Number and Numeration Practice Questions — JAMB UTME — Page 6

If $\log_x 64 - 2\log_4 x = 1$ , what is the sum of all possible real values of $x$ ?

\frac{33}{8}

\frac{31}{8}

4

-2

Show answer & explanation

Answer: $\frac{33}{8}$

Answer

The sum of all possible real values of

x

is

\frac{33}{8}

.

Using the change of base formula,

\log_x 64 = \frac{6}{\log_2 x}

and

2\log_4 x = \log_2 x

. Letting

u = \log_2 x

yields

\frac{6}{u} - u = 1

, which rearranges to

u^2 + u - 6 = 0

. Factoring gives

u = 2

or

u = -3

, yielding solutions

x = 2^2 = 4

and

x = 2^{-3} = \frac{1}{8}

. Summing these values gives

4 + \frac{1}{8} = \frac{33}{8}

.

Step-by-Step Solution

1

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

\log_x 64 = \frac{\log_2 64}{\log_2 x} = \frac{6}{\log_2 x}

and

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

, so

2\log_4 x = \log_2 x

.

Converting all terms to a common base allows substitution into a simple algebraic equation.

2

Substitute

u = \log_2 x

into the original equation.

\frac{6}{u} - u = 1.

This simplifies the logarithmic equation into a single-variable algebraic form.

3

Clear the fraction by multiplying through by

u

and rearrange into standard quadratic form.

6 - u^2 = u \implies u^2 + u - 6 = 0.

Multiplying by

u

(where

u \neq 0

) forms a quadratic equation.

4

Factor and solve the quadratic equation for

u

.

(u + 3)(u - 2) = 0 \implies u = 2 \text{ or } u = -3.

Finding the roots of the quadratic equation provides the possible values for

\log_2 x

.

5

Solve for

x

using the exponential form

x = 2^u

.

For

u = 2

:

x = 2^2 = 4

. For

u = -3

:

x = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}

.

Both

x = 4

and

x = \frac{1}{8}

are positive real numbers not equal to 1, so both are valid logarithmic bases.

6

Calculate the sum of all valid solutions for

x

.

4 + \frac{1}{8} = \frac{32}{8} + \frac{1}{8} = \frac{33}{8}.

The question asks for the sum of all real values of

x

satisfying the equation.

Key Concept

Logarithms and Change of Base

Estimated Time:2m 0s

Question 102Question →

If $x$ and $y$ are real numbers greater than $1$ satisfying the system of equations $\log_x y + \log_y x = \frac{5}{2}$ and $xy = 64$ with $x > y$ , find the value of $x - y$ .

Show answer & explanation

Answer: 12

Answer

The value of

x - y

is 12.

Using the change of base identity

\log_y x = \frac{1}{\log_x y}

, the equation

\log_x y + \log_y x = \frac{5}{2}

converts to

u + \frac{1}{u} = \frac{5}{2}

for

u = \log_x y

. Solving the quadratic equation

2u^2 - 5u + 2 = 0

yields

u = \frac{1}{2}

or

u = 2

. Because

x > y > 1

, we must have

\log_x y < 1

, selecting

u = \frac{1}{2} \implies x = y^2

. Substituting into

xy = 64

gives

y^3 = 64 \implies y = 4

and

x = 16

. Therefore,

x - y = 16 - 4 = 12

.

Step-by-Step Solution

1

Apply the reciprocal change of base identity

Rewrite

\log_y x

as

\frac{1}{\log_x y}

, yielding

\log_x y + \frac{1}{\log_x y} = \frac{5}{2}

.

According to the change of base formula,

\log_y x = \frac{\log_x x}{\log_x y} = \frac{1}{\log_x y}

.

2

Solve the quadratic equation in terms of

u = \log_x y

Substituting

u = \log_x y

gives

u + \frac{1}{u} = \frac{5}{2} \implies 2u^2 - 5u + 2 = 0

, which factors into

(2u - 1)(u - 2) = 0

, yielding

u = \frac{1}{2}

or

u = 2

.

Multiplying through by

2u

clears fractions and forms a standard quadratic equation.

