entries of the product matrix.

First element:

(1)(4) + (2)(1) = 4 + 2 = 6

. Second element:

(1)(5) + (2)(2) = 5 + 4 = 9

.

Multiply elements of row 1 of

A

by corresponding elements of columns 1 and 2 of

B

and sum them.

3

Calculate row 2 entries of the product matrix.

Third element:

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

. Fourth element:

(0)(5) + (3)(2) = 0 + 6 = 6

.

Multiply elements of row 2 of

A

by corresponding elements of columns 1 and 2 of

B

and sum them.

4

Assemble the product matrix.

AB = \begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix}

.

Combine all computed row and column entries into a single

2 \times 2

matrix.

Anahtar Kavram

Matrix Multiplication (

2 \times 2

Matrix Product)

Soru 142Soru →

If $\log_2 (x + 3) + \log_2 4 = 5$ , what is the value of $x$ ?

25

5

18

3

Cevabı ve açıklamayı göster

Cevap: $5$

Cevap

The value of

x

is

5

.

Using the logarithmic addition property

\log_b (M) + \log_b (N) = \log_b (M \cdot N)

, the equation

\log_2(x+3) + \log_2 4 = 5

becomes

\log_2[4(x+3)] = 5

. Converting to index form gives

4(x+3) = 2^5 = 32

. Dividing by

4

yields

x + 3 = 8

, so

x = 5

.

Adım Adım Çözüm

1

Apply the product rule of logarithms:

\log_b A + \log_b B = \log_b (A \cdot B)

\log_2 [4(x + 3)] = 5

Logarithms with the same base that are added together can be combined by multiplying their arguments.

2

Convert the logarithmic equation into its equivalent exponential form

4(x + 3) = 2^5 = 32

If

\log_b Y = c

, then

Y = b^c

.

3

Divide both sides by 4 and solve for

x

x + 3 = 8 \implies x = 5

Linear algebraic simplification.

Anahtar Kavram

Product Law of Logarithms and Log-to-Exponential Conversion

Soru 143Soru →

What is the simplified form of the expression $\frac{12}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$ ?

3\sqrt{2} + 2\sqrt{3} - \sqrt{30}

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

2\sqrt{2} + 3\sqrt{3} + \sqrt{30}

3\sqrt{2} + 2\sqrt{3} + \sqrt{10}

Cevabı ve açıklamayı göster

Cevap: $3\sqrt{2} + 2\sqrt{3} + \sqrt{30}$

Cevap

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

By grouping the denominator as

(\sqrt{2}+\sqrt{3}) - \sqrt{5}

and multiplying by its conjugate

(\sqrt{2}+\sqrt{3}) + \sqrt{5}

, the denominator reduces to

2\sqrt{6}

. Multiplying the resulting fraction by

\sqrt{6}/\sqrt{6}

yields

\sqrt{12} + \sqrt{18} + \sqrt{30}

, which simplifies directly to

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

.

Adım Adım Çözüm

1

Group the terms in the denominator as

((\sqrt{2} + \sqrt{3}) - \sqrt{5})

and multiply the numerator and denominator by its conjugate

((\sqrt{2} + \sqrt{3}) + \sqrt{5})

.

The fraction becomes

\frac{12((\sqrt{2} + \sqrt{3}) + \sqrt{5})}{((\sqrt{2} + \sqrt{3}) - \sqrt{5})((\sqrt{2} + \sqrt{3}) + \sqrt{5})}

.

Applying the difference of two squares to eliminate the outer radical.

2

Expand the denominator using

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

.

Denominator =

(\sqrt{2} + \sqrt{3})^2 - (\sqrt{5})^2 = (2 + 2\sqrt{6} + 3) - 5 = 2\sqrt{6}

.

Simplifying the algebraic square of a binomial surd.

3

Divide the numerator by the constant factor of the denominator.

\frac{12(\sqrt{2} + \sqrt{3} + \sqrt{5})}{2\sqrt{6}} = \frac{6(\sqrt{2} + \sqrt{3} + \sqrt{5})}{\sqrt{6}}.

Simplifying numerical coefficients before further rationalization.

4

Rationalize the remaining monomial radical in the denominator by multiplying numerator and denominator by

\sqrt{6}

.

