Number and Numeration Alıştırma Soruları — JAMB UTME — Sayfa 8

If $2^{2x + 1} - 5(2^x) + 2 = 0$ , what are the possible values of $x$ ?

-1

or

1

\frac{1}{2}

or

2

-1

or

2

-2

or

1

Cevabı ve açıklamayı göster

Cevap: $-1$ or $1$

Cevap

-1

or

1

Using the addition law of indices,

2^{2x+1}

can be rewritten as

2^1 \cdot 2^{2x} = 2 \cdot (2^x)^2

. Substituting

y = 2^x

produces the quadratic equation

2y^2 - 5y + 2 = 0

, which factorizes into

(2y - 1)(y - 2) = 0

. This gives

y = \frac{1}{2}

or

y = 2

. Converting back to exponential equations gives

2^x = 2^{-1} \implies x = -1

and

2^x = 2^1 \implies x = 1

. Thus, the solutions for

x

are

-1

or

1

.

Adım Adım Çözüm

1

Apply index laws to express the equation in terms of

2^x

2^{2x+1} = 2^{2x} \cdot 2^1 = 2 \cdot (2^x)^2

, so the equation becomes

2 \cdot (2^x)^2 - 5(2^x) + 2 = 0

Splitting the exponent using

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

reveals a quadratic structure in

2^x

.

2

Substitute

y = 2^x

and solve the resulting quadratic equation

2y^2 - 5y + 2 = 0 \implies (2y - 1)(y - 2) = 0 \implies y = \frac{1}{2} \text{ or } y = 2

Factoring the quadratic expression yields the values for the substitution variable

y

.

3

Equate

2^x

to each solution of

y

to solve for

x

For

y = \frac{1}{2}

:

2^x = 2^{-1} \implies x = -1

. For

y = 2

:

2^x = 2^1 \implies x = 1

.

Using the negative index rule

a^{-n} = \frac{1}{a^n}

allows matching exponents when bases are identical.

Anahtar Kavram

Quadratic Equations Reducible to Index Form

Soru 142Soru →

What is the simplified form of the surd expression $\sqrt{45} - \sqrt{20} + \sqrt{80}$ ?

5\sqrt{5}

\sqrt{105}

9\sqrt{5}

3\sqrt{5}

Cevabı ve açıklamayı göster

Cevap: $5\sqrt{5}$

Cevap

The simplified form of the expression is

5\sqrt{5}

.

Each radical is decomposed into a product involving a perfect square:

\sqrt{45} = 3\sqrt{5}

,

\sqrt{20} = 2\sqrt{5}

, and

\sqrt{80} = 4\sqrt{5}

. Combining the coefficients

(3 - 2 + 4)\sqrt{5}

yields

5\sqrt{5}

.

Adım Adım Çözüm

1

Simplify each individual surd into basic radical form by finding perfect square factors.

\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}

,

\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}

, and

\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}

.

Expressing surds in terms of identical basic radicals allows like terms to be combined.

2

Substitute the simplified surds back into the original expression and combine like terms.

3\sqrt{5} - 2\sqrt{5} + 4\sqrt{5} = (3 - 2 + 4)\sqrt{5} = 5\sqrt{5}

.

Perform addition and subtraction on the coefficients of similar surds.

Anahtar Kavram

Simplification of Surds

Soru 143Soru →

Three water pipes, $A$ , $B$ , and $C$ , are used to fill a storage tank. The filling rates of pipes $A$ and $B$ are in the ratio $3 : 2$ , while the filling rates of pipes $B$ and $C$ are in the ratio $4 : 3$ . Pipes $A$ and $B$ are opened together for $5\text{ hours}$ , after which pipe $A$ is closed. Pipes $B$ and $C$ are then opened together for $10\text{ hours}$ to completely fill the remaining capacity of the tank. How many hours would pipe $A$ alone take to fill the entire empty tank?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

Pipe

A

alone will take

20\text{ hours}

to fill the empty tank.