3

Select the valid root using given inequality constraints

Since

x > y > 1

, taking the logarithm base

x

yields

\log_x x > \log_x y \implies 1 > \log_x y

. Thus

u = \frac{1}{2}

, which means

y = x^{1/2}

or

x = y^2

.

The condition

x > y

restricts the logarithm of

y

base

x

to be strictly less than

1

.

4

Substitute into the product equation to find

x

and

y

Substituting

x = y^2

into

xy = 64

gives

y^3 = 64 \implies y = 4

. Consequently,

x = 4^2 = 16

.

Combining the relation

x = y^2

with

xy = 64

enables single-variable cubic solution.

5

Calculate the required difference

x - y

16 - 4 = 12

.

Direct subtraction of the derived values

x = 16

and

y = 4

.

Key Concept

Logarithmic Change of Base Reciprocal Property

Question 103Question →

A binary operation $\Delta$ defined on the set of integers modulo $11$ is given by $a \Delta b \equiv (2a^2 - 3ab + b^2) \pmod{11}$ . Find the smallest non-negative integer $x$ that satisfies the equation $4 \Delta x \equiv 5 \pmod{11}$ .

Show answer & explanation

Answer: 3

Answer

The smallest non-negative integer

x

that satisfies

4 \Delta x \equiv 5 \pmod{11}

is 3.

Evaluating

4 \Delta x

gives

32 - 12x + x^2 \equiv x^2 - x + 10 \pmod{11}

. Setting this congruent to

5 \pmod{11}

yields

x^2 - x + 5 \equiv 0 \pmod{11}

, which converts to

x^2 - x - 6 \equiv 0 \pmod{11}

. Factoring gives

(x - 3)(x + 2) \equiv 0 \pmod{11}

, which yields solutions

x \equiv 3 \pmod{11}

and

x \equiv 9 \pmod{11}

. The smallest non-negative integer among these solutions is

3

.

Step-by-Step Solution

1

Substitute the given value

a = 4

into the operation definition.

4 \Delta x = 2(4)^2 - 3(4)x + x^2 = 32 - 12x + x^2

This establishes the explicit algebraic polynomial in terms of

x

.

2

Reduce coefficients modulo

11

.

32 \equiv 10 \pmod{11}

and

-12x \equiv -x \pmod{11}

, giving

x^2 - x + 10 \pmod{11}

.

Simplifying coefficients reduces computational complexity during equation solving.

3

Form the modular quadratic equation and set it to zero.

x^2 - x + 10 \equiv 5 \pmod{11} \implies x^2 - x + 5 \equiv 0 \pmod{11} \implies x^2 - x - 6 \equiv 0 \pmod{11}

.

Expressing

5 \equiv -6 \pmod{11}

allows standard integer factorisation.

4

Factor the quadratic polynomial and solve for

x

.

(x - 3)(x + 2) \equiv 0 \pmod{11} \implies x \equiv 3 \text{ or } x \equiv -2 \equiv 9 \pmod{11}

.

Since

11

is prime, a product congruent to

0 \pmod{11}

implies at least one factor is congruent to

0 \pmod{11}

.

5

Select the smallest non-negative integer from the valid solution set

\{3, 9\}

.

x = 3

3

is non-negative and strictly smaller than

9

.

Key Concept

Modular Arithmetic Binary Operations and Quadratic Congruences

Estimated Time:3m 0s

Question 104Question →

What is the value of $\log_2 81 \cdot \log_3 32$ ?

Show answer & explanation

Answer: 20

Answer

The value of the logarithmic expression is 20.

Rewriting

81

as

3^4

and

32

as

2^5

allows extraction of coefficients

4

and

5

. Utilizing the change of base property

\log_2 3 \cdot \log_3 2 = 1

reduces the expression to

4 \times 5 = 20

.

Step-by-Step Solution

1

Rewrite arguments as prime powers

\log_2 81 = \log_2(3^4)

and

\log_3 32 = \log_3(2^5)

Simplifies terms to prime base components

2

Apply the power law of logarithms

4\log_2 3 \cdot 5\log_3 2

Brings powers out as numerical coefficients

3

Apply the change of base identity

\log_b a \cdot \log_a b = 1

4 \cdot 5 \cdot (\log_2 3 \cdot \log_3 2) = 20 \cdot 1 = 20

Reciprocal logarithm bases multiply to give 1

Key Concept

Logarithms and Change of Base

Estimated Time:45s

Question 105Question →

Given that

x

and

y

are real numbers satisfying the simultaneous exponential equations

3^x \times 9^y = 81

and

8^x \times 4^y = 256

find the value of

x^2 + y^2

.

Show answer & explanation

Answer: 5

Answer

The value of

x^2 + y^2

is 5.

Converting all terms to their prime bases (

3

for the first equation and

2

for the second equation) produces the simultaneous linear equations

x + 2y = 4

and

3x + 2y = 8

. Solving these gives

x = 2

and

y = 1

. Substituting these values into

x^2 + y^2

gives

2^2 + 1^2 = 5

.

Step-by-Step Solution

1

Convert all terms in the first equation to powers of base 3.

x + 2y = 4

Since

9 = 3^2

and

81 = 3^4

, applying the product law of indices

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

gives

3^{x+2y} = 3^4

. Equating the exponents gives

x + 2y = 4

.

2

Convert all terms in the second equation to powers of base 2.

3x + 2y = 8

Since

8 = 2^3

,

4 = 2^2

, and

256 = 2^8

, applying the laws of indices yields

2^{3x} \times 2^{2y} = 2^8 \implies 2^{3x+2y} = 2^8

. Equating exponents gives

3x + 2y = 8

.

3

Solve the system of simultaneous linear equations for

x

and

y

.

x = 2

and

y = 1

Subtracting

x + 2y = 4

from

3x + 2y = 8

yields

2x = 4 \implies x = 2

. Substituting

x = 2

into

x + 2y = 4

gives

2 + 2y = 4 \implies y = 1

.

4

Evaluate the target expression

x^2 + y^2

.

5

Substitute

x = 2

and

y = 1

into

x^2 + y^2

to obtain

2^2 + 1^2 = 4 + 1 = 5

.

Key Concept

Converting exponential terms to common prime bases to reduce exponential equations into linear equations.

Question 106Question →

A student measured the width of a textbook as $8.2\text{ cm}$ instead of its actual width of $8.0\text{ cm}$ . What is the percentage error in the measurement?

0.2\%

2.44\%

2.5\%

20\%

Show answer & explanation

Answer: $2.5\%$

Answer

2.5\%

The correct answer is

2.5\%

. The absolute error is

0.2\text{ cm}

. Dividing the absolute error (

0.2\text{ cm}

) by the actual measurement (

8.0\text{ cm}

) yields

\frac{0.2}{8.0} = 0.025

. Expressing this decimal as a percentage gives

0.025 \times 100\% = 2.5\%

.

Step-by-Step Solution

1

Find the absolute error in measurement

\text{Error} = |8.2\text{ cm} - 8.0\text{ cm}| = 0.2\text{ cm}

Percentage error requires knowing the difference between measured value and true value.

2

Apply the percentage error formula

\text{Percentage Error} = \frac{\text{Error}}{\text{True Value}} \times 100\%

Percentage error is always calculated relative to the true (actual) value.

3

Substitute values and evaluate

\frac{0.2}{8.0} \times 100\% = \frac{1}{40} \times 100\% = 2.5\%

Simplifying

\frac{0.2}{8.0}

yields

\frac{1}{40}

, which equals

2.5\%

.

Key Concept

Percentage Error Calculation

Estimated Time:45s

Question 107Question →

What is the value of $(-17 \times 5) \pmod{6}$ expressed in standard non-negative remainder form?

-1

1

3

5

Show answer & explanation

Answer: $5$

Answer

5

Multiplying

-17

by

5

yields

-85

. When

-85

is divided by

6

, the closest multiple of

6

below

-85

is

-90

. The non-negative remainder is

-85 - (-90) = 5

, which belongs to the canonical set

\{0, 1, 2, 3, 4, 5\}

.

Step-by-Step Solution

1

Multiply the numbers inside the parentheses

-17 \times 5 = -85

Perform standard multiplication before applying the modulo operation.

2

Express the negative number in terms of the modulus 6

-85 = 6 \times (-15) + 5

Find the largest multiple of 6 less than or equal to -85, which is -90.

