This problem involves ordering people by their heights based on given comparative statements. We need to deduce the complete or partial order of heights to answer the final question.
"Only C is taller than B."
This statement is crucial. It implies that B is taller than everyone else except C. So, C is the tallest, and B is the second tallest. The order starts as:
C > B > (E, F, H)
"E is taller than H."
This gives us a direct comparison between E and H:
E > H
"Only two people have heights between F and C."
We know C is the tallest. If there are two people between F and C, then the order must be C > _ > _ > F. Since B is the second tallest, B must be the first person after C. So, C > B > _ > F. This means there is one person between B and F. The only remaining people are E and H. Therefore, the order must be C > B > E > F or C > B > H > F. However, we know E > H. So, the order must be C > B > E > F.
Now we have:
C > B > E > F
The only person left is H. Since E > H, and H must be shorter than E, H must be the shortest person overall to fit into the sequence without violating the conditions. If H was between E and F, then E > H > F, which would place H between E and F, but F is already the fourth tallest. If H was between B and E, then C > B > H > E > F, which would violate E > H. Therefore, H must be the shortest.
The complete order from tallest to shortest is:
C > B > E > F > H
Answer the question: "How many people have heights between H and B?"
Looking at the derived order C > B > E > F > H, the people between H and B are E and F.
There are two people (E and F) between H and B.
A) Two — As determined from the complete height order, E and F are the two people whose heights are between H and B.
B) One — This would imply only one person is between H and B, which is incorrect as both E and F are between them.
C) Three — This would imply three people are between H and B, which is incorrect.
D) Four — This would imply four people are between H and B, which is incorrect.
To solve this problem, we will use trigonometric identities, specifically the complementary angle identities and the Pythagorean identity. The key identities are:
We have \(\sin^2 63^\circ\). We know that \(63^\circ = 90^\circ - 27^\circ\). So, \(\sin 63^\circ = \sin(90^\circ - 27^\circ) = \cos 27^\circ\). Therefore, \(\sin^2 63^\circ = \cos^2 27^\circ\).
Similarly, for \(\cos^2 73^\circ\), we know that \(73^\circ = 90^\circ - 17^\circ\). So, \(\cos 73^\circ = \cos(90^\circ - 17^\circ) = \sin 17^\circ\). Therefore, \(\cos^2 73^\circ = \sin^2 17^\circ\).
The given expression is:
\[ \sin^2 63^\circ + \sin^2 27^\circ + \cos^2 17^\circ + \cos^2 73^\circ + \cos^2 90^\circ \]Substitute the rewritten terms:
\[ (\cos^2 27^\circ) + \sin^2 27^\circ + \cos^2 17^\circ + (\sin^2 17^\circ) + \cos^2 90^\circ \]Group the terms that form the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ (\cos^2 27^\circ + \sin^2 27^\circ) + (\cos^2 17^\circ + \sin^2 17^\circ) + \cos^2 90^\circ \]This simplifies to:
\[ 1 + 1 + \cos^2 90^\circ \]We know that \(\cos 90^\circ = 0\). Therefore, \(\cos^2 90^\circ = 0^2 = 0\).
Substitute the value of \(\cos^2 90^\circ\) back into the expression:
\[ 1 + 1 + 0 = 2 \]The value of the expression is 2.
B) 2
The question asks for the number of trusted volunteers who accompanied Mahatma Gandhi on his famous 'Salt March'. This is a factual question related to a significant event in Indian history, specifically the Dandi March.
Correct Option: D) 78
To solve this problem, we will first assume a base salary for B. Then, we will calculate A's salary based on the given percentage increase. Finally, we will determine the percentage by which B's salary is less than A's salary using the formula for percentage decrease.
Assume B's salary: Let B's salary be \(₹100\).
Calculate A's salary: A's salary is 72% more than B's salary. So, A's salary will be:
\[ \text{A's Salary} = \text{B's Salary} + 72\% \text{ of B's Salary} \] \[ \text{A's Salary} = 100 + \left(\frac{72}{100} \times 100\right) \] \[ \text{A's Salary} = 100 + 72 = ₹172 \]Calculate the difference between A's and B's salary: The difference is:
\[ \text{Difference} = \text{A's Salary} - \text{B's Salary} \] \[ \text{Difference} = 172 - 100 = ₹72 \]Calculate the percentage by which B's salary is less than A's salary: To find how much percent B's salary is less than A's salary, we use A's salary as the base for comparison.
\[ \text{Percentage Less} = \left(\frac{\text{Difference}}{\text{A's Salary}} \times 100\right)\% \] \[ \text{Percentage Less} = \left(\frac{72}{172} \times 100\right)\% \] \[ \text{Percentage Less} = \left(\frac{7200}{172}\right)\% \] \[ \text{Percentage Less} \approx 41.86046\% \]Round to two decimal places: Rounding 41.86046% to two decimal places gives 41.86%.
Correct Option: B) 41.86%
To identify the correct reactants for preparing bleaching powder, recall the industrial method for its production. Bleaching powder, chemically known as calcium hypochlorite, is formed by the reaction of chlorine gas with dry slaked lime.
Correct Option: C) Chlorine gas and dry calcium hydroxide
The question asks for the process that reduces the DNA content by half in reproductive cells (gametes) compared to non-reproductive cells (somatic cells). This reduction is crucial for sexual reproduction to maintain the species' chromosome number across generations.
Therefore, meiosis is the specific process responsible for reducing the DNA content by half in reproductive cells.
Correct Option: B) Meiosis is a specialized type of cell division that reduces the chromosome number by half, creating four haploid cells, each genetically distinct from the parent cell. This reduction is essential for the formation of gametes (reproductive cells) to maintain the species' chromosome number after fertilization.
The question asks to identify the meaning of 'chaitya' in the context of Buddhist architecture. This requires knowledge of common terms used in ancient Indian architecture, specifically those related to Buddhism.
C) A hall for prayer and assembly — A chaitya-griha (chaitya hall) is a Buddhist prayer hall or assembly hall. These structures are typically rock-cut and feature a long, rectangular hall with an apse at one end, where a stupa is placed for veneration. They are central to Buddhist worship and congregational activities.
This question asks for the product formed when ethanol reacts with acetic acid. This is a classic example of an esterification reaction, where an alcohol reacts with a carboxylic acid to form an ester and water.
Correct Option: B) ester
The question asks for the chemical name of baking soda, a common household item. This requires knowledge of common chemical names and their corresponding household names.
Correct Option: C) sodium bicarbonate is the chemical name for baking soda. It is also known as bicarbonate of soda.
The pH scale is used to specify the acidity or basicity of an aqueous solution. It is defined as the negative base-10 logarithm of the hydrogen ion activity. For dilute solutions, the activity is approximated by the concentration of hydrogen ions \([\text{H}^+]\). A neutral solution is characterized by an equal concentration of hydrogen ions \([\text{H}^+]\) and hydroxide ions \([\text{OH}^-]\).
The ion product of water, \(\text{K}_\text{w}\), at \(25^\circ\text{C}\) is \(1.0 \times 10^{-14}\). Thus,
\[ \text{K}_\text{w} = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \]If \([\text{H}^+] = [\text{OH}^-]\), let's denote this concentration as \(x\). Then,
\[ x \cdot x = 1.0 \times 10^{-14} \]\[ x^2 = 1.0 \times 10^{-14} \]\[ x = \sqrt{1.0 \times 10^{-14}} \]\[ x = 1.0 \times 10^{-7} \text{ M} \]So, for a neutral solution, \([\text{H}^+] = 1.0 \times 10^{-7} \text{ M}\).
Therefore, when a solution contains an equal number of \(\text{H}^+\) and \(\text{OH}^-\) ions, its pH is 7.
Correct Option: D) pH = 7. This is the definition of a neutral solution, where the concentrations of hydrogen ions and hydroxide ions are equal, leading to a pH of 7 at standard temperature.