All practice questions

Question 721Question →

When the polynomial $P(x) = 3x^4 - 2x^3 + ax^2 + bx - 12$ is divided by $(x^2 - 4)$ , the remainder is $5x - 4$ . What is the value of $a + b$ ?

Show answer & explanation

Answer: 3

Answer

The value of

a + b

is

3

.

By using the Remainder Theorem for the quadratic divisor

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

, evaluating

P(2) = 6

gives

2a + b = -7

, and evaluating

P(-2) = -14

gives

2a - b = -33

. Solving these linear equations simultaneously yields

a = -10

and

b = 13

, which sums to

a + b = 3

.

Step-by-Step Solution

1

Set up the polynomial division relation.

P(x) = (x - 2)(x + 2)Q(x) + (5x - 4)

By the Remainder Theorem and Division Algorithm, dividing by

(x^2 - 4)

yields a remainder of

R(x) = 5x - 4

.

2

Find the values of

P(2)

and

P(-2)

from the remainder.

P(2) = 6

and

P(-2) = -14

Substituting the roots of the divisor

x = 2

and

x = -2

eliminates the quotient term

(x^2 - 4)Q(x)

.

3

Substitute

x = 2

into the polynomial

P(x)

and set equal to

6

.

2a + b = -7

3(16) - 2(8) + 4a + 2b - 12 = 20 + 4a + 2b = 6 \implies 4a + 2b = -14

.

4

Substitute

x = -2

into the polynomial

P(x)

and set equal to

-14

.

2a - b = -33

3(16) - 2(-8) + 4a - 2b - 12 = 52 + 4a - 2b = -14 \implies 4a - 2b = -66

.

5

Solve the system of equations for

a

and

b

.

a = -10

,

b = 13

, and

a + b = 3

Adding the two linear equations gives

4a = -40 \implies a = -10

. Substituting

a = -10

into

2a + b = -7

gives

b = 13

.

Key Concept

Polynomial Remainder Theorem for Non-Linear Divisors

Estimated Time:2m 30s

Question 722Question →

An arithmetic progression (A.P.) has a first term of $7$ and a common difference of $5$ . What is the $12^{\text{th}}$ term of the progression?

Show answer & explanation

Answer: 62

Answer

The

12^{\text{th}}

term of the progression is

62

.

Using the formula for the

n^{\text{th}}

term of an arithmetic progression,

T_n = a + (n - 1)d

, with

a = 7

,

d = 5

, and

n = 12

, we calculate

T_{12} = 7 + (12 - 1) \times 5 = 7 + 55 = 62

.

Step-by-Step Solution

1

Identify known parameters from the question

a = 7

,

d = 5

,

n = 12

These parameters are given in the problem statement.

2

Use the general formula for the

n^{\text{th}}

term of an arithmetic progression

T_n = a + (n - 1)d

This formula connects the first term, common difference, and term number to the value of the term.

3

Substitute values and evaluate

T_{12} = 7 + (12 - 1) \times 5 = 7 + 55 = 62

Performing basic arithmetic gives the final result.

Key Concept

Finding the

n^{\text{th}}

term of an Arithmetic Progression

Estimated Time:45s

Question 723Question →

An isosceles triangle has a perimeter of $36\text{ cm}$ and a base of length $16\text{ cm}$ . Calculate the area of the triangle in $\text{cm}^2$ .

Show answer & explanation

Answer: 48

Answer

The area of the isosceles triangle is

48\text{ cm}^2

.

The two equal sides of the isosceles triangle measure

\frac{36 - 16}{2} = 10\text{ cm}

each. An altitude dropped perpendicularly to the base bisects the

16\text{ cm}

base into two

8\text{ cm}

segments. By the Pythagorean theorem, the perpendicular height is

h = \sqrt{10^2 - 8^2} = 6\text{ cm}

. Therefore, the area is

\frac{1}{2} \times 16 \times 6 = 48\text{ cm}^2

.

Step-by-Step Solution

1

Determine the length of the two equal sides

Each equal side is

10\text{ cm}

Subtract the base length from the total perimeter (

36 - 16 = 20\text{ cm}

) and divide by

2

.

2

Calculate the perpendicular height (altitude) to the base

Height

h = 6\text{ cm}

The perpendicular altitude bisects the base into two

8\text{ cm}

segments, creating right-angled triangles with hypotenuse

10\text{ cm}

. Use Pythagoras' theorem:

h = \sqrt{10^2 - 8^2} = 6\text{ cm}

.

3

Calculate the area of the triangle

Area =

48\text{ cm}^2

Multiply half the base by the perpendicular height:

\frac{1}{2} \times 16 \times 6 = 48\text{ cm}^2

.

Key Concept

Perimeter and Area of Isosceles Triangles using Pythagorean Theorem

Estimated Time:1m 30s

Question 724Question →

Two straight paths diverge from a junction $J$ at an angle of $120^\circ$ . A person walks $5\text{ km}$ along the first path to point $A$ , and another person walks $16\text{ km}$ along the second path to point $B$ . What is the direct distance between $A$ and $B$ in kilometres?

Show answer & explanation

Answer: 19

Answer

The direct distance between points

A

and

B

is

19\text{ km}

.

Using the Cosine Rule

c^2 = a^2 + b^2 - 2ab \cos C

with sides

5\text{ km}

and

16\text{ km}

and included angle

120^\circ

yields

c^2 = 5^2 + 16^2 - 2(5)(16)(-0.5) = 25 + 256 + 80 = 361

. Taking the square root gives

19\text{ km}

.

Step-by-Step Solution

1

Identify given values and setup the Cosine Rule formula

Side

a = 5

, side

b = 16

, and included angle

\theta = 120^\circ

The scenario provides two sides and the included angle (SAS configuration), which requires the Cosine Rule to find the third side.

2

Substitute the values into

c^2 = a^2 + b^2 - 2ab \cos \theta

c^2 = 5^2 + 16^2 - 2(5)(16) \cos(120^\circ)

Populating the formula allows evaluation of the unknown distance squared.

3

Evaluate the trigonometric term and simplify

c^2 = 25 + 256 - 160(-0.5) = 281 + 80 = 361

The cosine of an obtuse angle in the second quadrant (

120^\circ

) is negative:

\cos(120^\circ) = -0.5

.

4

Take the square root to find the distance

c

c = \sqrt{361} = 19\text{ km}

Taking the positive square root gives the physical distance between the two points.

Key Concept

Applying the Cosine Rule to find the length of an unknown side in a non-right triangle given two sides and the included angle (SAS).

Question 725Question →

Evaluate the definite integral $\int_{0}^{3} (x^2 + 2) \, dx$ .

Show answer & explanation

Answer: 15

Answer

The value of the definite integral is 15.

Integrating

x^2 + 2

with respect to

x

yields

\left[\frac{x^3}{3} + 2x\right]_{0}^{3}

. Substituting the upper bound

x = 3

gives

\frac{27}{3} + 6 = 15

. Substituting the lower bound

x = 0

gives

0

. The net value is

15 - 0 = 15

.

Step-by-Step Solution

1

Find the antiderivative of

x^2 + 2

\frac{x^3}{3} + 2x

Apply the power rule of integration to each term.

2

Substitute the upper limit

x = 3

15

\frac{3^3}{3} + 2(3) = 9 + 6 = 15

3

Substitute the lower limit

x = 0

0

\frac{0^3}{3} + 2(0) = 0

4

Compute the difference between upper and lower limit values

15

15 - 0 = 15

Key Concept

Definite Integration of Polynomial Functions

Estimated Time:45s

Question 726Question →

A bag contains $5$ red balls, $3$ blue balls, and $n$ green balls. The theoretical probability of drawing a blue ball at random from the bag is $\frac{3}{20}$ . In a probability experiment, a ball is drawn at random from the bag and replaced $500$ times. If a green ball is observed $340$ times, calculate the absolute difference between the experimental probability and the theoretical probability of drawing a green ball.

Show answer & explanation

Answer: 0.08

Answer

The absolute difference between the experimental probability and the theoretical probability of drawing a green ball is

0.08

(or

\frac{2}{25}

).

The theoretical probability of drawing a blue ball establishes that the bag contains

20

total balls, meaning there are

12

green balls. The theoretical probability of picking a green ball is therefore

\frac{12}{20} = 0.60

. The experimental probability is

\frac{340}{500} = 0.68

. Taking the positive difference gives

|0.68 - 0.60| = 0.08

.

Step-by-Step Solution

1

Find the total number of balls and the value of

n

n = 12

, Total balls =

20

The theoretical probability of picking a blue ball is

\frac{3}{8+n} = \frac{3}{20}

, giving

8 + n = 20

.

2

Find the theoretical probability of drawing a green ball

P(\text{Green})_{\text{theo}} = 0.60

There are

12

green balls out of

20

total balls, so

\frac{12}{20} = 0.60

.

3

Find the experimental probability of drawing a green ball

P(\text{Green})_{\text{exp}} = 0.68

In

500

trials, a green ball was observed

340

times, so

\frac{340}{500} = 0.68

.

4

Calculate the absolute difference

|0.68 - 0.60| = 0.08

Subtract the theoretical probability from the experimental probability and take the absolute value.

Key Concept

Comparison of theoretical probability based on outcome sample space and experimental probability based on observed trial relative frequencies.

Question 727Question →

If $y = e^{-x} \tan x + \ln(2x + 1)$ , find the value of $\frac{dy}{dx}$ at $x = 0$ .

Show answer & explanation

Answer: 3

Answer

The value of

\frac{dy}{dx}

at

x = 0

is 3.

Applying the product rule to

e^{-x}\tan x

gives

-e^{-x}\tan x + e^{-x}\sec^2 x

, and applying the chain rule to

\ln(2x+1)

gives

\frac{2}{2x+1}

. Evaluating

\frac{dy}{dx} = -e^{-x}\tan x + e^{-x}\sec^2 x + \frac{2}{2x+1}

at

x = 0

yields

-e^0(0) + e^0(1) + \frac{2}{1} = 3

.

Step-by-Step Solution

1

Differentiate

u(x) = e^{-x} \tan x

using the product rule.

\frac{du}{dx} = -e^{-x} \tan x + e^{-x} \sec^2 x

The derivative of

e^{-x}

is

-e^{-x}

and the derivative of

\tan x

is \sec^2 x.

2

Differentiate

v(x) = \ln(2x + 1)

using the chain rule.

\frac{dv}{dx} = \frac{2}{2x + 1}

The derivative of

\ln(g(x))

is

\frac{g'(x)}{g(x)}

, where

g(x) = 2x + 1

and

g'(x) = 2

.

3

Sum the derivatives to find the complete expression for

\frac{dy}{dx}

.

\frac{dy}{dx} = -e^{-x} \tan x + e^{-x} \sec^2 x + \frac{2}{2x + 1}

The derivative of a sum is equal to the sum of the derivatives.

4

Evaluate the derivative at

x = 0

.

\frac{dy}{dx}\Big|_{x=0} = -e^0(0) + e^0(1)^2 + \frac{2}{1} = 3

Since

\tan(0) = 0

,

e^0 = 1

, and

\sec(0) = 1

, substituting

x = 0

simplifies the derivative to

0 + 1 + 2 = 3

.

Key Concept

Differentiation of Transcendental Functions (Product and Chain Rules)

Question 728Question →

Find the value of $x$ , in degrees, for $0^\circ \le x \le 90^\circ$ that satisfies the trigonometric equation $\sin 2x = \cos(x + 30^\circ)$ .

Show answer & explanation

Answer: 20

Answer

The value of

x

in the interval

0^\circ \le x \le 90^\circ

satisfying the equation is

20^\circ

.

Using the co-function identity

\cos \alpha = \sin(90^\circ - \alpha)

, we convert the right-hand side to

\sin(90^\circ - (x + 30^\circ)) = \sin(60^\circ - x)

. Equating the arguments gives

2x = 60^\circ - x

, which simplifies to

3x = 60^\circ

, yielding

x = 20^\circ

.

Step-by-Step Solution

1

Apply the co-function trigonometric identity

\cos(x + 30^\circ) = \sin(90^\circ - (x + 30^\circ)) = \sin(60^\circ - x)

Converting cosine to sine allows direct comparison of sine functions on both sides of the equation.

2

Set up the equation equating the angle expressions

2x = 60^\circ - x

Since

\sin(2x) = \sin(60^\circ - x)

and

x \in [0^\circ, 90^\circ]

, equating the principal angle arguments gives the primary solution.

3

Solve the linear equation for

x

3x = 60^\circ \implies x = 20^\circ

Adding

x

to both sides gives

3x = 60^\circ

, and dividing by 3 yields

x = 20^\circ

.

Key Concept

Co-function identities and simple trigonometric equations

Question 729Question →

Two independent weather forecasting stations, $A$ and $B$ , operate in a region. The probability that station $A$ makes an accurate forecast on any given day is $0.80$ , and the probability that station $B$ makes an accurate forecast is $0.75$ . What is the probability that at least one of the two stations makes an accurate forecast on a given day?

Show answer & explanation

Answer: 0.95

Answer

The probability that at least one of the two stations makes an accurate forecast is

0.95

.

The probability of at least one event occurring is given by

P(A \cup B) = P(A) + P(B) - P(A \cap B)

. Because events

A

and

B

are independent,

P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.75 = 0.60

. Substituting into the addition rule yields

0.80 + 0.75 - 0.60 = 0.95

. Alternatively, using the complement rule:

1 - P(A')P(B') = 1 - (1 - 0.80)(1 - 0.75) = 1 - (0.20 \times 0.25) = 1 - 0.05 = 0.95

.

Step-by-Step Solution

1

Calculate the probability of both events occurring simultaneously using the multiplication law for independent events.

P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.75 = 0.60

Since the two forecasting stations operate independently, the joint probability is the product of their individual probabilities.

2

Apply the general addition law of probability to calculate the probability of at least one station making an accurate forecast.

P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.80 + 0.75 - 0.60 = 0.95

The probability of compound event 'at least one' corresponds to the union of the two events.

Key Concept

Addition and Multiplication Laws of Probability for Independent Events

Question 730Question →

What is the simplified numerical value of $\frac{\sqrt{12}}{\sqrt{3} + 1} + \frac{\sqrt{12}}{\sqrt{3} - 1}$ ?

Show answer & explanation

Answer: 6

Answer

The simplified numerical value of the given expression is 6.

Combining the fractions using their common conjugate denominator

(\sqrt{3}+1)(\sqrt{3}-1) = 2

leads to a numerator of

2\sqrt{3}[(\sqrt{3}-1)+(\sqrt{3}+1)] = 2\sqrt{3}(2\sqrt{3}) = 12

. Dividing 12 by 2 yields the final answer of 6.

Step-by-Step Solution

1

Simplify the surd in the numerator

\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Simplifying surds into basic form makes subsequent calculations simpler.

2

Combine the fractions by finding a common denominator

\frac{2\sqrt{3}(\sqrt{3}-1) + 2\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}+1)(\sqrt{3}-1)}

Multiplying denominators forms a conjugate pair, which rationalizes the combined denominator.

3

Simplify the numerator and the denominator

\text{Numerator} = 2\sqrt{3}(\sqrt{3}-1 + \sqrt{3}+1) = 2\sqrt{3}(2\sqrt{3}) = 12; \quad \text{Denominator} = (\sqrt{3})^2 - 1^2 = 3 - 1 = 2

Expanding the numerator combines like surd terms, and using the difference of squares simplifies the denominator to a rational integer.

4

Perform the final division

\frac{12}{2} = 6

Dividing the simplified numerator by the rationalized denominator gives the final integer value.

Key Concept

Rationalization of binomial denominators using conjugate surds

Question 731Question →

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 732Question →

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 733Question →

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 734Question →

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 735Question →

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 736Question →

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 737Question →

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 738Question →

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 739Question →

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 740Question →

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

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept