All practice questions

Question 7541Question →

Find the total area of the region bounded by the curve $y = 3x^2 - 6x$ , the $x$ -axis, and the vertical lines $x = 0$ and $x = 3$ .

Show answer & explanation

Answer: 8

Answer

The total area bounded by the curve and the x-axis between x = 0 and x = 3 is 8 square units.

To find the total geometric area bounded by a curve and the x-axis, we must split the integral at any x-intercepts within the domain. For

y = 3x^2 - 6x

, the x-intercepts are

x = 0

and

x = 2

. Between

x = 0

and

x = 2

, the curve lies below the x-axis, giving an area magnitude of

4

. Between

x = 2

and

x = 3

, the curve lies above the x-axis, giving an area magnitude of

4

. Summing these positive magnitudes gives a total area of

8

.

Step-by-Step Solution

1

Find the roots of the curve

y = 3x^2 - 6x

within the given interval

[0, 3]

.

Setting

3x^2 - 6x = 0

yields

3x(x - 2) = 0

, giving

x = 0

and

x = 2

.

Roots inside the integration boundaries indicate where the curve crosses the x-axis, changing the sign of

y

.

2

Determine the position of the curve relative to the x-axis on each sub-interval.

On

[0, 2]

,

y \le 0

(below the x-axis). On

[2, 3]

,

y \ge 0

(above the x-axis).

Geometric area must be non-negative, so regions below the x-axis require integrating

-y

or taking the absolute value of the integral.

3

Evaluate the area

A_1

for the region below the x-axis from

x = 0

to

x = 2

.

A_1 = \int_{0}^{2} (6x - 3x^2) \, dx = \left[ 3x^2 - x^3 \right]_{0}^{2} = (3(4) - 8) - 0 = 4

.

Integrating

-y = 6x - 3x^2

yields the positive magnitude of the area below the x-axis.

4

Evaluate the area

A_2

for the region above the x-axis from

x = 2

to

x = 3

.

A_2 = \int_{2}^{3} (3x^2 - 6x) \, dx = \left[ x^3 - 3x^2 \right]_{2}^{3} = (3^3 - 3(3^2)) - (2^3 - 3(2^2)) = 0 - (-4) = 4

.

Direct integration of

y

on

[2, 3]

gives the area above the x-axis.

5

Combine the areas of both sub-regions.

Total Area

= A_1 + A_2 = 4 + 4 = 8

.

The total geometric area is the sum of the magnitudes of the areas of all separate bounded regions.

Key Concept

Calculating area under curves crossing the x-axis by splitting definite integrals at real roots

Question 7542Question →

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

\begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}

\begin{pmatrix} 3 & -2 \\ 0 & 20 \end{pmatrix}

\begin{pmatrix} -1 & -7 \\ 10 & 20 \end{pmatrix}

\begin{pmatrix} 3 & 11 \\ 2 & 16 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}$

Answer

The product matrix

PQ

is

\begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}

.

The matrix containing elements 3, -1 in the top row and 2, 16 in the bottom row correctly applies the row-by-column multiplication rule for matrices

P

and

Q

.

Step-by-Step Solution

1

Set up the row-by-column matrix multiplication

PQ

PQ = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix} \begin{pmatrix} 1 & -2 \\ 0 & 5 \end{pmatrix}

Matrix multiplication requires computing dot products of rows of the first matrix with columns of the second matrix.

2

Calculate the first row elements of the product matrix

Row 1, Column 1:

3(1) + 1(0) = 3 + 0 = 3

; Row 1, Column 2:

3(-2) + 1(5) = -6 + 5 = -1

Multiply elements of Row 1 of

P

by corresponding elements of Columns 1 and 2 of

Q

and sum them.

3

Calculate the second row elements of the product matrix

Row 2, Column 1:

2(1) + 4(0) = 2 + 0 = 2

; Row 2, Column 2:

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

Multiply elements of Row 2 of

P

by corresponding elements of Columns 1 and 2 of

Q

and sum them.

4

Assemble the resulting

2 \times 2

matrix

PQ = \begin{pmatrix} 3 & -1 \\ 2 & 16 \end{pmatrix}

Combine all computed entry values into matrix form.

Key Concept

Matrix Multiplication (Row-by-Column Rule)

Estimated Time:1m 30s

Question 7543Question →

The table below presents the cumulative frequency distribution of delay times (in minutes) for $50$ regional flights recorded at an airport:

Delay Time (minutes)	Cumulative Frequency
$10 - 19$	$8$
$20 - 29$	$20$
$30 - 39$	$40$
$40 - 49$	$50$

What is the $70\text{th}$ percentile of the delay times?

37.0

minutes

37.5

minutes

32.0

minutes

35.0

minutes

Show answer & explanation

Answer: $37.0$ minutes

Answer

The

70\text{th}

percentile of the delay times is

37.0

minutes.

The

70\text{th}

percentile position is found by taking

70\%

of

50

, which yields

35

. Looking at the cumulative frequency table, the 35th item falls in the

30 - 39

interval (class boundaries

29.5 - 39.5

). Applying the percentile interpolation formula:

P_{70} = 29.5 + \left(\frac{35 - 20}{20}\right) \times 10 = 29.5 + 7.5 = 37.0\text{ minutes}

.

Step-by-Step Solution

1

Determine the position of the 70th percentile rank.

Position rank

= \frac{70}{100} \times 50 = 35\text{th value}

.

The 70th percentile corresponds to 70% of the total frequency

N = 50

.

2

Identify the 70th percentile class interval and its parameters.

Interval:

30 - 39

; Lower class boundary

L = 29.5

, cumulative frequency preceding class

F = 20

, frequency of percentile class

f = 20

, class width

c = 10

.

The cumulative frequency reaches 20 at

29.5

and 40 at

39.5

, so the 35th value lies in the

30 - 39

class interval.

3

Apply the percentile interpolation formula

P_{70} = L + \left(\frac{\frac{70N}{100} - F}{f}\right) \times c

.

P_{70} = 29.5 + \left(\frac{35 - 20}{20}\right) \times 10 = 29.5 + \left(\frac{15}{20}\right) \times 10 = 29.5 + 7.5 = 37.0\text{ minutes}

.

Linear interpolation distributes the cumulative frequencies evenly across the class interval.

Key Concept

Calculating percentiles from grouped cumulative frequency data using class boundaries and linear interpolation.

Question 7544Question →

Three business partners, $X$ , $Y$ , and $Z$ , invest in a venture. $X$ contributes $\frac{1}{3}$ of the total capital for $\frac{1}{4}$ of the total duration of the business. $Y$ contributes $\frac{1}{4}$ of the capital for $\frac{1}{2}$ of the duration, while $Z$ contributes the remaining capital for the entire duration. If the total profit realized from the business is $\mathcal{N}78,000$ , what is $Z$ 's share of the profit?

\mathcal{N}32,500

\mathcal{N}26,000

\mathcal{N}52,000

\mathcal{N}60,000

Show answer & explanation

Answer: $\mathcal{N}52,000$

Answer

\mathcal{N}52,000

Profit distribution in compound partnerships depends on both capital invested and investment time duration.

X

's product is

\frac{1}{12}

,

Y

's product is

\frac{1}{8}

, and

Z

's product is

\frac{5}{12}

. Scaling these by 24 gives a ratio of

2 : 3 : 10

. With 15 total parts,

Z

receives

\frac{10}{15}

of

\mathcal{N}78,000

, which equals

\mathcal{N}52,000

.

Step-by-Step Solution

1

Determine

Z

's fraction of the total capital

Z

's capital fraction

= 1 - \left(\frac{1}{3} + \frac{1}{4}\right) = 1 - \frac{7}{12} = \frac{5}{12}

The sum of all capital contributions must equal 1.

2

Calculate the compound product of (Capital Share

\times

Time Duration) for each partner

X = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}

,

Y = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}

,

Z = \frac{5}{12} \times 1 = \frac{5}{12}

Profit sharing in compound partnerships is proportional to the product of capital invested and the time period.

3

Convert the fractions to an equivalent integer ratio

Multiplying

\frac{1}{12} : \frac{1}{8} : \frac{5}{12}

by 24 gives the ratio

2 : 3 : 10

Finding the least common multiple (24) simplifies the fractions into integer ratio parts.

4

Compute

Z

's profit share from the total profit

Total ratio parts

= 2 + 3 + 10 = 15

.

Z

's share

= \frac{10}{15} \times \mathcal{N}78,000 = \mathcal{N}52,000

Multiply total profit by

Z

's ratio fraction.

Key Concept

Compound Ratio and Partnership Profit Sharing

Question 7545Question →

An insulated neutral conductor gains $5.0 \times 10^{13}$ electrons during a electrostatic charging process. Given that the magnitude of the elementary charge is $e = 1.6 \times 10^{-19}\text{ C}$ , what is the magnitude of the net charge acquired by the conductor in microcoulombs ( $\mu\text{C}$ )?

Show answer & explanation

Answer: 8

Answer

The magnitude of the net electric charge acquired by the conductor is

8.0\ \mu\text{C}

.

According to the principle of charge quantization, the total magnitude of charge

Q

acquired by gaining

n

electrons is given by

Q = n e

. Substituting

n = 5.0 \times 10^{13}

and

e = 1.6 \times 10^{-19}\text{ C}

gives

Q = 8.0 \times 10^{-6}\text{ C}

. Expressed in microcoulombs,

8.0 \times 10^{-6}\text{ C} = 8.0\ \mu\text{C}

.

Step-by-Step Solution

1

Apply the principle of quantization of electric charge formula

Formula

Q = n e

established

Electric charge is quantized and exists in integer multiples of the elementary charge.

2

Multiply the number of electrons by the elementary charge value

Q = 8.0 \times 10^{-6}\text{ C}

Calculates total electrostatic charge in base SI units.

3

Convert the value from Coulombs to microcoulombs

8.0\ \mu\text{C}

The unit

1\ \mu\text{C}

equals

10^{-6}\text{ C}

.

Key Concept

Quantization of Electric Charge

Question 7546Question →

When differentiating the function $f(x) = x^2 - 3x$ from first principles, what is the simplified expression for the difference quotient $\frac{f(x+h) - f(x)}{h}$ for $h \neq 0$ before taking the limit as $h \to 0$ ?

Show answer & explanation

Answer: 2x + h - 3; 2x - 3 + h; h + 2x - 3; 2x+h-3; 2x-3+h

Answer

2x + h - 3

Expanding

f(x+h) = (x+h)^2 - 3(x+h) = x^2 + 2xh + h^2 - 3x - 3h

and subtracting

f(x) = x^2 - 3x

yields

2xh + h^2 - 3h

. Factoring and dividing by

h

gives the simplified difference quotient

2x + h - 3

.

Step-by-Step Solution

1

Set up the difference quotient expression

\frac{f(x+h) - f(x)}{h} = \frac{[(x+h)^2 - 3(x+h)] - (x^2 - 3x)}{h}

Substitute

(x+h)

into the function definition to evaluate

f(x+h)

.

2

Expand terms in the numerator

\frac{(x^2 + 2xh + h^2 - 3x - 3h) - x^2 + 3x}{h}

Expand

(x+h)^2 = x^2 + 2xh + h^2

and distribute

-3

across

(x+h)

as well as the negative sign across

f(x)

.

3

Combine like terms in the numerator

\frac{2xh + h^2 - 3h}{h}

The terms

x^2 - x^2 = 0

and

-3x + 3x = 0

cancel out.

4

Divide numerator terms by

h

2x + h - 3

Factor out

h

from the numerator to get

\frac{h(2x + h - 3)}{h} = 2x + h - 3

for

h \neq 0

.

Key Concept

Difference Quotient in First Principles Differentiation

Estimated Time:1m 30s

Question 7547Question →

What is the area of the region bounded by the curve $y = 6x - x^2$ and the line $y = 2x$ ?

\frac{32}{3}

square units

\frac{80}{3}

square units

\frac{160}{3}

square units

\frac{64}{3}

square units

Show answer & explanation

Answer: $\frac{32}{3}$ square units

Answer

\frac{32}{3}

square units

Equating

y = 6x - x^2

and

y = 2x

gives intersection points

x = 0

and

x = 4

. Integrating the upper curve minus the lower line,

\int_{0}^{4} (4x - x^2) \, dx = \left[ 2x^2 - \frac{x^3}{3} \right]_{0}^{4} = 32 - \frac{64}{3} = \frac{32}{3}

square units.

Step-by-Step Solution

1

Find the points of intersection between the curve and the line

x = 0

and

x = 4

Set

6x - x^2 = 2x \implies 4x - x^2 = 0 \implies x(4 - x) = 0

to find the integration bounds.

2

Set up the definite integral for the area between the two curves

A = \int_{0}^{4} ((6x - x^2) - 2x) \, dx = \int_{0}^{4} (4x - x^2) \, dx

The area between two functions

f(x)

and

g(x)

from

x=a

to

x=b

is given by

\int_{a}^{b} (f(x) - g(x)) \, dx

where

f(x) \ge g(x)

on

[a, b]

.

3

Compute the indefinite integral of the integrand

\int (4x - x^2) \, dx = 2x^2 - \frac{x^3}{3}

Apply the standard power rule of integration

\int x^n \, dx = \frac{x^{n+1}}{n+1}

.

4

Evaluate the definite integral using the limits

0

and

4

\left[ 2x^2 - \frac{x^3}{3} \right]_{0}^{4} = \left( 2(4)^2 - \frac{4^3}{3} \right) - 0 = 32 - \frac{64}{3} = \frac{32}{3}

square units

Substitute the upper limit

x = 4

and lower limit

x = 0

and simplify fractions.

Key Concept

Area Between Two Curves

Estimated Time:2m 0s

Question 7548Question →

A point $P(x, y)$ moves in a Cartesian plane such that the square of its distance from $A(3, 0)$ exceeds the square of its distance from $B(-1, 2)$ by $4$ units. Which of the following equations represents the locus of $P$ ?

2x - y = 0

2x + y - 4 = 0

x - 2y + 1 = 0

x^2 + y^2 - 4x + 2y - 4 = 0

Show answer & explanation

Answer: $2x - y = 0$

Answer

The equation representing the locus of

P

is

2x - y = 0

.

Using the distance formula,

PA^2 = (x - 3)^2 + y^2 = x^2 - 6x + 9 + y^2

and

PB^2 = (x + 1)^2 + (y - 2)^2 = x^2 + 2x + 1 + y^2 - 4y + 4

. Subtracting

PB^2

from

PA^2

yields

(x^2 + y^2 - 6x + 9) - (x^2 + y^2 + 2x - 4y + 5) = 4

, which simplifies to

-8x + 4y = 0

, or

2x - y = 0

.

Step-by-Step Solution

1

Express the square of the distance from

P(x, y)

to

A(3, 0)

using the distance formula

PA^2 = (x - 3)^2 + (y - 0)^2 = x^2 - 6x + 9 + y^2

The distance formula gives

PA^2 = (x - x_A)^2 + (y - y_A)^2

.

2

Express the square of the distance from

P(x, y)

to

B(-1, 2)

PB^2 = (x - (-1))^2 + (y - 2)^2 = (x + 1)^2 + (y - 2)^2 = x^2 + 2x + 1 + y^2 - 4y + 4 = x^2 + y^2 + 2x - 4y + 5

Expand both algebraic squares for

PB^2

.

3

Set up the locus condition

PA^2 - PB^2 = 4

and simplify

(x^2 + y^2 - 6x + 9) - (x^2 + y^2 + 2x - 4y + 5) = 4 \implies -8x + 4y + 4 = 4

Subtracting

PB^2

from

PA^2

cancels the quadratic

x^2

and

y^2

terms.

4

Rearrange into standard linear equation form

-8x + 4y = 0 \implies 2x - y = 0

Divide the entire equation by

-4

to simplify.

Key Concept

Locus defined by difference of squares of distances to two fixed points

Estimated Time:1m 30s

Question 7549Question →

Match each physical quantity listed on the left with its corresponding SI classification and base unit representation on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Electric current matches Fundamental quantity measured in amperes; Thermodynamic temperature matches Fundamental quantity measured in kelvins; Electric potential difference matches Derived quantity expressed as kg·m²·s⁻³·A⁻¹; Specific heat capacity matches Derived quantity expressed as m²·s⁻²·K⁻¹.

Electric current and thermodynamic temperature are basic SI fundamental quantities. Electric potential difference and specific heat capacity are derived quantities whose fundamental unit decompositions follow directly from their governing formulas.

Step-by-Step Solution

1

Identify fundamental physical quantities and their base units

Electric current and thermodynamic temperature are fundamental SI quantities with base units ampere (

\text{A}

) and kelvin (

\text{K}

) respectively.

Fundamental physical quantities are defined independently and serve as the foundation for the SI system.

2

Decompose electric potential difference into SI base units

Electric potential difference

V = \frac{\text{Work}}{\text{Charge}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s}} = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

Because it is expressed by combining fundamental quantities, it is a derived quantity.

3

Decompose specific heat capacity into SI base units

Specific heat capacity

c = \frac{\text{Energy}}{\text{Mass} \times \text{Temperature change}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{kg}\cdot\text{K}} = \text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

.

It is calculated from energy, mass, and temperature, making it a derived quantity.

Key Concept

Fundamental quantities are independent basic quantities, whereas derived quantities are formed through algebraic combination of fundamental quantities.

Estimated Time:1m 30s

Question 7550Question →

Given that the matrix $A = \begin{pmatrix} 1 & 2 & k \\ 0 & k & 3 \\ 2 & -1 & 1 \end{pmatrix}$ is singular, what is the positive value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The positive value of

k

is

3

.

A square matrix is singular when its determinant is zero. Expanding

\det(A)

along the first column yields

1 \cdot (k + 3) + 2 \cdot (6 - k^2) = -2k^2 + k + 15

. Equating this to zero gives the quadratic equation

2k^2 - k - 15 = 0

, which factors as

(2k + 5)(k - 3) = 0

. The roots are

k = -2.5

and

k = 3

. The positive value is

3

.

Step-by-Step Solution

1

Calculate the determinant of matrix A

\det(A) = -2k^2 + k + 15

Expanding along the first column simplifies the calculation because of the zero entry.

2

Set the determinant to zero for singularity

2k^2 - k - 15 = 0

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

3

Solve the quadratic equation for k

k = 3 or k = -2.5

Factoring

2k^2 - k - 15 = (2k + 5)(k - 3) = 0

yields two roots.

4

Choose the positive solution

k = 3

The question explicitly requires the positive value of

k

.

Key Concept

Singular matrix definition and 3x3 determinant evaluation

Question 7551Question →

In a hydrogen atom modeled according to Bohr's postulates, an electron undergoes a transition from an initial excited state $n$ to the ground state $n=1$ , emitting a photon whose wavelength is $\lambda = \frac{16}{15R}$ , where $R$ is the Rydberg constant. What is the orbital angular momentum of the electron in its initial excited state prior to the transition?

\frac{h}{2\pi}

\frac{h}{\pi}

\frac{2h}{\pi}

\frac{4h}{\pi}

Show answer & explanation

Answer: $\frac{2h}{\pi}$

Answer

\frac{2h}{\pi}

Using the Rydberg formula

\frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{n^2} \right)

with

\lambda = \frac{16}{15R}

, we find

\frac{15}{16} = 1 - \frac{1}{n^2}

, which gives

n = 4

. By Bohr's angular momentum quantization condition,

L = \frac{nh}{2\pi} = \frac{4h}{2\pi} = \frac{2h}{\pi}

.

Step-by-Step Solution

1

Apply the Rydberg formula for hydrogen to determine the principal quantum number

n

of the initial state.

\frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{n^2} \right) \implies \frac{15R}{16} = R \left( 1 - \frac{1}{n^2} \right) \implies 1 - \frac{1}{n^2} = \frac{15}{16} \implies \frac{1}{n^2} = \frac{1}{16} \implies n = 4

.

The Rydberg formula links the photon wavelength emitted during a transition between energy levels to the principal quantum numbers.

2

Calculate the orbital angular momentum

L

for

n=4

using Bohr's quantization condition.

L = \frac{n h}{2\pi} = \frac{4 h}{2\pi} = \frac{2 h}{\pi}

.

Bohr's quantization postulate dictates that orbital angular momentum is an integral multiple of

\frac{h}{2\pi}

.

Key Concept

Bohr's Atomic Model: Spectral transitions and angular momentum quantization

Estimated Time:2m 0s

Question 7552Question →

If $a$ and $b$ are real numbers satisfying the simultaneous equations $a + b = 10$ and $a^2 - b^2 = 40$ , what is the value of $a$ ?

Show answer & explanation

Answer: 7

Answer

The value of

a

is 7.

Using the difference of squares identity,

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

. Substituting

a+b = 10

into

a^2 - b^2 = 40

gives

10(a-b) = 40

, which simplifies to

a-b = 4

. Adding the two linear equations

a+b = 10

and

a-b = 4

eliminates

b

, giving

2a = 14

, hence

a = 7

.

Step-by-Step Solution

1

Factorize the quadratic expression

a^2 - b^2

.

(a + b)(a - b) = 40

Apply the difference of two squares identity.

2

Substitute

a + b = 10

into the factorized equation.

10(a - b) = 40 \implies a - b = 4

Simplifying yields a second linear equation.

3

Solve the system of linear equations

a + b = 10

and

a - b = 4

for

a

.

(a + b) + (a - b) = 10 + 4 \implies 2a = 14 \implies a = 7

Adding the two equations eliminates

b

directly.

Key Concept

Simultaneous linear and quadratic equations involving difference of squares

Question 7553Question →

Given the matrices $A = \begin{pmatrix} x & 1 \\ 4 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 \\ 1 & x \end{pmatrix}$ , what is the sum of all real values of $x$ for which the matrix $C = AB - B^T$ is singular?

\frac{5}{2}

\frac{13}{2}

1

-\frac{5}{2}

Show answer & explanation

Answer: $\frac{5}{2}$

Answer

The sum of all real values of

x

is

\frac{5}{2}

.

Evaluating

AB

yields

\begin{pmatrix} 2x + 1 & 4x \\ 10 & 2x + 12 \end{pmatrix}

and subtracting

B^T = \begin{pmatrix} 2 & 1 \\ 3 & x \end{pmatrix}

gives matrix

C = \begin{pmatrix} 2x - 1 & 4x - 1 \\ 7 & x + 12 \end{pmatrix}

. Setting

\det(C) = (2x - 1)(x + 12) - 7(4x - 1) = 0

results in the quadratic equation

2x^2 - 5x - 5 = 0

. Since the discriminant

\Delta = 65 > 0

, both roots are real, and their sum is given by

-\frac{b}{a} = \frac{5}{2}

.

Step-by-Step Solution

1

Compute the matrix product

AB

AB = \begin{pmatrix} x & 1 \\ 4 & 2 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 1 & x \end{pmatrix} = \begin{pmatrix} 2x + 1 & 4x \\ 10 & 2x + 12 \end{pmatrix}

Matrix multiplication requires taking the dot product of rows of the first matrix with columns of the second matrix.

2

Find the transpose of matrix

B

, denoted

B^T

B^T = \begin{pmatrix} 2 & 1 \\ 3 & x \end{pmatrix}

The transpose of a matrix is formed by swapping its rows and columns.

3

Compute matrix

C = AB - B^T

C = \begin{pmatrix} 2x + 1 & 4x \\ 10 & 2x + 12 \end{pmatrix} - \begin{pmatrix} 2 & 1 \\ 3 & x \end{pmatrix} = \begin{pmatrix} 2x - 1 & 4x - 1 \\ 7 & x + 12 \end{pmatrix}

Perform element-wise subtraction of matrix

B^T

from matrix

AB

.

4

Set the determinant of matrix

C

to zero for singularity

\det(C) = (2x - 1)(x + 12) - 7(4x - 1) = 2x^2 + 23x - 12 - (28x - 7) = 2x^2 - 5x - 5 = 0

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

5

Find the sum of the real roots of the quadratic equation

2x^2 - 5x - 5 = 0

Discriminant

\Delta = (-5)^2 - 4(2)(-5) = 65 > 0

. By Vieta's formula,

x_1 + x_2 = -\frac{b}{a} = -\frac{-5}{2} = \frac{5}{2}

.

Since the discriminant is positive, two distinct real roots exist, and their sum is given by

-\frac{b}{a}

.

Key Concept

Singular matrix condition, matrix multiplication, transpose operations, and Vieta's formulas.

Question 7554Question →

The parallel lines $L_1: 3x - 4y + 25 = 0$ and $L_2: 3x - 4y = 0$ are intersected by a straight line $L_3$ with gradient $m > 1$ . If the length of the line segment of $L_3$ intercepted between $L_1$ and $L_2$ is $5\sqrt{5}$ units, find the value of $m$ .

Show answer & explanation

Answer: 2

Answer

The value of the gradient

m

is 2.

The perpendicular distance between the parallel lines

L_1: 3x - 4y + 25 = 0

and

L_2: 3x - 4y = 0

is

d = \frac{25}{\sqrt{3^2 + (-4)^2}} = 5

units. The acute angle

\phi

between

L_3

and the parallel lines satisfies

\sin(\phi) = \frac{5}{5\sqrt{5}} = \frac{1}{\sqrt{5}}

, which gives

\tan(\phi) = \frac{1}{2}

. The gradient of

L_1

and

L_2

is

m_1 = \frac{3}{4}

. Using the tangent formula for the angle between two lines,

\tan(\phi) = \left|\frac{m - m_1}{1 + m m_1}\right| \implies \frac{1}{2} = \left|\frac{4m - 3}{4 + 3m}\right|

, which yields

m = 2

or

m = \frac{2}{11}

. Under the constraint

m > 1

, the unique value of

m

is

2

.

Step-by-Step Solution

1

Calculate the perpendicular distance

d

between the parallel lines

L_1

and

L_2

d = \frac{|25 - 0|}{\sqrt{3^2 + (-4)^2}} = \frac{25}{5} = 5

units

The perpendicular distance between two parallel lines

Ax + By + C_1 = 0

and

Ax + By + C_2 = 0

is given by

\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}

.

2

Determine the trigonometric relationship between the perpendicular distance, intercepted segment, and intersection angle

\phi

\sin(\phi) = \frac{\text{perpendicular distance}}{\text{intercepted segment}} = \frac{5}{5\sqrt{5}} = \frac{1}{\sqrt{5}}

The perpendicular distance forms the opposite side of a right triangle whose hypotenuse is the intercepted segment.

3

Calculate

\tan(\phi)

using right-triangle trigonometry

Since

\sin(\phi) = \frac{1}{\sqrt{5}}

,

\cos(\phi) = \sqrt{1 - \sin^2(\phi)} = \frac{2}{\sqrt{5}}

, so

\tan(\phi) = \frac{1}{2}

The angle between lines formula requires

\tan(\phi)

.

4

Apply the angle between two lines formula and solve for

m

\tan(\phi) = \left| \frac{m - m_1}{1 + m m_1} \right| \implies \frac{1}{2} = \left| \frac{4m - 3}{4 + 3m} \right|

where

m_1 = \frac{3}{4}

. Case 1:

\frac{4m - 3}{4 + 3m} = \frac{1}{2} \implies 8m - 6 = 4 + 3m \implies 5m = 10 \implies m = 2

. Case 2:

\frac{4m - 3}{4 + 3m} = -\frac{1}{2} \implies 8m - 6 = -4 - 3m \implies 11m = 2 \implies m = \frac{2}{11}

.

Evaluating the absolute value produces two potential solutions.

5

Apply the domain constraint

m > 1

m = 2

The problem restricts

m > 1

, which excludes

m = \frac{2}{11}

.

Key Concept

Distance between parallel lines and angle of intersection between straight lines

Question 7555Question →

Three of the exterior angles of a convex polygon are each $50^\circ$ , while the remaining exterior angles are each $35^\circ$ . How many sides does the polygon have?

6

8

9

10

Show answer & explanation

Answer: 9

Answer

The total number of sides of the polygon is 9.

The sum of all exterior angles of a convex polygon is

360^\circ

. The sum of the first three angles is

3 \times 50^\circ = 150^\circ

. Subtracting this from

360^\circ

leaves

210^\circ

for the remaining angles. Dividing

210^\circ

by

35^\circ

yields

6

remaining angles. Adding the initial

3

angles gives

3 + 6 = 9

sides in total.

Step-by-Step Solution

1

State the formula for the sum of exterior angles of a convex polygon.

The sum of exterior angles is always

360^\circ

.

This fundamental geometric property applies to all convex polygons regardless of the number of sides.

2

Set up an algebraic equation using the given exterior angle measures.

Let

k

be the number of remaining exterior angles measuring

35^\circ

. Then

3(50^\circ) + k(35^\circ) = 360^\circ

.

The total sum is composed of three

50^\circ

angles and

k

remaining

35^\circ

angles.

3

Solve for

k

.

150^\circ + 35^\circ k = 360^\circ \implies 35^\circ k = 210^\circ \implies k = 6

.

Subtracting

150^\circ

from both sides isolates the term containing

k

.

4

Calculate the total number of sides

n

.

n = 3 + 6 = 9

.

The polygon has a total number of sides equal to the total number of exterior angles (

3

initial angles

+ 6

remaining angles).

Key Concept

The sum of the exterior angles of any convex polygon is equal to 360 degrees.

Estimated Time:1m 15s

Question 7556Question →

In a survey of $180$ cloud computing engineers regarding their proficiency in three major platforms—AWS ( $A$ ), Azure ( $B$ ), and Google Cloud ( $C$ )—it was found that $95$ are proficient in AWS, $80$ in Azure, and $75$ in Google Cloud. Furthermore, $40$ are proficient in both AWS and Azure, $35$ in both Azure and Google Cloud, $30$ in both AWS and Google Cloud, and $15$ are not proficient in any of the three platforms. How many engineers are proficient in exactly one of these platforms?

Show answer & explanation

Answer: 100

Answer

100 engineers are proficient in exactly one platform.

Using the 3-set inclusion-exclusion formula, the number of engineers proficient in all three platforms is solved as 20. Subtracting the relevant overlap regions from each set gives 45 for AWS only, 25 for Azure only, and 30 for Google Cloud only. Adding these single-set values yields a total of 100 engineers proficient in exactly one platform.

Step-by-Step Solution

1

Determine the cardinality of the union of all three sets.

n(A \cup B \cup C) = 180 - 15 = 165

.

Subtracting the engineers who are not proficient in any of the three platforms from the universal set.

2

Apply the Principle of Inclusion-Exclusion to calculate the triple intersection

n(A \cap B \cap C)

.

n(A \cap B \cap C) = 20

.

Substituting known values gives

165 = 250 - 105 + n(A \cap B \cap C)

, which simplifies to

n(A \cap B \cap C) = 20

.

3

Calculate the counts for regions representing exactly two platforms.

AWS & Azure only = 20, Azure & GCP only = 15, AWS & GCP only = 10.

Subtracting the triple intersection count (20) from each pairwise intersection.

4

Calculate the single-set exclusive regions.

AWS only = 45, Azure only = 25, GCP only = 30.

Subtracting all multi-platform overlap regions from each total set size.

5

Sum the single-set exclusive regions.

Total =

45 + 25 + 30 = 100

.

Combining the counts of engineers proficient in AWS only, Azure only, and Google Cloud only.

Key Concept

Principle of Inclusion-Exclusion for Three Sets and Venn Diagram Region Partitioning

Question 7557Question →

In triangle $PQR$ , the side length $p = 12\text{ cm}$ , side length $q = 18\text{ cm}$ , and $\sin P = 0.4$ . What is the exact value of $\sin Q$ ?

Show answer & explanation

Answer: 0.6

Answer

The value of

\sin Q

is 0.6.

Applying the Sine Rule

\frac{p}{\sin P} = \frac{q}{\sin Q}

with

p = 12\text{ cm}

,

q = 18\text{ cm}

, and

\sin P = 0.4

gives

\frac{12}{0.4} = \frac{18}{\sin Q}

. Evaluating

\frac{12}{0.4} = 30

leads to

30 = \frac{18}{\sin Q}

, which yields

\sin Q = \frac{18}{30} = 0.6

.

Step-by-Step Solution

1

State the Sine Rule equation for the given triangle sides and angles.

\frac{p}{\sin P} = \frac{q}{\sin Q}

The Sine Rule relates the lengths of the sides of a triangle to the sines of its opposite angles.

2

Substitute the known numerical values into the Sine Rule equation.

\frac{12}{0.4} = \frac{18}{\sin Q}

Substituting

p = 12

,

q = 18

, and

\sin P = 0.4

sets up an equation with a single unknown.

3

Simplify the left side of the equation and solve for

\sin Q

.

\sin Q = \frac{18}{30} = 0.6

Dividing 12 by 0.4 yields 30, so rearranging gives

\sin Q = \frac{18}{30} = 0.6

.

Key Concept

Using the Sine Rule to calculate an unknown sine ratio

Question 7558Question →

When the polynomial $P(x) = x^3 - 3x^2 + kx + 12$ is divided by $(x - 2)$ , the remainder is $6$ . What is the value of the constant $k$ ?

Show answer & explanation

Answer: -1

Answer

The value of the constant

k

is

-1

.

By the Remainder Theorem, the remainder when

P(x)

is divided by

(x - 2)

is

P(2)

. Substituting

x = 2

into

P(x) = x^3 - 3x^2 + kx + 12

gives

P(2) = 8 - 12 + 2k + 12 = 8 + 2k

. Setting

8 + 2k = 6

yields

2k = -2

, so

k = -1

.

Step-by-Step Solution

1

Apply the Remainder Theorem

P(2) = 6

Dividing

P(x)

by

(x - 2)

leaves a remainder equal to evaluating

P(x)

at

x = 2

.

2

Substitute

x = 2

into

P(x)

and set equal to

6

(2)^3 - 3(2)^2 + 2k + 12 = 6

Set the evaluated polynomial equal to the given remainder.

3

Simplify the arithmetic terms

8 + 2k = 6

Calculate powers and products:

8 - 12 + 12 = 8

.

4

Solve the linear equation for

k

k = -1

Subtract 8 from both sides to get

2k = -2

, then divide by 2.

Key Concept

Polynomial Remainder Theorem

Question 7559Question →

If $\frac{2 + \sqrt{3}}{2 - \sqrt{3}} = x + y\sqrt{3}$ , where $x$ and $y$ are integers, what is the value of $x + y$ ?

Show answer & explanation

Answer: 11

Answer

The value of

x + y

is 11.

Multiplying the given fraction by

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

rationalises the denominator to 1 and simplifies the numerator to

7 + 4\sqrt{3}

. Comparing coefficients yields

x = 7

and

y = 4

, making

x + y = 11

.

Step-by-Step Solution

1

Multiply numerator and denominator by the conjugate of the denominator

\frac{(2 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}

Rationalising the denominator eliminates the surd from the denominator using the identity (a-b)(a+b) = a^2 - b^2.

2

Expand both the numerator and the denominator

\frac{4 + 4\sqrt{3} + 3}{4 - 3} = \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3}

Simplifying algebraic surd multiplication gives integer and surd terms.

3

Compare terms with x + y\sqrt{3} and solve for x and y

x = 7, y = 4 \implies x + y = 11

Matching rational components and coefficients of \sqrt{3} yields x and y.

Key Concept

Rationalisation of Binomial Denominators containing Surds

Estimated Time:1m 30s

Question 7560Question →

Given the $2 \times 2$ matrix $M = \begin{pmatrix} 7 & -2 \\ 4 & 3 \end{pmatrix}$ , what is the value of $\det(M)$ ?

Show answer & explanation

Answer: 29

Answer

The determinant of matrix

M

is 29.

The determinant of a

2 \times 2

matrix

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

is calculated as

ad - bc

. For the matrix

M = \begin{pmatrix} 7 & -2 \\ 4 & 3 \end{pmatrix}

, we compute

(7)(3) - (-2)(4) = 21 - (-8) = 29

.

Step-by-Step Solution

1

Identify the entries

a, b, c, d

from the matrix

M = \begin{pmatrix} 7 & -2 \\ 4 & 3 \end{pmatrix}

.

a = 7

,

b = -2

,

c = 4

, and

d = 3

.

These entries correspond to the standard

2 \times 2

matrix representation

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

.

2

Compute the determinant using the formula

\det(M) = ad - bc

.

\det(M) = (7)(3) - (-2)(4) = 21 + 8 = 29

.

Multiplying the main diagonal entries and subtracting the product of the off-diagonal entries yields the determinant.

Key Concept

Determinant of a 2x2 Matrix

Answer

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