All practice questions

Question 7301Question →

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 7302Question →

The test scores of six students in a mathematics quiz are $12$ , $18$ , $25$ , $31$ , $47$ , and $50$ . What is the range of these scores?

38

50

30.5

28

Show answer & explanation

Answer: $38$

Answer

The range of the scores is

38

.

The range is a measure of dispersion defined as the difference between the maximum value (

50

) and the minimum value (

12

). Subtracting

12

from

50

gives

38

.

Step-by-Step Solution

1

Identify the maximum and minimum values in the dataset.

Maximum score =

50

, Minimum score =

12

.

Range measures the spread between the highest and lowest values in a distribution.

2

Calculate the range using the formula

\text{Range} = \text{Maximum} - \text{Minimum}

.

\text{Range} = 50 - 12 = 38

.

Subtracting the minimum value from the maximum value yields the measure of dispersion known as range.

Key Concept

Range of a Dataset

Question 7303Question →

Let the universal set be $\mathcal{U} = \{x \in \mathbb{Z} : 1 \le x \le 40\}$ . Subsets $P$ , $Q$ , and $R$ of $\mathcal{U}$ are defined as follows:
$P = \{x \in \mathcal{U} : x \text{ is a multiple of } 3\}$ ,
$Q = \{x \in \mathcal{U} : x \text{ is a multiple of } 4\}$ , and
$R = \{x \in \mathcal{U} : x \text{ is a perfect square\}}$ .
What is the cardinality of the set $(P \cup Q)' \cap R'$ ?

18

22

20

17

Show answer & explanation

Answer: 18

Answer

18

By De Morgan's Law, the expression

(P \cup Q)' \cap R'

is equivalent to

(P \cup Q \cup R)'

. The universal set contains 40 integers. The set

P \cup Q \cup R

consists of all integers from 1 to 40 that are multiples of 3, multiples of 4, or perfect squares, giving a total of 22 distinct elements. Subtracting this from 40 leaves 18 elements that belong to neither

P

,

Q

, nor

R

.

Step-by-Step Solution

1

Apply De Morgan's Laws to simplify the required set expression

(P \cup Q)' \cap R' = (P \cup Q \cup R)'

The intersection of the complements of two sets is equal to the complement of their union.

2

List and count the elements belonging to sets

P

,

Q

, and

R

within

\mathcal{U} = \{1, 2, \dots, 40\}

P

has 13 elements (multiples of 3);

Q

adds 7 elements not in

P

(multiples of 4 excluding 12, 24, 36);

R

adds 2 elements not in

P \cup Q

(

1

and

25

)

Counting unique elements avoids double-counting shared elements across sets.

3

Calculate the cardinality of

P \cup Q \cup R

n(P \cup Q \cup R) = 13 + 7 + 2 = 22

Summing the non-overlapping subsets gives the total size of the union.

4

Subtract the union cardinality from the universal set cardinality

n(\mathcal{U}) = 40

n((P \cup Q \cup R)') = 40 - 22 = 18

The cardinality of the complement is the size of the universal set minus the size of the set itself.

Key Concept

De Morgan's Laws and Set Complements

Question 7304Question →

What is the gradient of the normal to the curve $y = x^2 - 4x + 5$ at the point where $x = 3$ ?

-\frac{1}{2}

2

\frac{1}{2}

-2

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

The gradient of the normal to the curve is

-\frac{1}{2}

.

Differentiating

y = x^2 - 4x + 5

gives

\frac{dy}{dx} = 2x - 4

. Substituting

x = 3

yields a tangent gradient of

2

. Since the normal line is perpendicular to the tangent line, its gradient is the negative reciprocal,

-\frac{1}{2}

.

Step-by-Step Solution

1

Differentiate the equation of the curve to determine the gradient function.

\frac{dy}{dx} = 2x - 4

The first derivative represents the gradient of the tangent to the curve at any point

x

.

2

Evaluate the derivative at

x = 3

to find the tangent gradient

m_t

.

m_t = 2(3) - 4 = 2

Substituting

x = 3

gives the slope of the tangent line at the given point.

3

Calculate the gradient of the normal

m_n

using

m_n = -\frac{1}{m_t}

.

m_n = -\frac{1}{2}

The normal line is perpendicular to the tangent line, so its gradient is the negative reciprocal of the tangent's gradient.

Key Concept

Gradient of a Normal Line

Question 7305Question →

A solid right circular cylinder has a height of $10\text{ cm}$ and a total surface area of $192\pi\text{ cm}^2$ . What is the volume of the cylinder in $\text{cm}^3$ expressed in terms of $\pi$ ?

Show answer & explanation

Answer: 360

Answer

360

The total surface area of a closed cylinder is

A = 2\pi r^2 + 2\pi r h

. Given

A = 192\pi

and

h = 10\text{ cm}

, we have

2\pi r^2 + 20\pi r = 192\pi

. Dividing by

2\pi

yields

r^2 + 10r - 96 = 0

. Factoring the quadratic gives

(r + 16)(r - 6) = 0

, so

r = 6\text{ cm}

(since radius must be positive). Substituting

r = 6\text{ cm}

and

h = 10\text{ cm}

into the volume formula

V = \pi r^2 h

yields

V = \pi (6^2)(10) = 360\pi\text{ cm}^3

. The numerical coefficient of

\pi

is

360

.

Step-by-Step Solution

1

Set up the equation for the total surface area of the cylinder

2\pi r^2 + 2\pi r(10) = 192\pi

The total surface area of a solid cylinder comprises the areas of the top and bottom circular bases (

2\pi r^2

) plus the curved surface area (

2\pi r h

).

2

Simplify and solve the resulting quadratic equation for radius

r

r^2 + 10r - 96 = 0 \implies (r + 16)(r - 6) = 0 \implies r = 6\text{ cm}

Dividing the equation by

2\pi

yields

r^2 + 10r - 96 = 0

. Since radius must be positive,

r = 6\text{ cm}

.

3

Calculate the volume of the cylinder using the radius and height

V = \pi r^2 h = \pi (6)^2 (10) = 360\pi\text{ cm}^3

The volume of a cylinder is calculated using the formula

V = \pi r^2 h

.

Key Concept

Surface Area and Volume of 3D Solids

Question 7306Question →

A piece of wire of length $88\text{ cm}$ is bent to form the perimeter of a sector of a circle of radius $21\text{ cm}$ . What is the area of the sector in $\text{cm}^2$ ?

Show answer & explanation

Answer: 483

Answer

The area of the sector is

483\text{ cm}^2

.

The total length of the wire forms the perimeter of the sector, which consists of two radii and the arc length (

P = 2r + l

). With

P = 88\text{ cm}

and

r = 21\text{ cm}

, the arc length is

l = 88 - 2(21) = 46\text{ cm}

. Using the sector area formula

A = \frac{1}{2} r l

, the area is

\frac{1}{2} \times 21 \times 46 = 483\text{ cm}^2

.

Step-by-Step Solution

1

Determine the arc length of the sector from the given perimeter

l = 46\text{ cm}

The total perimeter of a sector includes two straight radii and the curved arc length:

P = 2r + l

. Subtracting twice the radius (

2 \times 21 = 42\text{ cm}

) from the total wire length of

88\text{ cm}

gives an arc length of

46\text{ cm}

.

2

Calculate the area of the sector

A = 483\text{ cm}^2

The area of a sector can be computed using the formula

A = \frac{1}{2} r l

. Substituting

r = 21\text{ cm}

and

l = 46\text{ cm}

yields

A = \frac{1}{2} \times 21 \times 46 = 483\text{ cm}^2

.

Key Concept

Perimeter and Area of a Sector of a Circle

Question 7307Question →

Evaluate the limit:

\lim_{x \to 2} \frac{x^3 - 8}{\sqrt{x + 2} - 2}

What is the numerical value of this limit?

Show answer & explanation

Answer: 48

Answer

The numerical value of the limit is 48.

Evaluating the limit of

\frac{x^3 - 8}{\sqrt{x + 2} - 2}

as

x \to 2

gives an indeterminate form

\frac{0}{0}

. Factorizing the numerator gives

(x - 2)(x^2 + 2x + 4)

, and rationalizing the denominator by multiplying numerator and denominator by

(\sqrt{x + 2} + 2)

converts the denominator to

x - 2

. Canceling

(x - 2)

leaves

(x^2 + 2x + 4)(\sqrt{x + 2} + 2)

. Evaluating at

x = 2

gives

(4 + 4 + 4)(\sqrt{4} + 2) = 12 \times 4 = 48

.

Step-by-Step Solution

1

Identify the limit form via direct substitution

Substituting

x = 2

yields

\frac{0}{0}

.

Direct evaluation results in an indeterminate form, requiring algebraic manipulation to eliminate the zero factor.

2

Factorize the numerator using the difference of cubes formula

x^3 - 8 = (x - 2)(x^2 + 2x + 4)

Exposing the factor

(x - 2)

is essential to resolving the zero denominator.

3

Rationalize the denominator using its algebraic conjugate

Multiply top and bottom by

(\sqrt{x + 2} + 2)

to get denominator

(x + 2) - 4 = x - 2

.

Applying

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

eliminates the square root from the denominator.

4

Cancel the common factor and compute the final value

\lim_{x \to 2} (x^2 + 2x + 4)(\sqrt{x + 2} + 2) = (12)(4) = 48.

With

(x - 2)

cancelled for

x \neq 2

, direct substitution now yields a defined real number.

Key Concept

Limits of Indeterminate Forms using Difference of Cubes and Surd Rationalization

Estimated Time:2m 30s

Question 7308Question →

If $y = 3e^{2x} + \ln(\cos x)$ , calculate the value of $\frac{dy}{dx}$ at $x = 0$ .

Show answer & explanation

Answer: 6

Answer

The value of the derivative of

y = 3e^{2x} + \ln(\cos x)

at

x = 0

is 6.

Differentiating each transcendental term individually using the chain rule yields

\frac{dy}{dx} = 6e^{2x} - \tan x

. Substituting

x = 0

gives

6e^0 - \tan 0 = 6 - 0 = 6

.

Step-by-Step Solution

1

Differentiate the exponential term

3e^{2x}

6e^{2x}

Applying the derivative rule for exponential functions

\frac{d}{dx}[a e^{kx}] = a k e^{kx}

.

2

Differentiate the logarithmic term

\ln(\cos x)

-\tan x

Applying the chain rule

\frac{d}{dx}[\ln(u)] = \frac{1}{u}\frac{du}{dx}

where

u = \cos x

gives

\frac{-\sin x}{\cos x} = -\tan x

.

3

Combine terms and evaluate at

x = 0

6

Substituting

x = 0

into

\frac{dy}{dx} = 6e^{2x} - \tan x

yields

6e^0 - \tan 0 = 6(1) - 0 = 6

.

Key Concept

Differentiation of Trigonometric, Exponential, and Logarithmic Functions

Question 7309Question →

Determine the number of integer values of $x$ that satisfy the quadratic inequality $3x^2 - 14x - 5 < 0$ .

Show answer & explanation

Answer: 5

Answer

The total number of integer values satisfying the inequality is 5.

Factoring

3x^2 - 14x - 5 < 0

gives

(3x + 1)(x - 5) < 0

. The critical roots are

x = -\frac{1}{3}

and

x = 5

. Since the parabola opens upward, the expression is negative between the roots, yielding the interval

-\frac{1}{3} < x < 5

. The integer values contained in this interval are

0, 1, 2, 3,

and

4

, which total 5 values.

Step-by-Step Solution

1

Factor the quadratic expression

(3x + 1)(x - 5) < 0

Factoring identifies the critical values where the quadratic expression changes sign.

2

Determine the critical points

x = -\frac{1}{3}

and

x = 5

Setting each factor to zero gives the boundary roots of the equation.

3

Formulate the solution interval

-\frac{1}{3} < x < 5

Since the coefficient of

x^2

is positive, the quadratic curve is below the x-axis strictly between the two roots.

4

List and count the integer solutions

The integers are

0, 1, 2, 3, 4

, yielding 5 integer solutions.

Counting integers strictly greater than

-\frac{1}{3}

and strictly less than

5

.

Key Concept

Solving quadratic inequalities and finding integer solution counts

Estimated Time:1m 30s

Question 7310Question →

A solid wooden block is shaped as a frustum of a right circular cone with a top base radius of $4\text{ cm}$ , a bottom base radius of $12\text{ cm}$ , and a vertical height of $15\text{ cm}$ . A cylindrical hole of radius $4\text{ cm}$ is drilled vertically through the center of the frustum from the top base straight down to the bottom base. What is the volume of the remaining wooden solid in $\text{cm}^3$ ?

800\pi\text{ cm}^3

2880\pi\text{ cm}^3

560\pi\text{ cm}^3

80\pi\text{ cm}^3

Show answer & explanation

Answer: $800\pi\text{ cm}^3$

Answer

The volume of the remaining wooden solid is

800\pi\text{ cm}^3

.

The volume of a frustum of a cone with radii

R=12\text{ cm}

,

r=4\text{ cm}

, and height

h=15\text{ cm}

is

\frac{1}{3}\pi(15)(12^2 + 4^2 + 12 \times 4) = 1040\pi\text{ cm}^3

. Subtracting the volume of the central cylindrical hole of radius

4\text{ cm}

and height

15\text{ cm}

, which is

\pi(4^2)(15) = 240\pi\text{ cm}^3

, yields

1040\pi - 240\pi = 800\pi\text{ cm}^3

.

Step-by-Step Solution

1

Calculate the total volume of the conical frustum before drilling.

V_{\text{frustum}} = \frac{1}{3}\pi h (R^2 + r^2 + R r) = \frac{1}{3}\pi (15)(12^2 + 4^2 + 12 \times 4) = 5\pi(144 + 16 + 48) = 1040\pi\text{ cm}^3

.

The total volume of a frustum of a cone is determined by its vertical height and the radii of its top and bottom circular bases.

2

Calculate the volume of the cylindrical hole drilled through the solid.

V_{\text{cylinder}} = \pi r_{\text{hole}}^2 h = \pi (4^2)(15) = 240\pi\text{ cm}^3

.

The drilled hole forms a right circular cylinder of radius

4\text{ cm}

and height equal to the full height of the frustum (

15\text{ cm}

).

3

Subtract the cylinder's volume from the frustum's volume to find the remaining volume.

V_{\text{remaining}} = V_{\text{frustum}} - V_{\text{cylinder}} = 1040\pi - 240\pi = 800\pi\text{ cm}^3

.

Removing material by drilling decreases the overall volume of the original solid by the exact volume of the cylindrical bore.

Key Concept

Volume of Composite Solids and Conical Frustums

Estimated Time:2m 30s

Question 7311Question →

The monthly operating cost $C$ (in naira) of a commercial power generator is partly constant and partly varies directly as the square of its daily operating time $h$ (in hours). When the generator operates for $3$ hours per day, the monthly cost is ₦ $5,500$ . When it operates for $5$ hours per day, the monthly cost is ₦ $13,500$ . What is the daily operating time, in hours, when the monthly operating cost is ₦ $33,000$ ?

8 hours

7 hours

9 hours

10 hours

Show answer & explanation

Answer: 8 hours

Answer

8 hours

The relationship is modeled by partial variation

C = k_1 + k_2 h^2

. Solving the simultaneous equations derived from

h = 3

(

C = 5500

) and

h = 5

(

C = 13500

) yields

k_2 = 500

and

k_1 = 1000

. Substituting

C = 33000

into

33000 = 1000 + 500 h^2

gives

500 h^2 = 32000

, so

h^2 = 64

, which yields

h = 8

hours.

Step-by-Step Solution

1

Set up the general formula for partial variation.

C = k_1 + k_2 h^2

, where

k_1

and

k_2

are constants.

The cost consists of a constant part (

k_1

) and a part that varies directly as the square of daily hours (

k_2 h^2

).

2

Form simultaneous linear equations using the given data points.

Equation (1):

5500 = k_1 + 9 k_2

Equation (2):

13500 = k_1 + 25 k_2

Substitute

h = 3, C = 5500

and

h = 5, C = 13500

into the variation equation.

3

Solve for the variation constants

k_1

and

k_2

.

Subtract Equation (1) from Equation (2):

8000 = 16 k_2 \implies k_2 = 500

.
Substitute

k_2 = 500

into Equation (1):

5500 = k_1 + 9(500) \implies k_1 = 1000

.

Eliminating

k_1

gives

k_2

, which is then used to find the constant part

k_1

.

4

Calculate

h

when

C = 33000

.

33000 = 1000 + 500 h^2 \implies 32000 = 500 h^2 \implies h^2 = 64 \implies h = 8

hours.

Substitute the values of

k_1, k_2,

and target

C

into the relation equation to solve for

h

.

Key Concept

Partial Variation with Simultaneous Equations

Estimated Time:2m 30s

Question 7312Question →

Find the smallest integer $x$ that satisfies the linear inequality $5 - 3x \le -7$ .

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

x

satisfying the inequality is

4

.

Subtracting 5 from both sides of

5 - 3x \le -7

gives

-3x \le -12

. Dividing both sides by

-3

requires reversing the inequality sign to obtain

x \ge 4

. Therefore, the smallest integer value in the solution set is 4.

Step-by-Step Solution

1

Subtract 5 from both sides of the inequality to isolate the variable term

-3x \le -12

Subtracting a constant from both sides maintains the inequality direction.

2

Divide both sides by

-3

and flip the inequality sign

x \ge 4

Dividing an inequality by a negative number reverses the direction of the inequality symbol.

3

Identify the smallest integer satisfying the condition

4

The solution set contains all real numbers greater than or equal to 4, making 4 the minimum integer value.

Key Concept

Solving Linear Inequalities with Negative Coefficients

Estimated Time:45s

Question 7313Question →

If $y = 2\sin(3x) + e^{4x}$ , what is the value of $\frac{dy}{dx}$ at $x = 0$ ?

Show answer & explanation

Answer: 10

Answer

10

Differentiating

y = 2\sin(3x) + e^{4x}

yields

\frac{dy}{dx} = 6\cos(3x) + 4e^{4x}

. Evaluating this expression at

x = 0

gives

6\cos(0) + 4e^{0} = 6(1) + 4(1) = 10

.

Step-by-Step Solution

1

Differentiate each term of

y = 2\sin(3x) + e^{4x}

with respect to

x

.

\frac{dy}{dx} = 6\cos(3x) + 4e^{4x}

Applying the chain rule gives

\frac{d}{dx}[2\sin(3x)] = 2 \times 3\cos(3x) = 6\cos(3x)

and

\frac{d}{dx}[e^{4x}] = 4e^{4x}

.

2

Evaluate the derivative at

x = 0

.

6\cos(0) + 4e^{0} = 6(1) + 4(1) = 10

Substituting

x = 0

gives

\cos(0) = 1

and

e^{0} = 1

.

Key Concept

Differentiation of trigonometric and exponential functions

Question 7314Question →

At what point on the curve $y = 2x^2 - 8x + 5$ is the normal line parallel to the straight line $x + 4y - 7 = 0$ ?

(3, -1)

(1, -1)

(3, 11)

(4, 5)

Show answer & explanation

Answer: $(3, -1)$

Answer

The point of contact on the curve is

(3, -1)

.

The line

x + 4y - 7 = 0

has a slope of

-\frac{1}{4}

. Since the normal line is parallel to this line, the normal gradient is

m_n = -\frac{1}{4}

. Consequently, the tangent gradient must be

m_t = 4

because

m_t \cdot m_n = -1

. Equating the derivative

\frac{dy}{dx} = 4x - 8

to

4

yields

x = 3

. Substituting

x = 3

into the curve equation

y = 2x^2 - 8x + 5

gives

y = -1

, yielding the point

(3, -1)

.

Step-by-Step Solution

1

Find the gradient of the given straight line.

Rearranging

x + 4y - 7 = 0

into slope-intercept form gives

y = -\frac{1}{4}x + \frac{7}{4}

, so the line's gradient is

m = -\frac{1}{4}

.

Parallel lines have equal gradients, so the normal line to the curve must have gradient

m_n = -\frac{1}{4}

.

2

Determine the required gradient of the tangent line.

Since

m_t \cdot m_n = -1

, we have

m_t = -\frac{1}{-\frac{1}{4}} = 4

.

The tangent and normal lines are perpendicular to each other.

3

Differentiate the curve's equation to find the

x

-coordinate.

\frac{dy}{dx} = 4x - 8

. Setting

\frac{dy}{dx} = 4

gives

4x - 8 = 4 \implies 4x = 12 \implies x = 3

.

The derivative represents the slope of the tangent line at any point

x

.

4

Substitute

x = 3

back into the curve's equation to find

y

.

y = 2(3)^2 - 8(3) + 5 = 2(9) - 24 + 5 = 18 - 24 + 5 = -1

.

The point of contact lies on the original curve.

Key Concept

Relationship between gradients of parallel lines, tangent lines, and normal lines to a curve

Question 7315Question →

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 7316Question →

The mean score of $9$ students in a mathematics test was calculated as $12$ . It was later discovered that a score of $5$ was incorrectly recorded as $14$ , and an additional student's score of $31$ was omitted entirely. What is the correct mean score of all $10$ students?

13

13.9

14.4

14.8

Show answer & explanation

Answer: $13$

Answer

The correct mean score of all 10 students is

13

.

The original sum of

9

scores is

108

. Subtracting the incorrect value (

14

) and adding the true value (

5

) reduces the sum of the

9

scores to

99

. Adding the

10^{\text{th}}

score of

31

gives a total sum of

130

. Dividing

130

by

10

gives the correct mean of

13

.

Step-by-Step Solution

1

Calculate the initial total sum of the original 9 scores.

Initial sum

= 9 \times 12 = 108

.

Mean is defined as total sum divided by number of items, so total sum equals mean times count.

2

Adjust the total sum for the misread score.

Corrected sum of 9 scores

= 108 - 14 + 5 = 99

.

Subtract the incorrect value (

14

) and add the actual value (

5

).

3

Add the omitted 10th score to the sum.

New total sum

= 99 + 31 = 130

.

Including the omitted score increases the total score sum.

4

Divide the new total sum by the updated total number of students.

Correct mean

= \frac{130}{10} = 13

.

The total number of students increased from 9 to 10.

Key Concept

Correcting the mean of ungrouped data after data entry errors or additions

Estimated Time:1m 30s

Question 7317Question →

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 7318Question →

Evaluate the limit:

\lim_{x \to \infty} (\sqrt{x^2 + 6x} - x)

What is the numerical value of this limit?

3

6

0

Undefined

Show answer & explanation

Answer: 3

Answer

3

To evaluate the limit of

\sqrt{x^2 + 6x} - x

as

x \to \infty

, multiply and divide by its conjugate

\sqrt{x^2 + 6x} + x

. The numerator simplifies to

(x^2 + 6x) - x^2 = 6x

. Dividing both the numerator and denominator by

x

yields

\frac{6}{\sqrt{1 + 6/x} + 1}

. Taking the limit as

x \to \infty

reduces

\frac{6}{x}

to

0

, resulting in

\frac{6}{\sqrt{1} + 1} = 3

.

Step-by-Step Solution

1

Identify the indeterminate form

Direct evaluation gives

\infty - \infty

, which is an indeterminate form.

Substitution cannot be applied directly when subtracting infinite limits.

2

Multiply and divide by the algebraic conjugate

\lim_{x \to \infty} \frac{(\sqrt{x^2 + 6x} - x)(\sqrt{x^2 + 6x} + x)}{\sqrt{x^2 + 6x} + x} = \lim_{x \to \infty} \frac{(x^2 + 6x) - x^2}{\sqrt{x^2 + 6x} + x} = \lim_{x \to \infty} \frac{6x}{\sqrt{x^2 + 6x} + x}

The identity

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

eliminates the square root in the numerator.

3

Factor

x

out of the denominator

\lim_{x \to \infty} \frac{6x}{x \left(\sqrt{1 + \frac{6}{x}} + 1\right)} = \lim_{x \to \infty} \frac{6}{\sqrt{1 + \frac{6}{x}} + 1}

Dividing the numerator and denominator by

x

allows evaluation at infinity.

4

Compute the limit as

x \to \infty

Since

\lim_{x \to \infty} \frac{6}{x} = 0

, the expression becomes

\frac{6}{\sqrt{1 + 0} + 1} = \frac{6}{2} = 3

.

Terms with

x

in the denominator approach zero as

x

grows arbitrarily large.

Key Concept

Limits at infinity involving radical indeterminate forms of type

\infty - \infty

Estimated Time:2m 0s

Question 7319Question →

Find the sum of all integer values of $x$ that satisfy the system of inequalities $3x + 1 > 7$ and $x^2 - 4x - 21 \le 0$ .

Show answer & explanation

Answer: 25

Answer

25

Solving

3x + 1 > 7

gives

x > 2

. Solving

x^2 - 4x - 21 \le 0

gives

-3 \le x \le 7

. The combined condition is

2 < x \le 7

. The integer values satisfying this condition are 3, 4, 5, 6, and 7. The sum of these integers is

3 + 4 + 5 + 6 + 7 = 25

.

Step-by-Step Solution

1

Solve the linear inequality

3x + 1 > 7

.

x > 2

Subtracting 1 from both sides gives

3x > 6

, and dividing by 3 yields

x > 2

.

2

Solve the quadratic inequality

x^2 - 4x - 21 \le 0

.

-3 \le x \le 7

Factoring the quadratic expression gives

(x - 7)(x + 3) \le 0

. The roots are

-3

and

7

, and the quadratic curve is non-positive between these roots.

3

Determine the intersection of the solution sets

x > 2

and

-3 \le x \le 7

.

2 < x \le 7

The lower bound is determined by the strict linear constraint

x > 2

, while the upper bound is determined by the inclusive quadratic constraint

x \le 7

.

4

List the integer values of

x

within

2 < x \le 7

and compute their sum.

Integers are

3, 4, 5, 6, 7

; their sum is

25

.

The strict inequality excludes

2

, while the inclusive inequality includes

7

.

Key Concept

System of linear and quadratic inequalities

Question 7320Question →

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

Answer

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