All practice questions

Question 7701Question →

A bag contains $x$ red balls, $12$ blue balls, and $8$ green balls. The theoretical probability of selecting a green ball at random from the bag is $\frac{1}{4}$ . In an experiment where a ball is drawn with replacement $240$ times, a blue ball is recorded $102$ times. What is the positive difference between the experimental relative frequency and the theoretical probability of drawing a blue ball?

\frac{1}{20}

\frac{7}{40}

\frac{3}{8}

\frac{17}{40}

Show answer & explanation

Answer: $\frac{1}{20}$

Answer

The positive difference between the experimental relative frequency and the theoretical probability of drawing a blue ball is

\frac{1}{20}

.

The total number of balls in the container is determined from the theoretical probability of green balls:

\frac{8}{N} = \frac{1}{4} \implies N = 32

. The theoretical probability of drawing a blue ball is

\frac{12}{32} = \frac{3}{8} = \frac{15}{40}

. The experimental relative frequency is

\frac{102}{240} = \frac{17}{40}

. Taking the positive difference yields

\frac{17}{40} - \frac{15}{40} = \frac{2}{40} = \frac{1}{20}

.

Step-by-Step Solution

1

Calculate total number of balls in the bag

Total balls

N = 32

Given theoretical

P(\text{Green}) = \frac{8}{N} = \frac{1}{4}

, solving for

N

yields

N = 32

.

2

Determine theoretical probability of selecting a blue ball

P(\text{Blue}) = \frac{3}{8} = \frac{15}{40}

There are

12

blue balls out of

32

total balls, so

P(\text{Blue}) = \frac{12}{32} = \frac{3}{8}

.

3

Calculate experimental relative frequency of selecting a blue ball

Relative frequency =

\frac{17}{40}

Blue was drawn

102

times in

240

trials, so

\frac{102}{240} = \frac{17}{40}

.

4

Find the positive difference between experimental relative frequency and theoretical probability

Difference =

\frac{1}{20}

Subtracting

\frac{15}{40}

from

\frac{17}{40}

gives

\frac{2}{40} = \frac{1}{20}

.

Key Concept

Experimental relative frequency vs theoretical probability comparison

Estimated Time:2m 0s

Question 7702Question →

Given the simultaneous equations $2x + y = 5$ and $x^2 + xy - y^2 = -5$ , let $(x_1, y_1)$ and $(x_2, y_2)$ be the real solution pairs. What is the value of $x_1 x_2 + y_1 y_2$ ?

-5

13

5

95

Show answer & explanation

Answer: $-5$

Answer

-5

Substituting

y = 5 - 2x

into

x^2 + xy - y^2 = -5

yields

-5x^2 + 25x - 20 = 0

, which simplifies to

x^2 - 5x + 4 = 0

. The roots are

x_1 = 1

and

x_2 = 4

, giving corresponding

y

-values

y_1 = 3

and

y_2 = -3

. Evaluating

x_1 x_2 + y_1 y_2

gives

(1)(4) + (3)(-3) = 4 - 9 = -5

.

Step-by-Step Solution

1

Express

y

in terms of

x

using the linear equation.

y = 5 - 2x

Substitution method requires isolating one variable from the linear equation.

2

Substitute

y = 5 - 2x

into the quadratic equation

x^2 + xy - y^2 = -5

.

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

Form a single quadratic equation in terms of

x

.

3

Expand and simplify the quadratic equation.

x^2 + 5x - 2x^2 - (25 - 20x + 4x^2) = -5 \implies -5x^2 + 25x - 20 = 0 \implies x^2 - 5x + 4 = 0

Convert the equation to standard quadratic form

ax^2 + bx + c = 0

.

4

Factorize the quadratic equation to find the values of

x

.

(x - 1)(x - 4) = 0 \implies x_1 = 1, x_2 = 4

Obtain the two roots for

x

.

5

Find the corresponding

y

-values using

y = 5 - 2x

.

For

x_1 = 1

:

y_1 = 5 - 2(1) = 3

. For

x_2 = 4

:

y_2 = 5 - 2(4) = -3

. Solution pairs are

(1, 3)

and

(4, -3)

.

Calculate corresponding coordinate values for each root.

6

Compute the required expression

x_1 x_2 + y_1 y_2

.

x_1 x_2 + y_1 y_2 = (1)(4) + (3)(-3) = 4 - 9 = -5

Perform the final calculation requested in the stem.

Key Concept

Simultaneous Linear and Quadratic Equations

Question 7703Question →

An electron in an excited atom transitions from an energy level of $-2.5\text{ eV}$ to a lower energy level of $-8.5\text{ eV}$ . What is the energy of the emitted photon in Joules? ( $1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$ )

9.6 \times 10^{-19}\text{ J}

6.0 \times 10^{-19}\text{ J}

1.76 \times 10^{-18}\text{ J}

-9.6 \times 10^{-19}\text{ J}

Show answer & explanation

Answer: $9.6 \times 10^{-19}\text{ J}$

Answer

The energy of the emitted photon is

9.6 \times 10^{-19}\text{ J}

.

The energy of the emitted photon is given by

\Delta E = E_i - E_f = -2.5\text{ eV} - (-8.5\text{ eV}) = 6.0\text{ eV}

. Converting this to Joules gives

6.0 \times 1.6 \times 10^{-19}\text{ J} = 9.6 \times 10^{-19}\text{ J}

.

Step-by-Step Solution

1

Calculate the energy difference

\Delta E

between the initial and final energy levels.

\Delta E = E_{\text{initial}} - E_{\text{final}} = -2.5\text{ eV} - (-8.5\text{ eV}) = 6.0\text{ eV}

.

When an electron drops to a lower energy state, it emits a photon with energy equal to the difference between the two energy levels.

2

Convert the energy from electron-volts (eV) to Joules (J).

E = 6.0\text{ eV} \times 1.6 \times 10^{-19}\text{ J/eV} = 9.6 \times 10^{-19}\text{ J}

.

Standard SI unit calculations require multiplying the value in eV by

1.6 \times 10^{-19}\text{ J/eV}

.

Key Concept

Photon Emission during Atomic Transitions

Estimated Time:45s

Question 7704Question →

Given that $(x - 2)$ is a factor of the polynomial $P(x) = x^3 + kx^2 - 5x + 6$ , find the remainder when $P(x)$ is divided by $(x + 3)$ .

Show answer & explanation

Answer: -15

Answer

The remainder when

P(x)

is divided by

(x + 3)

is

-15

.

According to the Factor Theorem, since

(x - 2)

is a factor of

P(x) = x^3 + kx^2 - 5x + 6

, setting

x = 2

yields

P(2) = 0

. This gives

8 + 4k - 10 + 6 = 0

, which simplifies to

4k + 4 = 0

, giving

k = -1

. The polynomial is therefore

P(x) = x^3 - x^2 - 5x + 6

. By the Remainder Theorem, dividing

P(x)

by

(x + 3)

produces a remainder of

P(-3)

. Evaluating

P(-3) = (-3)^3 - (-3)^2 - 5(-3) + 6 = -27 - 9 + 15 + 6 = -15

.

Step-by-Step Solution

1

Apply the Factor Theorem to determine the unknown constant

k

.

k = -1

If

(x - 2)

is a factor of

P(x)

, then

P(2) = 0

. Substituting

x = 2

gives

2^3 + k(2)^2 - 5(2) + 6 = 0 \implies 4k + 4 = 0 \implies k = -1

.

2

Substitute

k = -1

into the original polynomial to get the full expression.

P(x) = x^3 - x^2 - 5x + 6

Replacing

k

with

-1

defines

P(x)

completely.

3

Apply the Remainder Theorem to find the remainder when

P(x)

is divided by

(x + 3)

.

Remainder is

-15

By the Remainder Theorem, dividing

P(x)

by

(x + 3)

leaves a remainder equal to

P(-3)

. Calculating

P(-3) = (-3)^3 - (-3)^2 - 5(-3) + 6 = -27 - 9 + 15 + 6 = -15

.

Key Concept

Factor and Remainder Theorems for Polynomials

Estimated Time:1m 30s

Question 7705Question →

What is the simplified value of $\frac{6}{\sqrt{7} - 1} - \frac{6}{\sqrt{7} + 1}$ ?

2

2\sqrt{7}

12

12\sqrt{7}

Show answer & explanation

Answer: 2

Answer

2

Combining the fractions over the common denominator

(\sqrt{7} - 1)(\sqrt{7} + 1) = 6

yields a numerator of

6(\sqrt{7} + 1) - 6(\sqrt{7} - 1) = 12

. Dividing 12 by 6 gives 2.

Step-by-Step Solution

1

Find a common denominator for the two fractions

The common denominator is

(\sqrt{7} - 1)(\sqrt{7} + 1) = (\sqrt{7})^2 - 1^2 = 7 - 1 = 6

.

Multiplying conjugate surds eliminates the radical in the denominator.

2

Combine the numerators over the common denominator

\frac{6(\sqrt{7} + 1) - 6(\sqrt{7} - 1)}{6}

Adjust each numerator by multiplying by the conjugate of its denominator.

3

Expand and simplify the numerator

6\sqrt{7} + 6 - 6\sqrt{7} + 6 = 12

The

6\sqrt{7}

terms cancel out:

6\sqrt{7} - 6\sqrt{7} = 0

, leaving

6 - (-6) = 12

.

4

Divide the simplified numerator by the denominator

\frac{12}{6} = 2

Simplify the final fraction to obtain an integer value.

Key Concept

Rationalisation and Subtraction of Surd Expressions

Estimated Time:1m 0s

Question 7706Question →

A curve has a gradient function defined by $\frac{dy}{dx} = 6x^2 + 8\sin(4x) + 3$ . If the curve passes through the point $(0, 10)$ , what is the value of the constant of integration $C$ ?

Show answer & explanation

Answer: 12

Answer

The constant of integration

C

is

12

.

Integrating

6x^2 + 8\sin(4x) + 3

with respect to

x

gives

y = 2x^3 - 2\cos(4x) + 3x + C

. Substituting

(0, 10)

into the integrated equation gives

10 = 2(0) - 2(1) + 3(0) + C

, which leads directly to

10 = -2 + C

, so

C = 12

.

Step-by-Step Solution

1

Integrate the gradient function to obtain the general equation of the curve.

y = 2x^3 - 2\cos(4x) + 3x + C

The antiderivative of

\sin(kx)

is

-\frac{1}{k}\cos(kx)

and the antiderivative of

x^n

is

\frac{x^{n+1}}{n+1}

.

2

Apply the given initial condition

(x, y) = (0, 10)

to solve for

C

.

10 = 2(0)^3 - 2\cos(0) + 3(0) + C \implies 10 = -2 + C

Evaluating at

x = 0

requires evaluating

\cos(0) = 1

, which leaves

-2

from the trigonometric term.

3

Solve the linear equation for

C

.

C = 12

Adding

2

to both sides of

10 = -2 + C

gives

C = 12

.

Key Concept

Indefinite Integration with Boundary Conditions

Question 7707Question →

A curve $y = F(x)$ has a gradient function given by $\frac{dy}{dx} = 9x^2 - 8x + 6\cos(3x) - 4\sin(2x)$ . Given that $y(0) = 7$ , determine the value of the constant of integration $C$ when the antiderivative is expressed in the standard form $y = 3x^3 - 4x^2 + 2\sin(3x) + 2\cos(2x) + C$ .

Show answer & explanation

Answer: 5

Answer

The value of the constant of integration

C

is 5.

Integrating

\frac{dy}{dx} = 9x^2 - 8x + 6\cos(3x) - 4\sin(2x)

yields

y = 3x^3 - 4x^2 + 2\sin(3x) + 2\cos(2x) + C

. Substituting

x = 0

into the expression gives

y(0) = 0 - 0 + 0 + 2(1) + C = 2 + C

. Equating to

y(0) = 7

gives

2 + C = 7

, which solves to

C = 5

.

Step-by-Step Solution

1

Integrate the gradient function term-by-term with respect to

x

y = 3x^3 - 4x^2 + 2\sin(3x) + 2\cos(2x) + C

The antiderivative of

9x^2

is

3x^3

, of

-8x

is

-4x^2

, of

6\cos(3x)

is

2\sin(3x)

, and of

-4\sin(2x)

is

2\cos(2x)

.

2

Apply the initial boundary condition

y(0) = 7

3(0)^3 - 4(0)^2 + 2\sin(0) + 2\cos(0) + C = 7 \implies 2 + C = 7

At

x = 0

,

\sin(0) = 0

and

\cos(0) = 1

, making the non-zero constant contribution equal to

2(1) = 2

.

3

Solve for the constant of integration

C

C = 5

Subtracting 2 from both sides of

2 + C = 7

yields

C = 5

.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions with Boundary Conditions

Question 7708Question →

Two candidates, $X$ and $Y$ , sit for an entrance examination independently. If the probability that candidate $X$ passes is $\frac{4}{7}$ and the probability that candidate $Y$ passes is $\frac{1}{3}$ , what is the probability that at least one of them passes the examination?

\frac{19}{21}

\frac{5}{7}

\frac{4}{21}

\frac{8}{21}

Show answer & explanation

Answer: $\frac{5}{7}$

Answer

The probability that at least one candidate passes the examination is

\frac{5}{7}

.

