The question asks to identify the specific region of Earth where life exists. This requires knowledge of Earth's major spheres and their definitions.
A) Biosphere is the correct answer because it is defined as the sum of all ecosystems, representing the zone of life on Earth. It includes parts of the atmosphere, hydrosphere, and lithosphere where living organisms are found.
The question asks to identify the award that is exclusively associated with music. We need to evaluate each option to determine its primary field of recognition.
Based on this evaluation, only the Grammy Award is exclusively associated with music.
Correct Option: A) Grammy is exclusively associated with achievements in the music industry.
This is a coding-decoding question where letters are assigned numerical values based on their alphabetical position. We need to identify the pattern used to convert the given words (AMBER, BROWN) into numbers and then apply the same pattern to find the numerical value for GREEN.
Correct Option: B) 36
The question asks about the first person to attempt to calculate national income in India. This is a factual question related to the economic history of India, specifically focusing on early estimations of national income.
Correct Option: D) Dadabhai Naoroji was the first to make an attempt to calculate national income in India. He estimated India's national income for the year 1867-68 and published it in his book "Poverty and Un-British Rule in India."
The question asks to identify the gas law that describes the inverse relationship between the volume and pressure of a gas at a constant temperature. We need to recall the definitions of the given gas laws.
Correct Option: C) Boyle's Law
This is a syllogism problem. We need to analyze the given statements using Venn diagrams or logical deductions to find which conclusion logically follows. We will represent each category (Pilots, Amateurs, Players) as sets and determine their relationships based on the given statements.
Correct Option: A) Some amateurs are not players
The question asks for the location of the headquarters of the United Nations Security Council. This is a factual question related to international organizations and their administrative centers. Knowledge of the United Nations' structure and main offices is key to answering this.
Correct Option: C) New York — The headquarters of the United Nations, including the United Nations Security Council, is located in New York City, USA. The UN Headquarters complex in Manhattan is an internationally recognized territory.
The question asks about the administrative control of Jan Shikshan Sansthan (JSS). To answer this, one needs to know which government ministry is responsible for skill development and vocational training initiatives in India, as JSS primarily focuses on these areas.
Correct Option: C) Jan Shikshan Sansthan (JSS) operates under the aegis of the Ministry of Skill Development and Entrepreneurship. This ministry is responsible for coordinating and implementing skill development initiatives across India, which perfectly aligns with the mission of JSS to provide vocational training and skills to the underprivileged.
If x = a sin t, y = b tan t, then \(\frac{a^2}{x^2}\) - \(\frac{b^2}{y^2}\) is:
The problem asks us to evaluate the expression \[\frac{a^2}{x^2} - \frac{b^2}{y^2}\] given the parametric equations \(x = a \sin t\) and \(y = b \tan t\). We will substitute the given expressions for \(x\) and \(y\) into the expression and simplify using trigonometric identities.
Given the equations:
\[ x = a \sin t \]\[ y = b \tan t \]From the first equation, we can find \(\frac{a}{x}\):
\[ \frac{x}{a} = \sin t \]\[ \frac{a}{x} = \frac{1}{\sin t} = \csc t \]Square both sides of the expression for \(\frac{a}{x}\):
\[ \frac{a^2}{x^2} = \csc^2 t \]From the second equation, we can find \(\frac{b}{y}\):
\[ \frac{y}{b} = \tan t \]\[ \frac{b}{y} = \frac{1}{\tan t} = \cot t \]Square both sides of the expression for \(\frac{b}{y}\):
\[ \frac{b^2}{y^2} = \cot^2 t \]Now, substitute these squared expressions into the given expression \[\frac{a^2}{x^2} - \frac{b^2}{y^2}\]:
\[ \frac{a^2}{x^2} - \frac{b^2}{y^2} = \csc^2 t - \cot^2 t \]Recall the fundamental trigonometric identity:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]Applying this identity, we get:
\[ \csc^2 t - \cot^2 t = 1 \]Therefore, the value of the expression is 1.
D) 1 — By substituting the given parametric equations into the expression and using the trigonometric identity \(\csc^2 t - \cot^2 t = 1\), we find the value of the expression to be 1.
The problem requires us to find the value of \(\theta\) given the trigonometric equation \(\cos(2\theta) = \sin(\theta)\) and the condition that \(\theta\) lies between 0° and 90°. We will use trigonometric identities to transform the equation into a solvable form, typically involving a single trigonometric function or a quadratic equation.
Given the equation:
\[ \cos(2\theta) = \sin(\theta) \]We know the double angle identity for cosine: \(\cos(2\theta) = 1 - 2\sin^2(\theta)\). Substitute this into the equation:
\[ 1 - 2\sin^2(\theta) = \sin(\theta) \]Rearrange the equation to form a quadratic equation in terms of \(\sin(\theta)\):
\[ 2\sin^2(\theta) + \sin(\theta) - 1 = 0 \]Let \(x = \sin(\theta)\). The equation becomes:
\[ 2x^2 + x - 1 = 0 \]Solve this quadratic equation for \(x\) using factorization or the quadratic formula. By factorization:
\[ 2x^2 + 2x - x - 1 = 0 \]Group terms:
\[ 2x(x + 1) - 1(x + 1) = 0 \]Factor out \((x + 1)\):
\[ (2x - 1)(x + 1) = 0 \]This gives two possible values for \(x\):
\[ 2x - 1 = 0 \implies x = \frac{1}{2} \]or
\[ x + 1 = 0 \implies x = -1 \]Substitute back \(x = \sin(\theta)\):
\[ \sin(\theta) = \frac{1}{2} \]or
\[ \sin(\theta) = -1 \]Consider the given range for \(\theta\): \(0° < \theta < 90°\). In this range, \(\sin(\theta)\) must be positive. Therefore, \(\sin(\theta) = -1\) is not a valid solution.
For \(\sin(\theta) = \frac{1}{2}\) in the range \(0° < \theta < 90°\), the only solution is:
\[ \theta = 30° \]C) 30° is the correct answer because it satisfies the equation \(\cos(2\theta) = \sin(\theta)\) and lies within the specified range of 0° to 90°.