Combining the given ratios

A : B = 3 : 2

and

B : C = 4 : 3

gives

A : B : C = 6 : 4 : 3

. Assigning relative rates of

6r

,

4r

, and

3r

per hour, the work completed in the first phase is

5 \times (6r + 4r) = 50r

, and in the second phase is

10 \times (4r + 3r) = 70r

. The total capacity of the tank is

50r + 70r = 120r

. Dividing the total capacity by pipe

A

's filling rate (

6r

) gives

120r / 6r = 20\text{ hours}

.

Adım Adım Çözüm

1

Find the combined ratio of filling rates for pipes A, B, and C

A : B : C = 6 : 4 : 3

Given

A : B = 3 : 2 = 6 : 4

and

B : C = 4 : 3

, aligning the common term

B = 4

yields

A : B : C = 6 : 4 : 3

.

2

Express rates in terms of a constant

r

and calculate work done in Phase 1

Work in Phase 1 =

50r

Rates are

A = 6r

,

B = 4r

,

C = 3r

. In

5\text{ hours}

, pipes

A

and

B

fill

5 \times (6r + 4r) = 50r

.

3

Calculate work done in Phase 2

Work in Phase 2 =

70r

In

10\text{ hours}

, pipes

B

and

C

fill

10 \times (4r + 3r) = 70r

.

4

Determine total capacity of the tank

Total Capacity =

120r

Total work required to fill the tank is

50r + 70r = 120r

.

5

Calculate time taken by pipe A alone

20\text{ hours}

Time taken by pipe

A = \frac{\text{Total Capacity}}{\text{Rate of } A} = \frac{120r}{6r} = 20\text{ hours}

.

Anahtar Kavram

Compound Ratio and Rates of Work

Soru 144Soru →

If $\log_3(x + 5) - \log_{\frac{1}{3}}(x - 1) = 3$ , find the real value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The real value of

x

is 4.

Using the change of base formula,

\log_{\frac{1}{3}}(x - 1) = -\log_3(x - 1)

. Substituting this back into the equation transforms it into

\log_3(x + 5) + \log_3(x - 1) = 3

. Combining the logarithms using the product property gives

\log_3[(x + 5)(x - 1)] = 3

, which means

(x + 5)(x - 1) = 3^3 = 27

. Expanding leads to

x^2 + 4x - 32 = 0

, factoring into

(x + 8)(x - 4) = 0

. Since logarithms require positive arguments (

x > 1

),

x = -8

is invalid, giving the final answer

x = 4

.

Adım Adım Çözüm

1

Apply change of base to express the equation in a single base

\log_{\frac{1}{3}}(x - 1) = \frac{\log_3(x - 1)}{\log_3(1/3)} = -\log_3(x - 1)

Change of base rule

\log_b a = \frac{\log_c a}{\log_c b}

with base

c = 3

, where

\log_3(1/3) = -1

.

2

Substitute back into the original equation and combine terms

\log_3(x + 5) - [-\log_3(x - 1)] = \log_3(x + 5) + \log_3(x - 1) = 3

Subtracting a negative logarithm equals adding the positive logarithm.

3

Apply the product rule of logarithms

\log_3[(x + 5)(x - 1)] = 3

Product rule:

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

.

4

Convert logarithmic equation to quadratic form and solve

(x + 5)(x - 1) = 3^3 = 27 \implies x^2 + 4x - 32 = 0 \implies (x + 8)(x - 4) = 0

Logarithmic definition

\log_b Y = k \implies Y = b^k

.

5

Verify domain constraints

x = 4

Logarithmic domain requires

x - 1 > 0 \implies x > 1

. Therefore,

x = -8

is extraneous.

Anahtar Kavram

Logarithmic Change of Base and Algebraic Reduction

Soru 145Soru →

An artisan deposited $\text{₦}20,000$ into a savings account that pays a compound interest rate of $10\%$ per annum compounded annually. What is the total compound interest earned at the end of $2$ years?

\text{₦}4,200

\text{₦}24,200

\text{₦}4,000

\text{₦}2,000

Cevabı ve açıklamayı göster

Cevap: $\text{₦}4,200$

Cevap

The total compound interest earned after 2 years is

\text{₦}4,200

.

The correct answer is obtained by calculating the total amount after 2 years at

10\%

compound interest, which is

20,000 \times (1.10)^2 = \text{₦}24,200

, and then subtracting the principal of

\text{₦}20,000

to get

\text{₦}4,200

.

Adım Adım Çözüm

1

Calculate the total accumulated amount (

A

) after 2 years using the compound interest formula

A = P\left(1 + \frac{R}{100}\right)^n

.

A = 20,000 \left(1 + \frac{10}{100}\right)^2 = 20,000 \times (1.1)^2 = 20,000 \times 1.21 = \text{₦}24,200

.

Determines the total value of the investment at the end of the 2-year period.

2

Subtract the principal (

P

) from the total accumulated amount (

A

) to find the interest earned (

I = A - P

).

I = 24,200 - 20,000 = \text{₦}4,200

.

Isolates the interest earned from the initial principal amount.

Anahtar Kavram

Compound Interest Calculation

Tahmini Süre:45s

Soru 146Soru →

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

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The value of

x

is 27.

Cross-multiplying yields

\sqrt{x} + \sqrt{3} = 2\sqrt{x} - 2\sqrt{3}

. Rearranging terms gives

\sqrt{x} = 3\sqrt{3}

. Squaring both sides produces

x = 3^2 \times 3 = 27

.

Adım Adım Çözüm

1

Multiply both sides by the denominator

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

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

Clear the rational surd expression to form a linear relation in terms of radicals

2

Expand the terms and isolate

\sqrt{x}

\sqrt{x} = 3\sqrt{3}

Group terms involving

\sqrt{x}

on one side and constant surds on the other side

3

Square both sides of the simplified equation

x = (3\sqrt{3})^2 = 9 \times 3 = 27

Eliminate the radical over

x

to obtain the integer solution

Anahtar Kavram

Solving algebraic surd equations using rationalization concepts and properties of radicals

Tahmini Süre:1m 30s

Soru 147Soru →

A school kitchen uses $150$ loaves of bread to feed $300$ students. If $120$ additional students join the school, how many loaves of bread in total will be needed to feed all the students at the same consumption rate?

60

loaves

210

loaves

270

loaves

375

loaves

Cevabı ve açıklamayı göster

Cevap: $210$ loaves

Cevap

210

loaves of bread

The rate of bread consumption is

150 \div 300 = 0.5

loaves per student. With

120

additional students, the total number of students becomes

300 + 120 = 420

. Multiplying

420

students by

0.5

loaves per student gives

210

loaves.

Adım Adım Çözüm

1

Calculate the rate of consumption per student.

Rate =

\frac{150\text{ loaves}}{300\text{ students}} = 0.5\text{ loaves per student}

.

Determining the unit rate simplifies finding the total quantity needed for any number of students.

2

Find the total number of students.

Total students =

300 + 120 = 420\text{ students}

.

The question asks for the total loaves required to feed all students after the addition.

3

Calculate the total loaves needed by multiplying the total number of students by the unit rate.

Total loaves =

420 \times 0.5 = 210\text{ loaves}

.

Applying direct proportion yields the total amount of bread required.

Anahtar Kavram

Direct Proportion and Rate

Soru 148Soru →

In a survey of $120$ town residents regarding three newspapers—*The Herald* ( $H$ ), *The Express* ( $E$ ), and *The Nation* ( $N$ )—it was found that $52$ read *The Herald*, $45$ read *The Express*, and $60$ read *The Nation*. Furthermore, $15$ read both *The Herald* and *The Express*, $22$ read both *The Express* and *The Nation*, and $18$ read both *The Herald* and *The Nation*. If the number of residents who read none of these three newspapers is twice the number of those who read all three newspapers, how many residents read exactly two of the newspapers?

31

37

43

55

Cevabı ve açıklamayı göster

Cevap: $37$

Cevap

The number of residents who read exactly two newspapers is

37

.

Using the principle of inclusion-exclusion,

n(H \cup E \cup N) = 52 + 45 + 60 - (15 + 22 + 18) + x = 102 + x

, where

x

is the number of residents reading all three newspapers. Since the number of residents reading none is

2x

, the total universe equation is

102 + x + 2x = 120

, giving

3x = 18

and

x = 6

. The number of residents reading exactly two newspapers is

(15 - 6) + (22 - 6) + (18 - 6) = 9 + 16 + 12 = 37

.

Adım Adım Çözüm

1

Formulate the principle of inclusion-exclusion for three sets.

n(H \cup E \cup N) = n(H) + n(E) + n(N) - n(H \cap E) - n(E \cap N) - n(H \cap N) + n(H \cap E \cap N)

To express the total number of residents reading at least one newspaper in terms of the unknown number of residents who read all three.

2

Substitute the given values and set up the equation for the universal set.

n(H \cup E \cup N) = 52 + 45 + 60 - 15 - 22 - 18 + x = 102 + x

. Let

n(H \cup E \cup N)' = 2x

. Then

102 + x + 2x = 120 \implies 102 + 3x = 120 \implies 3x = 18 \implies x = 6

.

The sum of elements in the union and its complement must equal the universal set size of

120

.

3

Calculate the number of residents reading exactly two newspapers.

(n(H \cap E) - x) + (n(E \cap N) - x) + (n(H \cap N) - x) = (15 - 6) + (22 - 6) + (18 - 6) = 9 + 16 + 12 = 37

Subtracting the triple intersection

x

from each pairwise intersection isolates the regions corresponding to exactly two newspapers.

Anahtar Kavram

Three-set inclusion-exclusion principle and Venn diagram cardinal region decomposition

Tahmini Süre:1m 30s

Soru 149Soru →

Convert the base 2 fractional number $110.101_2$ to its equivalent value in base 10 (decimal). What is the decimal value?

Cevabı ve açıklamayı göster

Cevap: 6.625

Cevap

The decimal value of

110.101_2

is

6.625

.

To convert

110.101_2

to base 10, expand both the integer and fractional parts using powers of 2. The integer component

110_2 = (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 4 + 2 + 0 = 6

. The fractional component

0.101_2 = (1 \times 2^{-1}) + (0 \times 2^{-2}) + (1 \times 2^{-3}) = 0.5 + 0 + 0.125 = 0.625

. Summing these values gives

6 + 0.625 = 6.625

.

Adım Adım Çözüm

1

Convert the integer part

110_2

to base 10

1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6

Each position to the left of the binary point corresponds to an increasing non-negative power of 2 (

2^0, 2^1, 2^2

).

2

Convert the fractional part

0.101_2

to base 10

1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = \frac{1}{2} + 0 + \frac{1}{8} = 0.5 + 0.125 = 0.625

Each position to the right of the binary point corresponds to a negative power of 2 (

2^{-1}, 2^{-2}, 2^{-3}

).

3

Combine the integer and fractional results

6 + 0.625 = 6.625

The total value in base 10 is the sum of the expanded integer and fractional components.

Anahtar Kavram

Conversion of Fractional Non-Decimal Numbers to Decimal

Soru 150Soru →

A worker's monthly salary was increased from $\text{N}50,000$ to $\text{N}55,000$ . What is the percentage increase in the worker's salary?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The percentage increase in the worker's salary is

10\%

.

The salary increase is

\text{N}55,000 - \text{N}50,000 = \text{N}5,000

. Evaluating this change relative to the initial salary gives

\frac{5,000}{50,000} \times 100\% = 10\%

.

Adım Adım Çözüm

1

Calculate the absolute increase in salary

Increase =

\text{N}5,000

Subtract the initial salary from the new salary:

\text{N}55,000 - \text{N}50,000 = \text{N}5,000

.

2

Compute the percentage increase relative to the original salary

Percentage Increase =

10\%

Divide the increase by the original value and multiply by

100\%

:

\frac{5,000}{50,000} \times 100\% = 10\%

.

Anahtar Kavram

Percentage Increase

Tahmini Süre:45s

Soru 151Soru →

Convert the base 5 number $234_5$ to a base 10 (decimal) number. What is the value in base 10?

Cevabı ve açıklamayı göster

Cevap: 69

Cevap

The decimal (base 10) equivalent of

234_5

is 69.

Expanding

234_5

gives

(2 \times 5^2) + (3 \times 5^1) + (4 \times 5^0) = 50 + 15 + 4 = 69

.

Adım Adım Çözüm

1

Write out the positional expansion of

234_5

using powers of 5 starting from position 0 on the right.

234_5 = (2 \times 5^2) + (3 \times 5^1) + (4 \times 5^0)

Converting from any non-decimal base to base 10 involves multiplying each digit by its corresponding positional base power.

2

Evaluate the powers of 5 and multiply by the respective digits.

2 \times 25 = 50

,

3 \times 5 = 15

,

4 \times 1 = 4

Apply basic arithmetic exponents:

5^2=25

,

5^1=5

,

5^0=1

.

3

Sum all calculated values together.

50 + 15 + 4 = 69

Adding the positional values yields the total value in base 10.

Anahtar Kavram

Converting a non-decimal number to base 10 using expansion by powers of the base.

Soru 152Soru →

If $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} = p + q\sqrt{3}$ , where $p$ and $q$ are rational numbers, what is the value of $p + q$ ?

1

2

3

5

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

p + q

is 3.

Multiplying both numerator and denominator by the conjugate

(\sqrt{6} + \sqrt{2})

yields

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

. Matching terms with

p + q\sqrt{3}

gives

p = 2

and

q = 1

, so

p + q = 3

.

Adım Adım Çözüm

1

Multiply the numerator and denominator by the conjugate of the denominator,

(\sqrt{6} + \sqrt{2})

.

\frac{(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})}

To eliminate radicals from the denominator.

2

Expand both numerator and denominator.

Numerator:

6 + 2\sqrt{12} + 2 = 8 + 4\sqrt{3}

. Denominator:

6 - 2 = 4

.

Using algebraic expansion

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

and difference of two squares.

3

Simplify the resulting fraction.

\frac{8 + 4\sqrt{3}}{4} = 2 + \sqrt{3}

Dividing each term in the numerator by 4.

4

Equate to

p + q\sqrt{3}

to determine

p

and

q

, then find

p + q

.

p = 2

,

q = 1 \implies p + q = 2 + 1 = 3

.

Comparing rational and irrational parts separately.

Anahtar Kavram

Rationalization of Denominators and Surd Conjugates

Soru 153Soru →

A contractor estimates that $20$ workers, working $8\text{ hours}$ per day, can complete a road construction project in $30\text{ days}$ . All workers work at the same constant rate. After working for $10\text{ days}$ , $4$ workers leave the site. To make up for the loss, the daily working duration for each remaining worker is increased to $10\text{ hours}$ per day. However, due to fatigue, the work efficiency of each remaining worker drops by $20\%$ . How many additional days will be required to complete the remaining work?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

It will take 25 additional days to complete the remaining work.

The total project requires

4,800

worker-hours (

20 \times 8 \times 30

). In the first

10

days,

1,600

worker-hours are completed, leaving

3,200

worker-hours. After

4

workers leave,

16

workers remain. Working

10

hours a day at

80\%

efficiency, the

16

workers produce

16 \times 10 \times 0.8 = 128

effective worker-hours each day. Dividing the remaining

3,200

worker-hours by

128

gives exactly

25

additional days.

Adım Adım Çözüm

1

Calculate total work units required for the entire project

Total work =

4800\text{ worker-hours}

Work rate is proportional to workers multiplied by total hours worked.

2

Calculate completed work and remaining work

Work done =

1600\text{ worker-hours}

, Remaining work =

3200\text{ worker-hours}

Subtracting completed work from total work gives the remaining required effort.

3

Determine the effective rate per day after conditions change

Effective daily output =

128\text{ worker-hours/day}

16 remaining workers working 10 hours daily at 80% efficiency yield

16 \times 10 \times 0.8 = 128

worker-hours per day.

4

Divide remaining work by the new effective daily rate

Number of additional days =

25\text{ days}

Dividing

3200\text{ worker-hours}

by

128\text{ worker-hours/day}

gives

25\text{ days}

.

Anahtar Kavram

Compound Proportion and Work Efficiency

Soru 154Soru →

Given the matrices $A = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -1 \\ 2 & 5 \end{pmatrix}$ , find the determinant of the matrix $C = 2A - B$ .

Cevabı ve açıklamayı göster

Cevap: -15

Cevap

The determinant of the matrix

C = 2A - B

is

-15

.

Scalar multiplication gives

2A = \begin{pmatrix} 4 & 2 \\ 8 & 6 \end{pmatrix}

. Subtracting

B

yields

C = \begin{pmatrix} 3 & 3 \\ 6 & 1 \end{pmatrix}

. Evaluating the determinant gives

\det(C) = (3)(1) - (3)(6) = 3 - 18 = -15

.

Adım Adım Çözüm

1

Multiply matrix

A

by scalar

2

2A = \begin{pmatrix} 4 & 2 \\ 8 & 6 \end{pmatrix}

Scalar multiplication requires multiplying each entry of matrix

A

by

2

.

2

Subtract matrix

B

from matrix

2A

entry-wise to find matrix

C

C = \begin{pmatrix} 4 - 1 & 2 - (-1) \\ 8 - 2 & 6 - 5 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 6 & 1 \end{pmatrix}

Subtract corresponding entries of matrix

B

from

2A

.

3

Calculate the determinant of matrix

C

\det(C) = (3)(1) - (3)(6) = 3 - 18 = -15

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is

ad - bc

.

Anahtar Kavram

Matrix Operations and Determinants

Soru 155Soru →

If $\log_2 x + \log_4 x + \log_{16} x = 7$ , what is the value of $x$ ?

16

4

2

8

Cevabı ve açıklamayı göster

Cevap: $16$

Cevap

16

Using the change of base property,

\log_4 x = \frac{1}{2}\log_2 x

and

\log_{16} x = \frac{1}{4}\log_2 x

. Combining like terms yields

\left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x

. Setting

\frac{7}{4}\log_2 x = 7

gives

\log_2 x = 4

, which leads to

x = 2^4 = 16

.

Adım Adım Çözüm

1

Apply the change of base formula to express all logarithmic terms in base 2.

\log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{1}{2}\log_2 x

and

\log_{16} x = \frac{\log_2 x}{\log_2 16} = \frac{1}{4}\log_2 x

.

Logarithms with different bases must be converted to a common base to combine them.

2

Substitute the transformed terms back into the equation and factor out

\log_2 x

.

\log_2 x + \frac{1}{2}\log_2 x + \frac{1}{4}\log_2 x = \left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x.

Factoring allows summing the coefficients of

\log_2 x

.

3

Solve for

\log_2 x

by equating the simplified expression to 7.

\frac{7}{4}\log_2 x = 7 \implies \log_2 x = 7 \times \frac{4}{7} = 4.

Isolating the logarithmic term gives its numerical value.

4

Convert the logarithmic equation to exponential form to solve for

x

.

x = 2^4 = 16.

By definition,

\log_b a = c \iff a = b^c

.

Anahtar Kavram

Change of Base Formula for Logarithms

Tahmini Süre:1m 30s

Soru 156Soru →

If matrix $A$ has dimension $2 \times 3$ and matrix $B$ has dimension $3 \times 4$ , what is the dimension of the matrix product $AB$ ?

3 \times 3

2 \times 4

4 \times 2

2 \times 3

Cevabı ve açıklamayı göster

Cevap: $2 \times 4$

Cevap

The dimension of the matrix product

AB

is

2 \times 4

.

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. When a matrix of size

m \times n

is multiplied by a matrix of size

n \times p

, the resulting product matrix has size

m \times p

. For

A_{2 \times 3}

and

B_{3 \times 4}

, the outer dimensions give

2 \times 4

.

Adım Adım Çözüm

1

Identify the dimensions of the given matrices

A

and

B

Matrix

A

is of order

2 \times 3

(

m = 2, n = 3

) and Matrix

B

is of order

3 \times 4

(

n = 3, p = 4

).

Matrix multiplication requirement requires the number of columns of the first matrix to match the number of rows of the second matrix.

2

Apply the matrix multiplication dimension rule

The product matrix

AB

has

m

rows and

p

columns, resulting in dimension

2 \times 4

.

When multiplying an

m \times n

matrix by an

n \times p

matrix, the resulting matrix has dimensions

m \times p

.

Anahtar Kavram

Matrix Multiplication Dimension Compatibility and Resulting Order

Tahmini Süre:45s

Soru 157Soru →

On an international flight carrying $150$ passengers, each passenger was offered three meal options: Chicken ( $C$ ), Fish ( $F$ ), and Vegetarian ( $V$ ). A survey of their choices showed that $75$ passengers chose Chicken, $60$ chose Fish, and $50$ chose Vegetarian. Additionally, $15$ passengers chose both Chicken and Fish, $12$ chose both Fish and Vegetarian, $18$ chose both Chicken and Vegetarian, while $8$ passengers chose all three meals. How many passengers chose none of the three meal options?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2 passengers chose none of the meal options.

Using the inclusion-exclusion principle for three overlapping sets, the total number of passengers taking at least one meal is calculated as

|C \cup F \cup V| = 75 + 60 + 50 - (15 + 12 + 18) + 8 = 148

. Subtracting this value from the total count of

150

passengers yields

150 - 148 = 2

passengers who selected none of the meal options.

Adım Adım Çözüm

1

Identify the cardinalities of the individual sets, pairwise intersections, triple intersection, and the universal set.

N(U) = 150

,

|C| = 75

,

|F| = 60

,

|V| = 50

,

|C \cap F| = 15

,

|F \cap V| = 12

,

|C \cap V| = 18

, and

|C \cap F \cap V| = 8

.

Organizing the given information allows for direct application of set cardinality formulas.

2

Calculate the total number of passengers who selected at least one meal using the Principle of Inclusion-Exclusion for three sets.

|C \cup F \cup V| = 75 + 60 + 50 - (15 + 12 + 18) + 8 = 185 - 45 + 8 = 148

.

Adding individual set totals overcounts elements in pairwise intersections, and subtracting pairwise intersections subtracts the triple intersection one too many times, so it must be added back.

3

Find the number of passengers who selected none of the meal choices by taking the complement of the union with respect to the universal set.

N(\text{None}) = N(U) - |C \cup F \cup V| = 150 - 148 = 2

.

Passengers choosing none of the meal options correspond to the region outside all three sets within the universal set.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets and Complement of Union

Tahmini Süre:1m 30s

Soru 158Soru →

If

x > 0

satisfies the exponential equation

3^{x+1} + 3^{1-x} = 10

find the value of

8^x + 4^{x-1}

.

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The value of

8^x + 4^{x-1}

is 9.

Applying index laws to

3^{x+1} + 3^{1-x} = 10

yields

3(3^x) + \frac{3}{3^x} = 10

. Substituting

u = 3^x

gives

3u^2 - 10u + 3 = 0

, which factors to

(3u - 1)(u - 3) = 0

, yielding

u = 3

or

u = \frac{1}{3}

. Thus

x = 1

or

x = -1

. Given

x > 0

,

x = 1

. Substituting

x = 1

into

8^x + 4^{x-1}

gives

8^1 + 4^0 = 8 + 1 = 9

.

Adım Adım Çözüm

1

Apply the product and negative power laws of indices to separate the terms in the given equation.

3^{x+1} = 3^1 \cdot 3^x = 3(3^x)

and

3^{1-x} = 3^1 \cdot 3^{-x} = \frac{3}{3^x}

, making the equation

3(3^x) + \frac{3}{3^x} = 10

.

According to the laws of indices,

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

and

a^{-n} = \frac{1}{a^n}

.

2

Substitute

u = 3^x

into the equation and clear the fraction to form a standard quadratic equation.

3u + \frac{3}{u} = 10 \implies 3u^2 - 10u + 3 = 0

.

Multiplying through by

u

eliminates the fraction and forms a quadratic in terms of

u

.

3

Factor the quadratic equation

3u^2 - 10u + 3 = 0

to find the values of

u

.

(3u - 1)(u - 3) = 0 \implies u = 3

or

u = \frac{1}{3}

.

Factoring by splitting the middle term gives the linear factors.

4

Equate

3^x

to the values of

u

and apply the constraint

x > 0

.

3^x = 3^1 \implies x = 1

and

3^x = 3^{-1} \implies x = -1

. Selecting the positive root gives

x = 1

.

Equating exponents with matching base 3 gives the solutions for

x

.

5

Substitute

x = 1

into the target expression

8^x + 4^{x-1}

and simplify.

8^1 + 4^{1-1} = 8 + 4^0 = 8 + 1 = 9

.

By the zero index law, any non-zero base raised to the power 0 equals 1 (

a^0 = 1

).

Anahtar Kavram

Solving quadratic-form exponential equations using index laws and applying the zero index rule.

Soru 159Soru →

If the determinant of the matrix $M = \begin{pmatrix} x & 3 \\ 2 & 5 \end{pmatrix}$ is $14$ , find the value of $x$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

x

is

4

.

For matrix

M = \begin{pmatrix} x & 3 \\ 2 & 5 \end{pmatrix}

, the determinant is calculated as

(x)(5) - (3)(2) = 5x - 6

. Equating this to

14

gives

5x - 6 = 14

, which simplifies to

5x = 20

, yielding

x = 4

.

Adım Adım Çözüm

1

Apply the

2 \times 2

determinant formula

\det = ad - bc

\det(M) = (x \times 5) - (3 \times 2) = 5x - 6

The determinant of a

2 \times 2

matrix is the product of the main diagonal minus the product of the anti-diagonal.

2

Set the determinant equal to the given value

14

5x - 6 = 14

The problem states that the determinant is equal to 14.

3

Solve the linear equation for

x

5x = 20 \implies x = 4

Adding 6 to both sides gives

5x = 20

, and dividing by 5 yields

x = 4

.

Anahtar Kavram

Determinant of a 2x2 Matrix

Soru 160Soru →

If $x$ is the least positive integer satisfying the modular congruence $4x + 9 \equiv 3 \pmod{11}$ , what is the value of $(x^2 - 5x + 2) \pmod{11}$ expressed in standard non-negative remainder form?

2

4

7

9

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The value of

(x^2 - 5x + 2) \pmod{11}

in standard non-negative remainder form is 9.

Solving

4x + 9 \equiv 3 \pmod{11}

yields

4x \equiv -6 \equiv 5 \pmod{11}

. Multiplying by 3 (the inverse of 4 mod 11) gives

x \equiv 15 \equiv 4 \pmod{11}

. Evaluating

(x^2 - 5x + 2)

at

x = 4

gives

16 - 20 + 2 = -2

. Converting

-2

into the standard non-negative remainder range

[0, 10]

gives

-2 + 11 = 9

.

Adım Adım Çözüm

1

Isolate the variable term in the linear modular congruence

4x \equiv 3 - 9 \equiv -6 \pmod{11}

Subtract 9 from both sides of the congruence.

2

Convert the negative right-hand side to a non-negative residue modulo 11

4x \equiv -6 + 11 \equiv 5 \pmod{11}

Add the modulus 11 to obtain the canonical non-negative equivalent.

3

Solve for

x

by multiplying by the multiplicative inverse of 4 modulo 11

Since

4 \times 3 = 12 \equiv 1 \pmod{11}

, multiply both sides by 3:

x \equiv 5 \times 3 = 15 \equiv 4 \pmod{11}

. Thus, the least positive integer is

x = 4

.

The modular inverse of 4 modulo 11 is 3.

4

Substitute

x = 4

into the expression

(x^2 - 5x + 2)

4^2 - 5(4) + 2 = 16 - 20 + 2 = -2

Evaluate the quadratic expression using the calculated value of

x

.

5

Express

-2

in canonical non-negative remainder form modulo 11

-2 + 11 = 9 \pmod{11}

Add the modulus 11 to convert the negative result to a non-negative remainder within

[0, 10]

.

Anahtar Kavram

Solving linear modular congruences and converting negative remainders to canonical form

Number and Numeration

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