All practice questions

Question 7601Question →

If $(u, v)$ is a pair of real numbers satisfying the simultaneous equations $u - 2v = 1$ and $u^2 - 3v^2 = 13$ , what is the sum of all possible values of $u$ ?

-6

-4

6

-8

Show answer & explanation

Answer: $-6$

Answer

The sum of all possible values of

u

is

-6

.

From the linear equation

u - 2v = 1

, we express

u

as

u = 2v + 1

. Substituting this into the quadratic equation

u^2 - 3v^2 = 13

gives

(2v + 1)^2 - 3v^2 = 13

, which expands and simplifies to

v^2 + 4v - 12 = 0

. Factoring gives

(v + 6)(v - 2) = 0

, so

v = 2

or

v = -6

. Substituting these back into

u = 2v + 1

yields

u = 5

(for

v = 2

) and

u = -11

(for

v = -6

). The sum of all possible values of

u

is

5 + (-11) = -6

.

Step-by-Step Solution

1

Express

u

in terms of

v

using the linear equation.

u = 2v + 1

Isolating

u

allows for straightforward substitution into the quadratic equation.

2

Substitute

u = 2v + 1

into the quadratic equation

u^2 - 3v^2 = 13

and expand.

(2v + 1)^2 - 3v^2 = 13 \implies (4v^2 + 4v + 1) - 3v^2 = 13 \implies v^2 + 4v - 12 = 0

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

v

.

3

Solve the quadratic equation for

v

by factorisation.

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

Finding the roots of the quadratic gives the

v

-coordinates of the solution pairs.

4

Calculate the corresponding values of

u

using

u = 2v + 1

.

For

v = 2

:

u = 2(2) + 1 = 5

.
For

v = -6

:

u = 2(-6) + 1 = -11

.

Each

v

value corresponds to a specific

u

value in the solution pairs.

5

Compute the sum of all possible values of

u

.

5 + (-11) = -6

The question asks specifically for the sum of the

u

-values.

Key Concept

Solving simultaneous linear and quadratic equations using substitution

Question 7602Question →

A solid rectangular wooden block measures $10\text{ cm}$ by $14\text{ cm}$ by $12\text{ cm}$ . A cylindrical hole of radius $3.5\text{ cm}$ is drilled straight through the block along its height of $12\text{ cm}$ . Taking $\pi = \frac{22}{7}$ , what is the volume of the remaining solid in $\text{cm}^3$ ?

Show answer & explanation

Answer: 1218

Answer

The volume of the remaining solid is

1218\text{ cm}^3

.

The initial total volume of the rectangular block is

10 \times 14 \times 12 = 1680\text{ cm}^3

. The volume of the cylindrical hole drilled through it is

\frac{22}{7} \times (3.5)^2 \times 12 = 462\text{ cm}^3

. Subtracting the removed cylindrical volume from the block gives

1680 - 462 = 1218\text{ cm}^3

.

Step-by-Step Solution

1

Calculate the total volume of the rectangular block.

Volume of block =

1680\text{ cm}^3

The initial volume of the cuboid before drilling is calculated by multiplying its length, width, and height:

10 \times 14 \times 12 = 1680\text{ cm}^3

.

2

Calculate the volume of the cylindrical hole removed from the block.

Volume of cylinder =

462\text{ cm}^3

The cylindrical hole has radius

r = 3.5\text{ cm} = \frac{7}{2}\text{ cm}

and height

h = 12\text{ cm}

. Using

V = \pi r^2 h

, we get

\frac{22}{7} \times \frac{49}{4} \times 12 = 462\text{ cm}^3

.

3

Subtract the volume of the cylindrical hole from the total volume of the block.

Remaining volume =

1218\text{ cm}^3

Because material is removed by drilling, the remaining volume is

1680 - 462 = 1218\text{ cm}^3

.

Key Concept

Volume of a composite solid (cuboid with a cylindrical cavity)

Question 7603Question →

A delivery van travels from Town A to Town B, a distance of $120\text{ km}$ , at an average speed of $60\text{ km/h}$ . On the return journey along the same route, traffic congestion reduces its average speed by one-third. What is the average speed of the van for the entire round trip?

48\text{ km/h}

50\text{ km/h}

30\text{ km/h}

24\text{ km/h}

Show answer & explanation

Answer: $48\text{ km/h}$

Answer

The average speed for the entire round trip is

48\text{ km/h}

.

