Step 1 : Understanding the Question

The problem asks us to determine the percentage error in the calculated area of a circle given that the measurement of its radius has a known percentage error. This is a classic application of error propagation in physical measurements or mathematical approximations. In mathematics and physics, when a quantity is derived from a measured value raised to a power, the relative error in the derived quantity is proportional to that power. Here, we must relate the change in area to the change in radius using calculus-based error analysis.


Step 2 : Key Formulas and approach

1. The formula for the area of a circle is \( A = \pi r^2 \).

2. For any function \( y = f(x) \), the absolute error is approximated by \( \Delta y \approx f'(x) \Delta x \).

3. For power functions of the form \( Q = x^n \), the relative error is given by \( \frac{\Delta Q}{Q} = n \frac{\Delta x}{x} \).

4. Percentage error is defined as \( \left( \frac{\Delta value}{value} \times 100 \right) \).

5. Our approach will involve taking the natural logarithm of the area formula or using differentiation to find the relationship between the relative error of the area and the radius.


Step 3 : Detailed Explanation and Final Answer


We start with the standard geometric formula for the area of a circle: \( A = \pi r^2 \).

To find the error relationship, we differentiate both sides with respect to \( r \). This gives us \( \frac{dA}{dr} = 2\pi r \), which can be written in differential form as \( dA = 2\pi r \, dr \).

To find the relative error, we divide the change in area (\( dA \)) by the total area (\( A \)): \( \frac{dA}{A} = \frac{2\pi r \, dr}{\pi r^2} \).

Simplifying the expression on the right, the constant \( \pi \) cancels out and one factor of \( r \) cancels out, leaving us with: \( \frac{dA}{A} = 2 \frac{dr}{r} \).

To convert this relative error into a percentage error, we multiply both sides of the equation by 100: \( \left( \frac{dA}{A} \times 100 \right) = 2 \times \left( \frac{dr}{r} \times 100 \right) \).

The problem states that the percentage error in the radius is 3, which means \( \frac{dr}{r} \times 100 = 3 \).

Substituting this value into our derived error equation: \( Percentage error in Area = 2 \times 3 = 6 \).

This demonstrates that because the area depends on the square of the radius, any small error in the radius is doubled in the resulting area calculation.



Therefore, the final answer for the percentage error in the area is: \[ \boxed{6} \]

Hence, the correct option is (C).
Quick Tip: For any formula involving powers like \( Z = A^m B^n \), the total percentage error is the sum of the individual percentage errors multiplied by their respective powers. For a circle, \( Area = \pi r^2 \), so the power is 2. This means the error in area will always be exactly twice the error in the radius for small percentage changes.

Step 1 : Understanding the Question

The objective is to calculate the total geometric area enclosed between the cubic curve \( y = x^3 \), the horizontal x-axis (\( y=0 \)), and the vertical boundary lines \( x = -2 \) and \( x = 4 \). A critical aspect of "Area Under Curves" problems is recognizing that "Area" is always a non-negative quantity. Since the cubic function \( y = x^3 \) takes both negative and positive values over the interval \([-2, 4]\), we cannot simply perform a single definite integral from \(-2\) to \(4\), as the negative and positive regions would cancel each other out algebraically. We must treat the regions separately.


Step 2 : Key Formulas and approach

1. Total Area \( A = \int_{a}^{b} |f(x)| \, dx \).

2. Integration of power functions: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

3. Splitting the interval: If \( f(x) \) changes sign at \( c \), then \( \int_{a}^{b} |f(x)| \, dx = |\int_{a}^{c} f(x) \, dx| + |\int_{c}^{b} f(x) \, dx| \).

4. For \( y = x^3 \), the function is negative when \( x < 0 \) and positive when \( x > 0 \). Thus, we split the integral at the origin (\( x = 0 \)).


Step 3 : Detailed Explanation and Final Answer


First, we identify the root of the function within the given limits. For \( y = x^3 \), setting \( y = 0 \) gives \( x = 0 \). This is our split point.

We calculate the area of the first region (\( A_1 \)) from \( x = -2 \) to \( x = 0 \). In this interval, the curve lies below the x-axis.

\( A_1 = | \int_{-2}^{0} x^3 \, dx | = | [\frac{x^4}{4}]_{-2}^{0} | \).