\frac{6(\sqrt{12} + \sqrt{18} + \sqrt{30})}{6} = \sqrt{12} + \sqrt{18} + \sqrt{30}.

Eliminating

\sqrt{6}

from the denominator.

5

Simplify each radical to its simplest surd form.

\sqrt{12} = 2\sqrt{3}, \quad \sqrt{18} = 3\sqrt{2}, \quad \text{so } \sqrt{12} + \sqrt{18} + \sqrt{30} = 3\sqrt{2} + 2\sqrt{3} + \sqrt{30}.

Factoring out perfect square components from radical terms.

Anahtar Kavram

Rationalisation of trinomial surd denominators using repeated conjugate multiplication

Tahmini Süre:2m 0s

Soru 144Soru →

What is the sum of all real values of $x$ that satisfy the logarithmic equation $\log_3(9^x + 81) - x = \log_3 30$ ?

4

10

30

6

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Cevap: 4

Cevap

The sum of all real values of

x

satisfying the equation is 4.

Using logarithmic properties,

\log_3(9^x + 81) - x = \log_3 30

simplifies to

\frac{9^x + 81}{3^x} = 30

. Substituting

u = 3^x

produces the quadratic equation

u^2 - 30u + 81 = 0

, which yields

u = 3

and

u = 27

. Solving

3^x = 3

and

3^x = 27

gives

x = 1

and

x = 3

. Their sum is

1 + 3 = 4

.

Adım Adım Çözüm

1

Express

x

as

\log_3(3^x)

and apply the logarithm quotient rule.

\log_3\left(\frac{9^x + 81}{3^x}\right) = \log_3 30

Using

\log_b A - \log_b B = \log_b\left(\frac{A}{B}\right)

combines terms into a single logarithm.

2

Equate the logarithmic arguments and split the fraction.

\frac{9^x + 81}{3^x} = 30 \implies 3^x + \frac{81}{3^x} = 30

Logarithmic equality with identical bases implies equality of their arguments.

3

Substitute

u = 3^x

to form and solve a quadratic equation.

u + \frac{81}{u} = 30 \implies u^2 - 30u + 81 = 0 \implies (u - 3)(u - 27) = 0 \implies u = 3 \text{ or } u = 27

Since

9^x = (3^x)^2 = u^2

, substituting

u = 3^x

converts the equation into a standard quadratic equation.

4

Solve for

x

from

3^x = u

and calculate the sum of the solutions.

3^x = 3 \implies x = 1; \quad 3^x = 27 \implies x = 3. \quad \text{Sum} = 1 + 3 = 4

Matching exponential bases gives

x = 1

and

x = 3

.

Anahtar Kavram

Logarithmic quotient identity and quadratic substitution for exponential equations

Tahmini Süre:2m 0s

Soru 145Soru →

An infinite geometric progression of positive terms has a sum to infinity of $16$ , and the sum of its first two terms is $12$ . An arithmetic progression has its first term equal to the first term of this geometric progression, and its $5^{\text{th}}$ term equal to the sum to infinity of the geometric progression. Calculate the $10^{\text{th}}$ term of the arithmetic progression.

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Cevap: 26

Cevap

The 10th term of the arithmetic progression is 26.

By solving the geometric progression system, we find the common ratio

r = \frac{1}{2}

and first term

a = 8

. Using

a = 8

as the first term of the arithmetic progression and setting its

5^{\text{th}}

term

A_5 = 16

, we determine the common difference

d = 2

. Calculating

A_{10} = 8 + 9(2)

yields

26

.

Adım Adım Çözüm

1

Formulate equations for the geometric progression using the sum to infinity and sum of the first two terms

a = 16(1 - r)

and

a(1 + r) = 12

The standard formula for the sum to infinity of a GP is

S_\infty = \frac{a}{1-r}

and the sum of the first two terms is

S_2 = a + ar = a(1+r)

.

2

Solve for the common ratio

r

and first term

a

of the geometric progression

r = 0.5

and

a = 8

Substituting

a = 16(1-r)

yields

16(1-r^2) = 12 \implies r^2 = \frac{1}{4} \implies r = \frac{1}{2}

. Then

a = 16(1 - 0.5) = 8

.

