All practice questions

Question 7581Question →

A binary operation $\star$ on the set of real numbers $\mathbb{R} \setminus \{1\}$ is defined by $a \star b = a + b - ab$ . If $y^{-1}$ denotes the inverse of an element $y$ under $\star$ , find the value of $x$ such that $(x \star 3)^{-1} = 2$ .

Show answer & explanation

Answer: 0.5

Answer

The value of

x

is

0.5

.

To solve for

x

, first calculate the identity element

e = 0

using

a \star e = a

. Next, use

y \star y^{-1} = 0

to get the inverse formula

y^{-1} = \frac{y}{y - 1}

. Substituting

y = x \star 3 = 3 - 2x

into the inverse formula yields

\frac{3 - 2x}{2 - 2x} = 2

, which simplifies to

2x = 1

, giving

x = 0.5

.

Step-by-Step Solution

1

Find the identity element

e

of the operation

\star

.

e = 0

By definition of the identity element,

a \star e = a \implies a + e - ae = a \implies e(1 - a) = 0

, so

e = 0

for all

a \neq 1

.

2

Derive the general expression for the inverse element

y^{-1}

of

y

.

y^{-1} = \frac{y}{y - 1}

By definition of inverse element,

y \star y^{-1} = e \implies y + y^{-1} - y y^{-1} = 0 \implies y^{-1}(1 - y) = -y \implies y^{-1} = \frac{y}{y - 1}

.

3

Evaluate the inner operation

x \star 3

.

x \star 3 = 3 - 2x

Using the operational rule

a \star b = a + b - ab

, we obtain

x \star 3 = x + 3 - 3x = 3 - 2x

.

4

Set up and solve the equation

(x \star 3)^{-1} = 2

.

x = 0.5

Letting

y = 3 - 2x

, its inverse is

y^{-1} = \frac{3 - 2x}{(3 - 2x) - 1} = \frac{3 - 2x}{2 - 2x}

. Equating this to

2

gives

\frac{3 - 2x}{2 - 2x} = 2 \implies 3 - 2x = 4 - 4x \implies 2x = 1 \implies x = 0.5

.

Key Concept

Identity and Inverse Elements in Binary Operations

Question 7582Question →

In $\Delta ABC$ , side $a = 4\text{ cm}$ , side $b = 4\sqrt{2}\text{ cm}$ , and $\angle A = 30^\circ$ . If $\angle B$ is an obtuse angle, what is the measure of $\angle B$ ?

45^\circ

135^\circ

105^\circ

150^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

135^\circ

Applying the Sine Rule gives

\frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin B}

, which simplifies to

\sin B = \frac{\sqrt{2}}{2}

. The inverse sine operation yields an acute angle of

45^\circ

and an obtuse angle of

180^\circ - 45^\circ = 135^\circ

. Since the stem specifies that angle B is obtuse, the correct value is

135^\circ

.

Step-by-Step Solution

1

Apply the Sine Rule formula relating sides

a

,

b

and their opposite angles

A

,

B

.

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.

2

Substitute the known values into the equation:

a = 4

,

b = 4\sqrt{2}

, and

A = 30^\circ

.

\frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin B} \implies \frac{4}{0.5} = \frac{4\sqrt{2}}{\sin B} \implies 8 = \frac{4\sqrt{2}}{\sin B}

\sin 30^\circ = 0.5

.

3

Solve for

\sin B

.

\sin B = \frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2}

Rearranging the equation yields the value for

\sin B

.

4

Find the obtuse angle whose sine is

\frac{\sqrt{2}}{2}

.

\angle B = 180^\circ - 45^\circ = 135^\circ

Sine is positive in both the first and second quadrants. The acute reference angle is

\arcsin\left(\frac{\sqrt{2}}{2}\right) = 45^\circ

, so the supplementary obtuse angle is

180^\circ - 45^\circ = 135^\circ

.

Key Concept

Sine Rule and the Ambiguous Case

Question 7583Question →

A binary operation $\oplus$ is defined on the set of real numbers $\mathbb{R}$ by $a \oplus b = 2a + 3b - 5$ . What is the value of $(4 \oplus 1) \oplus 2$ ?

Show answer & explanation

Answer: 13

Answer

The value of

(4 \oplus 1) \oplus 2

is 13.