The correct answer is found by dividing the total round-trip distance of

240\text{ km}

by the total journey time of

5\text{ hours}

(

2\text{ hours}

outbound and

3\text{ hours}

return), yielding

48\text{ km/h}

.

Step-by-Step Solution

1

Calculate the time taken for the outbound journey.

\text{Time}_{\text{out}} = \frac{120\text{ km}}{60\text{ km/h}} = 2\text{ hours}

.

Time equals distance divided by speed.

2

Calculate the return speed and return time.

\text{Return speed} = 60 - \left(\frac{1}{3} \times 60\right) = 40\text{ km/h}. \text{Time}_{\text{return}} = \frac{120\text{ km}}{40\text{ km/h}} = 3\text{ hours}.

Speed reduction of one-third leaves two-thirds of the initial speed.

3

Compute total distance and total time for the round trip.

\text{Total distance} = 120 + 120 = 240\text{ km}; \text{Total time} = 2 + 3 = 5\text{ hours}.

A round trip covers the distance twice.

4

Divide total distance by total time to find average speed.

\text{Average speed} = \frac{240\text{ km}}{5\text{ hours}} = 48\text{ km/h}.

Average rate over unequal sub-intervals requires total quantity over total time.

Key Concept

Average Rate for Round-Trip Journeys

Question 7604Question →

What is the area of the region bounded by the curve $y = \sin x$ and the straight line $y = \frac{2}{\pi}x$ in the first quadrant for $0 \leq x \leq \frac{\pi}{2}$ ?

1 - \frac{\pi}{4}

square units

1 + \frac{\pi}{4}

square units

1 - \frac{\pi}{2}

square units

\frac{\pi}{4}

square units$

Show answer & explanation

Answer: $1 - \frac{\pi}{4}$ square units

Answer

The area of the enclosed region is

1 - \frac{\pi}{4}

square units.

The bounded area is calculated by taking the definite integral of the upper curve minus the lower curve over the interval

[0, \pi/2]

. The upper boundary is

y = \sin x

and the lower boundary is

y = \frac{2}{\pi}x

. Computing

\int_{0}^{\pi/2} (\sin x - \frac{2}{\pi}x) dx

gives

[-\cos x - \frac{x^2}{\pi}]_{0}^{\pi/2} = (0 - \frac{\pi}{4}) - (-1 - 0) = 1 - \frac{\pi}{4}

square units.

Step-by-Step Solution

1

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

A = \int_{0}^{\pi/2} \left(\sin x - \frac{2}{\pi}x\right) dx

In the interval

[0, \pi/2]

, the curve

y = \sin x

lies above the straight line

y = \frac{2}{\pi}x

.

2

Find the antiderivative of each term

\int \sin x \, dx = -\cos x \quad \text{and} \quad \int \frac{2}{\pi}x \, dx = \frac{x^2}{\pi}

Integration rules for basic trigonometric and power functions.

3

Evaluate the definite integral from lower limit

x = 0

to upper limit

x = \frac{\pi}{2}

A = \left[-\cos x - \frac{x^2}{\pi}\right]_{0}^{\pi/2} = \left(-\cos\frac{\pi}{2} - \frac{(\pi/2)^2}{\pi}\right) - \left(-\cos 0 - \frac{0^2}{\pi}\right)

Apply the Fundamental Theorem of Calculus:

F(b) - F(a)

.

4

Simplify the numeric expression

A = \left(0 - \frac{\pi}{4}\right) - (-1 - 0) = -\frac{\pi}{4} + 1 = 1 - \frac{\pi}{4}

Since

\cos(\pi/2) = 0

and

\cos(0) = 1

.

Key Concept

Area bounded between curves using definite integration

Estimated Time:2m 0s

Question 7605Question →

What is the indefinite integral $\int (5x^4 - 3\sin(x)) \, dx$ ?

x^5 + 3\cos(x) + C

x^5 - 3\cos(x) + C

x^5 + 3\cos(x)

20x^3 - 3\cos(x) + C

Show answer & explanation

Answer: $x^5 + 3\cos(x) + C$

Answer

x^5 + 3\cos(x) + C

Integrating

5x^4

gives

x^5

, and integrating

-3\sin(x)

gives

+3\cos(x)

since

\int \sin(x) \, dx = -\cos(x)

. Adding the arbitrary constant of integration

C

results in

x^5 + 3\cos(x) + C

.