3

Determine the common difference

d

of the arithmetic progression

d = 2

The first term of the AP is

A_1 = a = 8

and the 5th term is

A_5 = S_\infty = 16

. Using

A_5 = A_1 + 4d \implies 8 + 4d = 16 \implies d = 2

.

4

Calculate the 10th term of the arithmetic progression

A_{10} = 26

Using the AP

n^{\text{th}}

term formula

A_n = A_1 + (n-1)d

:

A_{10} = 8 + 9(2) = 26

.

Anahtar Kavram

Combining geometric progression parameters (sum to infinity and sum of terms) with arithmetic progression term formulas

Soru 146Soru →

If $\log_{10} 2 = p$ and $\log_{10} 3 = q$ , which of the following expressions represents $\log_{10} 18$ in terms of $p$ and $q$ ?

p + 2q

2pq

2p + q

p + q^2

Cevabı ve açıklamayı göster

Cevap: $p + 2q$

Cevap

The expression for

\log_{10} 18

in terms of

p

and

q

is

p + 2q

.

Expanding

\log_{10} 18

as

\log_{10}(2 \times 3^2)

gives

\log_{10} 2 + 2\log_{10} 3

, which simplifies directly to

p + 2q

.

Adım Adım Çözüm

1

Factorize 18 into prime factors.

18 = 2 \times 3^2

Decomposing 18 into prime factors allows the application of the given logarithm values for 2 and 3.

2

Apply the product rule of logarithms:

\log_{10}(a \times b) = \log_{10} a + \log_{10} b

.

\log_{10} 18 = \log_{10}(2 \times 3^2) = \log_{10} 2 + \log_{10}(3^2)

The logarithm of a product equals the sum of the logarithms of individual factors.

3

Apply the power rule of logarithms:

\log_{10}(b^n) = n \log_{10} b

.

\log_{10}(3^2) = 2 \log_{10} 3

The logarithm of a power expression allows bringing the exponent to the front as a multiplier.

4

Substitute the defined values

p = \log_{10} 2

and

q = \log_{10} 3

.

\log_{10} 18 = p + 2q

Replaces logarithmic terms with their algebraic representations.

Anahtar Kavram

Logarithm Expansion Laws (Product and Power Rules)

Soru 147Soru →

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 12\}$ . Consider the subsets $P = \{x : x \text{ is a factor of } 12\}$ and $Q = \{x : x \text{ is an even number}, 1 \le x \le 12\}$ . Which of the following sets represents $(P \cup Q)'$ ?

\{5, 7, 9, 11\}

\{2, 4, 6, 12\}

\{5, 7, 11\}

\{1, 3, 5, 7, 9, 11\}

Cevabı ve açıklamayı göster

Cevap: $\{5, 7, 9, 11\}$

Cevap

\{5, 7, 9, 11\}

The set of factors of 12 within the given range is {1, 2, 3, 4, 6, 12} and the set of even numbers up to 12 is {2, 4, 6, 8, 10, 12}. Their union contains all numbers that are either even or factors of 12, which is {1, 2, 3, 4, 6, 8, 10, 12}. Subtracting this union from the universal set {1, 2, ..., 12} yields {5, 7, 9, 11}.

Adım Adım Çözüm

1

List all elements of the universal set and the subsets

P

and

Q

.

\mathcal{U} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

,

P = \{1, 2, 3, 4, 6, 12\}

, and

Q = \{2, 4, 6, 8, 10, 12\}

.

Identify the element listings explicitly from the given definitions.

2

Find the union of sets

P

and

Q

, denoted by

P \cup Q

.

P \cup Q = \{1, 2, 3, 4, 6, 8, 10, 12\}

.

Combine all unique elements belonging to either

P

,

Q

, or both.

3

Find the complement

(P \cup Q)'

relative to the universal set

\mathcal{U}

.

(P \cup Q)' = \mathcal{U} \setminus (P \cup Q) = \{5, 7, 9, 11\}

.

Select all elements in

\mathcal{U}

that do not appear in

P \cup Q

.