Evaluating the limits: \( A_1 = | \frac{0^4}{4} - \frac{(-2)^4}{4} | = | 0 - \frac{16}{4} | = | -4 | = 4 \) square units.

Next, we calculate the area of the second region (\( A_2 \)) from \( x = 0 \) to \( x = 4 \). Here, the curve is above the x-axis.

\( A_2 = \int_{0}^{4} x^3 \, dx = [\frac{x^4}{4}]_{0}^{4} \).

Evaluating the limits: \( A_2 = \frac{4^4}{4} - \frac{0^4}{4} = \frac{256}{4} - 0 = 64 \) square units.

The total area is the arithmetic sum of these two magnitude values: \( A = A_1 + A_2 = 4 + 64 \).

This results in a total area of 68 square units. It is important to note that if we had integrated directly from \(-2\) to \(4\), we would have obtained \( 64 - 4 = 60 \), which is the "net area" or integral value, but not the geometric area.



The final result for the bounded area is: \[ \boxed{68} \]

Thus, the correct option is (D).
Quick Tip: Always sketch the function or check for intercepts with the x-axis before integrating for area. For odd functions like \( x^3, \sin x, or x \), the integral over a symmetric interval is zero, but the total area is double the integral of one side. In non-symmetric intervals like this one, calculate the absolute value of each segment separately to avoid sign cancellation.

Step 1 : Understanding the Question

This is a probability problem that involves identifying the composition of a set based on an observed outcome and a posterior probability. We are given two boxes, each with 10 balls. We know that a black ball was drawn, and we know the probability that this ball originated specifically from the second box is \( 1/5 \). We need to work backwards to find the possible number of black balls in the first box. This scenario is a perfect application of Bayes' Theorem, which relates conditional and marginal probabilities of random events.


Step 2 : Key Formulas and approach

1. Let \( B_1 \) and \( B_2 \) be the events of selecting Box 1 and Box 2 respectively. Since the box is chosen at random, \( P(B_1) = P(B_2) = 1/2 \).

2. Let \( E \) be the event that the drawn ball is black.

3. Bayes' Theorem: \( P(B_2|E) = \frac{P(E|B_2)P(B_2)}{P(E|B_1)P(B_1) + P(E|B_2)P(B_2)} \).

4. Let \( n_1 \) be the number of black balls in Box 1 and \( n_2 \) be the number of black balls in Box 2. Then \( P(E|B_1) = n_1/10 \) and \( P(E|B_2) = n_2/10 \).

5. We will substitute the given values into the formula and solve for the relationship between \( n_1 \) and \( n_2 \).


Step 3 : Detailed Explanation and Final Answer


We are given \( P(B_2|E) = 1/5 \). By Bayes' Theorem: \( \frac{1}{5} = \frac{(n_2/10)(1/2)}{(n_1/10)(1/2) + (n_2/10)(1/2)} \).

The term \( (1/2) \) and the denominator \( 10 \) cancel out from the entire fraction, simplifying the equation to: \( \frac{1}{5} = \frac{n_2}{n_1 + n_2} \).

Cross-multiplying gives us: \( n_1 + n_2 = 5n_2 \), which simplifies further to \( n_1 = 4n_2 \).

Since the number of balls in a box must be an integer between 0 and 10, we test possible integer values for \( n_2 \).

Case 1: If \( n_2 = 1 \), then \( n_1 = 4(1) = 4 \).

Case 2: If \( n_2 = 2 \), then \( n_1 = 4(2) = 8 \).

Case 3: If \( n_2 = 3 \), then \( n_1 = 4(3) = 12 \). However, the total balls per box is 10, so this is not possible.

Thus, the possible values for the number of black balls in the first box are 4 or 8. If we consider the wording "few are black and rest are white" and the potential for the boxes to be swapped in identification, these two solutions satisfy the probability condition.



The possible counts of black balls in the first box are: \[ \boxed{4 or 8} \]

Hence, the correct option is (C).
Quick Tip: In Bayes' Theorem problems where the prior probabilities are equal (like choosing between two boxes), the posterior probability simplifies to the ratio of the specific likelihood to the sum of all likelihoods. Essentially, \( P(B_2|E) = \frac{Black balls in Box 2}{Total Black balls in both boxes} \). This shortcut allows for very quick mental calculations in multiple-choice questions.

Step 1 : Understanding the Question

The problem asks for the count of real-valued solutions among all possible values of the complex expression \( (-1 - \sqrt{3}i)^{3/4} \). In complex analysis, raising a number to a fractional power results in multiple distinct values. Specifically, an expression of the form \( z^{p/q} \) generally has \( q \) distinct roots if \( p/q \) is in its simplest form. We need to identify how many of these four roots lie on the real axis of the complex plane. A complex number is real if its imaginary part is zero, which occurs when its argument is an integer multiple of \( \pi \).


Step 2 : Key Formulas and approach

1. Express the complex number \( z = -1 - \sqrt{3}i \) in polar form: \( z = r(\cos \theta + i\sin \theta) \).

2. Modulus \( r = \sqrt{x^2 + y^2} \). Argument \( \theta = \tan^{-1}(y/x) \) (considering the quadrant).

3. General polar form: \( z = r[ \cos(\theta + 2k\pi) + i\sin(\theta + 2k\pi) ] \).

4. De Moivre’s Theorem: \( z^{n} = r^{n} [ \cos(n\phi) + i\sin(n\phi) ] \).

5. A value is real if the argument \( n\phi = m\pi \) for some integer \( m \).


Step 3 : Detailed Explanation and Final Answer


For \( z = -1 - \sqrt{3}i \), we find \( r = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2 \).

The point \((-1, -\sqrt{3})\) is in the 3rd quadrant. The reference angle is \( \tan \alpha = |-\sqrt{3} / -1| = \sqrt{3} \), so \( \alpha = \pi/3 \). The principal argument is \( \theta = \pi + \pi/3 = 4\pi/3 \).

The general argument is \( \frac{4\pi}{3} + 2k\pi = \frac{(4+6k)\pi}{3} \), for \( k = 0, 1, 2, 3 \).

Now we apply the power \( 3/4 \). The argument of the values becomes: \( Arg(z^{3/4}) = \frac{3}{4} \times \frac{(4+6k)\pi}{3} = \frac{(4+6k)\pi}{4} = \frac{(2+3k)\pi}{2} \).

We check the values of this argument for \( k = 0, 1, 2, 3 \):

For \( k=0 \): Arg = \( \pi \). Since \( \sin(\pi) = 0 \), this value is real.

For \( k=1 \): Arg = \( 5\pi/2 \). This is equivalent to \( \pi/2 \). It is purely imaginary.

For \( k=2 \): Arg = \( 8\pi/2 = 4\pi \). Since \( \sin(4\pi) = 0 \), this value is real.

For \( k=3 \): Arg = \( 11\pi/2 \). This is equivalent to \( 3\pi/2 \). It is purely imaginary.

We find exactly two values where the argument is a multiple of \( \pi \), leading to zero imaginary parts.



The number of real values is: \[ \boxed{2} \]

Hence, the correct option is (C).
Quick Tip: To quickly find real roots of complex powers, look at the argument. If the initial argument of \( z^p \) is an integer multiple of \( \pi / q \), there's a high chance some roots will be real. Specifically, check if \( \frac{p \times (\theta + 2k\pi)}{q} = m\pi \). For \( z^{3/4} \), we are looking for arguments like \( 0, \pi, 2\pi, \dots \) after the transformation.

Step 1 : Understanding the Question

This problem asks for the number of distinct normal lines that can be constructed passing through a specific external point \( (2, 0) \) to the curve defined by the parabola \( y^2 = 7x \). In coordinate geometry, a normal at any point on a parabola is a line perpendicular to the tangent at that point. While a tangent usually has a unique solution from a point under specific conditions, a normal can have up to three real solutions from a single point in the plane, depending on the location of that point relative to the parabola's "evolute." We must solve the cubic equation generated by the normal's slope to find how many real normals exist.


Step 2 : Key Formulas and approach

1. Standard parabola \( y^2 = 4ax \). Here, \( 4a = 7 \), so \( a = 7/4 \).

2. The equation of a normal to the parabola \( y^2 = 4ax \) with slope \( m \) is \( y = mx - 2am - am^3 \).

3. Alternatively, using the parameter \( t \) where \( m = -t \), the equation is \( y = -tx + 2at + at^3 \).

4. Substitute the given point \( (h, k) = (2, 0) \) into the equation to get a cubic equation in \( t \) (or \( m \)).

5. The number of real roots of this cubic equation corresponds to the number of possible normals.


Step 3 : Detailed Explanation and Final Answer


We start with the given parabola \( y^2 = 7x \). Comparing it with \( y^2 = 4ax \), we find \( a = 7/4 \).

The general equation of the normal in terms of the slope \( m \) is: \( y = mx - 2(\frac{7}{4})m - (\frac{7}{4})m^3 \).

Simplifying the constants: \( y = mx - \frac{7}{2}m - \frac{7}{4}m^3 \).

Since the normal passes through the point \( (2, 0) \), we substitute \( x = 2 \) and \( y = 0 \):

\( 0 = m(2) - \frac{7}{2}m - \frac{7}{4}m^3 \).

Combine the terms involving \( m \): \( 0 = (2 - \frac{7}{2})m - \frac{7}{4}m^3 \).

\( 0 = -\frac{3}{2}m - \frac{7}{4}m^3 \).

Factor out \( m \): \( m ( -\frac{3}{2} - \frac{7}{4}m^2 ) = 0 \).

This gives two possibilities:

Case 1: \( m = 0 \). This is a valid real root, representing the x-axis (the axis of the parabola).

Case 2: \( -\frac{3}{2} - \frac{7}{4}m^2 = 0 \), which implies \( \frac{7}{4}m^2 = -\frac{3}{2} \).

This simplifies to \( m^2 = -\frac{6}{7} \). Since the square of a real number cannot be negative, there are no real values of \( m \) from this part.

Therefore, only one real slope \( m = 0 \) exists. This means only one real normal can be drawn.



The number of normals is: \[ \boxed{1} \]

Hence, the correct option is (A).
Quick Tip: For a parabola \( y^2 = 4ax \), three real normals can be drawn from a point \( (h, k) \) only if \( h > 2a \). In our case, \( a = 1.75 \), so \( 2a = 3.5 \). Since the given point has \( h = 2 \), and \( 2 < 3.5 \), it is mathematically impossible to have three real normals. Checking this condition first can save significant calculation time.

Step 1 : Understanding the Question

The problem involves calculating the probability of a specific sum appearing when three fair six-sided dice are rolled simultaneously. The sum must be a prime number. The range of possible sums for three dice is from 3 (when all dice show 1) to 18 (when all dice show 6). To solve this, we must identify all prime numbers in this range and then count the number of distinct combinations (permutations) of dice faces that produce those sums. The probability is then the ratio of these favorable outcomes to the total possible outcomes.


Step 2 : Key Formulas and approach

1. Total sample space size for \( n \) dice is \( 6^n \). For 3 dice, \( S = 6^3 = 216 \).

2. Prime numbers between 3 and 18 are: 3, 5, 7, 11, 13, and 17.

3. The number of ways to get a sum \( S \) with 3 dice can be calculated using the formula for combinations or by manual partitions.

4. Formula for sum \( S \) with \( n \) dice: The coefficient of \( x^S \) in \( (x + x^2 + x^3 + x^4 + x^5 + x^6)^3 \).

5. Probability \( P = \frac{Number of favorable outcomes}{Total outcomes} \).


Step 3 : Detailed Explanation and Final Answer


Total outcomes = 216. We calculate favorable ways for each prime sum:

Sum 3: Only (1,1,1). Ways = 1.

Sum 5: (1,1,3) in 3 ways, (1,2,2) in 3 ways. Total = 6.

Sum 7: (1,1,5) in 3 ways, (1,2,4) in 6 ways, (1,3,3) in 3 ways, (2,2,3) in 3 ways. Total = 15.

Sum 11: (1,4,6) in 6, (1,5,5) in 3, (2,3,6) in 6, (2,4,5) in 6, (3,3,5) in 3, (3,4,4) in 3. Total = 27.

Sum 13: (1,6,6) in 3, (2,5,6) in 6, (3,4,6) in 6, (3,5,5) in 3, (4,4,5) in 3. Total = 21.

Sum 17: (5,6,6) in 3 ways. Total = 3.

Summing these favorable cases: \( 1 + 6 + 15 + 27 + 21 + 3 = 73 \).

The probability is the total favorable outcomes divided by the total sample space: \( P = 73/216 \).

Note: Calculations for sums like 11 and 13 can also be cross-checked using the symmetry property where ways for sum \( S \) equals ways for sum \( (21-S) \). For example, ways for sum 11 equals ways for sum 10.



The calculated probability is: \[ \boxed{\frac{73}{216}} \]

Hence, the correct option is (B).
Quick Tip: To save time on sums of 3 dice, remember the frequency distribution peaks at 10 and 11 (27 ways each). For sums \( S \leq 8 \), the number of ways is \( \binom{S-1}{2} \). For larger sums, use the symmetry property: \( Ways for sum X = Ways for sum (3 \times 7 - X) \). This helps you quickly find outcomes for 13, 17, etc., by calculating for 8 and 4 instead.

Step 1 : Understanding the Question

The question asks for the characteristic spatial range over which the weak nuclear force operates. The weak nuclear force is one of the four fundamental interactions in nature, alongside gravity, electromagnetism, and the strong nuclear force. Unlike gravity and electromagnetism, which have infinite range, nuclear forces are highly localized. Specifically, the weak force is responsible for phenomena such as radioactive beta decay and the fusion processes that power stars. Determining its range helps us understand why its effects are only observed at subatomic scales.


Step 2 : Key Formulas and approach

1. The range of a force is inversely proportional to the mass of its exchange particles (gauge bosons) according to the Yukawa potential model.

2. For the weak force, the exchange particles are the \( W^+ \), \( W^- \), and \( Z^0 \) bosons.

3. These bosons are very massive (about 80–90 GeV/\(c^2\)), which limits the force to a very short range.

4. Approximate ranges of fundamental forces:

- Gravitational: Infinite.

- Electromagnetic: Infinite.

- Strong Nuclear: \( \sim 10^{-15} \) m (size of a nucleus).

- Weak Nuclear: \( \sim 10^{-16} \) to \( 10^{-17} \) m.


Step 3 : Detailed Explanation and Final Answer


Fundamental forces are categorized by their strength and their range. While gravity is the weakest but acts across the universe, the nuclear forces are strong but confined to extremely small distances.

The weak nuclear force is unique because its carrier particles are massive. According to the uncertainty principle, a massive particle can only exist for a very short time and thus can only travel a very short distance.

The range is approximately \( 0.1% \) of the diameter of a typical atomic nucleus. A nucleus is about \( 10^{-15} \) m in size (1 femtometer).

Empirical data and the Standard Model of particle physics place the effective range of the weak interaction at roughly \( 10^{-16} \) meters to \( 10^{-18} \) meters.

Among the given choices, \( 10^{-16} \) m represents the correct order of magnitude for this interaction.

It is significantly shorter than the range of the strong force (\( 10^{-15} \) m) and much smaller than the atomic scale (\( 10^{-10} \) m).



The range of the weak nuclear force is of the order: \[ \boxed{10^{-16} m} \]

Hence, the correct option is (D).
Quick Tip: A useful way to remember the scales: \( 10^{-10} \) m is the atom (Angstrom), \( 10^{-15} \) m is the nucleus (Femtometer/Strong force), and \( 10^{-16} \) m is the Weak force. As the mass of the carrier particle increases, the range of the force decreases. The W and Z bosons are nearly 100 times heavier than a proton, explaining the extremely short range of the weak force.

Step 1 : Understanding the Question

The problem involves finding a specific temperature where a defined algebraic relationship exists between two different thermometric scales: Celsius (\( ^\circ C \)) and Fahrenheit (\( ^\circ F \)). We are told that the Fahrenheit reading is "90% more" than the Celsius reading. This is a word problem that requires translating a percentage increase into a linear equation and then using the standard conversion formula between the two scales to solve for the unknown temperature. The final answer must be provided in the Fahrenheit scale.


Step 2 : Key Formulas and approach

1. The standard conversion formula is \( F = \frac{9}{5}C + 32 \) or \( F = 1.8C + 32 \).

2. The condition "F is 90% more than C" translates to: \( F = C + 90% of C \).

3. Mathematically, this is \( F = C + 0.9C = 1.9C \).

4. Our approach is to set the two expressions for \( F \) equal to each other, solve for \( C \), and then substitute \( C \) back to find the value of \( F \).


Step 3 : Detailed Explanation and Final Answer


Start with the given condition: \( F = 1.9C \).

Use the universal conversion formula: \( F = 1.8C + 32 \).

Equate the two expressions: \( 1.9C = 1.8C + 32 \).

Subtract \( 1.8C \) from both sides to isolate the variable: \( 1.9C - 1.8C = 32 \).

This gives \( 0.1C = 32 \).

Solving for \( C \), we divide by \( 0.1 \): \( C = \frac{32}{0.1} = 320^\circ C \).

Now, we need the value in Fahrenheit as per the options. We use the condition \( F = 1.9C \).

\( F = 1.9 \times 320 \).

Calculating the product: \( 1.9 \times 320 = 19 \times 32 = 608 \).

Thus, the temperature is \( 608^\circ F \). We can verify this with the conversion formula: \( 1.8(320) + 32 = 576 + 32 = 608 \). The result is consistent.



The required temperature in Fahrenheit is: \[ \boxed{608^\circ F} \]

Hence, the correct option is (A).
Quick Tip: In thermometry problems, "X% more" always means multiplying the base value by \( (1 + X/100) \). If the question says "90% more," use \( 1.9 \). If it says "90% of," use \( 0.9 \). Paying close attention to the preposition "more" or "less" is crucial to setting up the correct algebraic equation.

Step 1 : Understanding the Question

This question focuses on environmental chemistry and the physiological impacts of atmospheric pollutants. Specifically, it asks us to identify a gas that acts as a potent lung irritant and is particularly dangerous to children, causing acute respiratory distress. While many gases are harmful, they have distinct modes of toxicity. Some interfere with blood chemistry, while others directly damage the mucosal lining of the respiratory tract. We must evaluate the options based on their known medical and environmental effects.


Step 2 : Key Formulas and approach

1. Identify the chemical nature of each gas: CO (Carbon Monoxide), \(SO_2\) (Sulfur Dioxide), \(CO_2\) (Carbon Dioxide), and \(NO_2\) (Nitrogen Dioxide).

2. Analyze the biological mechanism: Lung irritants are usually oxidizing agents or acids that react with moisture in the lungs to cause inflammation.

3. Nitrogen oxides (\(NO_x\)), specifically \(NO_2\), are known to be brown, toxic gases that penetrate deep into the lungs.

4. Comparative analysis of options:

- CO: Binds to hemoglobin (asphyxiant, not primarily an irritant).

- \(SO_2\): Irritant, but primarily affects the upper respiratory tract.

- \(CO_2\): Greenhouse gas, non-toxic at standard concentrations.

- \(NO_2\): Deep lung irritant, linked to asthma and acute respiratory disease.


Step 3 : Detailed Explanation and Final Answer


Nitrogen dioxide (\(NO_2\)) is a prominent air pollutant primarily emitted from high-temperature combustion in vehicle engines and industrial plants.

It is a highly reactive oxidant. When inhaled, it reacts with the fluids lining the respiratory tract, leading to the formation of reactive oxygen species that damage cells.

In children, whose respiratory systems are still developing and who breathe at a faster rate than adults, the impact of \(NO_2\) is severe. It leads to increased susceptibility to respiratory infections and "acute respiratory disease."

Medical studies have consistently shown that high concentrations of \(NO_2\) trigger asthma attacks and reduce lung function growth in children living in urban environments.

While \(SO_2\) is also an irritant, \(NO_2\) is the specific pollutant highlighted in most environmental science curricula for its role in acute respiratory diseases in younger populations.

Carbon monoxide (CO) is dangerous because it forms carboxyhemoglobin, preventing oxygen transport, but it does not "irritate" the lung tissue in the same inflammatory way that \(NO_2\) does.



The correct pollutant is: \[ \boxed{NO_2} \]

Hence, the correct option is (D).
Quick Tip: Remember the "Nitrogen - Negative" rule for health: \( NO_2 \) is the brown gas that causes lung inflammation and is a major component of photochemical smog. CO affects the blood (hemoglobin), while \( NO_2 \) affects the breath (lungs). This distinction helps in answering many environmental chemistry questions regarding health impacts.