Number and Numeration Practice Questions — JAMB UTME

Question 1Question →

If $(125)_x + (32)_x = (201)_x$ , where $x$ represents the base of the numerals, what is the value of $x$ ?

6

7

5

8

Show answer & explanation

Answer: 6

Answer

The base

x

is equal to

6

.

Expanding the base-

x

numbers into base 10 polynomials yields

(x^2 + 2x + 5) + (3x + 2) = 2x^2 + 1

. Simplifying this equation gives

x^2 - 5x - 6 = 0

, which factors as

(x - 6)(x + 1) = 0

. Since a number base must be positive and greater than the highest digit in the expression (which is 5), the only valid solution is 6.

Step-by-Step Solution

1

Expand all numbers in base

x

into positional power notation (base 10 equivalent)

(125)_x = 1 \cdot x^2 + 2 \cdot x^1 + 5 \cdot x^0 = x^2 + 2x + 5

,

(32)_x = 3 \cdot x^1 + 2 \cdot x^0 = 3x + 2

, and

(201)_x = 2 \cdot x^2 + 0 \cdot x^1 + 1 \cdot x^0 = 2x^2 + 1

Converting all terms to base 10 allows setup of a algebraic equation in

x

.

2

Substitute the expanded terms back into the original equation and simplify

(x^2 + 2x + 5) + (3x + 2) = 2x^2 + 1 \implies x^2 + 5x + 7 = 2x^2 + 1

Combining like terms on the left-hand side prepares the expression for quadratic rearrangement.

3

Rearrange into standard quadratic form

ax^2 + bx + c = 0

2x^2 - x^2 - 5x + 1 - 7 = 0 \implies x^2 - 5x - 6 = 0

Subtracting

(x^2 + 5x + 7)

from both sides sets the quadratic equation to zero.

4

Solve the quadratic equation by factoring and evaluate valid base conditions

(x - 6)(x + 1) = 0 \implies x = 6

or

x = -1

. Valid base:

x = 6

A number base must be a positive integer greater than any single digit present in the given numbers (digits up to 5 appear, so

x > 5

).

Key Concept

Conversion of numbers in arbitrary base

x

to base 10 via place-value expansion to solve polynomial equations.

Question 2Question →

What value of $x$ satisfies the surd equation $\sqrt{x + 7} - \sqrt{x} = 1$ ?

9

3

16

7

Show answer & explanation

Answer: $9$

Answer

The value of

x

that satisfies the equation is

9

.

By rearranging the equation to

\sqrt{x+7} = \sqrt{x} + 1

and squaring both sides, we get

x + 7 = x + 2\sqrt{x} + 1

. Subtracting

x + 1

from both sides gives

6 = 2\sqrt{x}

, which yields

\sqrt{x} = 3

. Squaring both sides produces

x = 9

, which correctly satisfies the original equation.

Step-by-Step Solution

1

Isolate one of the radical terms on one side of the equation.

\sqrt{x + 7} = \sqrt{x} + 1

Isolating a square root allows squaring both sides to eliminate the outer radical.

2

Square both sides of the equation.

x + 7 = (\sqrt{x} + 1)^2 = x + 2\sqrt{x} + 1

Expanding the right-hand side using

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

removes the radical from the left side.

3

Subtract

x

and

1

from both sides to isolate the remaining radical term.

6 = 2\sqrt{x} \implies \sqrt{x} = 3

Simplifying the linear terms leaves a simple square root equation.

4

Square both sides to find

x

.

x = 3^2 = 9

Squaring

\sqrt{x}

isolates

x

completely.

Key Concept

Solving Surd Equations by Isolating Radicals and Squaring

Question 3Question →

Given that $123_x = 313_5$ , what is the value of the base $x$ ?

6

7

8

9

Show answer & explanation

Answer: 8

Answer

8

Converting

313_5

to base 10 yields

3(25) + 1(5) + 3(1) = 83

. Expanding

123_x

yields

x^2 + 2x + 3

. Setting them equal produces

x^2 + 2x + 3 = 83

, which simplifies to

x^2 + 2x - 80 = 0

. Solving

(x-8)(x+10) = 0

gives

x = 8

since a base must be positive.

Step-by-Step Solution

1

Convert the right-hand side

313_5

to base 10.

3(5^2) + 1(5^1) + 3(5^0) = 75 + 5 + 3 = 83_{10}

Converting all terms to base 10 allows forming a standard algebraic equation.

2

Expand the left-hand side

123_x

in powers of

x

.

1 \cdot x^2 + 2 \cdot x^1 + 3 \cdot x^0 = x^2 + 2x + 3

Expressing the number in terms of its base

x

positional values.

3

Set the two base 10 expressions equal and solve the quadratic equation.

x^2 + 2x + 3 = 83 \implies x^2 + 2x - 80 = 0 \implies (x + 10)(x - 8) = 0 \implies x = 8

The base

x

must be a positive integer greater than any digit in

123_x

(so

x > 3

), which leaves

x = 8

.

Key Concept

Solving equations with unknown number bases by expanding into base 10 polynomials.

Question 4Question →

If $\log_3 x - 2 \log_x 3 = 1$ , what is the sum of all valid solutions for $x$ ?

\frac{28}{3}

12

1

9

Show answer & explanation

Answer: $\frac{28}{3}$

Answer

The sum of all valid solutions for

x

is

\frac{28}{3}

.

By using the change of base identity

\log_x 3 = \frac{1}{\log_3 x}

and substituting

u = \log_3 x

, the given equation simplifies to

u - \frac{2}{u} = 1

. Multiplying by

u

gives the quadratic equation

u^2 - u - 2 = 0

, which factors as

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

. This yields

u = 2

or

u = -1

. Converting back to

x

gives

x = 3^2 = 9

and

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

. Both values satisfy domain constraints for logarithmic base (

x > 0

and

x \neq 1

). Adding these values gives

9 + \frac{1}{3} = \frac{28}{3}

.

Step-by-Step Solution

1

Apply the change of base formula to express the equation with a common logarithmic base

Using

\log_x 3 = \frac{1}{\log_3 x}

, the equation becomes

\log_3 x - \frac{2}{\log_3 x} = 1

.

Change of base allows all terms containing

x

to be represented in terms of a single logarithmic expression.

2

Substitute

u = \log_3 x

and clear the fraction to form a quadratic equation

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

.

Multiplying through by

u

(where

u \neq 0

) transforms the logarithmic relationship into a standard quadratic algebraic equation.

3

Factor the quadratic equation to find the values of

u

(u - 2)(u + 1) = 0 \implies u = 2 \text{ or } u = -1

.

Factoring isolates the linear roots for the substituted variable

u

.

4

Solve for

x

from

u = \log_3 x

and verify validity within the domain

For

u = 2

:

\log_3 x = 2 \implies x = 3^2 = 9

.
For

u = -1

:

\log_3 x = -1 \implies x = 3^{-1} = \frac{1}{3}

.
Both solutions are positive and

x \neq 1

, so both are valid.

Converting from logarithmic form to exponential form retrieves the original variable

x

.

5

Calculate the sum of all valid solutions

Sum =

9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3}

.

The question requires finding the total sum of all permissible real solutions for

x

.

Key Concept

Logarithmic Change of Base and Equations Reducible to Quadratics

Question 5Question →

Find the smallest non-negative integer $x$ that satisfies the linear modular congruence $3x \equiv 8 \pmod{11}$ .

Show answer & explanation

Answer: 10

Answer

The smallest non-negative integer

x

is 10.

Evaluating

3x \equiv 8 \pmod{11}

by testing multiples of

11

added to

8

gives

30

, which divided by

3

yields

x = 10

. Since

10 \in [0, 10]

, it is the canonical solution.

Step-by-Step Solution

1

Convert the modular congruence into an algebraic equation

3x = 8 + 11k

for an integer

k

By definition of congruence modulo

11

,

3x - 8

must be a multiple of

11

.

2

Find the smallest integer

k \ge 0

such that

8 + 11k

is divisible by 3

When

k = 2

,

8 + 11(2) = 30

30

is divisible by

3

(

30 / 3 = 10

).

3

Divide by 3 to isolate

x

x = 10

3(10) = 30 \equiv 8 \pmod{11}

.

Key Concept

Linear Modular Congruence

Estimated Time:1m 15s

Question 6Question →

Find the real value of $x$ that satisfies the exponential equation $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$ .

Show answer & explanation

Answer: 1.5

Answer

The value of

x

is

1.5

(or

\frac{3}{2}

).

By using index laws to group base-2 terms on the left side and base-3 terms on the right side, we obtain

2^{2x}\left(\frac{3}{2}\right) = 3^x\left(\frac{4}{\sqrt{3}}\right)

. Rearranging gives

\left(\frac{4}{3}\right)^x = \frac{8}{3\sqrt{3}} = \left(\frac{4}{3}\right)^{\frac{3}{2}}

. Equating indices gives

x = 1.5

.

Step-by-Step Solution

1

Group like exponential terms with base 2 and base 3 on opposite sides of the equation.

4^x + 2^{2x - 1} = 3^{x + \frac{1}{2}} + 3^{x - \frac{1}{2}}

Grouping terms with common prime bases allows for factoring exponential terms.

2

Apply the product and power laws of indices:

4^x = 2^{2x}

,

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

,

3^{x \pm \frac{1}{2}} = 3^x \cdot 3^{\pm \frac{1}{2}}

.

2^{2x} + \frac{1}{2} \cdot 2^{2x} = 3^x \cdot \sqrt{3} + 3^x \cdot \frac{1}{\sqrt{3}}

Separating the variable exponents from constant exponents prepares each side for factoring.

3

Factor out

2^{2x}

from the left side and

3^x

from the right side, then simplify arithmetic terms.

2^{2x}\left(\frac{3}{2}\right) = 3^x\left(\frac{4}{\sqrt{3}}\right)

Factoring isolates the variable terms

2^{2x}

and

3^x

from numerical constants.

4

Divide to form the ratio

\frac{4^x}{3^x} = \left(\frac{4}{3}\right)^x

and simplify the numerical fraction on the right.

\left(\frac{4}{3}\right)^x = \frac{8}{3\sqrt{3}}

Expressing both sides with unified variable bases facilitates solving for

x

by equating powers.

5

Rewrite

\frac{8}{3\sqrt{3}}

as a power of

\frac{4}{3}

and solve for

x

.

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{\frac{3}{2}} \implies x = \frac{3}{2} = 1.5

Since

\frac{8}{3\sqrt{3}} = \frac{4^{3/2}}{3^{3/2}} = (\frac{4}{3})^{3/2}

, equating exponents yields

x = 1.5

.

Key Concept

Solving mixed-base exponential equations by grouping, factoring, and converting to a unified base ratio.

Estimated Time:3m 0s

Question 7Question →

If the matrix $P = \begin{pmatrix} k & 3 & 1 \\ 2 & 1 & k \\ 4 & 2 & 0 \end{pmatrix}$ is singular, find the non-zero value of $k$ .

Show answer & explanation

Answer: 6

Answer

The non-zero value of

k

is 6.

For matrix P to be singular, its determinant must be 0. Expanding along row 3 yields 4(3k - 1) - 2(k^2 - 2) = 12k - 4 - 2k^2 + 4 = -2k^2 + 12k = 0. Factoring gives -2k(k - 6) = 0, which yields k = 0 or k = 6. The non-zero value is 6.

Step-by-Step Solution

1

Set the determinant of matrix P to 0

\det(P) = 0

A matrix is singular if and only if its determinant equals zero.

2

Evaluate the 3x3 determinant by expanding along the third row

4 \cdot (3k - 1) - 2 \cdot (k^2 - 2) + 0 = 0

Expanding along the third row takes advantage of the zero entry to simplify computation.

3

Expand and combine like terms

-2k^2 + 12k = 0

12k - 4 - 2k^2 + 4 reduces to -2k^2 + 12k.

4

Factor out common factors and solve for k

-2k(k - 6) = 0 \implies k = 0 \text{ or } k = 6

Applying the zero-product property.

5

Select the required root

k = 6

The question specifies the non-zero value of k.

Key Concept

Determinant of a 3x3 matrix and singular matrix condition

Estimated Time:2m 0s

Question 8Question →

If $\log_2 x + \log_4 x + \log_{16} x = \frac{21}{4}$ , find the value of $x$ .

Show answer & explanation

Answer: 8

Answer

8

Applying the change of base formula

\log_b x = \frac{\log_2 x}{\log_2 b}

allows

\log_4 x

and

\log_{16} x

to be rewritten as

\frac{1}{2}\log_2 x

and

\frac{1}{4}\log_2 x

. Summing

(1 + \frac{1}{2} + \frac{1}{4})\log_2 x

yields

\frac{7}{4}\log_2 x = \frac{21}{4}

, which simplifies to

\log_2 x = 3

. Converting to exponential form gives

x = 2^3 = 8

.

Step-by-Step Solution

1

Express all logarithmic terms in terms of base 2 using the change of base formula

\log_4 x = \frac{1}{2}\log_2 x \text{ and } \log_{16} x = \frac{1}{4}\log_2 x

Converting all terms to a common base allows for algebraic simplification.

2

Substitute the expressions back into the equation and factor out \log_2 x

\left(1 + \frac{1}{2} + \frac{1}{4}\right)\log_2 x = \frac{7}{4}\log_2 x = \frac{21}{4}

Combining the fractional coefficients simplifies the left side of the equation.

3

Solve for \log_2 x and evaluate x using the definition of logarithm

\log_2 x = 3 \implies x = 2^3 = 8

Multiplying both sides by \frac{4}{7} isolates \log_2 x, and converting to exponential form gives the value of x.

Key Concept

Logarithms and Change of Base

Question 9Question →

Given that $\log_a b = 2$ and $\log_b c = 3$ , what is the value of $\log_{a b c} (a^3 b^2 c)$ ?

\frac{13}{9}

\frac{3}{2}

\frac{11}{9}

\frac{11}{6}

Show answer & explanation

Answer: $\frac{13}{9}$

Answer

\frac{13}{9}

Using the change of base chain rule,

\log_a c = \log_a b \cdot \log_b c = 2 \times 3 = 6

. Changing the base of the target expression to base

a

gives

\frac{\log_a (a^3 b^2 c)}{\log_a (a b c)}

. Expanding both terms using product and power rules gives numerator

3(1) + 2(2) + 6 = 13

and denominator

1 + 2 + 6 = 9

, resulting in

\frac{13}{9}

.

Step-by-Step Solution

1

Express

\log_a c

using the change of base relationship.

\log_a c = \log_a b \cdot \log_b c = 2 \times 3 = 6

By the change of base rule (chain rule of logarithms),

\log_a b \cdot \log_b c = \log_a c

.

2

Apply the change of base formula to convert

\log_{a b c} (a^3 b^2 c)

to base

a

.

\log_{a b c} (a^3 b^2 c) = \frac{\log_a (a^3 b^2 c)}{\log_a (a b c)}

The change of base formula states that

\log_B X = \frac{\log_a X}{\log_a B}

.

3

Expand the numerator and denominator using logarithmic product and power rules.

Numerator:

3\log_a a + 2\log_a b + \log_a c = 3(1) + 2(2) + 6 = 13

. Denominator:

\log_a a + \log_a b + \log_a c = 1 + 2 + 6 = 9

.

\log_a (X Y Z) = \log_a X + \log_a Y + \log_a Z

and

\log_a (X^k) = k \log_a X

.

4

Divide the expanded numerator by the denominator.

\frac{13}{9}

Substituting the computed values yields the simplified fraction

\frac{13}{9}

.

Key Concept

Change of Base Rule and Logarithmic Expansion Laws

Estimated Time:2m 0s

Question 10Question →

Given that $245_x - 157_x = 66_x$ , find the value of the base $x$ .

Show answer & explanation

Answer: 8

Answer

The value of the base

x

is 8.

Expanding all terms in powers of

x

gives

2x^2 + 4x + 5 - (x^2 + 5x + 7) = 6x + 6

. Simplifying gives

x^2 - 7x - 8 = 0

, which factors into

(x - 8)(x + 1) = 0

. Since a number base must be a positive integer greater than any digit present in the equation (the maximum digit here is 7), the only valid base is

x = 8

.

Step-by-Step Solution

1

Convert each positional number into its base 10 polynomial expansion.

245_x = 2x^2 + 4x + 5

,

157_x = x^2 + 5x + 7

, and

66_x = 6x + 6

.

A number

d_2 d_1 d_0

in base

x

represents

d_2 x^2 + d_1 x^1 + d_0 x^0

in base 10.

2

Substitute the expanded expressions into the given subtraction equation.

(2x^2 + 4x + 5) - (x^2 + 5x + 7) = 6x + 6

This translates the base

x

relationship into a standard base 10 equation.

3

Simplify and rearrange into standard quadratic form.

x^2 - 7x - 8 = 0

Expanding the subtraction yields

x^2 - x - 2 = 6x + 6

. Subtracting

(6x + 6)

from both sides produces a quadratic set to zero.

4

Factorize the quadratic expression.

(x - 8)(x + 1) = 0 \implies x = 8 \text{ or } x = -1

Finding the roots provides potential mathematical values for

x

.

5

Apply number base constraints to select the valid root.

x = 8

A valid base must be a positive integer strictly greater than any individual digit in the expression. Since 7 appears in

157_x

,

x > 7

, ruling out

-1

and confirming

x = 8

.

Key Concept

Converting numbers from an unknown base

x

to base 10 polynomials to solve algebraic equations.

Question 11Question →

Evaluate the numerical value of each of the following sequence and series expressions, and arrange the items in ascending order (from smallest to largest value):

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is: the sum to infinity of

18 + 6 + 2 + \dots

(

27

), followed by the 4th term of the geometric progression (

48

), then the 15th term of the arithmetic progression (

59

), and finally the sum of the first 5 terms of the arithmetic progression (

70

).

Evaluating each expression gives values of

27

,

48

,

59

, and

70

. Arranging these from smallest to largest places the sum to infinity (

27

) first, the GP term (

48

) second, the 15th AP term (

59

) third, and the AP sum (

70

) last.

Step-by-Step Solution

1

Calculate the sum to infinity of the given geometric progression

For

18 + 6 + 2 + \dots

, first term

a = 18

and common ratio

r = \frac{6}{18} = \frac{1}{3}

. Using

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

, we get

S_\infty = \frac{18}{1 - 1/3} = \frac{18}{2/3} = 27

.

The sum to infinity formula for a convergent geometric series is

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

where

|r| < 1

.

2

Calculate the 4th term of the geometric progression

Given

T_2 = ar = 12

and

T_5 = ar^4 = 96

. Dividing

T_5

by

T_2

yields

r^3 = \frac{96}{12} = 8 \implies r = 2

. Thus

a = \frac{12}{2} = 6

. The 4th term is

T_4 = ar^3 = 6 \times 2^3 = 48

.

Using the

n

-th term formula of a GP,

T_n = ar^{n-1}

, to solve simultaneous equations for

a

and

r

.

3

Calculate the 15th term of the arithmetic progression

Given

T_3 = a + 2d = 11

and

T_8 = a + 7d = 31

. Subtracting the equations gives

5d = 20 \implies d = 4

. Substituting back yields

a = 11 - 2(4) = 3

. The 15th term is

T_{15} = a + 14d = 3 + 14(4) = 59

.

Using the

n

-th term formula of an AP,

T_n = a + (n-1)d

, to find the first term

a

and common difference

d

.

4

Calculate the sum of the first 5 terms of the arithmetic progression

With

a = 4

,

d = 5

, and

n = 5

,

S_5 = \frac{5}{2}[2(4) + (5-1)5] = \frac{5}{2}[8 + 20] = \frac{5}{2}(28) = 70

.

Applying the AP sum formula

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

.

5

Order the evaluated quantities from smallest to largest

Comparing numerical results

27 < 48 < 59 < 70

gives the sequence: item 1, item 2, item 4, item 3.

Sorting the calculated values in ascending numerical order.

Key Concept

Evaluation and comparison of terms and sums in Arithmetic and Geometric Progressions

Estimated Time:2m 30s

Question 12Question →

If $2^{2x + 1} - 9(2^x) + 4 = 0$ , what is the product of all real values of $x$ that satisfy the equation?

-2

2

1

-1

Show answer & explanation

Answer: $-2$

Answer

The product of all real values of

x

satisfying the equation is

-2

.

Rewriting

2^{2x+1}

as

2(2^x)^2

allows substitution of

u = 2^x

, giving

2u^2 - 9u + 4 = 0

. Solving for

u

gives

u = \frac{1}{2}

and

u = 4

. Converting back to

x

via

2^x = 2^{-1}

and

2^x = 2^2

yields

x = -1

and

x = 2

. The product of these roots is

(-1) \times 2 = -2

.

Step-by-Step Solution

1

Apply the law of indices to rewrite the first term.

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

. Thus, the equation becomes

2(2^x)^2 - 9(2^x) + 4 = 0

.

Splitting the index using

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

enables transformation into a quadratic form.

2

Substitute

u = 2^x

and solve the resulting quadratic equation.

2u^2 - 9u + 4 = 0 \implies (2u - 1)(u - 4) = 0 \implies u = \frac{1}{2} \text{ or } u = 4

.

Algebraic substitution simplifies the exponential equation into a standard quadratic equation.

3

Convert the values of

u

back to

x

using index laws.

For

u = \frac{1}{2}

,

2^x = 2^{-1} \implies x_1 = -1

. For

u = 4

,

2^x = 2^2 \implies x_2 = 2

.

Equating bases (

2^x = 2^k \implies x = k

) isolates the unknown variable

x

.

4

Compute the product of the solutions

x_1

and

x_2

.

Product =

x_1 \cdot x_2 = (-1) \times 2 = -2

.

The question specifically requests the product of all real values of

x

.

Key Concept

Solving quadratic-form exponential equations using index laws and variable substitution.

Estimated Time:2m 0s

Question 13Question →

If $\log_9 x + \log_3 x = 3$ , what is the value of $x$ ?

9

3

27

81

Show answer & explanation

Answer: $9$

Answer

The value of

x

is

9

.

Using the change of base formula,

\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{1}{2}\log_3 x

. Adding

\log_3 x

yields

\frac{3}{2}\log_3 x = 3

, which simplifies to

\log_3 x = 2

. Converting to exponential form gives

x = 3^2 = 9

.

Step-by-Step Solution

1

Express

\log_9 x

in base

3

using the change of base formula.

\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{\log_3 x}{2} = \frac{1}{2}\log_3 x

Converting all logarithmic terms to a common base (base 3) allows combining terms.

2

Substitute

\frac{1}{2}\log_3 x

into the given equation.

\frac{1}{2}\log_3 x + \log_3 x = 3 \implies \frac{3}{2}\log_3 x = 3

Combine like logarithmic terms.

3

Solve for

\log_3 x

.

\log_3 x = 3 \times \frac{2}{3} = 2

Multiply both sides by

\frac{2}{3}

to isolate

\log_3 x

.

4

Convert from logarithmic form to exponential form.

x = 3^2 = 9

By definition,

\log_b a = c

is equivalent to

b^c = a

.

Key Concept

Logarithms and Change of Base

Question 14Question →

If $\log_4(x + 2) - \log_{16}(x + 2) = 1$ , what is the value of $x$ ?

Show answer & explanation

Answer: 14

Answer

The value of

x

is 14.

Applying the change of base formula converts

\log_{16}(x + 2)

into

\frac{1}{2}\log_4(x + 2)

. The given equation reduces to

\frac{1}{2}\log_4(x + 2) = 1

, which means

\log_4(x + 2) = 2

. Writing this in exponential form yields

x + 2 = 4^2 = 16

, giving

x = 14

.

Step-by-Step Solution

1

Change the base of

\log_{16}(x + 2)

to base 4

\log_{16}(x + 2) = \frac{\log_4(x + 2)}{\log_4 16} = \frac{1}{2}\log_4(x + 2)

Using the change of base identity

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

allows all logarithmic terms to share base 4.

2

Substitute and simplify the expression on the left-hand side

\log_4(x + 2) - \frac{1}{2}\log_4(x + 2) = \frac{1}{2}\log_4(x + 2) = 1

Subtracting half of the term from the whole term leaves half of the term.

3

Isolate the logarithmic expression

\log_4(x + 2) = 2

Multiplying both sides of the equation by 2.

4

Convert from logarithmic to index form

x + 2 = 4^2 = 16

By definition,

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

.

5

Solve for the variable

x

x = 14

Subtracting 2 from both sides gives the final value of

x

.

Key Concept

Logarithms and Change of Base

Question 15Question →

If $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 1 \\ 0 & -2 \end{pmatrix}$ , what is the product matrix $AB$ ?

\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}

\begin{pmatrix} 10 & 3 \\ 0 & -8 \end{pmatrix}

\begin{pmatrix} 11 & 19 \\ -2 & -8 \end{pmatrix}

\begin{pmatrix} 10 & 8 \\ 5 & -7 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}$

Answer

The matrix product

AB

is

\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}

.

Matrix multiplication is evaluated using the row-by-column rule. Evaluating each entry: top-left is

2(5)+3(0)=10

, top-right is

2(1)+3(-2)=-4

, bottom-left is

1(5)+4(0)=5

, and bottom-right is

1(1)+4(-2)=-7

. Thus, the product matrix is

\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}

.

Step-by-Step Solution

1

Calculate the top-left element (Row 1 of A

\cdot

Column 1 of B)

2(5) + 3(0) = 10 + 0 = 10

Matrix multiplication takes the dot product of the corresponding row of the first matrix and column of the second matrix.

2

Calculate the top-right element (Row 1 of A

\cdot

Column 2 of B)

2(1) + 3(-2) = 2 - 6 = -4

Combine products of elements in row 1 of A and column 2 of B.

3

Calculate the bottom-left element (Row 2 of A

\cdot

Column 1 of B)

1(5) + 4(0) = 5 + 0 = 5

Combine products of elements in row 2 of A and column 1 of B.

4

Calculate the bottom-right element (Row 2 of A

\cdot

Column 2 of B)

1(1) + 4(-2) = 1 - 8 = -7

Combine products of elements in row 2 of A and column 2 of B.

Key Concept

Matrix Multiplication (Row-by-Column Rule)

Question 16Question →

If $214_x = 312_5$ , where $x$ represents a positive integer base, find the value of $x$ .

Show answer & explanation

Answer: 6

Answer

The value of the base

x

is 6.

Expanding

312_5

to base 10 gives

3(25) + 1(5) + 2(1) = 82

. Expanding

214_x

gives

2x^2 + x + 4

. Equating both expressions yields

2x^2 + x - 78 = 0

. Factoring as

(2x + 13)(x - 6) = 0

gives the positive integer root

x = 6

.

Step-by-Step Solution

1

Convert the right side of the equation from base 5 to base 10

312_5 = 3 \times 5^2 + 1 \times 5^1 + 2 \times 5^0 = 75 + 5 + 2 = 82_{10}

Converting known non-decimal bases to base 10 provides a standard baseline for algebraic manipulation.

2

Expand the left side expression in terms of powers of

x

214_x = 2x^2 + x + 4

Positional values in base

x

correspond to powers of

x

(

x^2, x^1, x^0

).

3

Formulate and rearrange the resulting quadratic equation in standard form

2x^2 + x + 4 = 82 \implies 2x^2 + x - 78 = 0

Setting the base 10 expansions equal forms a quadratic equation.

4

Solve the quadratic equation for positive integer values of

x

x = 6

Factoring

(2x + 13)(x - 6) = 0

gives

x = 6

or

x = -6.5

. A base must be a positive integer larger than all individual digits appearing in the number (digits are 2, 1, 4, so

x > 4

).

Key Concept

Converting non-decimal numbers to base 10 using positional expansion to solve polynomial equations involving unknown bases.

Question 17Question →

A container holds a liquid mixture where water constitutes $\frac{3}{8}$ of the total volume. When $15\text{ liters}$ of pure water is added to the mixture, water then accounts for $50\%$ of the new total volume. What was the initial total volume of the mixture in liters?

Show answer & explanation

Answer: 60

Answer

The initial total volume of the mixture was 60 liters.

The correct answer is 60 liters. By modeling the initial volume of water as

\frac{3}{8}V

, adding 15 liters yields a new water volume of

\frac{3}{8}V + 15

out of a total volume of

V + 15

. Setting

\frac{3}{8}V + 15 = 0.5(V + 15)

and solving gives

V = 60

.

Step-by-Step Solution

1

Define the variable and write an algebraic expression for the initial amount of water.

If

V

is the initial total volume in liters, initial water volume =

\frac{3}{8}V

.

Water makes up

\frac{3}{8}

of the total initial volume.

2

Account for the addition of 15 liters of pure water to both water volume and total volume.

New water volume =

\frac{3}{8}V + 15

; New total volume =

V + 15

.

Adding pure water increases both the specific water volume and the total mixture volume by 15 liters.

3

Formulate an equation relating new water volume to 50% of the new total volume.

\frac{3}{8}V + 15 = 0.5(V + 15)

Water constitutes 50% (or

\frac{1}{2}

) of the updated mixture.

4

Solve the linear equation for

V

.

7.5 = \frac{1}{8}V \implies V = 60

Subtracting

\frac{3}{8}V

and

7.5

from both sides isolates

\frac{1}{8}V

on one side.

Key Concept

Solving multi-step fraction and percentage mixture problems

Question 18Question →

What is the value of the expression $\frac{0.00048 \times 0.025}{0.0016}$ expressed in standard form?

7.5 \times 10^{-3}

7.5 \times 10^{-2}

7.5 \times 10^{-4}

1.2 \times 10^{-2}

Show answer & explanation

Answer: $7.5 \times 10^{-3}$

Answer

7.5 \times 10^{-3}

The expression evaluates step-by-step to

0.75 \times 10^{-2}

, which when written in standard form

A \times 10^n

(where

1 \le A < 10

) gives

7.5 \times 10^{-3}

.

Step-by-Step Solution

1

Convert each decimal in the expression to standard form (powers of 10).

0.00048 = 4.8 \times 10^{-4}

,

0.025 = 2.5 \times 10^{-2}

, and

0.0016 = 1.6 \times 10^{-3}

.

Converting decimals into standard form simplifies multiplication and division using laws of indices.

2

Simplify the numerator.

(4.8 \times 10^{-4}) \times (2.5 \times 10^{-2}) = (4.8 \times 2.5) \times 10^{-4 + (-2)} = 12 \times 10^{-6} = 1.2 \times 10^{-5}

.

Multiply the numerical coefficients together and add the exponents of 10.

3

Divide the numerator by the denominator.

\frac{1.2 \times 10^{-5}}{1.6 \times 10^{-3}} = \left(\frac{1.2}{1.6}\right) \times 10^{-5 - (-3)} = 0.75 \times 10^{-2}

.

Divide the coefficients and subtract the exponent of the denominator from that of the numerator.

4

Convert the result into standard form (

A \times 10^n

where

1 \le A < 10

).

0.75 \times 10^{-2} = (7.5 \times 10^{-1}) \times 10^{-2} = 7.5 \times 10^{-3}

.

Standard form requires the leading coefficient to be between 1 and 10.

Key Concept

Fractions, Decimals, and Standard Form Conversions

Question 19Question →

A trader buys a quantity of goods. He sells $\frac{1}{4}$ of the total goods at a profit of $20\%$ , and $\frac{1}{2}$ of the remaining goods at a loss of $10\%$ . If the rest of the goods are sold at cost price, resulting in an overall net profit of $\text{N}1,500$ , what was the total cost price of the goods in Naira?

Show answer & explanation

Answer: 120000

Answer

The total cost price of the goods is 120,000 Naira.

Let

C

be the total cost price of the goods. The first portion sold is

\frac{1}{4}C

at a profit of

20\%

, giving a gain of

0.20 \times \frac{1}{4}C = 0.05C

. The remaining portion is

1 - \frac{1}{4} = \frac{3}{4}C

. Half of this remainder is

\frac{1}{2} \times \frac{3}{4}C = \frac{3}{8}C

, which is sold at a

10\%

loss, causing a loss of

0.10 \times \frac{3}{8}C = 0.0375C

. The final remaining

\frac{3}{8}C

is sold at cost price (0 profit). Thus, the net profit is

0.05C - 0.0375C = 0.0125C

. Setting

0.0125C = 1,500

and solving for

C

gives

C = \frac{1,500}{0.0125} = 120,000

Naira.

Step-by-Step Solution

1

Define the unknown total cost price

Let

C

represent the total cost price of the goods in Naira.

Establishing a variable allows for algebraic formulation of profits and losses.

2

Calculate profit from the first portion

\text{Profit}_1 = 20\% \text{ of } \frac{1}{4}C = 0.20 \times 0.25C = +0.05C

The trader sells a quarter of the total value at a 20% gain.

3

Determine the remaining quantity and calculate loss from the second portion

\text{Remaining} = C - \frac{1}{4}C = \frac{3}{4}C; \quad \text{Second Portion} = \frac{1}{2} \times \frac{3}{4}C = \frac{3}{8}C = 0.375C; \quad \text{Loss}_2 = 10\% \text{ of } 0.375C = -0.0375C

The second sale applies to half of what was left after the first sale, incurred as a loss.

4

Equate net profit to given numerical value and solve for total cost price

\text{Net Profit} = 0.05C - 0.0375C = 0.0125C = 1,500 \implies C = \frac{1,500}{0.0125} = 120,000

The rest of the goods were sold at cost price (zero profit/loss), so net profit equals gain from portion 1 minus loss from portion 2.

Key Concept

Fractions and Percentages of Quantities

Estimated Time:1m 30s

Question 20Question →

A metal rod of length $12.5\text{ cm}$ is divided into two parts in the ratio $2 : 3$ . If the length of the smaller part is measured as $5.2\text{ cm}$ , what is the percentage error in the measurement of the smaller part?

4.0\%

3.8\%

1.6\%

2.7\%

Show answer & explanation

Answer: $4.0\%$

Answer

The percentage error in the measurement of the smaller part is

4.0\%

.

The true length of the smaller part of the rod is

\frac{2}{5} \times 12.5 = 5.0\text{ cm}

. Subtracting this from the measured length

5.2\text{ cm}

gives an error of

0.2\text{ cm}

. Dividing the error

0.2\text{ cm}

by the true value

5.0\text{ cm}

and multiplying by

100\%

yields

4.0\%

.

Step-by-Step Solution

1

Calculate the actual length of the smaller part using the given ratio.

Total ratio parts =

2 + 3 = 5

. Actual length of smaller part =

\frac{2}{5} \times 12.5\text{ cm} = 5.0\text{ cm}

.

The total length is shared according to the ratio

2 : 3

, so the smaller piece corresponds to 2 out of 5 equal parts.

2

Find the absolute error in measurement.

Absolute error =

|5.2\text{ cm} - 5.0\text{ cm}| = 0.2\text{ cm}

.

Absolute error is the magnitude of the difference between the measured value and the true value.

3

Compute the percentage error.

Percentage error =

\frac{\text{Absolute Error}}{\text{True Value}} \times 100\% = \frac{0.2}{5.0} \times 100\% = 4.0\%

.

Percentage error is defined as the absolute error divided by the true value, multiplied by 100%.

Key Concept

Ratio partitioning and Percentage Error calculation

Estimated Time:1m 30s

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