Anahtar Kavram

Complement of a Set Union

Tahmini Süre:50s

Soru 148Soru →

The $1^{\text{st}}$ , $2^{\text{nd}}$ , and $5^{\text{th}}$ terms of an arithmetic progression (A.P.) with a non-zero common difference form three consecutive terms of a geometric progression (G.P.). If the sum of the first $4$ terms of the A.P. is $40$ , what is the $5^{\text{th}}$ term of the G.P.?

\frac{405}{2}

\frac{1215}{2}

162

\frac{45}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{405}{2}$

Cevap

\frac{405}{2}

By setting up the geometric mean property

(a+d)^2 = a(a+4d)

, we find

d = 2a

, which establishes that the G.P. has a common ratio

r = 3

. Substituting

d = 2a

into the A.P. sum formula

S_4 = 2[2a + 3d] = 16a = 40

gives

a = \frac{5}{2}

. Finally, evaluating the

5^{\text{th}}

term of the G.P. using

a r^4 = \frac{5}{2} \times 3^4

yields

\frac{405}{2}

.

Adım Adım Çözüm

1

Express the given A.P. terms in terms of first term

a

and common difference

d

, and set up the G.P. condition.

The terms are

T_1 = a

,

T_2 = a + d

, and

T_5 = a + 4d

. Since they form a G.P.,

(a + d)^2 = a(a + 4d)

.

Three terms

x, y, z

form a G.P. if

y^2 = xz

.

2

Solve for the relationship between

d

and

a

.

a^2 + 2ad + d^2 = a^2 + 4ad \implies d^2 = 2ad \implies d = 2a

(since

d \neq 0

).

Expanding and simplifying the equation yields the ratio of

d

to

a

.

3

Determine the common ratio

r

of the G.P.

r = \frac{T_2}{T_1} = \frac{a + d}{a} = \frac{a + 2a}{a} = 3

.

The common ratio is the quotient of consecutive terms of the G.P.

4

Use the sum of the first

4

terms of the A.P. to find

a

.

S_4 = \frac{4}{2}[2a + (4-1)d] = 2[2a + 3(2a)] = 16a = 40 \implies a = \frac{40}{16} = \frac{5}{2}

.

Applying

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

with

S_4 = 40

allows solving for

a

.

5

Calculate the

5^{\text{th}}

term of the G.P.

G_5 = g_1 \cdot r^{5-1} = a \cdot r^4 = \frac{5}{2} \cdot 3^4 = \frac{5}{2} \cdot 81 = \frac{405}{2}

.

The

n^{\text{th}}

term of a G.P. is

g_n = g_1 r^{n-1}

.

Anahtar Kavram

Connecting Arithmetic and Geometric Progressions using term definitions and sum formulas.

Soru 149Soru →

If $\frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} = k\sqrt{6}$ , find the value of the integer $k$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of the integer

k

is 4.

Combining the fractions over the common denominator

(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = 3-2 = 1

yields a numerator of

(3+2+2\sqrt{6}) - (3+2-2\sqrt{6}) = 4\sqrt{6}

. Thus,

k\sqrt{6} = 4\sqrt{6}

, which gives

k = 4

.

Adım Adım Çözüm

1

Combine the fractions using their common denominator

\frac{(\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}

Subtracting algebraic fractions requires finding the least common denominator, which is the product of the conjugate pair.

2

Expand the terms in the numerator

(\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \text{ and } (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6}

Use the perfect square expansion formula

(a \pm b)^2 = a^2 \pm 2ab + b^2

.

3

Subtract the expanded terms in the numerator and simplify the denominator

\text{Numerator: } (5 + 2\sqrt{6}) - (5 - 2\sqrt{6}) = 4\sqrt{6}, \text{ Denominator: } (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1

Apply the difference of two squares identity

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

to the denominator and carefully distribute the negative sign across terms in the numerator.

4

Equate the simplified expression to

k\sqrt{6}

and solve for

k

k = 4

Comparing

\frac{4\sqrt{6}}{1} = 4\sqrt{6}

with

k\sqrt{6}

yields

k = 4

.