Step-by-Step Solution

1

Integrate the polynomial term

5x^4

\int 5x^4 \, dx = \frac{5x^{4+1}}{4+1} = x^5

Apply the power rule for integration:

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

.

2

Integrate the trigonometric term

-3\sin(x)

\int -3\sin(x) \, dx = -3(-\cos(x)) = 3\cos(x)

The integral of

\sin(x)

with respect to

x

is

-\cos(x)

.

3

Combine the terms and add the constant of integration

x^5 + 3\cos(x) + C

An indefinite integral requires an arbitrary constant

C

.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions

Question 7606Question →

Match each type of thermometer with its corresponding physical thermometric property.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Liquid-in-glass thermometer matches change in length or volume of a liquid column; Constant-volume gas thermometer matches change in pressure of a gas; Resistance thermometer matches change in electrical resistance; Thermocouple matches electromotive force (e.m.f.) produced across junctions.

Each thermometer is accurately matched to the physical property that undergoes a measurable change as temperature varies.

Step-by-Step Solution

1

Identify the defining physical property that varies with temperature for each instrument.

Each thermometer operates on a distinct physical property that changes predictably when heated or cooled.

Thermometric properties must be reproducible and continuously measurable across a temperature range.

2

Pair each instrument with its specific thermometric property.

Liquid-in-glass pairs with liquid column expansion/length; constant-volume gas thermometer pairs with gas pressure; resistance thermometer pairs with electrical resistance; thermocouple pairs with thermoelectric e.m.f.

These pairs represent standard physical principles used in thermometry.

Key Concept

Thermometric properties and operating principles of thermometers

Question 7607Question →

Match each physical quantity to its correct physical quantity classification and corresponding SI base unit expression.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Luminous intensity matches with fundamental quantity in

\text{cd}

; Electric potential matches with derived quantity in

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

; Specific heat capacity matches with derived quantity in

\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

; Thermodynamic temperature matches with fundamental quantity in

\text{K}

.

Luminous intensity and thermodynamic temperature are fundamental SI quantities with base units cd and K. Electric potential and specific heat capacity are derived quantities whose definitions reduce to

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

and

\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

respectively.

Step-by-Step Solution

1

Identify fundamental physical quantities

Luminous intensity and thermodynamic temperature are base SI quantities measured in candela (cd) and kelvin (K) respectively.

Base quantities cannot be defined in terms of other physical quantities.

2

Decompose derived quantities into base SI units

Electric potential

V = \frac{W}{q}

yields

\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}

. Specific heat capacity

c = \frac{Q}{m\Delta T}

yields

\text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

.

Derived quantities originate from mathematical combinations of fundamental quantities.

Key Concept

Classification of physical quantities into fundamental (base) and derived categories and resolution into SI base units

Question 7608Question →

If $(x_1, y_1)$ and $(x_2, y_2)$ are the real solution pairs to the simultaneous equations $3x - y = 5$ and $x^2 + 2xy - y^2 = 7$ , with $x_1 < x_2$ , what is the value of $y_2 - y_1$ ?

18

6

20

14

Show answer & explanation

Answer: 18

Answer

18

Expressing

y

as

3x - 5

and substituting into the quadratic equation yields

x^2 - 10x + 16 = 0

. Solving gives

x_1 = 2

and

x_2 = 8

. Evaluating

y = 3x - 5

for both values gives

y_1 = 1

and

y_2 = 19

. The difference

y_2 - y_1 = 19 - 1 = 18

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation

y = 3x - 5

Isolation of

y

facilitates substitution into the non-linear equation.

2

Substitute

y = 3x - 5

into the quadratic equation

x^2 + 2xy - y^2 = 7

x^2 + 2x(3x - 5) - (3x - 5)^2 = 7

Reduces the system to a single quadratic equation in terms of

x

.

3

Expand and simplify the algebraic expression

x^2 + 6x^2 - 10x - (9x^2 - 30x + 25) = 7 \implies -2x^2 + 20x - 32 = 0 \implies x^2 - 10x + 16 = 0

Simplifies the equation to standard quadratic form

ax^2 + bx + c = 0

.

4

Solve the quadratic equation

x^2 - 10x + 16 = 0

by factorization

(x - 2)(x - 8) = 0 \implies x_1 = 2

and

x_2 = 8

Identifies the two roots with

x_1 < x_2

as specified by the condition.

5

Determine the corresponding

y

-values using

y = 3x - 5

y_1 = 3(2) - 5 = 1

and

y_2 = 3(8) - 5 = 19

Obtains the complete coordinate solution pairs

(2, 1)

and

(8, 19)

.

6

Calculate the required difference

y_2 - y_1

y_2 - y_1 = 19 - 1 = 18

Computes the final required target value.

Key Concept

Solving Simultaneous Linear and Quadratic Equations by Substitution

Question 7609Question →

If $y = x^3 \sin(2x)$ , what is $\frac{dy}{dx}$ ?

3x^2 \sin(2x) + 2x^3 \cos(2x)

3x^2 \sin(2x) + x^3 \cos(2x)

6x^2 \cos(2x)

3x^2 \sin(2x) - 2x^3 \cos(2x)

Show answer & explanation

Answer: $3x^2 \sin(2x) + 2x^3 \cos(2x)$

Answer

The derivative

\frac{dy}{dx}

is

3x^2 \sin(2x) + 2x^3 \cos(2x)

.

Applying the product rule

\frac{d}{dx}[uv] = u \frac{dv}{dx} + v \frac{du}{dx}

with

u = x^3

and

v = \sin(2x)

yields

x^3(2\cos(2x)) + \sin(2x)(3x^2)

, which simplifies to

3x^2 \sin(2x) + 2x^3 \cos(2x)

.

Step-by-Step Solution

1

Identify the two component functions for the Product Rule

Let

u = x^3

and

v = \sin(2x)

, so

y = uv

.

The function is a product of an algebraic term and a trigonometric term.

2

Find the derivative of each component function

\frac{du}{dx} = 3x^2 and \frac{dv}{dx} = 2\cos(2x) using the chain rule for \sin(2x).

The derivative of \sin(2x) with respect to 2x is \cos(2x), multiplied by the derivative of 2x which is 2.

3

Apply the Product Rule formula \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}

\frac{dy}{dx} = x^3 \cdot 2\cos(2x) + \sin(2x) \cdot 3x^2 = 3x^2 \sin(2x) + 2x^3 \cos(2x).

Combining the differentiated terms according to the product rule gives the final derivative.

Key Concept

Product Rule combined with Chain Rule for trigonometric functions

Estimated Time:1m 30s

Question 7610Question →

Match each composite physical quantity or ratio on the left with its correct fundamental (SI base) unit decomposition on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Electric potential gradient matches

\text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}

; Coefficient of dynamic viscosity matches

\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}

; Ratio of Planck's constant to moment of inertia matches

\text{s}^{-1}

; Specific latent heat divided by spatial temperature gradient matches

\text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}

.

Each physical quantity or ratio is systematically reduced to its SI base quantities (mass in kg, length in m, time in s, electric current in A, thermodynamic temperature in K) by substituting fundamental definitions of derived units.

Step-by-Step Solution

1

Decompose electric potential gradient into fundamental SI base units

\text{Electric potential gradient} = \frac{\text{Electric Potential}}{\text{Distance}} = \frac{\text{Work}}{\text{Charge} \times \text{Distance}} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s} \times \text{m}} = \text{kg}\cdot\text{m}\cdot\text{s}^{-3}\cdot\text{A}^{-1}

.

Electric potential is defined as energy per unit charge, and potential gradient is its spatial rate of change.

2

Decompose coefficient of dynamic viscosity into fundamental SI base units

\eta = \frac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}} = \frac{(\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times \text{m}}{\text{m}^2 \times (\text{m}\cdot\text{s}^{-1})} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}

.

Newton's law of viscosity relates shear force to surface area and velocity gradient.

3

Determine the base unit ratio of Planck's constant to moment of inertia

\frac{h}{I} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}}{\text{kg}\cdot\text{m}^2} = \text{s}^{-1}.

Planck's constant carries dimensions of angular momentum, while moment of inertia is mass multiplied by distance squared.

4

Decompose specific latent heat divided by spatial temperature gradient

\frac{L}{\frac{\Delta T}{\Delta x}} = \frac{\text{J}\cdot\text{kg}^{-1}}{\text{K}\cdot\text{m}^{-1}} = \frac{\text{m}^2\cdot\text{s}^{-2}}{\text{K}\cdot\text{m}^{-1}} = \text{m}^3\cdot\text{s}^{-2}\cdot\text{K}^{-1}.

Specific heat quantities represent thermal energy per unit mass, whereas temperature gradient represents thermal variation per unit displacement.

Key Concept

Fundamental SI base unit decomposition of derived physical quantities

Question 7611Question →

The frequency distribution table below shows the masses (in $\text{kg}$ ) of $40$ cassava tubers harvested from an agricultural test plot:

Mass ( $\text{kg}$ )	Frequency ( $f$ )
$10 - 14$	$4$
$15 - 19$	$8$
$20 - 24$	$12$
$25 - 29$	$10$
$30 - 34$	$6$

What is the estimated mean mass of the harvested cassava tubers?

20.75\text{ kg}

22.40\text{ kg}

22.75\text{ kg}

24.75\text{ kg}

Show answer & explanation

Answer: $22.75\text{ kg}$

Answer

22.75\text{ kg}

The value

22.75\text{ kg}

is correct because the mean for grouped data is obtained by multiplying each class midpoint by its frequency, summing these products to get

910

, and dividing by the total frequency of

40

.

Step-by-Step Solution

1

Find the class midpoint (

x

) for each class interval using

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

.

Midpoints are:

12

for

10-14

,

17

for

15-19

,

22

for

20-24

,

27

for

25-29

, and

32

for

30-34

.

Grouped data calculations require representative central values for each class interval.

2

Multiply each class midpoint (

x

) by its corresponding frequency (

f

) to get

fx

, then sum all

fx

values.

\sum fx = (4 \times 12) + (8 \times 17) + (12 \times 22) + (10 \times 27) + (6 \times 32) = 48 + 136 + 264 + 270 + 192 = 910

.

This calculates the total estimated mass of all harvested cassava tubers.

3

Sum the frequencies to get the total number of items

\sum f

.

\sum f = 4 + 8 + 12 + 10 + 6 = 40.

The mean formula divides the total sum by the total frequency.

4

Compute the grouped mean using

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

.

\bar{x} = \frac{910}{40} = 22.75\text{ kg}.

Dividing the sum of products by the total frequency yields the estimated mean.

Key Concept

Grouped Mean Calculation

Question 7612Question →

The frequency of oscillation $f$ of a small liquid droplet executing spherical oscillations depends on the surface tension $\gamma$ of the liquid, its mass density $\rho$ , and the radius $r$ of the droplet according to the relationship $f = k \gamma^a \rho^b r^c$ , where $k$ is a dimensionless constant. Which of the following represents the correct value of the exponent $c$ ?

-\frac{3}{2}

-\frac{1}{2}

\frac{1}{2}

\frac{3}{2}

Show answer & explanation

Answer: $-\frac{3}{2}$

Answer

The correct value of the exponent

c

is

-\frac{3}{2}

.

By writing the dimensional equation

[M^0 L^0 T^{-1}] = [M T^{-2}]^a [M L^{-3}]^b [L]^c

, we solve for the exponents:

-2a = -1

gives

a = 1/2

,

a + b = 0

gives

b = -1/2

, and

-3b + c = 0

gives

3/2 + c = 0 \implies c = -3/2

. Thus, the option equal to

-3/2

is correct.

Step-by-Step Solution

1

Express each physical quantity in terms of its base dimensions

[M]

,

[L]

, and

[T]

.

Frequency

f = [T^{-1}]

, Surface tension

\gamma = \frac{\text{Force}}{\text{Length}} = [M T^{-2}]

, Density

\rho = [M L^{-3}]

, and Radius

r = [L]

.

Dimensional analysis requires reducing derived physical quantities to their base units.

2

Substitute dimensions into the given formula

f = k \gamma^a \rho^b r^c

.

[M^0 L^0 T^{-1}] = [M T^{-2}]^a [M L^{-3}]^b [L]^c = M^{a+b} L^{-3b+c} T^{-2a}

.

The principle of dimensional homogeneity requires both sides of a physical equation to have matching dimensions.

3

Equate exponents for each base dimension

M

,

L

, and

T

.

For

T

:

-2a = -1 \implies a = \frac{1}{2}

. For

M

:

a + b = 0 \implies b = -a = -\frac{1}{2}

. For

L

:

-3b + c = 0 \implies -3\left(-\frac{1}{2}\right) + c = 0

.

Matching powers across orthogonal base dimensions provides a system of linear equations.

4

Solve for the target exponent

c

.

\frac{3}{2} + c = 0 \implies c = -\frac{3}{2}.

Subtracting

3/2

from both sides gives the exact value of exponent

c

.

Key Concept

Dimensional Homogeneity and Dimensional Analysis

Estimated Time:2m 0s

Question 7613Question →

What is the period, in degrees, of the trigonometric function $y = 7\sin(5x) - 2$ ?

Show answer & explanation

Answer: 72

Answer

The period of the trigonometric function is 72 degrees.

For any function of the form

y = A\sin(Bx) + D

, the period

T

in degrees is calculated using

T = \frac{360^\circ}{|B|}

. For the given equation

y = 7\sin(5x) - 2

, the value of

B

is 5. Substituting this into the formula gives

T = \frac{360^\circ}{5} = 72^\circ

.

Step-by-Step Solution

1

Identify the coefficient

B

of the variable

x

in the given function

y = 7\sin(5x) - 2

.

Here,

A = 7

,

B = 5

, and

D = -2

.

The period of a sine function depends on the angular frequency parameter

B

multiplying the input variable

x

.

2

Apply the standard formula for finding the period

T

of a sine function in degrees:

T = \frac{360^\circ}{|B|}

.

T = \frac{360^\circ}{5} = 72^\circ

.

Dividing the standard full revolution of

360^\circ

by the multiplier

5

determines the angle needed for one full cycle.

Key Concept

Period of a Trigonometric Graph

Estimated Time:45s

Question 7614Question →

During a quality control inspection at a manufacturing plant, two independent automated scanners, $S_1$ and $S_2$ , inspect items along a production line. The probability that scanner $S_1$ detects a surface flaw on a defective item is $0.90$ , and the probability that scanner $S_2$ detects the flaw is $0.80$ . What is the probability that a surface flaw is detected by exactly one of the two scanners?

Show answer & explanation

Answer: 0.26

Answer

The probability that a surface flaw is detected by exactly one of the two scanners is 0.26.

The probability of a surface flaw being detected by exactly one scanner is calculated by combining two mutually exclusive compound events: scanner

S_1

detects it while scanner

S_2

fails (

0.90 \times 0.20 = 0.18

), and scanner

S_1

fails while scanner

S_2

detects it (

0.10 \times 0.80 = 0.08

). Adding these two values yields

0.18 + 0.08 = 0.26

.

Step-by-Step Solution

1

Determine the probabilities of non-detection for each scanner.

P(S_1') = 0.10, P(S_2') = 0.20

The probability of an event's complement is 1 minus the probability of the event.

2

Calculate the probability that scanner S_1 detects the flaw while scanner S_2 misses it.

P(S_1 \cap S_2') = 0.90 \times 0.20 = 0.18

Because the scanners operate independently, joint probability is the product of their individual probabilities.

3

Calculate the probability that scanner S_1 misses the flaw while scanner S_2 detects it.

P(S_1' \cap S_2) = 0.10 \times 0.80 = 0.08

Independence allows multiplication of individual probabilities.

4

Sum the probabilities of the two mutually exclusive outcomes.

P(\text{exactly one}) = 0.18 + 0.08 = 0.26

The addition law applies because detecting the flaw with only S_1 and only S_2 are mutually exclusive events.

Key Concept

Compound probability laws for independent and mutually exclusive events

Question 7615Question →

The interior angles of a convex polygon form an arithmetic progression. If the smallest interior angle is $120^\circ$ and the common difference between consecutive interior angles is $5^\circ$ , how many sides does the polygon have?

Show answer & explanation

Answer: 9

Answer

The polygon has 9 sides.

Equating the interior angle sum formula

(n-2) \times 180^\circ

with the sum formula for an arithmetic progression

\frac{n}{2}[2(120^\circ) + (n-1)5^\circ]

yields the quadratic equation

n^2 - 25n + 144 = 0

. Solving gives

n = 9

and

n = 16

. For

n = 16

, the largest angle would be

195^\circ

, which is impossible for a convex polygon. Thus, the only valid number of sides is 9.

Step-by-Step Solution

1

Formulate expressions for the sum of the interior angles.

The interior angle sum of an

n

-sided polygon is

S = (n-2) \times 180^\circ

. As an arithmetic sequence with

a = 120^\circ

and

d = 5^\circ

, the sum is

S = \frac{n}{2}[2(120^\circ) + (n-1)5^\circ]

.

Both methods express the total sum of all interior angles of the polygon.

2

Equate the two sum formulas and simplify into a quadratic equation.

(n-2) \times 180 = \frac{n}{2}(235 + 5n) \implies n^2 - 25n + 144 = 0

.

