All practice questions

Evaluate the limit $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ . What is the numerical value of this limit?

Show answer & explanation

Answer: 4

Answer

The value of the limit is

4

.

Direct substitution of

x = 2

produces the indeterminate form

\frac{0}{0}

. Factoring the numerator gives

\frac{(x-2)(x+2)}{x-2}

. Canceling the non-zero factor

(x-2)

simplifies the expression to

x+2

. Evaluating the limit as

x

approaches

2

yields

2 + 2 = 4

.

Step-by-Step Solution

1

Check the form by direct substitution of

x = 2

Obtained the indeterminate form

\frac{0}{0}

Direct substitution results in division by zero, requiring algebraic simplification.

2

Factor the polynomial in the numerator

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

Difference of two squares factorization allows canceling common terms.

3

Cancel the common factor

(x - 2)

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

For

x \neq 2

, division by

(x - 2)

is valid.

4

Evaluate the simplified limit as

x \to 2

2 + 2 = 4

Substitute

x = 2

directly into the continuous polynomial

x + 2

.

Key Concept

Evaluating indeterminate limits of the form

\frac{0}{0}

via factorization

Question 142Question →

Three business partners share a total profit of ₦60,000 in the ratio $1 : 2 : 3$ . What is the share of the partner who receives the largest portion, in naira?

Show answer & explanation

Answer: 30000

Answer

The share of the partner receiving the largest portion is ₦30,000.

Sum the parts of the ratio (

1 + 2 + 3 = 6

). The largest share corresponds to 3 parts out of 6. Calculating

\frac{3}{6} \times 60,000

yields 30,000 naira.

Step-by-Step Solution

1

Calculate the sum of all ratio parts

1 + 2 + 3 = 6 parts

The total amount is divided into equal parts represented by the sum of the ratio numbers.

2

Determine the monetary value of a single part

₦60,000 / 6 = ₦10,000 per part

Dividing the total sum by the total number of parts gives the value of one unit part.

3

Calculate the largest share corresponding to 3 parts

3 × ₦10,000 = ₦30,000

The largest portion of the ratio is 3 parts.

Key Concept

Direct Ratio Sharing

Question 143Question →

Two independent events, $A$ and $B$ , have probabilities $P(A) = 0.4$ and $P(B) = 0.5$ . What is the probability that both event $A$ and event $B$ occur, $P(A \cap B)$ ?

Show answer & explanation

Answer: 0.2

Answer

The probability that both events occur is 0.2.

For independent events, the joint probability of both events occurring simultaneously is found by multiplying their individual probabilities:

P(A \cap B) = P(A) \times P(B) = 0.4 \times 0.5 = 0.2

.

Step-by-Step Solution

1

Identify event independence and the required probability operation

Events

A

and

B

are independent, and the question requires calculating their intersection

P(A \cap B)

.

The problem explicitly states that the events are independent.

2

Apply the multiplication law for independent events

P(A \cap B) = P(A) \times P(B)

For any two independent events, the probability of both occurring together is the product of their individual probabilities.

3

Perform the multiplication

P(A \cap B) = 0.4 \times 0.5 = 0.2

Multiplying 0.4 by 0.5 gives 0.2.

Key Concept

Multiplication Law of Probability for Independent Events

Question 144Question →

The sum of the first $n$ terms of an arithmetic progression is given by $S_n = 2n^2 + 3n$ . What is the value of the $7^{\text{th}}$ term of the progression?

Show answer & explanation

Answer: 29

Answer

The

7^{\text{th}}

term of the arithmetic progression is

29

.

For any sequence, the

n

-th term is found using

T_n = S_n - S_{n-1}

. Substituting

n = 7

gives

S_7 = 2(7)^2 + 3(7) = 119

and

S_6 = 2(6)^2 + 3(6) = 90

. Thus,

T_7 = 119 - 90 = 29

.

Step-by-Step Solution

1

State the relationship between the

n

-th term

T_n

and the sum of first

n

terms

S_n

T_n = S_n - S_{n-1}

The sum of the first

n

terms minus the sum of the first

n-1

terms equals the

n

-th term.