First, evaluate the inner expression

4 \oplus 1

using

a = 4

and

b = 1

, which gives

2(4) + 3(1) - 5 = 8 + 3 - 5 = 6

. Then, substitute this result into the outer expression to evaluate

6 \oplus 2

using

a = 6

and

b = 2

, yielding

2(6) + 3(2) - 5 = 12 + 6 - 5 = 13

.

Step-by-Step Solution

1

Evaluate the inner operation

4 \oplus 1

6

Substitute

a = 4

and

b = 1

into the operational rule

a \oplus b = 2a + 3b - 5

.

2

Evaluate the outer operation using the result from Step 1:

6 \oplus 2

13

Substitute

a = 6

and

b = 2

into the operational rule

a \oplus b = 2a + 3b - 5

.

Key Concept

Evaluation of Binary Operations

Question 7584Question →

A thermometric property $X$ of a system has a value of $15.0\,\text{units}$ at the ice point ( $0^\circ\text{C}$ ) and $75.0\,\text{units}$ at the steam point ( $100^\circ\text{C}$ ). If the thermometric property is measured as $33.0\,\text{units}$ when immersed in a chemical bath, what is the temperature of the bath on the absolute thermodynamic scale?

303\,\text{K}

317\,\text{K}

297\,\text{K}

30\,\text{K}

Show answer & explanation

Answer: $303\,\text{K}$

Answer

303\,\text{K}

The temperature of the bath on the Celsius scale is found by taking the ratio of the change in thermometric property from the ice point to the total fundamental interval:

\frac{33.0 - 15.0}{75.0 - 15.0} \times 100^\circ\text{C} = \frac{18}{60} \times 100 = 30^\circ\text{C}

. Converting this temperature to the absolute (Kelvin) scale requires adding

273\,\text{K}

, yielding

30 + 273 = 303\,\text{K}

.

Step-by-Step Solution

1

Identify the given thermometric values for the fixed points and the unknown state

X_0 = 15.0\,\text{units}

(ice point,

0^\circ\text{C}

),

X_{100} = 75.0\,\text{units}

(steam point,

100^\circ\text{C}

), and

X_T = 33.0\,\text{units}

Linear thermometric interpolation requires defining the fixed reference points and the measured property value.

2

Calculate the temperature on the Celsius scale using the linear scale formula

\theta = \frac{X_T - X_0}{X_{100} - X_0} \times 100^\circ\text{C} = \frac{33.0 - 15.0}{75.0 - 15.0} \times 100 = \frac{18.0}{60.0} \times 100 = 30^\circ\text{C}

The temperature change relative to the fundamental interval determines the position on the Celsius scale.

3

Convert the temperature from degrees Celsius to Kelvin

T = \theta + 273 = 30 + 273 = 303\,\text{K}

The absolute thermodynamic scale (Kelvin) is shifted from the Celsius scale by adding

273\,\text{K}

.

Key Concept

Linear interpolation of temperature using thermometric properties and conversion to the absolute scale

Estimated Time:2m 0s

Question 7585Question →

Two point charges, $q_1 = +4.0 \times 10^{-8}\text{ C}$ and $q_2 = -9.0 \times 10^{-8}\text{ C}$ , are fixed in a vacuum at a distance of $0.50\text{ m}$ apart. A third point charge $q_3 = +2.0 \times 10^{-8}\text{ C}$ is placed along the line passing through $q_1$ and $q_2$ such that the net electrostatic force acting on it is zero. What is the distance of $q_3$ from $q_1$ ?

0.20\text{ m}

0.40\text{ m}

1.00\text{ m}

1.50\text{ m}

Show answer & explanation

Answer: $1.00\text{ m}$

Answer

The distance of the third charge from

q_1

is

1.00\text{ m}

(located on the side of

q_1

opposite to

q_2

).

The correct distance of

1.00\text{ m}

is determined by recognizing that zero net force on a test charge

q_3

occurs outside the opposite charges

q_1

and

q_2

, specifically on the side of the smaller charge

q_1

. Equating electrostatic forces gives

\frac{4}{d^2} = \frac{9}{(d+0.50)^2}

, which yields

d = 1.00\text{ m}

.

Step-by-Step Solution

1

Determine the equilibrium region for the third charge.

Because

q_1 = +4.0 \times 10^{-8}\text{ C}

and

q_2 = -9.0 \times 10^{-8}\text{ C}

have opposite charges, electrostatic forces on

q_3

point in opposite directions only outside the segment connecting them. To balance the forces,

q_3

must be closer to the smaller magnitude charge

q_1

, placing it at distance

d

to the left of

q_1

.

Between two opposite charges, the force from the positive charge and the force from the negative charge act in the same direction, so net zero force is impossible between them.

2

Set up Coulomb's Law equilibrium equation.

\frac{k |q_1 q_3|}{d^2} = \frac{k |q_2 q_3|}{(d + 0.50)^2} \implies \frac{4.0 \times 10^{-8}}{d^2} = \frac{9.0 \times 10^{-8}}{(d + 0.50)^2}

Equilibrium requires the force magnitude exerted by

q_1

on

q_3

to equal the force magnitude exerted by

q_2

on

q_3

.

3

Solve the equation for distance

d

.

\frac{4}{d^2} = \frac{9}{(d + 0.50)^2} \implies \frac{2}{d} = \frac{3}{d + 0.50} \implies 2(d + 0.50) = 3d \implies d = 1.00\text{ m}

.

Taking the square root of both sides simplifies the inverse-square relation to a solvable linear relation.

Key Concept

Electrostatic equilibrium and vector force cancellation for point charges

Question 7586Question →

If $5^{2x - 1} \times 25^{x + 1} = 125^{x + 2}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

The value of

x

is

5

.

Converting all terms to base

5

gives

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

. Simplifying the exponents yields

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

, which reduces to

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

. Setting the exponents equal to each other gives

4x + 1 = 3x + 6

, resulting in

x = 5

.

Step-by-Step Solution

1

Express all bases in terms of prime base 5

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

All terms must share the same base to combine exponents using index laws.

2

Apply power of a power rule

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

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

Multiplication of inner and outer powers simplifies composite exponent expressions.

3

Apply the product rule

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

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

Adding the exponents on the left-hand side produces a single exponential term.

4

Equate exponents and solve for

x

x = 5

Equal bases imply equal exponents:

4x + 1 = 3x + 6

.

Key Concept

Solving exponential equations using common base conversion and laws of indices

Estimated Time:1m 30s

Question 7587Question →

A plane contains $12$ distinct points such that exactly $5$ of them lie on the same straight line, while no other subset of three points is collinear. How many distinct straight lines can be formed by joining pairs of these points?

Show answer & explanation

Answer: 57

Answer

The total number of distinct straight lines that can be formed is 57.

The total number of distinct straight lines is obtained by taking all possible pairs of points from 12, which is

^{12}C_2 = 66

, subtracting the pairs formed among the 5 collinear points (

^5C_2 = 10

), and adding 1 back to account for the single straight line on which those 5 points lie. This gives

66 - 10 + 1 = 57

.

Step-by-Step Solution

1

Calculate the total possible lines by selecting any 2 points out of 12.

^{12}C_2 = \frac{12 \times 11}{2} = 66

Any pair of non-collinear points uniquely determines a straight line.

2

Calculate the combinations of 2 points selected from the 5 collinear points.

^5C_2 = \frac{5 \times 4}{2} = 10

These 10 point pairs all lie on the exact same single straight line instead of forming 10 distinct lines.

3

Adjust the count to find the net number of distinct straight lines.

66 - 10 + 1 = 57

Subtracting the 10 overcounted combinations and adding back 1 for the single line passing through all 5 collinear points gives the correct total.

Key Concept

Combinations with Collinear Constraints

Question 7588Question →

A rectangle has an area of $10\text{ cm}^2$ and a length of $(\sqrt{7} + \sqrt{2})\text{ cm}$ . What is the width of the rectangle in simplified surd form?

(2\sqrt{7} - 2\sqrt{2})\text{ cm}

(2\sqrt{7} + 2\sqrt{2})\text{ cm}

2\sqrt{5}\text{ cm}

\frac{10\sqrt{7} - 10\sqrt{2}}{9}\text{ cm}

Show answer & explanation

Answer: $(2\sqrt{7} - 2\sqrt{2})\text{ cm}$

Answer

The width of the rectangle in simplified surd form is

(2\sqrt{7} - 2\sqrt{2})\text{ cm}

.