Expanding and dividing by 5 reduces the equality to standard quadratic form.

3

Solve for

n

and test the validity of the roots based on polygon convexity.

(n - 9)(n - 16) = 0 \implies n = 9

or

n = 16

. Testing

n = 16

gives a largest angle of

120^\circ + 15(5^\circ) = 195^\circ

(invalid as it exceeds

180^\circ

). Testing

n = 9

gives a largest angle of

120^\circ + 8(5^\circ) = 160^\circ

(valid).

A convex polygon cannot have any interior angle greater than or equal to

180^\circ

.

Key Concept

Polygon interior angle sum and arithmetic progressions

Question 7616Question →

A binary operation $\circ$ on the set of real numbers $\mathbb{R}$ is defined by $a \circ b = a + b - \frac{1}{2}ab$ . What is the inverse of $3$ under this operation?

6

-6

-3

\frac{1}{3}

Show answer & explanation

Answer: $6$

Answer

The inverse of

3

under the defined binary operation is

6

.

To find the inverse of

3

, we first find the identity element

e

using

a \circ e = a \implies a + e - \frac{1}{2}ae = a

, which gives

e = 0

. Then, setting

3 \circ x = 0

gives

3 + x - \frac{3}{2}x = 0 \implies 3 = \frac{1}{2}x \implies x = 6

.

Step-by-Step Solution

1

Find the identity element

e

of the operation.

e = 0

By definition of identity element,

a \circ e = a

. Substituting into the operation formula gives

a + e - \frac{1}{2}ae = a \implies e\left(1 - \frac{1}{2}a\right) = 0 \implies e = 0

for all

a \neq 2

.

2

Set up the inverse equation for the element

3

.

3 \circ x = 0

Let

x

be the inverse of

3

. By definition of inverse element,

3 \circ x = e

, where

e = 0

.

3

Apply the binary operation definition to the left-hand side.

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

Substitute

a = 3

and

b = x

into

a \circ b = a + b - \frac{1}{2}ab

.

4

Solve the linear equation for

x

.

x = 6

Simplify the equation:

3 + x - \frac{3}{2}x = 0 \implies 3 - \frac{1}{2}x = 0 \implies \frac{1}{2}x = 3 \implies x = 6

.

Key Concept

Identity and Inverse Elements of Binary Operations

Question 7617Question →

Given the matrix $P = \begin{pmatrix} x & 1 & 2 \\ 3 & 0 & x \\ 1 & 2 & 1 \end{pmatrix}$ , for what positive value of $x$ is the determinant of $P$ equal to $3$ ?

Show answer & explanation

Answer: 2

Answer

The positive value of

x

is

2

.

Expanding the determinant of

P

along the first row yields

\det(P) = -2x^2 + x + 9

. Setting

\det(P) = 3

leads to

-2x^2 + x + 9 = 3

, which rearranges to

2x^2 - x - 6 = 0

. Factoring gives

(2x + 3)(x - 2) = 0

, leading to roots

x = -1.5

and

x = 2

. The positive value is

2

.

Step-by-Step Solution

1

Calculate the determinant of matrix

P

in terms of

x

\det(P) = -2x^2 + x + 9

Expanding along the first row gives

x(0 \cdot 1 - 2x) - 1(3 \cdot 1 - x \cdot 1) + 2(3 \cdot 2 - 0 \cdot 1) = -2x^2 + x + 9

.

2

Set up the quadratic equation using

\det(P) = 3

2x^2 - x - 6 = 0

Subtracting

3

from both sides gives

-2x^2 + x + 6 = 0

, which simplifies to

2x^2 - x - 6 = 0

.

3

Solve the quadratic equation for

x

x = 2 \text{ or } x = -1.5

Factoring

(2x + 3)(x - 2) = 0

yields

x = -\frac{3}{2}

and

x = 2

.

4

Select the positive solution as required by the problem statement

x = 2

The problem specifically requests the positive real value of

x

.

Key Concept

3x3 Determinant Evaluation and Quadratic Solving

Question 7618Question →

A quantity $Q$ is partly constant and partly varies inversely as $p$ . Given that $Q = 11$ when $p = 2$ , and $Q = 5$ when $p = 5$ , what is the value of $Q$ when $p = 10$ ?

2

3

4

7

Show answer & explanation

Answer: $3$

Answer

The value of

Q

when

p = 10

is

3

.

The relationship for partial inverse variation is

Q = k_1 + \frac{k_2}{p}

. Substituting

(p=2, Q=11)

gives

2k_1 + k_2 = 22

, and substituting

(p=5, Q=5)

gives

5k_1 + k_2 = 25

. Solving these simultaneous equations gives

k_1 = 1

and

k_2 = 20

. Substituting

p = 10

into

Q = 1 + \frac{20}{p}

results in

Q = 1 + 2 = 3

.

Step-by-Step Solution

1

Set up the general formula for partial variation.

Q = k_1 + \frac{k_2}{p}

, where

k_1

and

k_2

are constants.

Partial variation consists of a constant part and a part that varies inversely with

p

.

2

Substitute the given values to form two simultaneous linear equations.

For

p = 2, Q = 11 \implies 11 = k_1 + \frac{k_2}{2} \implies 2k_1 + k_2 = 22

For

p = 5, Q = 5 \implies 5 = k_1 + \frac{k_2}{5} \implies 5k_1 + k_2 = 25

Substituting known data points provides equations to solve for the variation constants.

3

Solve the simultaneous equations for

k_1

and

k_2

.

Subtracting the first equation from the second gives

3k_1 = 3 \implies k_1 = 1

.
Substituting

k_1 = 1

into

2(1) + k_2 = 22

yields

k_2 = 20

.

Determining

k_1

and

k_2

establishes the explicit relationship between

Q

and

p

.

4

Calculate

Q

when

p = 10

.

Q = 1 + \frac{20}{10} = 1 + 2 = 3

.

Substitute

p = 10

into the established variation equation

Q = 1 + \frac{20}{p}

.

Key Concept

Partial Variation and Simultaneous Equations

Estimated Time:1m 30s

Question 7619Question →

Given that $x$ is a positive real number and the determinant of the matrix $A = \begin{pmatrix} x & 1 & 0 \\ 2 & x & 3 \\ 0 & 1 & 4 \end{pmatrix}$ is $2$ , find the value of $x$ .

Show answer & explanation

Answer: 2

Answer

The positive value of

x

is

2

.

Expanding the determinant along the first row yields

\det(A) = x(4x - 3) - 1(8 - 0) = 4x^2 - 3x - 8

. Equating this to

2

gives

4x^2 - 3x - 10 = 0

. Factoring the quadratic gives

(4x + 5)(x - 2) = 0

, which yields

x = 2

or

x = -1.25

. Because

x

must be positive, the correct value is

2

.

Step-by-Step Solution

1

Expand the

3 \times 3

determinant along the first row

\det(A) = x(4x - 3) - 1(8 - 0) + 0 = 4x^2 - 3x - 8

Expanding along the first row leverages the zero entry to simplify calculation of the determinant.

2

Equate the determinant expression to the given determinant value

4x^2 - 3x - 10 = 0

Setting the calculated determinant equal to

2

creates a quadratic equation in terms of

x

.

3

Factor the quadratic equation to find the candidate values for

x

(4x + 5)(x - 2) = 0 \implies x = 2 \text{ or } x = -1.25

Factoring determines all algebraic solutions that satisfy the determinant equation.

4

Apply the positivity constraint given in the problem statement

x = 2

The question restricts

x

to positive real numbers, discarding the negative root.

Key Concept

Determinant of a 3x3 matrix and quadratic equations

Estimated Time:2m 0s

Question 7620Question →

What is the value of the definite integral $\int_{0}^{2} (3x^2 + 2) \, dx$ ?

Show answer & explanation

Answer: 12

Answer

12

To evaluate

\int_{0}^{2} (3x^2 + 2) \, dx

, integrate

3x^2 + 2

to get

x^3 + 2x

. Substituting the limits gives

(2^3 + 2(2)) - (0^3 + 2(0)) = (8 + 4) - 0 = 12

.

Step-by-Step Solution

1

Integrate the function term by term

\int (3x^2 + 2) \, dx = x^3 + 2x

By the power rule of integration, \int 3x^2 \, dx = x^3 and \int 2 \, dx = 2x.

2

Apply the limits of integration from 0 to 2

[x^3 + 2x]_0^2 = (2^3 + 2(2)) - (0^3 + 2(0)) = 12 - 0 = 12

Substitute the upper limit 2 into the antiderivative and subtract the value obtained by substituting the lower limit 0.

Key Concept

Evaluation of Definite Integrals

Answer

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