The probability that at least one candidate passes is given by the union of independent events

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

. Substituting the given values yields

\frac{4}{7} + \frac{1}{3} - \left(\frac{4}{7} \cdot \frac{1}{3}\right) = \frac{15}{21} = \frac{5}{7}

. Alternatively, using the complement rule:

1 - P(X')P(Y') = 1 - \left(1 - \frac{4}{7}\right)\left(1 - \frac{1}{3}\right) = 1 - \left(\frac{3}{7} \cdot \frac{2}{3}\right) = 1 - \frac{2}{7} = \frac{5}{7}

.

Step-by-Step Solution

1

Identify given probabilities and state independence condition

P(X) = \frac{4}{7}

and

P(Y) = \frac{1}{3}

. Since

X

and

Y

are independent events,

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

.

Independent events allow the joint probability of both events occurring to be calculated as the product of their individual probabilities.

2

Calculate the intersection probability

P(X \cap Y)

P(X \cap Y) = \frac{4}{7} \times \frac{1}{3} = \frac{4}{21}

.

The intersection gives the probability that both candidate

X

and candidate

Y

pass.

3

Apply the general addition law of probability

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

P(X \cup Y) = \frac{4}{7} + \frac{1}{3} - \frac{4}{21} = \frac{12}{21} + \frac{7}{21} - \frac{4}{21} = \frac{15}{21} = \frac{5}{7}

.

The union of two events represents the event that at least one of them occurs.

Key Concept

Probability Laws for Compound and Independent Events

Alternative Method

Using the complementary law of probability:

P(\text{at least one passes}) = 1 - P(\text{neither passes}) = 1 - P(X')P(Y') = 1 - \left(1 - \frac{4}{7}\right)\left(1 - \frac{1}{3}\right) = 1 - \left(\frac{3}{7} \times \frac{2}{3}\right) = 1 - \frac{2}{7} = \frac{5}{7}

.

Estimated Time:1m 30s

Question 7709Question →

Determine the number of distinct solutions to the trigonometric equation $2\cos^2 \theta + \sin \theta - 1 = 0$ within the interval $0^\circ \le \theta \le 360^\circ$ .

Show answer & explanation

Answer: 3

Answer

The total number of distinct solutions in the given interval is 3.

Substituting

\cos^2 \theta = 1 - \sin^2 \theta

yields the quadratic equation

2\sin^2 \theta - \sin \theta - 1 = 0

. Factoring gives

\sin \theta = 1

and

\sin \theta = -\frac{1}{2}

. Within

0^\circ \le \theta \le 360^\circ

,

\sin \theta = 1

gives one solution (

90^\circ

), while

\sin \theta = -\frac{1}{2}

gives two solutions (

210^\circ

and

330^\circ

). In total, there are 3 distinct solutions.

Step-by-Step Solution

1

Substitute the identity

\cos^2 \theta = 1 - \sin^2 \theta

into the original equation

2(1 - \sin^2 \theta) + \sin \theta - 1 = 0

Converting the equation into a single trigonometric ratio allows for algebraic solving.

2

Simplify and arrange into quadratic form

2\sin^2 \theta - \sin \theta - 1 = 0

This puts the expression into standard quadratic form

au^2 + bu + c = 0

where

u = \sin \theta

.

3

Factor the quadratic equation

(2\sin \theta + 1)(\sin \theta - 1) = 0

Factoring determines the roots for

\sin \theta

.

4

Solve for possible values of

\sin \theta

\sin \theta = 1

or

\sin \theta = -\frac{1}{2}

By the zero-product property, at least one factor must equal zero.

5

Find the angles for each ratio in the interval

0^\circ \le \theta \le 360^\circ

\theta = 90^\circ, 210^\circ, 330^\circ

\sin \theta = 1

has one solution (

90^\circ

) and

\sin \theta = -0.5

has two solutions in the 3rd and 4th quadrants (

210^\circ

and

330^\circ

).

6

Count the solutions

3

There are 3 distinct values of

\theta

satisfying the condition.

Key Concept

Solving quadratic trigonometric equations using basic identities and quadrant analysis

Estimated Time:1m 30s

Question 7710Question →

Find the acute angle $\theta$ , in degrees, that satisfies the trigonometric equation $\sqrt{3}\tan \theta - 3 = 0$ .

Show answer & explanation

Answer: 60

Answer

The acute angle

\theta

is

60^\circ

.

Rearranging the equation

\sqrt{3}\tan \theta - 3 = 0

gives

\sqrt{3}\tan \theta = 3

, so

\tan \theta = \frac{3}{\sqrt{3}} = \sqrt{3}

. For an acute angle (

0^\circ < \theta < 90^\circ

), the angle with a tangent equal to

\sqrt{3}

is

60^\circ

.

Step-by-Step Solution

1

Isolate the trigonometric ratio

\tan \theta

\tan \theta = \sqrt{3}

Add

3

to both sides and divide by

\sqrt{3}

, giving

\frac{3}{\sqrt{3}} = \sqrt{3}

.

2

Determine the value of the acute angle

\theta

\theta = 60^\circ

From special angle exact values,

\tan(60^\circ) = \sqrt{3}

.

Key Concept

Solving Simple Trigonometric Equations

Question 7711Question →

A straight line $L$ has a $y$ -intercept of $-4$ and is perpendicular to the line segment connecting the points $P(-3, 1)$ and $Q(5, 5)$ . If the line $L$ intersects the $x$ -axis at $(a, 0)$ , what is the value of $a$ ?

-8

-2

2

8

Show answer & explanation

Answer: $-2$

Answer

The value of

a

is

-2

.

The gradient of

PQ

is calculated as

m_1 = \frac{5 - 1}{5 - (-3)} = \frac{1}{2}

. Because line

L

is perpendicular to

PQ

, its slope is

m_2 = -2

. Combining this with the given

y

-intercept of

-4

yields the equation

y = -2x - 4

. Setting

y = 0

to determine the

x

-intercept gives

0 = -2a - 4

, which solves to

a = -2

.

Step-by-Step Solution

1

Find the gradient

m_1

of the line segment

PQ

m_1 = \frac{5 - 1}{5 - (-3)} = \frac{4}{8} = \frac{1}{2}

The formula for the gradient between two points

(x_1, y_1)

and

(x_2, y_2)

is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Calculate the gradient

m_2

of line

L

m_2 = -\frac{1}{m_1} = -\frac{1}{1/2} = -2

Perpendicular lines have gradients satisfying

m_1 \cdot m_2 = -1

.

3

Determine the equation of line

L

y = -2x - 4

Using the slope-intercept form

y = mx + c

, where

m = -2

and

c = -4

.

4

Find the

x

-intercept by substituting

y = 0

and

x = a

0 = -2a - 4 \implies 2a = -4 \implies a = -2

At the

x

-axis, the

y

-coordinate is always equal to

0

.

Key Concept

Perpendicular Line Slopes and Coordinate Intercepts

Question 7712Question →

The couple per unit twist $C$ (torque per unit angle of twist) of a solid wire of length $L$ , radius $r$ , and shear modulus $\eta$ is modeled by the equation:

C = \frac{\pi \eta r^x}{2 L}

Using dimensional analysis, determine the numerical value of the exponent $x$ .

Show answer & explanation

Answer: 4

Answer

The numerical value of the exponent x is 4.

Applying the principle of dimensional homogeneity requires the dimensions of couple per unit twist

[\text{M L}^2 \text{T}^{-2}]

to equal the dimensions of

\frac{\eta r^x}{L}

, which simplifies to

[\text{M L}^{x-2} \text{T}^{-2}]

. Equating exponents of length gives

2 = x - 2

, yielding

x = 4

.

Step-by-Step Solution

1

Determine the dimensional formula of couple per unit twist

C

.

[C] = \text{M L}^2 \text{T}^{-2}

Couple (torque) is force multiplied by perpendicular distance, which has dimensions

[\text{M L T}^{-2}][\text{L}] = [\text{M L}^2 \text{T}^{-2}]

. The angle of twist (in radians) is dimensionless.

2

Determine the dimensional formula of shear modulus

\eta

.

[\eta] = \text{M L}^{-1} \text{T}^{-2}

Shear modulus is defined as shear stress divided by shear strain. Stress has dimensions of force per unit area

[\text{M L T}^{-2}]/[\text{L}^2] = [\text{M L}^{-1} \text{T}^{-2}]

, while strain is dimensionless.

3

Set up the dimensional equation for the relation

C = \frac{\pi \eta r^x}{2 L}

.

[\text{M L}^2 \text{T}^{-2}] = \frac{[\text{M L}^{-1} \text{T}^{-2}][\text{L}]^x}{[\text{L}]} = [\text{M L}^{x-2} \text{T}^{-2}]

Pure numerical constants such as

\pi

and

2

are dimensionless. Length

L

and radius

r

both have dimension

[\text{L}]

.

4

Equate the exponents of length

\text{L}

on both sides of the dimensional equation.

x = 4

Comparing powers of

\text{L}

on both sides gives

2 = x - 2

, which solves to

x = 4

.

Key Concept

Dimensional Homogeneity in Mechanics

Question 7713Question →

Match each physical quantity listed on the left with its corresponding expression in fundamental SI base units on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Pressure matches with

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

, Surface Tension matches with

\text{kg}\cdot\text{s}^{-2}

, Specific Heat Capacity matches with

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

, and Electric Capacitance matches with

\text{kg}^{-1}\cdot\text{m}^{-2}\cdot\text{s}^4\cdot\text{A}^2

.

Each quantity on the left is matched correctly to its fundamental SI base unit equivalent derived directly from its defining physical formula.

Step-by-Step Solution

1

Derive base SI units for Pressure

P = \frac{F}{A} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2} = \text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}

Pressure is force per unit area.

2

Derive base SI units for Surface Tension

\gamma = \frac{F}{L} = \frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}} = \text{kg}\cdot\text{s}^{-2}

Surface tension is force per unit length.

3

Derive base SI units for Specific Heat Capacity

c = \frac{Q}{m\Delta T} = \frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{kg}\cdot\text{K}} = \text{m}^2\cdot\text{s}^{-2}\cdot\text{K}^{-1}

Specific heat capacity is thermal energy per unit mass per kelvin.

4

Derive base SI units for Electric Capacitance

C = \frac{Q}{V} = \frac{\text{A}\cdot\text{s}}{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}} = \text{kg}^{-1}\cdot\text{m}^{-2}\cdot\text{s}^4\cdot\text{A}^2

Capacitance is electric charge divided by electric potential difference.

Key Concept

Expressing derived SI units in terms of fundamental base SI units

Question 7714Question →

A parallel-plate capacitor with air between its plates has a capacitance of $12\text{ }\mu\text{F}$ . A dielectric slab of relative permittivity $\varepsilon_r = 4$ and thickness $t = \frac{d}{3}$ , where $d$ is the total plate separation, is inserted between the plates parallel to them. What is the effective capacitance of the modified capacitor?

16\text{ }\mu\text{F}

48\text{ }\mu\text{F}

162\text{ }\mu\text{F}

6\text{ }\mu\text{F}

Show answer & explanation

Answer: $16\text{ }\mu\text{F}$

Answer

The effective capacitance of the modified capacitor is

16\text{ }\mu\text{F}

.

Inserting a dielectric slab of thickness

t = d/3

creates a system equivalent to two series capacitors: an air-filled region of thickness

2d/3

(

C_1 = 1.5 C_0 = 18\text{ }\mu\text{F}

) and a dielectric-filled region of thickness

d/3

(

C_2 = 12 C_0 = 144\text{ }\mu\text{F}

). Combining them via the series reciprocal formula gives

C_{\text{eq}} = \frac{18 \times 144}{18 + 144} = 16\text{ }\mu\text{F}

.

Step-by-Step Solution

1

Model the partially filled capacitor as two capacitors connected in series.

Air layer of thickness

d_1 = d - t = \frac{2}{3}d

forms capacitor

C_1

. Dielectric layer of thickness

d_2 = t = \frac{1}{3}d

forms capacitor

C_2

.

Dividing the plate gap vertically into two distinct media creates two capacitive regions sharing the same electric flux path.

2

Calculate the individual capacitances

C_1

and

C_2

in terms of initial air capacitance

C_0 = 12\text{ }\mu\text{F}

.

C_1 = \frac{\varepsilon_0 A}{\frac{2}{3}d} = \frac{3}{2}C_0 = \frac{3}{2}(12) = 18\text{ }\mu\text{F}

and

C_2 = \frac{\varepsilon_r \varepsilon_0 A}{\frac{1}{3}d} = 3 \varepsilon_r C_0 = 3(4)(12) = 144\text{ }\mu\text{F}

.

Capacitance is inversely proportional to plate distance and directly proportional to relative permittivity.

3

Calculate the equivalent capacitance

C_{\text{eq}}

for two series capacitors.

\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} = \frac{1}{18} + \frac{1}{144} = \frac{8 + 1}{144} = \frac{9}{144} = \frac{1}{16}\text{ }\mu\text{F}^{-1} \implies C_{\text{eq}} = 16\text{ }\mu\text{F}

.

Capacitors connected in series combine reciprocally.

Key Concept

Partially filled parallel-plate capacitors act as series combinations of distinct capacitive layers.

Estimated Time:3m 0s

Question 7715Question →

The length of the mercury column in an uncalibrated thermometer is $4.0\,\text{cm}$ at the ice point ( $0^\circ\text{C}$ ) and $24.0\,\text{cm}$ at the steam point ( $100^\circ\text{C}$ ). What is the temperature in degrees Celsius when the length of the mercury column is $19.0\,\text{cm}$ ?

Show answer & explanation

Answer: 75

Answer

The temperature corresponding to a mercury column length of

19.0\,\text{cm}

is

75^\circ\text{C}

.

The temperature on the Celsius scale is determined by the ratio of the length change above the ice point to the total length change between the ice and steam points:

T = \frac{L_T - L_0}{L_{100} - L_0} \times 100^\circ\text{C}

. Substituting

L_0 = 4.0\,\text{cm}

,

L_{100} = 24.0\,\text{cm}

, and

L_T = 19.0\,\text{cm}

gives

T = \frac{15.0}{20.0} \times 100 = 75^\circ\text{C}

.

Step-by-Step Solution

1

Identify given thermometric length values at fixed points and at the unknown temperature

L_0 = 4.0\,\text{cm}

,

L_{100} = 24.0\,\text{cm}

, and

L_T = 19.0\,\text{cm}

These represent the length at the lower fixed point (

0^\circ\text{C}

), upper fixed point (

100^\circ\text{C}

), and intermediate temperature

T

respectively.

2

Set up the linear interpolation equation on the Celsius scale

T = \frac{L_T - L_0}{L_{100} - L_0} \times 100^\circ\text{C}

Thermometric expansion is assumed to vary linearly with temperature over the operational range.

3

Substitute the given values and perform arithmetic calculation

T = \frac{19.0 - 4.0}{24.0 - 4.0} \times 100 = \frac{15.0}{20.0} \times 100 = 75^\circ\text{C}

Simplifying

\frac{15.0}{20.0}

gives

0.75

, which multiplied by

100

equals

75

.

Key Concept

Temperature measurement using linear variation of thermometric properties

Question 7716Question →

The mean of a set of $7$ numbers arranged in ascending order is $16$ . If the median of the set is $16$ and the mean of the smallest $3$ numbers is $11$ , what is the mean of the largest $3$ numbers?

Show answer & explanation

Answer: 21

Answer

The mean of the largest 3 numbers is 21.

For a set of 7 ordered numbers, the median is the 4th number, which is given as 16. The total sum of all 7 numbers is 7 × 16 = 112. The sum of the 3 smallest numbers is 3 × 11 = 33. Subtracting the 3 smallest numbers and the median from the total sum leaves the sum of the 3 largest numbers: 112 - 33 - 16 = 63. Dividing 63 by 3 gives a mean of 21.

Step-by-Step Solution

1

Find the total sum of the set of 7 numbers

Sum = 7 × 16 = 112

The mean of a set is the total sum divided by the number of elements.

2

Determine the median value and the sum of the 3 smallest numbers

Median (4th number) = 16, Sum of 3 smallest = 3 × 11 = 33

For an ordered set of 7 numbers, the 4th number is the median, and the mean of the first 3 numbers gives their sum when multiplied by 3.

3

Calculate the sum of the 3 largest numbers

Sum of 3 largest = 112 - 33 - 16 = 63

The total sum is the sum of the 3 smallest numbers, the median (4th number), and the 3 largest numbers.

4

Compute the mean of the 3 largest numbers

Mean = 63 / 3 = 21

Dividing the sum of the 3 largest numbers by 3 gives their arithmetic mean.

Key Concept

Arithmetic Mean and Median of Ungrouped Data Subgroups

Question 7717Question →

Which of the following observations occurs when a beam of cathode rays passes between two oppositely charged parallel metal plates?

The beam is deflected towards the positively charged plate.

The beam is deflected towards the negatively charged plate.

The beam passes straight through without any deflection.

The beam splits into two components moving toward both plates.

Show answer & explanation

Answer: The beam is deflected towards the positively charged plate.

Answer

The beam is deflected towards the positively charged plate.

Cathode rays consist of negatively charged electrons. When placed in an electric field between two oppositely charged plates, the electrostatic attraction pulls the negatively charged electrons toward the positively charged plate.

Step-by-Step Solution

1

Identify the nature and electric charge of cathode rays.

Cathode rays are streams of fast-moving electrons, which carry a negative electric charge.

Determining the sign of the charge is required to find the direction of the electrostatic force.

2

Apply electrostatic force principles to the beam in an electric field.

Opposite charges attract, so negatively charged electrons experience a force toward the positively charged plate.

An electric field exerts an attractive force on negative charges directed toward the positive region.

Key Concept

Deflection of Cathode Rays in an Electric Field

Question 7718Question →

A committee of $5$ members is to be selected from $6$ doctors and $4$ nurses. In how many ways can this committee be formed if it must include at least $3$ doctors?

186

120

720

3600

Show answer & explanation

Answer: 186

Answer

186 ways

The required committee must contain at least 3 doctors out of 5 total members. The three possible scenarios are: 3 doctors and 2 nurses (

^6C_3 \times ^4C_2 = 120

), 4 doctors and 1 nurse (

^6C_4 \times ^4C_1 = 60

), and 5 doctors and 0 nurses (

^6C_5 \times ^4C_0 = 6

). Summing these gives

120 + 60 + 6 = 186

.

Step-by-Step Solution

1

Identify the possible valid committee compositions given the condition 'at least 3 doctors'.

Three mutually exclusive cases exist for a 5-member committee: (3 doctors, 2 nurses), (4 doctors, 1 nurse), or (5 doctors, 0 nurses).

The total size of the committee is 5, so selecting more doctors reduces the required number of nurses.

2

Calculate the combinations for Case 1: 3 doctors and 2 nurses.

^6C_3 \times ^4C_2 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} \times \frac{4 \times 3}{2 \times 1} = 20 \times 6 = 120

Order of selection within each group does not matter.

3

Calculate the combinations for Case 2: 4 doctors and 1 nurse.

^6C_4 \times ^4C_1 = \frac{6 \times 5}{2 \times 1} \times 4 = 15 \times 4 = 60

Selecting 4 doctors out of 6 is equivalent to choosing which 2 doctors to exclude.

4

Calculate the combinations for Case 3: 5 doctors and 0 nurses.

^6C_5 \times ^4C_0 = 6 \times 1 = 6

Choosing 5 doctors out of 6 yields 6 possibilities, and choosing 0 nurses yields 1.

5

Sum the number of ways from all valid cases.

120 + 60 + 6 = 186

By the addition principle of counting, the total ways to form the committee is the sum of ways across mutually exclusive cases.

Key Concept

Combinations with constraints (addition and multiplication principles)

Question 7719Question →

What is the gradient of a straight line that is perpendicular to the line passing through the points $(1, -2)$ and $(5, 6)$ ?

-\frac{1}{2}

2

-2

\frac{1}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

-\frac{1}{2}

The slope of the given line segment is computed as

\frac{6 - (-2)}{5 - 1} = 2

. The perpendicular gradient is the negative reciprocal of

2

, which gives

-\frac{1}{2}

.

Step-by-Step Solution

1

Calculate the gradient of the line passing through the two points

(1, -2)

and

(5, 6)

.

Using

m_1 = \frac{y_2 - y_1}{x_2 - x_1}

, we obtain

m_1 = \frac{6 - (-2)}{5 - 1} = \frac{8}{4} = 2

.

The slope of a straight line through two points is defined as the change in

y

divided by the change in

x

.

2

Apply the perpendicularity condition to find the perpendicular slope

m_2

.

Since

m_1 \cdot m_2 = -1

,

m_2 = -\frac{1}{m_1} = -\frac{1}{2}

.

Two non-vertical lines are perpendicular if and only if the product of their gradients is

-1

.

Key Concept

Perpendicular Line Gradients

Estimated Time:45s

Question 7720Question →

An aluminium rod of initial length $2.0\text{ m}$ at $20^\circ\text{C}$ expands by $0.96\text{ mm}$ when heated. If the linear expansivity of aluminium is $2.4 \times 10^{-5}\text{ K}^{-1}$ , what is the rise in temperature of the rod?

Show answer & explanation

Answer: 20

Answer

The rise in temperature of the aluminium rod is

20\text{ K}

.

The fractional change in length depends on linear expansivity and temperature change through

\Delta L = L_0 \alpha \Delta T

. Substituting the converted expansion

\Delta L = 9.6 \times 10^{-4}\text{ m}

, initial length

L_0 = 2.0\text{ m}

, and linear expansivity

\alpha = 2.4 \times 10^{-5}\text{ K}^{-1}

gives

\Delta T = \frac{9.6 \times 10^{-4}}{2.0 \times 2.4 \times 10^{-5}} = 20\text{ K}

.

Step-by-Step Solution

1

Convert change in length from millimeters to meters

\Delta L = 9.6 \times 10^{-4}\text{ m}

Units must be consistent with initial length in meters.

2

Rearrange the linear thermal expansion formula \Delta L = L_0 \alpha \Delta T for temperature change \Delta T

\Delta T = \frac{\Delta L}{L_0 \alpha}

To isolate the unknown quantity \Delta T.

3

Substitute values into the rearranged formula and compute \Delta T

\Delta T = \frac{9.6 \times 10^{-4}}{2.0 \times (2.4 \times 10^{-5})} = 20\text{ K}

Evaluating the mathematical expression yields the required temperature rise.

Key Concept

Linear Expansivity and Thermal Expansion of Solids

Answer

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