The area formula for a rectangle gives

\text{width} = \frac{10}{\sqrt{7} + \sqrt{2}}

. Multiplying both numerator and denominator by the conjugate

(\sqrt{7} - \sqrt{2})

produces

\frac{10(\sqrt{7} - \sqrt{2})}{7 - 2} = \frac{10(\sqrt{7} - \sqrt{2})}{5} = 2\sqrt{7} - 2\sqrt{2}

.

Step-by-Step Solution

1

Set up the formula for the width of the rectangle

\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{10}{\sqrt{7} + \sqrt{2}}

The area of a rectangle is length multiplied by width.

2

Rationalize the denominator by multiplying top and bottom by the conjugate

(\sqrt{7} - \sqrt{2})

\text{Width} = \frac{10(\sqrt{7} - \sqrt{2})}{(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})}

Multiplying by the conjugate creates a difference of squares in the denominator, eliminating radicals.

3

Simplify the denominator using

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

(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5

Squaring a square root yields the underlying rational number.

4

Divide the numerator by the simplified denominator

\frac{10(\sqrt{7} - \sqrt{2})}{5} = 2(\sqrt{7} - \sqrt{2}) = 2\sqrt{7} - 2\sqrt{2}

Dividing

10

by

5

gives

2

, which is then distributed across the terms inside the parentheses.

Key Concept

Rationalization of Denominators with Binomial Surds

Question 7589Question →

A search-and-rescue helicopter leaves a central station $P$ and flies $16\text{ km}$ on a bearing of $050^\circ$ to reach a waypoint $Q$ . It then changes course and flies $12\text{ km}$ on a bearing of $140^\circ$ to reach a mountain rescue site $R$ . From the central station $P$ , the angle of elevation to the helicopter hovering vertically above point $R$ is $45^\circ$ . What is the vertical height of the helicopter above the horizontal plane of station $P$ , in kilometers?

Show answer & explanation

Answer: 20

Answer

The vertical height of the helicopter above the horizontal plane of station

P

is

20\text{ km}

.

The horizontal journey forms a right-angled triangle

PQR

with side lengths

16\text{ km}

and

12\text{ km}

, yielding a hypotenuse (horizontal distance

PR

) of

20\text{ km}

. Since the angle of elevation from

P

to the hovering helicopter is

45^\circ

, the vertical height is equal to

20 \times \tan(45^\circ) = 20\text{ km}

.

Step-by-Step Solution

1

Find the back bearing of station

P

from waypoint

Q

Back bearing =

050^\circ + 180^\circ = 230^\circ

To determine the enclosed interior angle at point

Q

, the reverse direction from

Q

to

P

must be calculated.

2

Calculate the interior angle

\angle PQR

\angle PQR = 230^\circ - 140^\circ = 90^\circ

Subtracting the forward bearing of

R

from the back bearing of

P

yields the right angle between the two paths.

3

Calculate the horizontal displacement distance

PR

PR = \sqrt{16^2 + 12^2} = 20\text{ km}

Since

\triangle PQR

is right-angled at

Q

, the distance

PR

is obtained using the Pythagorean theorem.

4

Determine the vertical altitude using trigonometry

\text{Height} = PR \times \tan(45^\circ) = 20 \times 1 = 20\text{ km}

In the vertical right-angled triangle formed by

P

, the ground projection of

R

, and the helicopter,

\tan(45^\circ) = \frac{\text{Height}}{\text{Horizontal Distance}}

.

Key Concept

Combining 3-point bearings in 2D with right-triangle trigonometry for 3D angles of elevation.

Question 7590Question →

If $\log_2 x + \log_8 x = 4$ , what is the value of $x$ ?

4

6

8

64

Show answer & explanation

Answer: 8

Answer

8

By applying the change of base formula,

\log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{1}{3}\log_2 x

. Rewriting the equation gives

\log_2 x + \frac{1}{3}\log_2 x = \frac{4}{3}\log_2 x = 4

. Solving for

\log_2 x

yields

\log_2 x = 3

, which in exponential form gives

x = 2^3 = 8

.

Step-by-Step Solution

1

Apply the change of base formula to express

\log_8 x

in base 2

\log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{\log_2 x}{3}

Logarithms must be converted to a common base before combining terms.

2

Substitute

\log_8 x = \frac{1}{3}\log_2 x

into the original equation

\log_2 x + \frac{1}{3}\log_2 x = 4 \implies \frac{4}{3}\log_2 x = 4

Combine like logarithmic terms.

3

Solve for

\log_2 x

\log_2 x = 4 \times \frac{3}{4} = 3

Isolate the logarithmic expression by multiplying both sides by

\frac{3}{4}

.

4

Convert the logarithmic equation to its exponential form to solve for

x

x = 2^3 = 8

Definition of logarithms states that

\log_b a = c \iff b^c = a

.

Key Concept

Logarithms and Change of Base

Question 7591Question →

An observer standing at the top of a vertical lighthouse observes two boats, $X$ and $Y$ , on the surrounding horizontal sea surface. Boat $X$ lies due South of the lighthouse at an angle of depression of $30^\circ$ , while boat $Y$ lies due East of the lighthouse at an angle of depression of $45^\circ$ . If the straight-line distance between boat $X$ and boat $Y$ is $80\text{ m}$ , what is the height of the lighthouse?

40\text{ m}

40\sqrt{3}\text{ m}

40\sqrt{2}\text{ m}

20\sqrt{3}\text{ m}

Show answer & explanation

Answer: $40\text{ m}$

Answer

The height of the lighthouse is

40\text{ m}

.

The height of the lighthouse is

40\text{ m}

. Since boat

X

is due South and boat

Y

is due East, the line segments connecting the base of the lighthouse to each boat form a right angle (

90^\circ

). Using basic trigonometry, the distance to boat

X

is

h\sqrt{3}

and to boat

Y

is

h

. Applying Pythagoras' theorem to the right triangle formed on the sea surface gives

(h\sqrt{3})^2 + h^2 = 80^2

, which simplifies to

4h^2 = 6400

, giving

h = 40\text{ m}

.

Step-by-Step Solution

1

Express the horizontal distance from the lighthouse base

L

to boat

X

in terms of height

h

.

LX = \frac{h}{\tan 30^\circ} = h\sqrt{3}\text{ m}

The angle of elevation from boat

X

to the top of the lighthouse is equal to the angle of depression (

30^\circ

).

2

Express the horizontal distance from the lighthouse base

L

to boat

Y

in terms of height

h

.

LY = \frac{h}{\tan 45^\circ} = h\text{ m}

The angle of elevation from boat

Y

to the top of the lighthouse is

45^\circ

.

3

Set up Pythagoras' theorem for right-angled triangle

XLY

on the horizontal plane.

XY^2 = LX^2 + LY^2 \implies 80^2 = (h\sqrt{3})^2 + h^2

Boat

X

is due South and boat

Y

is due East of the lighthouse base, making

\angle XLY = 90^\circ

.

4

Solve the algebraic equation for

h

.

6400 = 3h^2 + h^2 = 4h^2 \implies h^2 = 1600 \implies h = 40\text{ m}

Dividing

6400

by

4

gives

1600

, whose square root is

40

.

Key Concept

3D Geometry combining Angles of Elevation/Depression with Perpendicular Bearings

Question 7592Question →

The $3^{\text{rd}}$ term of a geometric progression is $18$ and the $6^{\text{th}}$ term is $486$ . What is the sum of the first $5$ terms of the progression?

242

162

80

728

Show answer & explanation

Answer: $242$

Answer

The sum of the first

5

terms is

242

.

Using

T_n = a r^{n-1}

, we set up

a r^2 = 18

and

a r^5 = 486

. Dividing these gives

r^3 = 27

, so

r = 3

, which leads to

a = 2

. Applying

S_5 = \frac{a(r^5 - 1)}{r - 1}

yields

\frac{2(243 - 1)}{2} = 242

.

Step-by-Step Solution

1

Express the given terms using the geometric progression

n^{\text{th}}

term formula

T_n = a r^{n-1}

.

T_3 = a r^2 = 18

and

T_6 = a r^5 = 486

.

This establishes a system of equations in terms of the first term

a

and common ratio

r

.

2

Divide the expression for

T_6

by

T_3

to determine

r

.

\frac{a r^5}{a r^2} = \frac{486}{18} \implies r^3 = 27 \implies r = 3.

Dividing the equations eliminates

a

and allows direct solution for the common ratio

r

.

3

Substitute

r = 3

back into

T_3 = a r^2 = 18

to solve for

a

.

a (3)^2 = 18 \implies 9a = 18 \implies a = 2.

Determining the first term

a

is required to evaluate the sum.

4

Calculate the sum of the first

5

terms using

S_n = \frac{a(r^n - 1)}{r - 1}

.

S_5 = \frac{2(3^5 - 1)}{3 - 1} = \frac{2(243 - 1)}{2} = 242.

Applying the GP sum formula yields the required value.

Key Concept

Geometric Progression nth term and sum formulas

Question 7593Question →

What is the value of $y$ at the local minimum stationary point of the curve $y = x^3 - 3x^2 - 9x + 15$ for $x > 0$ ?

Show answer & explanation

Answer: -12

Answer

The value of

y

at the local minimum stationary point is

-12

.

To find the local minimum point of

y = x^3 - 3x^2 - 9x + 15

, set the derivative

\frac{dy}{dx} = 3x^2 - 6x - 9

equal to

0

. Solving

3(x - 3)(x + 1) = 0

yields

x = 3

for

x > 0

. The second derivative

\frac{d^2y}{dx^2} = 6x - 6

equals

12 > 0

at

x = 3

, confirming a local minimum. Substituting

x = 3

back into the original curve equation yields

y = (3)^3 - 3(3)^2 - 9(3) + 15 = -12

.

Step-by-Step Solution

1

Differentiate

y = x^3 - 3x^2 - 9x + 15

with respect to

x

\frac{dy}{dx} = 3x^2 - 6x - 9

Stationary points occur where the slope (first derivative) is zero.

2

Set

\frac{dy}{dx} = 0

and solve for

x

x = 3 \text{ or } x = -1

Factoring

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

gives

(x - 3)(x + 1) = 0

.

3

Apply the second derivative test at

x = 3

(since

x > 0

)

\frac{d^2y}{dx^2} = 6(3) - 6 = 12 > 0

A positive second derivative confirms that

x = 3

is a local minimum.

4

Evaluate

y

at

x = 3

in the original curve equation

y = (3)^3 - 3(3)^2 - 9(3) + 15 = -12

Substituting

x = 3

into

y(x)

gives the

y

-coordinate of the minimum point.

Key Concept

Stationary Points and Local Minima of Polynomial Curves

Question 7594Question →

If $y = \int (4x^3 - 2\sin(x)) \, dx$ and $y = 5$ when $x = 0$ , what is the value of the constant of integration $C$ ?

Show answer & explanation

Answer: 3

Answer

The constant of integration

C

is 3.

Integrating

4x^3 - 2\sin(x)

gives

y = x^4 + 2\cos(x) + C

. Substituting

x = 0

and

y = 5

yields

5 = 0 + 2(1) + C

, so

C = 3

.

Step-by-Step Solution

1

Integrate the polynomial and trigonometric terms

y = x^4 + 2\cos(x) + C

The integral of

4x^3

with respect to

x

is

x^4

, and the integral of

-2\sin(x)

is

2\cos(x)

.

2

Apply the given initial condition

y(0) = 5

5 = 0^4 + 2\cos(0) + C \implies 5 = 2 + C

Evaluating the expression at

x = 0

uses the trigonometric property

\cos(0) = 1

.

3

Solve the linear equation for

C

C = 3

Subtracting 2 from both sides isolates

C

.

Key Concept

Determining the Constant of Integration from Initial Boundary Conditions

Question 7595Question →

A biased four-sided die numbered $1$ , $2$ , $3$ , and $4$ is rolled $200$ times in a probability experiment. The frequency of each outcome is recorded in the table below:

Outcome	Frequency
$1$	$45$
$2$	$60$
$3$	$55$
$4$	$40$

What is the experimental probability of rolling an odd number?

Show answer & explanation

Answer: 0.5

Answer

0.5

The correct experimental probability is calculated by dividing the total frequency of observed odd outcomes (

45 + 55 = 100

) by the total number of trials (

200

), which gives

\frac{100}{200} = 0.5

.

Step-by-Step Solution

1

Identify the favorable outcomes for rolling an odd number

The odd outcomes on the die are

1

and

3

.

Odd numbers are integers that are not divisible by

2

.

2

Calculate the total frequency of the favorable outcomes

Frequency of

1

is

45

and frequency of

3

is

55

. Total frequency

= 45 + 55 = 100

.

The total number of times an odd number was rolled is the sum of the frequencies of outcome

1

and outcome

3

.

3

Compute the experimental probability

\text{Experimental Probability} = \frac{\text{Frequency of odd outcomes}}{\text{Total number of trials}} = \frac{100}{200} = 0.5

.

Experimental probability measures the relative frequency of an event occurring in an actual experiment.

Key Concept

Experimental Probability

Question 7596Question →

In J. J. Thomson's plum pudding model of the atom, how is the positive charge distributed throughout the atom?

It is uniformly spread throughout a spherical volume containing embedded electrons.

It is concentrated in a tiny, dense central core called the nucleus.

It orbits around a central cluster of negative charges in fixed energy levels.

It is localized entirely on the outer surface of a hollow atomic shell.

Show answer & explanation

Answer: It is uniformly spread throughout a spherical volume containing embedded electrons.

Answer

The positive charge is uniformly spread throughout a spherical volume containing embedded electrons.

In J. J. Thomson's plum pudding model, the atom is envisioned as a continuous sphere of positive charge within which negatively charged electrons are embedded evenly to maintain overall electrical neutrality.

Step-by-Step Solution

1

Recall the fundamental proposal of Thomson's atomic model (1897).

J. J. Thomson envisioned the atom as a sphere of positive electrification.

This accounted for atomic neutrality following his discovery of the electron.

2

Identify the arrangement of electrons within this positive sphere.

Electrons were thought to be embedded throughout the positive sphere like plums in a pudding.

This balanced the positive charge uniformly across the atomic volume.

Key Concept

Thomson's Plum Pudding Atomic Model

Estimated Time:45s

Question 7597Question →

Consider the following four physical quantities:
I. Electric current
II. Electric charge
III. Thermodynamic temperature
IV. Heat capacity

Which of the following pairs correctly identifies a fundamental physical quantity followed by a derived physical quantity?

Electric current and Thermodynamic temperature

Electric charge and Heat capacity

Thermodynamic temperature and Heat capacity

Heat capacity and Electric current

Show answer & explanation

Answer: Thermodynamic temperature and Heat capacity

Answer

Thermodynamic temperature and Heat capacity

Thermodynamic temperature is one of the seven base SI fundamental quantities. Heat capacity is a derived quantity defined as thermal energy per temperature change (

J/K

). Thus, the pair of Thermodynamic temperature and Heat capacity strictly follows the order of a fundamental quantity followed by a derived quantity.

Step-by-Step Solution

1

Identify the fundamental physical quantities among the given list.

Electric current (I) and Thermodynamic temperature (III) are base SI quantities, so they are fundamental.

Fundamental quantities are independent physical quantities that cannot be defined in terms of other quantities.

2

Identify the derived physical quantities among the given list.

Electric charge (II, defined by

Q = I \cdot t

) and Heat capacity (IV, defined by

C = \frac{Q}{\Delta T}

) are derived quantities.

Derived quantities are defined mathematically from combinations of fundamental physical quantities.

3

Select the option that lists a fundamental quantity followed by a derived quantity.

Thermodynamic temperature (fundamental) and Heat capacity (derived) matches the required order.

Thermodynamic temperature is fundamental, and heat capacity is derived.

Key Concept

Classification of physical quantities into base (fundamental) and derived quantities in the SI system.

Question 7598Question →

Find the sum of all values of $x$ (in degrees) in the interval $0^\circ \le x \le 180^\circ$ that satisfy the trigonometric equation $\cos(3x - 45^\circ) = -\frac{\sqrt{2}}{2}$ .

Show answer & explanation

Answer: 330

Answer

The sum of all values of

x

satisfying the equation in the domain

0^\circ \le x \le 180^\circ

is 330.

Transforming the domain

0^\circ \le x \le 180^\circ

gives

-45^\circ \le 3x - 45^\circ \le 495^\circ

. The angles within this range where the cosine value equals

-\frac{\sqrt{2}}{2}

are

135^\circ

,

225^\circ

, and

495^\circ

. Solving

3x - 45^\circ

for each of these angles gives

x = 60^\circ

,

90^\circ

, and

180^\circ

. Summing these three roots yields

330^\circ

.