Anahtar Kavram

Rationalisation of surd denominators using conjugate pairs and difference of squares

Soru 150Soru →

The rate of heat transfer $Q$ across a building wall varies directly as the surface area $A$ of the wall and the temperature difference $\Delta T$ between the interior and exterior, and inversely as the wall thickness $d$ . When the surface area is $4\text{ m}^2$ , the temperature difference is $15^\circ\text{C}$ , and the thickness is $0.05\text{ m}$ , the heat transfer rate is $1200\text{ W}$ . What is the heat transfer rate in watts when the surface area is $6\text{ m}^2$ , the temperature difference is $20^\circ\text{C}$ , and the thickness is $0.08\text{ m}$ ?

Cevabı ve açıklamayı göster

Cevap: 1500

Cevap

The heat transfer rate is

1500\text{ W}

.

Establishing the variation constant

k = 1

using the initial given values and substituting the new parameters yields

Q = \frac{1 \times 6 \times 20}{0.08} = 1500\text{ W}

.

Adım Adım Çözüm

1

Formulate the variation equation

Q = \frac{k A \Delta T}{d}

Direct variation means multiplying by

A

and

\Delta T

, while inverse variation means dividing by

d

.

2

Calculate the constant of variation

k

k = 1

Substituting

Q=1200

,

A=4

,

\Delta T=15

, and

d=0.05

gives

1200 = \frac{60k}{0.05} = 1200k

, so

k = 1

.

3

Compute the target heat transfer rate

Q

1500\text{ W}

Substituting

k=1

,

A=6

,

\Delta T=20

, and

d=0.08

gives

Q = \frac{1 \times 6 \times 20}{0.08} = \frac{120}{0.08} = 1500\text{ W}

.

Anahtar Kavram

Joint and Inverse Variation

Soru 151Soru →

An arithmetic progression (A.P.) has a first term of $5$ and a common difference of $-3$ . What is the $7^{\text{th}}$ term of the progression?

-13

-16

-19

23

Cevabı ve açıklamayı göster

Cevap: $-13$

Cevap

-13

Using the

n^{\text{th}}

term formula

T_n = a + (n - 1)d

with

a = 5

,

d = -3

, and

n = 7

, we get

T_7 = 5 + (6)(-3) = -13

.

Adım Adım Çözüm

1

Identify the given parameters of the A.P.

First term

a = 5

, common difference

d = -3

, and position

n = 7

.

These parameters are directly specified in the problem.

2

Apply the

n^{\text{th}}

term formula

T_n = a + (n - 1)d

.

T_7 = 5 + (7 - 1)(-3) = 5 + 6(-3)

.

The common difference is added

(n-1)

times to the first term.

3

Perform the multiplication and addition.

T_7 = 5 - 18 = -13

.

Simplifying the arithmetic gives the exact value of the

7^{\text{th}}

term.

Anahtar Kavram

Calculating the

n^{\text{th}}

term of an Arithmetic Progression

Soru 152Soru →

If $(x - 1)$ is a factor of the polynomial $P(x) = x^3 + 2x^2 - 5x + k$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

is

2

.

According to the Factor Theorem, a linear polynomial

(x - a)

is a factor of

P(x)

if and only if

P(a) = 0

. For the factor

(x - 1)

, setting

x = 1

gives

P(1) = (1)^3 + 2(1)^2 - 5(1) + k = 0

. Simplifying yields

1 + 2 - 5 + k = 0

, which simplifies further to

-2 + k = 0

, giving

k = 2

.

Adım Adım Çözüm

1

Apply the Factor Theorem

P(1) = 0

Since

(x - 1)

is a factor, setting

x = 1

makes the polynomial equal to zero.

2

Substitute

x = 1

into

P(x)

1^3 + 2(1)^2 - 5(1) + k = 0

Evaluate the polynomial expression at

x = 1

.

3

Simplify and solve for

k

k = 2

Combine constants

1 + 2 - 5 = -2

and solve

-2 + k = 0

.

Anahtar Kavram

Factor Theorem

Soru 153Soru →

If $\sqrt{28 - 10\sqrt{3}} + \sqrt{7 + 4\sqrt{3}} = k$ , where $k$ is a rational number, find the value of $k$ .

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

k

is

7

.

Simplifying each nested surd into standard binomial surd form gives

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

and

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

. Combining these terms cancels out the irrational component

\sqrt{3}

, leaving the integer result

5 + 2 = 7

.

Adım Adım Çözüm

1

Simplify the nested surd

\sqrt{28 - 10\sqrt{3}}

into binomial surd form

\sqrt{28 - 10\sqrt{3}} = 5 - \sqrt{3}

Assuming a form

x - y\sqrt{3}

and squaring both sides gives

x^2 + 3y^2 - 2xy\sqrt{3} = 28 - 10\sqrt{3}

. Matching components leads to

xy = 5

and

x^2 + 3y^2 = 28

, which yields integer values

x = 5

and

y = 1

.

2

Simplify the nested surd

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

into binomial surd form

\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}

Assuming a form

u + v\sqrt{3}

and squaring both sides gives

u^2 + 3v^2 + 2uv\sqrt{3} = 7 + 4\sqrt{3}

. Matching components leads to

uv = 2

and

u^2 + 3v^2 = 7

, which yields integer values

u = 2

and

v = 1

.

3

Sum the simplified expressions to calculate

k

k = 7

Summing

(5 - \sqrt{3}) + (2 + \sqrt{3})

results in the irrational parts

-\sqrt{3}

and

\sqrt{3}

cancelling out, leaving

5 + 2 = 7

.

Anahtar Kavram

Square Root of a Binomial Surd Expression

Soru 154Soru →

The maximum safe load $L$ supported by a horizontal wooden beam varies directly as its width $w$ and the square of its depth $d$ , and inversely as its length $l$ . A beam of width $6\text{ cm}$ , depth $10\text{ cm}$ , and length $4\text{ m}$ can support a maximum safe load of $900\text{ kg}$ . What is the maximum safe load that can be supported by a beam of the same material having a width of $4\text{ cm}$ , depth $12\text{ cm}$ , and length $6\text{ m}$ ?

576\text{ kg}

480\text{ kg}

1,296\text{ kg}

864\text{ kg}

Cevabı ve açıklamayı göster

Cevap: $576\text{ kg}$

Cevap

576\text{ kg}

The relationship is modeled by

L = \frac{k w d^2}{l}

. Using the initial parameters (

w=6

,

d=10

,

l=4

,

L=900

), we find

k = 6

. Substituting

w=4

,

d=12

, and

l=6

into the equation gives

L = \frac{6 \times 4 \times 144}{6} = 576\text{ kg}

.

Adım Adım Çözüm

1

Formulate the variation equation

L = \frac{k \cdot w \cdot d^2}{l}

Direct variation means multiplying factors in the numerator, while inverse variation places the variable in the denominator.

2

Calculate the constant of variation

k

using initial conditions

900 = \frac{k \cdot 6 \cdot 10^2}{4} \implies 900 = \frac{600 k}{4} = 150 k \implies k = 6

Substitute

L = 900

,

w = 6

,

d = 10

, and

l = 4

to solve for

k

.

3

Calculate the new load

L

for the new dimensions

L = \frac{6 \cdot 4 \cdot 12^2}{6} = \frac{6 \cdot 4 \cdot 144}{6} = 576\text{ kg}

Substitute

k = 6

,

w = 4

,

d = 12

, and

l = 6

into the variation formula.

Anahtar Kavram

Joint and Inverse Variation

Tahmini Süre:1m 30s

Soru 155Soru →

If $(2x - 3)$ is a factor of the polynomial $P(x) = 2x^3 - 5x^2 + ax + 6$ , what is the value of the constant $a$ ?

-8

-5

-1

1

Cevabı ve açıklamayı göster

Cevap: -1

Cevap

The value of the constant

a

is

-1

.

According to the Factor Theorem, if

(2x - 3)

is a factor of

P(x)

, then

P\left(\frac{3}{2}\right) = 0

. Substituting

x = \frac{3}{2}

gives

\frac{27}{4} - \frac{45}{4} + \frac{3}{2}a + 6 = 0

, which simplifies to

\frac{3}{2} + \frac{3}{2}a = 0

, resulting in

a = -1

.

Adım Adım Çözüm

1

Apply the Factor Theorem by setting the linear factor equal to zero.

2x - 3 = 0 \implies x = \frac{3}{2}

If

(2x - 3)

is a factor of

P(x)

, then

P\left(\frac{3}{2}\right) = 0

.

2

Substitute

x = \frac{3}{2}

into the polynomial expression

P(x)

.

P\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 - 5\left(\frac{3}{2}\right)^2 + a\left(\frac{3}{2}\right) + 6 = 0

Setting the resulting expression equal to zero allows solving for

a

.

3

Simplify the powers and numerical terms.

2\left(\frac{27}{8}\right) - 5\left(\frac{9}{4}\right) + \frac{3}{2}a + 6 = 0 \implies \frac{27}{4} - \frac{45}{4} + \frac{3}{2}a + 6 = 0

Evaluate each fraction before combining terms.

4

Combine constant terms and solve for

a

.

-\frac{18}{4} + 6 + \frac{3}{2}a = 0 \implies -\frac{9}{2} + 6 + \frac{3}{2}a = 0 \implies \frac{3}{2} + \frac{3}{2}a = 0 \implies a = -1

Isolating

a

yields the correct constant value.

Anahtar Kavram

Factor Theorem for linear divisors of the form

(ax - b)

Soru 156Soru →

What is the remainder when the polynomial $P(x) = x^3 - 4x^2 + 5x - 2$ is divided by $(x + 1)$ ?

-12

0

-4

-2

Cevabı ve açıklamayı göster

Cevap: $-12$

Cevap

-12

According to the Remainder Theorem, dividing a polynomial

P(x)

by a linear divisor

(x - a)

yields a remainder equal to

P(a)

. Here, the divisor is

(x + 1)

, which corresponds to

a = -1

. Substituting

x = -1

into

P(x) = x^3 - 4x^2 + 5x - 2

gives

(-1)^3 - 4(-1)^2 + 5(-1) - 2 = -1 - 4 - 5 - 2 = -12

. Thus, the value

-12

is correct.

Adım Adım Çözüm

1

Apply the Remainder Theorem

To find the remainder when

P(x)

is divided by

(x + 1)

, evaluate

P(-1)

by setting

x + 1 = 0 \implies x = -1

.

By the Remainder Theorem, dividing

P(x)

by

(x - a)

leaves a remainder equal to

P(a)

.

2

Substitute

x = -1

into the polynomial

P(x) = x^3 - 4x^2 + 5x - 2

P(-1) = (-1)^3 - 4(-1)^2 + 5(-1) - 2

Replace each instance of

x

with

-1

.

3

Simplify the powers and terms

P(-1) = -1 - 4(1) - 5 - 2 = -1 - 4 - 5 - 2 = -12

Compute arithmetic operations following standard order of operations.

Anahtar Kavram

Remainder Theorem

Tahmini Süre:45s

Soru 157Soru →

If $y$ varies inversely as $x$ , and $y = 8$ when $x = 3$ , what is the value of $y$ when $x = 6$ ?

4

16

24

2

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

For inverse variation,

y = \frac{k}{x}

. Given

y = 8

when

x = 3

, the constant of variation

k = 8 \times 3 = 24

. Substituting

x = 6

gives

y = \frac{24}{6} = 4

.

Adım Adım Çözüm

1

Set up the equation for inverse variation

y = \frac{k}{x}

Inverse variation means

y

is inversely proportional to

x

with constant

k

.

2

Solve for the variation constant

k

using

y = 8

and

x = 3

k = 8 \times 3 = 24

Multiply both sides of the equation by

x

.

3

Calculate

y

when

x = 6

y = \frac{24}{6} = 4

Substitute the constant

k = 24

and

x = 6

into the inverse variation equation.

Anahtar Kavram

Inverse Variation

Soru 158Soru →

The electric power $P$ dissipated in a resistor varies directly as the square of the current $I$ flowing through it. If a current of $3\text{ A}$ produces a power of $45\text{ W}$ , what is the power dissipated, in watts, when the current is $5\text{ A}$ ?

Cevabı ve açıklamayı göster

Cevap: 125

Cevap

The power dissipated when the current is 5 A is 125 W.

Because electric power varies directly as the square of the current, the formula is

P = k I^2

. Substituting the initial conditions gives

45 = k(3^2) = 9k

, so

k = 5

. Evaluating at

I = 5\text{ A}

gives

P = 5(5^2) = 125\text{ W}

.

Adım Adım Çözüm

1

Set up the variation equation

P = k I^2

Power varies directly as the square of current.

2

Find the constant of variation

k

k = 5

Substitute

P = 45

and

I = 3

into the variation equation:

45 = k(3^2) \implies 45 = 9k \implies k = 5

.

3

Calculate the required power for

I = 5\text{ A}

P = 125\text{ W}

Substitute

k = 5

and

I = 5

into

P = k I^2

:

P = 5(5^2) = 125

.

Anahtar Kavram

Direct variation involving a squared quantity

Soru 159Soru →

The polynomial $P(x) = 2x^3 + ax^2 + bx + 6$ leaves a remainder of $12$ when divided by $(x - 1)$ and has $(x + 3)$ as a factor. What is the value of $a - b$ ?

6

8

4

-28

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The value of

a - b

is

6

.

Applying the Remainder Theorem with

P(1) = 12

gives

a + b = 4

. Applying the Factor Theorem with

P(-3) = 0

gives

3a - b = 16

. Solving these simultaneous linear equations gives

a = 5

and

b = -1

. Evaluating

a - b = 5 - (-1)

gives

6

.

Adım Adım Çözüm

1

Apply the Remainder Theorem for the divisor

(x - 1)

a + b = 4

According to the Remainder Theorem,

P(1) = 12

. Substituting

x = 1

gives

2(1)^3 + a(1)^2 + b(1) + 6 = 12 \implies a + b + 8 = 12 \implies a + b = 4

.

2

Apply the Factor Theorem for the factor

(x + 3)

3a - b = 16

According to the Factor Theorem,

P(-3) = 0

. Substituting

x = -3

gives

2(-3)^3 + a(-3)^2 + b(-3) + 6 = 0 \implies -54 + 9a - 3b + 6 = 0 \implies 9a - 3b = 48 \implies 3a - b = 16

.

3

Solve the system of linear equations for

a

and

b

a = 5

and

b = -1

Adding

(a + b = 4)

and

(3a - b = 16)

yields

4a = 20 \implies a = 5

. Substituting

a = 5

into

a + b = 4

gives

b = -1

.

4

Calculate the target value

a - b

a - b = 6

Subtracting

b

from

a

gives

5 - (-1) = 5 + 1 = 6

.

Anahtar Kavram

Polynomial Remainder and Factor Theorems

Soru 160Soru →

In a survey of 160 music enthusiasts, 80 listen to Afrobeat, 70 listen to Highlife, and 65 listen to Reggae. It was found that 30 listen to both Afrobeat and Highlife, 25 listen to both Highlife and Reggae, and 28 listen to both Afrobeat and Reggae. If 15 enthusiasts listen to none of these three genres, how many enthusiasts listen to exactly two of these genres?

Cevabı ve açıklamayı göster

Cevap: 44

Cevap

The number of enthusiasts who listen to exactly two of the three genres is 44.

The correct answer is 44. By subtracting the 15 non-listeners from 160, the union of the three sets contains 145 members. Solving the 3-set inclusion-exclusion equation yields 13 enthusiasts who listen to all three genres. Subtracting 13 from each given pairwise intersection gives 17, 12, and 15 for the regions representing listeners of exactly two genres. Summing these three counts yields 44.

Adım Adım Çözüm

1

Calculate the cardinality of the union of all three sets.

n(A \cup H \cup R) = 160 - 15 = 145

Subtracting the number of enthusiasts listening to none of the genres from the universal set.

2

Apply the Principle of Inclusion-Exclusion for 3 sets to find the triple intersection.

n(A \cap H \cap R) = 13

Substituting known cardinalities into

n(A \cup H \cup R) = n(A)+n(H)+n(R) - [n(A \cap H)+n(H \cap R)+n(A \cap R)] + n(A \cap H \cap R)

.

3

Calculate the count of enthusiasts in each 'exactly two genres' region.

Afrobeat and Highlife only = 17; Highlife and Reggae only = 12; Afrobeat and Reggae only = 15.

Subtracting the triple intersection count (

13

) from each pairwise intersection.

4

Sum the counts of the three distinct two-genre regions.

17 + 12 + 15 = 44

Combining all mutually exclusive regions representing enthusiasts of exactly two genres.

Anahtar Kavram

Principle of Inclusion-Exclusion for three sets and Venn diagram region partitioning.

Algebra

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