2

Calculate the sum of the first 7 terms (

S_7

)

S_7 = 2(7)^2 + 3(7) = 119

Substitute

n = 7

into the sum formula

S_n = 2n^2 + 3n

.

3

Calculate the sum of the first 6 terms (

S_6

)

S_6 = 2(6)^2 + 3(6) = 90

Substitute

n = 6

into the sum formula

S_n = 2n^2 + 3n

.

4

Compute the

7^{\text{th}}

term (

T_7

)

T_7 = 119 - 90 = 29

Subtract

S_6

from

S_7

.

Key Concept

Relationship between the nth term and the sum of first n terms of an AP

Question 145Question →

The mean mark of a student in $6$ tests is $14$ . When the highest and lowest marks, which differ by $12$ , are excluded, the mean mark of the remaining $4$ tests becomes $13.5$ . What is the highest mark?

Show answer & explanation

Answer: 21

Answer

The highest mark is 21.

The total sum of all 6 tests is

6 \times 14 = 84

. When the highest (

H

) and lowest (

L

) marks are removed, the total sum of the remaining 4 tests is

4 \times 13.5 = 54

. The sum of the excluded marks is

H + L = 84 - 54 = 30

. Knowing that their difference is

H - L = 12

, we add the two equations to get

2H = 42

, which gives

H = 21

.

Step-by-Step Solution

1

Calculate the total sum of all 6 test marks

Sum of 6 marks =

6 \times 14 = 84

The sum of data values equals the mean multiplied by the number of items.

2

Calculate the sum of the remaining 4 test marks

Sum of 4 marks =

4 \times 13.5 = 54

Multiplying the new mean by 4 gives the sum of the test scores excluding the highest and lowest values.

3

Determine the combined sum of the highest (H) and lowest (L) marks

H + L = 84 - 54 = 30

Subtracting the sum of the 4 remaining marks from the total initial sum yields the sum of the two excluded marks.

4

Solve for H using the simultaneous linear equations

H = 21

Adding H + L = 30 and H - L = 12 gives 2H = 42, which solves to H = 21.

Key Concept

Mean of Ungrouped Data and Handling Excluded Values

Question 146Question →

A convex polygon has $54$ diagonals. How many distinct triangles can be formed by joining any three of its vertices?

Show answer & explanation

Answer: 220

Answer

220 distinct triangles

Solving the equation for the number of diagonals

\frac{n(n-3)}{2} = 54

yields

n = 12

vertices. The number of triangles that can be formed by selecting any 3 of these 12 vertices is given by

\binom{12}{3} = \frac{12 \times 11 \times 10}{6} = 220

.

Step-by-Step Solution

1

Determine the number of vertices

n

of the polygon using the diagonals formula.

n = 12

The number of diagonals

D

in an

n

-sided convex polygon is given by

D = \binom{n}{2} - n = \frac{n(n-3)}{2}

. Setting

\frac{n(n-3)}{2} = 54

gives

n^2 - 3n - 108 = 0

. Factoring

(n - 12)(n + 9) = 0

yields

n = 12

since the number of vertices must be positive.

2

Calculate the number of distinct triangles formed by choosing 3 vertices from 12.

220

Each set of 3 distinct vertices forms one unique triangle. The order in which the vertices are chosen does not matter, so we use combinations:

\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220

.

Key Concept

Combinations applied to geometric figures and polygon properties

Estimated Time:2m 0s

Question 147Question →

Find the smallest positive integer $n$ that simultaneously satisfies the linear modular congruences $3n \equiv 5 \pmod{13}$ and $4n \equiv 2 \pmod{9}$ .

Show answer & explanation

Answer: 32

Answer

The smallest positive integer satisfying both congruences is 32.

Solving the first congruence

3n \equiv 5 \pmod{13}

gives

n \equiv 6 \pmod{13}

(since

3 \times 9 \equiv 1 \pmod{13}

and

5 \times 9 = 45 \equiv 6 \pmod{13}

), which means

n

can be written as

13k + 6

. Substituting this into the second congruence

4n \equiv 2 \pmod{9}

yields

4(13k + 6) \equiv 2 \pmod{9} \implies 52k + 24 \equiv 2 \pmod{9}

. Reducing the coefficients modulo 9 gives

7k + 6 \equiv 2 \pmod{9} \implies 7k \equiv -4 \equiv 5 \pmod{9}

. Multiplying by 4 (the modular inverse of 7 modulo 9) yields

k \equiv 20 \equiv 2 \pmod{9}

. Setting

k = 2

yields the smallest positive integer

n = 13(2) + 6 = 32

.