Step-by-Step Solution

1

Determine the interval of the transformed angle

\theta = 3x - 45^\circ

.

-45^\circ \le 3x - 45^\circ \le 495^\circ

Applying the linear transformation

3x - 45^\circ

to the given domain

0^\circ \le x \le 180^\circ

establishes the exact boundaries for the argument of the cosine function.

2

Find all values of

\theta

within

[-45^\circ, 495^\circ]

satisfying

\cos \theta = -\frac{\sqrt{2}}{2}

.

\theta \in \{135^\circ, 225^\circ, 495^\circ\}

Cosine is negative in Quadrants II and III. The reference angle is

45^\circ

, giving base solutions

135^\circ

and

225^\circ

. Adding

360^\circ

to

135^\circ

gives

495^\circ

, which lies exactly on the upper boundary.

3

Solve for

x

by setting

3x - 45^\circ

equal to each valid

\theta

.

x \in \{60^\circ, 90^\circ, 180^\circ\}

Isolating

x

yields

x = \frac{\theta + 45^\circ}{3}

. All three resulting values lie within

[0^\circ, 180^\circ]

.

4

Sum the valid solution values.

60^\circ + 90^\circ + 180^\circ = 330^\circ

The problem asks specifically for the sum of all solution angles in degrees.

Key Concept

Solving multi-angle trigonometric equations with phase shifts across a specified domain

Question 7599Question →

What is the positive value of $x$ that satisfies the simultaneous equations $y - x = 1$ and $y = x^2 - 5$ ?

2

3

4

5

Show answer & explanation

Answer: $3$

Answer

The positive value of

x

is

3

.

Substituting

y = x + 1

from the linear equation into the quadratic equation gives

x + 1 = x^2 - 5

. Rearranging terms yields

x^2 - x - 6 = 0

, which factors as

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

. The roots are

x = 3

and

x = -2

. Selecting the positive solution gives

3

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the linear equation.

y = x + 1

Isolating

y

simplifies substitution into the quadratic equation.

2

Substitute

y = x + 1

into the quadratic equation

y = x^2 - 5

.

x + 1 = x^2 - 5

This produces a quadratic equation in one variable,

x

.

3

Rearrange into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - x - 6 = 0

Standard quadratic form allows direct factorization.

4

Factorize the quadratic equation.

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

Factoring determines the roots of the equation.

5

Solve for

x

and identify the positive value.

x = 3

or

x = -2

; positive value is

3

.

The question specifies the positive real solution for

x

.

Key Concept

Solving simultaneous linear and quadratic equations using substitution

Question 7600Question →

If $x$ and $y$ satisfy the simultaneous equations $x - 2y = 1$ and $x^2 - xy - y^2 = 5$ , what is the positive value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The positive value of

x

is 3.

Substituting

x = 2y + 1

into the non-linear equation yields

(2y + 1)^2 - (2y + 1)y - y^2 = 5

, which simplifies to

y^2 + 3y - 4 = 0

. Solving this gives

y = 1

or

y = -4

. Substituting

y = 1

into

x = 2y + 1

gives

x = 3

, which is the positive value of

x

.

Step-by-Step Solution

1

Make

x

the subject of the linear equation

x = 2y + 1

Substitution is the standard method for solving linear-quadratic simultaneous systems.

2

Substitute

x = 2y + 1

into

x^2 - xy - y^2 = 5

(2y + 1)^2 - (2y + 1)y - y^2 = 5

Eliminates

x

to create a single quadratic equation in terms of

y

.

3

Expand and simplify to standard quadratic form

y^2 + 3y - 4 = 0

Simplifying algebraic expressions allows factorization.

4

Solve for

y

by factoring

y = 1

or

y = -4

Factors of

-4

that sum to

3

are

+4

and

-1

.

5

Calculate corresponding values for

x

x = 3

when

y = 1

, and

x = -7

when

y = -4

Substitute

y

back into the linear expression for

x

.

6

Select the positive value of

x

3

The question explicitly asks for the positive value of

x

.

Key Concept

Simultaneous Linear and Quadratic Equations

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept