We need to determine which option correctly represents the relationship "D is the father-in-law of A" based on the given coded relationships. We will decode each option step-by-step to establish the family tree and identify the relationship between D and A.
B) A@B \( C # D > E
This problem can be solved using a system of two linear equations with two variables. We are given the total number of heads and the total number of legs for chickens and goats. We know that each chicken has 1 head and 2 legs, and each goat has 1 head and 4 legs. We can set up equations based on this information and solve for the number of chickens.
Define variables:
Formulate equations based on heads:
Each animal has one head. The total number of heads is 42.
\[ C + G = 42 \quad \text{(Equation 1)} \]Formulate equations based on legs:
Each chicken has 2 legs, and each goat has 4 legs. The total number of legs is 138.
\[ 2C + 4G = 138 \quad \text{(Equation 2)} \]Solve the system of equations:
From Equation 1, we can express \(G\) in terms of \(C\):
\[ G = 42 - C \]Substitute this expression for \(G\) into Equation 2:
\[ 2C + 4(42 - C) = 138 \]Simplify and solve for \(C\):
\[ 2C + 168 - 4C = 138 \]\[ -2C = 138 - 168 \]\[ -2C = -30 \]\[ C = \frac{-30}{-2} \]\[ C = 15 \]Find the number of goats (optional, but good for verification):
Substitute \(C = 15\) back into Equation 1:
\[ 15 + G = 42 \]\[ G = 42 - 15 \]\[ G = 27 \]Verify the total number of legs:
Number of chicken legs \( = 15 \times 2 = 30 \)
Number of goat legs \( = 27 \times 4 = 108 \)
Total legs \( = 30 + 108 = 138 \)
This matches the given total number of legs, so our values are correct.
C) 15 — The calculation shows that there are 15 chickens.
The problem involves three positive numbers whose ratio is given, along with the sum of their squares. We need to find the smallest of these numbers. We can represent the numbers using a common multiple based on their ratio, form an equation using the sum of their squares, solve for the common multiple, and then find the smallest number.
B) The smallest number is 24.
The question asks to identify the first female student from IIT-Madras to win the President of India Prize in 2019. This is a factual recall question related to current affairs and achievements in education.
In 2019, Kavitha Gopal, a B.Tech student from the Department of Computer Science and Engineering, became the first girl student from IIT-Madras to win the prestigious President of India Prize. This award is given for outstanding academic performance and leadership qualities.
A) Kavitha Gopal is the correct answer. She was the first girl student from IIT-Madras to win the President of India Prize in 2019 for her exceptional academic record and leadership.
The question asks us to identify the odd one out from a given set of materials: Cotton, Nylon, Silk, and Cashmere. We need to categorize these materials based on their origin (natural or synthetic).
Correct Option: D) Nylon
Nylon is a synthetic fiber, while Cotton, Silk, and Cashmere are all natural fibers. This makes Nylon the odd one out.
The question asks to identify the city where the historical monument Charminar is located. This is a direct factual question related to Indian geography and history, often covered in general knowledge sections of competitive exams.
Correct Option: B) Hyderabad is the correct answer. The Charminar is an iconic monument and mosque located in Hyderabad, Telangana, India. It was constructed in 1591 by Muhammad Quli Qutb Shah, the fifth sultan of the Qutb Shahi dynasty, to commemorate the end of a deadly plague.
The question asks for the year Gandhiji initiated the Dandi March (Salt March) to protest the British salt monopoly. This is a direct historical fact question.
Correct Option: D) 1930
If sin(A - B) = \( \frac{1}{2} \) and cos(A + B) = \( \frac{1}{2} \) with 0° < (A + B) ≤ 90°, A > B, then find the measure of A and B.
We are given two trigonometric equations involving angles A and B, along with conditions for the range of (A + B) and the relationship between A and B. We need to find the specific values of A and B. We will use the standard values of trigonometric ratios for common angles to convert the given equations into a system of linear equations in terms of (A - B) and (A + B). Then, we will solve this system to find A and B.
Analyze the first equation:
Given: \( \sin(A - B) = \frac{1}{2} \)
We know that \( \sin(30^\circ) = \frac{1}{2} \).
Therefore,
\[ A - B = 30^\circ \quad \text{(Equation 1)} \]This is because the sine function is positive in the first quadrant, and \(30^\circ\) is the principal value.
Analyze the second equation:
Given: \( \cos(A + B) = \frac{1}{2} \)
We know that \( \cos(60^\circ) = \frac{1}{2} \).
Therefore,
\[ A + B = 60^\circ \quad \text{(Equation 2)} \]This is consistent with the given condition \( 0^\circ < (A + B) \le 90^\circ \).
Solve the system of linear equations:
We have two equations:
Equation 1: \( A - B = 30^\circ \)
Equation 2: \( A + B = 60^\circ \)
Add Equation 1 and Equation 2:
\[ (A - B) + (A + B) = 30^\circ + 60^\circ \]\[ 2A = 90^\circ \]
\[ A = \frac{90^\circ}{2} \]
\[ A = 45^\circ \]
Substitute the value of A into one of the equations to find B:
Using Equation 2:
\[ 45^\circ + B = 60^\circ \]\[ B = 60^\circ - 45^\circ \]
\[ B = 15^\circ \]
Verify the conditions:
We found \( A = 45^\circ \) and \( B = 15^\circ \).
Both conditions are satisfied.
Correct Option: B) 45°, 15°
The question asks to identify the 'Father of Modern Computers'. This requires knowledge of key figures in the history of computing and their specific contributions.
Correct Option: B) Alan Turing is widely regarded as the 'Father of Modern Computers' due to his foundational theoretical work on computability (Turing Machine) and his practical contributions to early computing devices during World War II. His work provided the abstract model that underlies all modern computers.
If the numerator of a fraction is increased by 30% and its denominator is decreased by 35%, the value of the fraction becomes \( \frac{3}{15} \). Find the original fraction.
\( \frac{3}{10} \)
\( \frac{1}{5} \)
\( \frac{1}{10} \)
\( \frac{3}{5} \)
To find the original fraction, we will represent the numerator and denominator with variables. Then, we will apply the given percentage changes to the numerator and denominator to form a new fraction. This new fraction will be equated to the given resulting fraction, and we will solve for the original fraction.
Let the original fraction be \( \frac{x}{y} \), where \(x\) is the numerator and \(y\) is the denominator.
The numerator is increased by 30%. An increase of 30% means the new numerator will be \( x + 0.30x = 1.30x \).
The denominator is decreased by 35%. A decrease of 35% means the new denominator will be \( y - 0.35y = 0.65y \).
The new fraction formed is \( \frac{1.30x}{0.65y} \).
We are given that this new fraction is equal to \( \frac{3}{15} \). So, we set up the equation:
\[ \frac{1.30x}{0.65y} = \frac{3}{15} \]Simplify the fraction on the right side:
\[ \frac{3}{15} = \frac{1}{5} \]Now, simplify the left side of the equation:
\[ \frac{1.30x}{0.65y} = \frac{130x}{65y} \]\[ \frac{130}{65} \times \frac{x}{y} = 2 \times \frac{x}{y} \]Substitute this back into the equation:
\[ 2 \times \frac{x}{y} = \frac{1}{5} \]To find the original fraction \( \frac{x}{y} \), divide both sides by 2:
\[ \frac{x}{y} = \frac{1}{5 \times 2} \]\[ \frac{x}{y} = \frac{1}{10} \]C) \( \frac{1}{10} \) is the original fraction, as derived from the calculations.