3

Extract the canonical non-negative remainder

5

In standard modular arithmetic, remainders must lie in the range

[0, n-1]

, where

n

is the modulus.

Key Concept

Modular Arithmetic and Non-Negative Remainders

Question 108Question →

Simplify the surd expression $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

\frac{7 + 2\sqrt{10}}{3}

\frac{\sqrt{7} + 2\sqrt{10}}{3}

\frac{7 + \sqrt{10}}{3}

\frac{7 + 2\sqrt{10}}{7}

Show answer & explanation

Answer: $\frac{7 + 2\sqrt{10}}{3}$

Answer

\frac{7 + 2\sqrt{10}}{3}

Multiplying both the numerator and denominator by the conjugate of the denominator,

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

, expands the numerator to

5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

and simplifies the denominator using the difference of squares to

5 - 2 = 3

, giving

\frac{7 + 2\sqrt{10}}{3}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

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

is

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

.

Rationalizing a binomial denominator requires multiplying by its conjugate to apply the difference of squares identity

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

.

2

Multiply both numerator and denominator by the conjugate

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

This maintains the value of the fraction while removing radical terms from the denominator.

3

Expand the numerator and denominator

Numerator:

(\sqrt{5})^2 + 2\sqrt{5}\sqrt{2} + (\sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

. Denominator:

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

.

Apply algebraic expansion

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

and difference of squares.

4

Combine terms to form the final simplified expression

\frac{7 + 2\sqrt{10}}{3}

The expression is now fully rationalized and in standard simplified surd form.

Key Concept

Rationalization of Binomial Denominators

Question 109Question →

Given that $\frac{\sqrt{x + 2} + \sqrt{x - 2}}{\sqrt{x + 2} - \sqrt{x - 2}} = 3$ , what is the value of $x$ ?

\frac{10}{3}

6

\frac{5}{2}

\frac{8}{3}

Show answer & explanation

Answer: $\frac{10}{3}$

Answer

The value of

x

is

\frac{10}{3}

.

The correct solution is found by cross-multiplying the equation to get

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

. Grouping similar surd terms gives

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

, which simplifies to

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

. Squaring both sides yields

4(x - 2) = x + 2

, simplifying to

4x - 8 = x + 2

, which gives

3x = 10

and

x = \frac{10}{3}

.

Step-by-Step Solution

1

Cross-multiply to clear the denominator

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

Clear the fraction to group like radical terms on opposite sides.

2

Rearrange and combine like terms

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

, which simplifies to

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

Isolate the radical expressions.

3

Square both sides of the equation

(2\sqrt{x - 2})^2 = (\sqrt{x + 2})^2 \implies 4(x - 2) = x + 2

Eliminate radicals by squaring both sides, ensuring the coefficient

2

is squared to

4

.

4

Solve the linear equation for

x

4x - 8 = x + 2 \implies 3x = 10 \implies x = \frac{10}{3}

Isolate

x

to find the final value.

Key Concept

Solving Radical and Surd Equations

Question 110Question →

If $27^{x - 1} = 9^{x + 1}$ , determine the value of $x$ .

Show answer & explanation

Answer: 5

Answer

The value of

x

is 5.

Rewriting 27 as

3^3

and 9 as

3^2

transforms the given equation into

3^{3(x - 1)} = 3^{2(x + 1)}

. Equating exponents gives

3x - 3 = 2x + 2

, which simplifies directly to

x = 5

.

Step-by-Step Solution

1

Express numbers in terms of a common base

(3^3)^{x - 1} = (3^2)^{x + 1}

Both 27 and 9 are powers of 3, allowing reduction to a single base.

2

Apply power of a power index law

3^{3x - 3} = 3^{2x + 2}

Multiply the base power by the expression in the exponent:

3 \times (x - 1) = 3x - 3

and

2 \times (x + 1) = 2x + 2

.

3

Equate exponents and solve for

x

3x - 3 = 2x + 2 \implies x = 5

Equal bases imply that the index powers must be equal.

Key Concept

Equating exponential expressions using a common base

Question 111Question →

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

Show answer & explanation

Answer: 13.6% decrease

Answer

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.

Step-by-Step Solution

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\%

.

Key Concept

Compound Percentage Change and Variable Multipliers

Question 112Question →

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

Show answer & explanation

Answer: 3

Answer

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.

Step-by-Step Solution

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.

Key Concept

Modular Arithmetic and Non-Negative Remainders

Question 113Question →

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

Show answer & explanation

Answer: 7

Answer

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

.

Step-by-Step Solution

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.

Key Concept

Expansion and conversion from Base x to Base 10

Estimated Time:45s

Question 114Question →

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

Show answer & explanation

Answer: 2

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

Solving exponential equations by expressing numbers with equal bases

Question 115Question →

If $1101_2 + 32_4 = x_5$ , what is the value of $x$ ?

Show answer & explanation

Answer: 102

Answer

The value of

x

is

102

.

To add numbers given in different bases, convert each term to base 10:

1101_2 = 13_{10}

and

32_4 = 14_{10}

. Summing these decimal values gives

13 + 14 = 27_{10}

. Converting

27_{10}

to base 5 by repeated division by 5 gives remainders 2, 0, 1 (from least significant to most significant digit), resulting in

102_5

. Therefore,

x = 102

.

Step-by-Step Solution

1

Convert

1101_2

to decimal (base 10)

13_{10}

Expand in powers of 2:

1(2^3) + 1(2^2) + 0(2^1) + 1(2^0) = 8 + 4 + 0 + 1 = 13

.

2

Convert

32_4

to decimal (base 10)

14_{10}

Expand in powers of 4:

3(4^1) + 2(4^0) = 12 + 2 = 14

.

3

Add the two base-10 numbers together

27_{10}

Perform standard addition:

13 + 14 = 27

.

4

Convert

27_{10}

into base 5

102_5

Divide 27 repeatedly by 5 and record remainders:

27 = 5 \times 5 + 2

,

5 = 1 \times 5 + 0

,

1 = 0 \times 5 + 1

. Reading remainders upward yields

102_5

.

Key Concept

Conversion between different number bases via base 10

Question 116Question →

What is the simplified form of the surd expression $\frac{4}{\sqrt{5} - 1}$ after rationalizing the denominator?

1 + \sqrt{5}

2

4 + 4\sqrt{5}

\sqrt{5} - 1

Show answer & explanation

Answer: $1 + \sqrt{5}$

Answer

1 + \sqrt{5}

The expression

1 + \sqrt{5}

is correct because multiplying the numerator and denominator by the conjugate

\sqrt{5} + 1

converts the denominator to

(\sqrt{5})^2 - 1^2 = 4

. Canceling the common factor of

4

in numerator and denominator simplifies the expression completely to

1 + \sqrt{5}

.

Step-by-Step Solution

1

Identify the conjugate of the denominator

The conjugate of

\sqrt{5} - 1

is

\sqrt{5} + 1

.

Multiplying a binomial surd by its conjugate eliminates the radical in the denominator using the difference of two squares identity

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

.

2

Multiply both the numerator and the denominator by the conjugate

\frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{4(\sqrt{5} + 1)}{(\sqrt{5})^2 - (1)^2}

Multiplying both numerator and denominator by the same expression preserves the value of the fraction.

3

Simplify the denominator and evaluate the fraction

\frac{4(\sqrt{5} + 1)}{5 - 1} = \frac{4(\sqrt{5} + 1)}{4} = 1 + \sqrt{5}

Dividing the numerator by

4

cancels out the factor of

4

.

Key Concept

Rationalization of Binomial Denominators

Question 117Question →

If $x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ and $y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$ , determine the numerical value of $x^2 + y^2$ .

Show answer & explanation

Answer: 23

Answer

The numerical value of

x^2 + y^2

is

23

.

Rationalizing

x

yields

\frac{5 + \sqrt{21}}{2}

and rationalizing

y

yields

\frac{5 - \sqrt{21}}{2}

. The sum

x + y

equals

5

and the product

xy

equals

1

. Substituting these into

x^2 + y^2 = (x + y)^2 - 2xy

gives

5^2 - 2(1) = 23

.

Step-by-Step Solution

1

Rationalize the denominators of

x

and

y

x = \frac{5 + \sqrt{21}}{2}

and

y = \frac{5 - \sqrt{21}}{2}

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

2

Calculate the sum

x + y

and the product

xy

x + y = 5

and

xy = 1

Summing conjugate surd expressions cancels the radical term, and multiplying them applies the difference of two squares.

3

Evaluate

x^2 + y^2

using the identity

(x + y)^2 - 2xy

x^2 + y^2 = 5^2 - 2(1) = 23

Substituting the known sum and product avoids having to square complex surd expressions directly.

Key Concept

Rationalization of binomial denominators and application of symmetric algebraic identities.

Question 118Question →

If $1.41_x = \frac{46}{25}_{10}$ , where $x$ is a positive integer base, find the value of $x$ .

Show answer & explanation

Answer: 5

Answer

The value of the base

x

is 5.

Expanding

1.41_x

yields

1 + \frac{4}{x} + \frac{1}{x^2}

. Setting this equal to

\frac{46}{25}

gives

\frac{4x + 1}{x^2} = \frac{21}{25}

. Cross-multiplying results in the quadratic equation

21x^2 - 100x - 25 = 0

, which factors as

(21x + 5)(x - 5) = 0

. Since a number base must be a positive integer greater than 4,

x = 5

is the only valid solution.

Step-by-Step Solution

1

Expand

1.41_x

using place value powers of

x

.

1 + \frac{4}{x} + \frac{1}{x^2}

Fractional digits to the right of the radix point represent negative powers of the base (

x^{-1}, x^{-2}, \dots

).

2

Equate the expanded form to

\frac{46}{25}

and simplify.

\frac{4x + 1}{x^2} = \frac{21}{25}

Subtracting 1 from both sides isolates the fractional place values.

3

Cross-multiply and solve the quadratic equation

21x^2 - 100x - 25 = 0

.

(21x + 5)(x - 5) = 0 \implies x = 5

A base must be a positive integer strictly greater than any individual digit in the number (digits present are 1 and 4).

Key Concept

Conversion of fractional numbers in non-decimal bases to base 10 and solving polynomial equations in unknown bases.

Question 119Question →

If the surd expression $\frac{12}{\sqrt{6}}$ is simplified to the form $k\sqrt{6}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

is 2.

Multiplying the numerator and denominator of

\frac{12}{\sqrt{6}}

by

\sqrt{6}

yields

\frac{12\sqrt{6}}{6}

. Dividing

12

by

6

simplifies the expression to

2\sqrt{6}

. Matching this with

k\sqrt{6}

gives

k = 2

.

Step-by-Step Solution

1

Rationalize the denominator of

\frac{12}{\sqrt{6}}

by multiplying the numerator and denominator by

\sqrt{6}

.

\frac{12\sqrt{6}}{6}

Multiplying by

\frac{\sqrt{6}}{\sqrt{6}}

removes the radical from the denominator without changing the value of the expression.

2

Divide the integer coefficient in the numerator by the denominator.

2\sqrt{6}

Simplifying

\frac{12}{6}

yields 2.

3

Compare

2\sqrt{6}

with

k\sqrt{6}

to determine the value of

k

.

k = 2

The coefficient of

\sqrt{6}

is 2.

Key Concept

Rationalization of monomial surd denominators

Question 120Question →

A student calculated the value of $\frac{0.048}{0.006}$ as $8.5$ . Calculate the percentage error in the student's calculation.

Show answer & explanation

Answer: 6.25

Answer

The percentage error is

6.25\%

.

Evaluating

0.048 \div 0.006

yields a true value of

8

. The error in estimation is

|8.5 - 8| = 0.5

. Dividing this error by the true value of

8

and multiplying by

100\%

gives a percentage error of

6.25\%

.

Step-by-Step Solution

1

Calculate the true value of the expression

True value =

\frac{0.048}{0.006} = 8

The exact result is required to establish the baseline for percentage error calculation.

2

Determine the magnitude of the error

Error =

|8.5 - 8| = 0.5

Error is defined as the absolute difference between the estimated value and the true value.

3

Compute the percentage error

Percentage error =

\frac{0.5}{8} \times 100\% = 6.25\%

Percentage error expresses the error as a percentage of the true value.

Key Concept

Percentage Error

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