Study the given pattern carefully and select the number from among the given options that can replace the question mark (?).
The problem involves identifying a pattern in a series of circles, each divided into four sections with numbers. The goal is to determine the number that replaces the question mark in the last circle.
Correct Option: C) 10
This problem involves understanding the definition of supplementary angles and setting up an algebraic equation based on the given relationship between an angle and its supplement. We will define the angle as a variable, express its supplementary angle in terms of that variable, and then use the given condition to solve for the variable.
Correct Option: C) 60°
If cosθ + secθ = \(\sqrt{3}\), then the value of \( \cos ^{3}\theta + \sec ^{3}\theta \) is:
2\(\sqrt{3}\)
\(\frac{1}{\sqrt{3}}\)
3\(\sqrt{3}\)
0
The problem asks for the value of \( \cos^3\theta + \sec^3\theta \) given that \( \cos\theta + \sec\theta = \sqrt{3} \). We can use the algebraic identity for the sum of cubes, \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \) or \( a^3 + b^3 = (a+b)^3 - 3ab(a+b) \). Since \( \sec\theta = \frac{1}{\cos\theta} \), the product \( \cos\theta \cdot \sec\theta \) simplifies to 1, which will be very useful.
Given the equation:
\[ \cos\theta + \sec\theta = \sqrt{3} \]We need to find the value of \( \cos^3\theta + \sec^3\theta \). Let \( a = \cos\theta \) and \( b = \sec\theta \). The expression becomes \( a^3 + b^3 \).
Use the algebraic identity for the sum of cubes: \( a^3 + b^3 = (a+b)^3 - 3ab(a+b) \).
Substitute \( a = \cos\theta \) and \( b = \sec\theta \) into the identity:
\[ \cos^3\theta + \sec^3\theta = (\cos\theta + \sec\theta)^3 - 3(\cos\theta \cdot \sec\theta)(\cos\theta + \sec\theta) \]We know that \( \sec\theta = \frac{1}{\cos\theta} \), so the product \( \cos\theta \cdot \sec\theta \) simplifies to:
\[ \cos\theta \cdot \sec\theta = \cos\theta \cdot \frac{1}{\cos\theta} = 1 \]Substitute the given value \( \cos\theta + \sec\theta = \sqrt{3} \) and the product \( \cos\theta \cdot \sec\theta = 1 \) into the equation from step 4:
\[ \cos^3\theta + \sec^3\theta = (\sqrt{3})^3 - 3(1)(\sqrt{3}) \]Calculate \( (\sqrt{3})^3 \):
\[ (\sqrt{3})^3 = \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = 3\sqrt{3} \]Substitute this back into the equation:
\[ \cos^3\theta + \sec^3\theta = 3\sqrt{3} - 3\sqrt{3} \]Perform the subtraction:
\[ \cos^3\theta + \sec^3\theta = 0 \]D) 0
To find the number of factors of a given number that satisfy certain divisibility conditions, we first express the conditions in terms of prime factorization. Then, we determine the possible powers for each prime factor based on these conditions and calculate the number of combinations.
The given number is \(N = 2^3 \times 3^3 \times 5^4 \times 7^2\).
A factor \(F\) of \(N\) can be written in the form \(F = 2^a \times 3^b \times 5^c \times 7^d\), where \(0 \le a \le 3\), \(0 \le b \le 3\), \(0 \le c \le 4\), and \(0 \le d \le 2\).
The condition "divisible by 50" means the factor \(F\) must be a multiple of \(50 = 2^1 \times 5^2\). This implies:
The condition "not divisible by 100" means the factor \(F\) must not be a multiple of \(100 = 2^2 \times 5^2\). This implies that it's NOT true that (\(a \ge 2\) AND \(c \ge 2\)).
Combining the conditions:
Let's analyze the possible values for \(a\) and \(c\) that satisfy \(1 \le a \le 3\) and \(2 \le c \le 4\):
Now, apply the "not divisible by 100" condition. This means we must exclude factors where \(a \ge 2\) AND \(c \ge 2\).
Therefore, the only allowed value for \(a\) is \(a=1\). The allowed values for \(c\) are \(2, 3, 4\).
The number of choices for each exponent are:
The total number of such factors is the product of the number of choices for each exponent:
\[ \text{Number of factors} = 1 \times 4 \times 3 \times 3 = 36 \]To maintain the same area of a rectangle when its length is increased, the width must be decreased proportionally. We will use the formula for the area of a rectangle and set the original area equal to the new area to find the required percentage of the original width.
Let the original length of the rectangle be \(L\) and the original width be \(W\).
The original area of the rectangle is given by:
\[ \text{Area}_{\text{original}} = L \times W \]The length is increased by 70%. So, the new length (\(L'\)) will be:
\[ L' = L + 0.70L = 1.70L \]Let the new width be \(W'\). We need to find \(W'\) such that the new area is equal to the original area.
The new area of the rectangle is:
\[ \text{Area}_{\text{new}} = L' \times W' = 1.70L \times W' \]To maintain the same area, we set the original area equal to the new area:
\[ L \times W = 1.70L \times W' \]Now, we solve for \(W'\) in terms of \(W\):
\[ W' = \frac{L \times W}{1.70L} \]\[ W' = \frac{W}{1.70} \]To express \(W'\) as a percentage of the original width \(W\), we calculate \(\frac{W'}{W} \times 100\%\):
\[ \frac{W'}{W} = \frac{1}{1.70} \]\[ \text{Percentage of original width} = \frac{1}{1.70} \times 100\% \]\[ \text{Percentage of original width} \approx 0.588235 \times 100\% \]\[ \text{Percentage of original width} \approx 58.8235\% \]Rounding to the nearest integer, the new width should be approximately 59% of the original width.
C) 59% is the correct answer. The calculation shows that the new width should be approximately 58.82% of the original width, which rounds to 59%.
The question asks to identify the gas produced during the process of photosynthesis. To answer this, recall the overall chemical equation for photosynthesis and the reactants and products involved.
Correct Option: B) Oxygen is the gas produced during the process of photosynthesis. It is released into the atmosphere as a byproduct, which is essential for the respiration of most living organisms.
The question asks to identify the Chief Election Commissioner of India as of October 2020. This requires knowledge of the individuals holding key constitutional positions during that specific period.
As of October 2020, Sunil Arora was serving as the Chief Election Commissioner of India. He assumed office on December 2, 2018, and demitted office on April 12, 2021.
B) Sunil Arora was the Chief Election Commissioner of India as of October 2020.
The question asks to identify the author of the famous novel 'A Passage to India'. This is a direct factual recall question related to English literature, often covered in general knowledge sections of competitive exams.
Correct Option: C) EM Forster wrote the novel 'A Passage to India'. Published in 1924, the novel explores the relationship between Indians and the British during the British Raj and addresses themes of colonialism, race, and friendship. It is considered one of his most significant works.
To answer this question, we need to recall the historical facts regarding the drafting and adoption of the Constitution of India. The process involved the Constituent Assembly, which took a specific duration from its first sitting to the final adoption of the Constitution.
Correct Option: C) 2 years, 11 months and 17 days
The question asks for the establishment year of the National Green Tribunal (NGT) by the Government of India. This is a factual question related to environmental governance in India.
Correct Option: D) 2010