Step-by-Step Solution

1

Solve the first modular congruence

3n \equiv 5 \pmod{13}

for

n

n \equiv 6 \pmod{13}

, which implies

n = 13k + 6

Multiplying both sides by the modular inverse of 3 modulo 13 (which is 9) isolates

n

.

2

Solve the second modular congruence

4n \equiv 2 \pmod{9}

for

n

n \equiv 5 \pmod{9}

Multiplying both sides by the modular inverse of 4 modulo 9 (which is 7) isolates

n

.

3

Substitute

n = 13k + 6

into

n \equiv 5 \pmod{9}

and simplify modulo 9

4k \equiv 8 \pmod{9}

Reducing 13 modulo 9 yields

4k

, and subtracting 6 from 5 yields

-1 \equiv 8 \pmod{9}

.

4

Solve for

k

and calculate the smallest positive integer

n

k \equiv 2 \pmod{9}

, giving

n = 13(2) + 6 = 32

Setting the integer parameter

k

to its minimum non-negative value

2

provides the smallest positive integer solution.

Key Concept

System of Linear Modular Congruences and Modular Inverses

Question 148Question →

What is the $10^{\text{th}}$ term of the arithmetic progression $3, 7, 11, 15, \dots$ ?

Show answer & explanation

Answer: 39

Answer

The

10^{\text{th}}

term of the arithmetic progression is

39

.

By applying the

n^{\text{th}}

term formula for an arithmetic progression

T_n = a + (n - 1)d

with first term

a = 3

, common difference

d = 4

, and term index

n = 10

, the calculation yields

T_{10} = 3 + (10 - 1) \times 4 = 3 + 36 = 39

.

Step-by-Step Solution

1

Identify the key parameters of the arithmetic progression from the given sequence.

First term

a = 3

, common difference

d = 7 - 3 = 4

, and number of terms

n = 10

.

These parameters are required to use the

n^{\text{th}}

term formula of an A.P.

2

Substitute the values into the formula

T_n = a + (n - 1)d

.

T_{10} = 3 + (10 - 1) \times 4

The formula relates the

n^{\text{th}}

term to the first term, common difference, and term position.

3

Evaluate the mathematical expression.

T_{10} = 3 + 9 \times 4 = 3 + 36 = 39

Perform multiplication before addition according to standard order of operations.

Key Concept

nth term of an Arithmetic Progression

Question 149Question →

What is the measure, in degrees, of each interior angle of a regular octagon (an 8-sided regular polygon)?

Show answer & explanation

Answer: 135

Answer

Each interior angle of a regular octagon measures 135 degrees.

The sum of the interior angles of a polygon with

n

sides is

(n - 2) \times 180^\circ

. For a regular octagon (

n = 8

), the total interior angle sum is

(8 - 2) \times 180^\circ = 1080^\circ

. Since all 8 interior angles of a regular octagon are congruent, dividing the total sum by 8 yields

135^\circ

for each interior angle.

Step-by-Step Solution

1

Find the sum of all interior angles of the regular octagon.

Sum of interior angles =

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

.

The sum of interior angles for any

n

-sided polygon is

(n - 2) \times 180^\circ

.

2

Calculate the measure of a single interior angle.

Interior angle =

\frac{1080^\circ}{8} = 135^\circ

.

In a regular polygon, all interior angles are equal in measure.

Key Concept

Interior Angle of a Regular Polygon

Question 150Question →

Given the simultaneous equations $x + 2y = 7$ and $x^2 + 3xy + y^2 = 19$ , find the product of all possible values of $y$ that satisfy the system.

Show answer & explanation

Answer: -30

Answer

The product of all possible values of y that satisfy the system is -30.

Rearranging the linear equation gives x = 7 - 2y. Substituting this into the quadratic equation x^2 + 3xy + y^2 = 19 produces (7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19. Expanding and combining like terms yields y^2 + 7y - 30 = 0. Solving for y gives y = 3 and y = -10. Multiplying these values together gives a product of -30.

Step-by-Step Solution

1

Isolate x in the linear equation

x = 7 - 2y

Expressing one variable in terms of the other enables substitution into the quadratic equation.

2

Substitute x into the quadratic equation and expand

(7 - 2y)^2 + 3(7 - 2y)y + y^2 = 19

This reduces the system to a single quadratic equation in terms of y.

3

Simplify the resulting expression into standard quadratic form

y^2 + 7y - 30 = 0

Expanding yields (49 - 28y + 4y^2) + (21y - 6y^2) + y^2 = 19, which reduces to y^2 + 7y - 30 = 0.

4

Calculate the product of the roots of y

y_1 * y_2 = -30

By Vieta's formulas, the product of roots for y^2 + ay + b = 0 is b/1 = -30 (or factoring gives y = 3 and y = -10, with product 3 * (-10) = -30).

Key Concept

Solving simultaneous linear and quadratic equations via substitution and applying quadratic root properties

Estimated Time:2m 0s

Question 151Question →

If $y = (2x + 1)^3$ , find the value of $\frac{dy}{dx}$ at $x = 1$ .

Show answer & explanation

Answer: 54

Answer

54

Applying the chain rule gives

\frac{dy}{dx} = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2

. Evaluating at

x = 1

gives

6(3)^2 = 54

.

Step-by-Step Solution

1

Differentiate

y = (2x + 1)^3

using the chain rule.

\frac{dy}{dx} = 6(2x + 1)^2

According to the chain rule,

\frac{d}{dx}[u^n] = n u^{n-1} \cdot \frac{du}{dx}

, where

u = 2x + 1

and

\frac{du}{dx} = 2

.

2

Evaluate the derivative at

x = 1

.

\frac{dy}{dx}\Big|_{x=1} = 54

Substituting

x = 1

into

6(2x + 1)^2

yields

6(3)^2 = 54

.

Key Concept

Chain Rule for Differentiation

Question 152Question →

Find the real value of

x

that satisfies the exponential equation

5^{2x - 1} \times 25^{x + 1} = 125^{x + 2}

Show answer & explanation

Answer: 5

Answer

The value of

x

is 5.

By converting all terms to base 5 (

25 = 5^2

and

125 = 5^3

), the equation becomes

5^{2x-1} \times 5^{2(x+1)} = 5^{3(x+2)}

. Simplifying exponents gives

5^{2x-1} \times 5^{2x+2} = 5^{3x+6}

. Adding the left-hand powers results in

5^{4x+1} = 5^{3x+6}

. Equating the exponents yields

4x + 1 = 3x + 6

, leading directly to

x = 5

.

Step-by-Step Solution

1

Express all terms with a common base of 5

5^{2x-1} \times 5^{2x+2} = 5^{3x+6}

Since

25 = 5^2

and

125 = 5^3

, using index laws

(a^m)^n = a^{mn}

allows all expressions to share base 5.

2

Apply the product rule of indices on the left side

5^{4x+1} = 5^{3x+6}

According to the product law

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

, the powers are added:

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

.

3

Equate powers of equal bases to solve for x

x = 5

Since bases are equal, exponents must be equal:

4x + 1 = 3x + 6 \implies 4x - 3x = 6 - 1 \implies x = 5

.

Key Concept

Indices and Laws of Indices

Question 153Question →

In a probability experiment, two fair six-sided dice were rolled $180$ times, yielding an experimental probability of $\frac{5}{18}$ for obtaining a sum divisible by $3$ . If $m$ additional consecutive rolls were conducted and every single one resulted in a sum divisible by $3$ , the updated overall experimental probability equaled the theoretical probability that the absolute difference between the numbers shown on two fair six-sided dice is at most $1$ . Calculate the value of $m$ .

Show answer & explanation

Answer: 54

Answer

54

The initial number of successful trials is

180 \times \frac{5}{18} = 50

. The theoretical probability of rolling two dice with an absolute difference of at most

1

is calculated by counting

6

outcomes with difference

0

and

10

outcomes with difference

1

, giving

\frac{16}{36} = \frac{4}{9}

. Equating the updated experimental probability

\frac{50 + m}{180 + m}

to

\frac{4}{9}

yields

9(50 + m) = 4(180 + m)

, which simplifies to

5m = 270

or

m = 54

.

Step-by-Step Solution

1

Calculate the initial number of successful trials from the given experimental probability.

Initial successful outcomes =

180 \times \frac{5}{18} = 50

.

Experimental probability is defined as the number of successful trials divided by the total number of trials.

2

Calculate the theoretical probability that the absolute difference between two rolled six-sided dice is at most 1.

Favorable outcomes = 16, so

P(\text{theoretical}) = \frac{16}{36} = \frac{4}{9}

.

Outcomes with difference 0:

(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

(6 outcomes). Outcomes with difference 1:

(1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5)

(10 outcomes). Total sample space

= 6 \times 6 = 36

.

3

Formulate the algebraic equation relating the updated experimental probability to the theoretical probability.

\frac{50 + m}{180 + m} = \frac{4}{9}

.

Adding

m

consecutive successful rolls increases both the number of successful outcomes (to

50 + m

) and the total number of trials (to

180 + m

).

4

Solve the equation for

m

.

9(50 + m) = 4(180 + m) \implies 450 + 9m = 720 + 4m \implies 5m = 270 \implies m = 54

.

Cross-multiplication converts the rational expression into a linear equation.

Key Concept

Experimental and Theoretical Probability Synthesis

Estimated Time:3m 0s

Question 154Question →

Pump A can fill a water reservoir in $6$ hours, while a drain pipe can empty the full reservoir in $15$ hours. Pump A is switched on to fill an empty reservoir while the drain pipe is accidentally left open. After $3$ hours, an identical pump, Pump B, is also switched on to assist Pump A while the drain pipe remains open. How many total hours will it take for the reservoir to become completely full?

Show answer & explanation

Answer: 5.625

Answer

The total time required to fill the reservoir completely is

5.625

hours.

To solve multi-stage work and rate problems involving opposing forces (filling vs. draining), calculate the net rate of change per unit of time for each stage. In stage one, Pump A adds

\frac{1}{6}

while the drain removes

\frac{1}{15}

, giving a net rate of

\frac{1}{10}

per hour. In 3 hours,

\frac{3}{10}

of the reservoir is filled, leaving

\frac{7}{10}

. In stage two, adding identical Pump B increases the filling rate to

2 \times \frac{1}{6} = \frac{1}{3}

. Subtracting the drain rate

\frac{1}{15}

yields a net rate of

\frac{4}{15}

per hour. Dividing the remaining

\frac{7}{10}

by

\frac{4}{15}

gives

2.625

hours. Adding the initial 3 hours yields a total of

5.625

hours.

Step-by-Step Solution

1

Determine individual hourly rates

Pump A rate =

+\frac{1}{6}

reservoir/hr, Drain rate =

-\frac{1}{15}

reservoir/hr

Rate is the reciprocal of the time required to complete the full job.

2

Calculate net rate and progress for the first 3 hours

Net rate =

\frac{1}{6} - \frac{1}{15} = \frac{1}{10}

reservoir/hr. Progress in 3 hours =

3 \times \frac{1}{10} = \frac{3}{10}

of the reservoir.

Only Pump A and the drain pipe are active during the initial 3-hour period.

3

Calculate remaining fraction of reservoir to be filled

Remaining portion =

1 - \frac{3}{10} = \frac{7}{10}

The total capacity of the reservoir is represented by

1

whole unit.

4

Calculate the combined rate after Pump B is added

New net rate =

\frac{1}{6} + \frac{1}{6} - \frac{1}{15} = \frac{1}{3} - \frac{1}{15} = \frac{4}{15}

reservoir/hr

Pump B is identical to Pump A, so its rate is also

\frac{1}{6}

reservoir/hr.

5

Find additional time needed and total time elapsed

Additional time =

\frac{7/10}{4/15} = \frac{7}{10} \times \frac{15}{4} = \frac{21}{8} = 2.625

hours. Total time =

3 + 2.625 = 5.625

hours.

Time equals remaining work divided by combined net rate, then added to elapsed time.

Key Concept

Work-Rate and Simultaneous Operations (Combined Filling and Emptying Rates)

Estimated Time:2m 30s

Question 155Question →

The sum of the first four terms of an arithmetic progression (A.P.) is $32$ , and the sum of the next four terms is $96$ . What is the common difference of the progression?

Show answer & explanation

Answer: 4

Answer

The common difference of the arithmetic progression is

4

.

The sum of the first four terms yields

2a + 3d = 16

. The sum of the first eight terms is

32 + 96 = 128

, which gives

2a + 7d = 32

. Subtracting these two linear equations gives

4d = 16

, leading to

d = 4

.

Step-by-Step Solution

1

Formulate an equation for the sum of the first 4 terms.

2a + 3d = 16

The sum of the first

n

terms of an A.P. is

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

. Substituting

n = 4

and

S_4 = 32

gives

2(2a + 3d) = 32

, which simplifies to

2a + 3d = 16

.

2

Formulate an equation for the sum of the first 8 terms.

2a + 7d = 32

The total sum of the first 8 terms is the sum of the first 4 terms plus the sum of the next 4 terms (

S_8 = 32 + 96 = 128

). Substituting

n = 8

gives

4(2a + 7d) = 128

, which simplifies to

2a + 7d = 32

.

3

Solve the system of simultaneous linear equations for

d

.

d = 4

Subtracting

(2a + 3d = 16)

from

(2a + 7d = 32)

eliminates

2a

, resulting in

4d = 16

, which yields

d = 4

.

Key Concept

Sum of an Arithmetic Progression

Question 156Question →

In an agricultural experiment, two crop varieties, $X$ and $Y$ , are tested independently for germination under drought conditions. The probability that variety $X$ germinates is $0.65$ , and the probability that at least one of the two varieties germinates is $0.86$ . What is the probability that variety $Y$ germinates?

Show answer & explanation

Answer: 0.6

Answer

The probability that variety Y germinates is 0.6

Using the law of addition for independent events

P(X \cup Y) = P(X) + P(Y) - P(X)P(Y)

, substituting

P(X) = 0.65

and

P(X \cup Y) = 0.86

gives

0.86 = 0.65 + 0.35 P(Y)

, which yields

P(Y) = 0.6

.

Step-by-Step Solution

1

Apply the general addition law for two probability events

P(X \cup Y) = P(X) + P(Y) - P(X \cap Y)

The addition law relates the union, individual probabilities, and intersection of compound events.

2

Express the intersection using the law of multiplication for independent events

P(X \cap Y) = P(X) \times P(Y) = 0.65 \times P(Y)

Because germination of variety X and variety Y are independent events.

3

Substitute given values into the combined probability formula and solve for

P(Y)

0.86 = 0.65 + P(Y) - 0.65 P(Y) \implies 0.21 = 0.35 P(Y) \implies P(Y) = 0.6

Isolating the unknown probability

P(Y)

yields the correct decimal value.

Key Concept

Compound Probability Laws and Independent Events

Question 157Question →

The table below shows the speed distribution (in km/h) recorded for $50$ vehicles passing a police checkpoint on a highway:

Speed Interval (km/h)	Frequency ( $f$ )
$40 - 49$	$6$
$50 - 59$	$12$
$60 - 69$	$18$
$70 - 79$	$10$
$80 - 89$	$4$

Calculate the mean speed of the vehicles.

Show answer & explanation

Answer: 63.3

Answer

The mean speed of the vehicles is

63.3\text{ km/h}

.

The mean of a grouped frequency distribution is computed using the formula

\bar{x} = \frac{\sum fx}{\sum f}

. The midpoints (

x

) for the intervals are

44.5, 54.5, 64.5, 74.5,

and

84.5

. Multiplying these midpoints by their respective frequencies yields products totaling

3165

. Dividing this total by the total number of vehicles (

50

) gives a mean speed of

63.3\text{ km/h}

.

Step-by-Step Solution

1

Find the class midpoints (

x

) for each speed interval

Midpoints are

x_1 = 44.5

,

x_2 = 54.5

,

x_3 = 64.5

,

x_4 = 74.5

, and

x_5 = 84.5

.

The midpoint of a grouped class interval is calculated as

\frac{\text{Lower limit} + \text{Upper limit}}{2}

.

2

Calculate the product

fx

for each class

6 \times 44.5 = 267

,

12 \times 54.5 = 654

,

18 \times 64.5 = 1161

,

10 \times 74.5 = 745

, and

4 \times 84.5 = 338

.

Multiplying class midpoint by frequency estimates the total contribution of all items within that interval.

3

Find total frequency

\sum f

and sum of products

\sum fx

\sum f = 6 + 12 + 18 + 10 + 4 = 50

and

\sum fx = 267 + 654 + 1161 + 745 + 338 = 3165

.

Summing frequencies gives total sample size, and summing

fx

gives estimated grand total.

4

Compute the estimated mean speed

\bar{x} = \frac{\sum fx}{\sum f} = \frac{3165}{50} = 63.3\text{ km/h}

.

The standard formula for the mean of grouped frequency data is

\bar{x} = \frac{\sum fx}{\sum f}

.

Key Concept

Grouped Mean Calculation

Question 158Question →

What is the determinant of the $2 \times 2$ matrix $P = \begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}$ ?

Show answer & explanation

Answer: 14

Answer

The determinant of matrix

P

is

14

.

For any

2 \times 2

matrix

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

, the determinant is given by

ad - bc

. Substituting

a=5

,

b=2

,

c=3

, and

d=4

yields

(5 \times 4) - (2 \times 3) = 20 - 6 = 14

.

Step-by-Step Solution

1

Apply the determinant formula for a

2 \times 2

matrix

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

, which is

\det(P) = ad - bc

.

\det(P) = (5)(4) - (2)(3)

The determinant of a

2 \times 2

matrix is defined as the product of the main diagonal elements minus the product of the off-diagonal elements.

2

Evaluate the arithmetic expression.

20 - 6 = 14

Perform multiplication followed by subtraction to get the final determinant value.

Key Concept

Determinant of a 2x2 Matrix

Estimated Time:45s

Question 159Question →

If $y = 5\sin x + e^{3x}$ , what is the value of $\frac{dy}{dx}$ at $x = 0$ ?

Show answer & explanation

Answer: 8

Answer

The value of the derivative at

x = 0

is 8.

Differentiating

y = 5\sin x + e^{3x}

yields

\frac{dy}{dx} = 5\cos x + 3e^{3x}

. Substituting

x = 0

gives

5\cos(0) + 3e^{0} = 5(1) + 3(1) = 8

.

Step-by-Step Solution

1

Differentiate each term of the function

y = 5\sin x + e^{3x}

with respect to

x

.

\frac{dy}{dx} = 5\cos x + 3e^{3x}

The derivative of

\sin x

is

\cos x

, and applying the chain rule to

e^{3x}

gives

3e^{3x}

.

2

Evaluate the derivative

\frac{dy}{dx}

at

x = 0

.

5\cos(0) + 3e^0 = 5(1) + 3(1) = 8

Evaluating trigonometric and exponential functions at zero gives

\cos(0) = 1

and

e^0 = 1

.

Key Concept

Differentiation of trigonometric and exponential functions and evaluation of derivatives at specific points.

Question 160Question →

A student measures the length of a metal rod as $4.0\text{ cm}$ . If the actual length of the rod is $5.0\text{ cm}$ , what is the percentage error in the measurement?

Show answer & explanation

Answer: 20

Answer

The percentage error in the measurement is 20%.

The absolute error is the difference between the true length (

5.0\text{ cm}

) and the measured length (

4.0\text{ cm}

), which is

1.0\text{ cm}

. Dividing

1.0\text{ cm}

by the true length

5.0\text{ cm}

gives a fractional error of

0.20

. Expressed as a percentage,

0.20 \times 100\% = 20\%

.

Step-by-Step Solution

1

Find the error (difference between measured value and actual value)

Error =

|5.0\text{ cm} - 4.0\text{ cm}| = 1.0\text{ cm}

Absolute error is defined as the magnitude of the difference between the actual value and the measured value.

2

Calculate the percentage error

Percentage error =

\frac{1.0\text{ cm}}{5.0\text{ cm}} \times 100\% = 20\%

Percentage error is the ratio of absolute error to the actual value, expressed as a percentage.

Key Concept

Percentage Error Calculation

Answer

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