If G is the G.M. (Geometric Mean) of the product of r sets of observations with geometric means G1, G2, G3, …, Gr respectively, then find the value of G?
log G1 + log G2 + … + log Gr
G1⋅G2⋅…⋅Gr
G1 + G2 + … + Gr
log G1 × log G2 × … × log Gr
The question gives r sets of observations with geometric means G₁, G₂, ..., Gᵣ and asks for the geometric mean G of the product of these sets (each observation formed by multiplying the corresponding observations of the sets). We need to use the definition of the geometric mean and show that the G.M. of a product equals the product of the individual G.M.s.
Identify the given information:
Write the definition of the geometric mean:
For a set with observations x₁, x₂, ..., xₙ:
\[ G_1=(x_1x_2\cdots x_n)^{\frac{1}{n}} \]Find the G.M. of the product (shown for two sets x and y):
\[ G=\left[(x_1y_1)(x_2y_2)\cdots(x_ny_n)\right]^{\frac{1}{n}} \] \[ =(x_1x_2\cdots x_n)^{\frac{1}{n}}\times(y_1y_2\cdots y_n)^{\frac{1}{n}} \] \[ =G_1\times G_2 \]Extend to r sets:
\[ G=G_1\cdot G_2\cdot G_3\cdots G_r \]Therefore, the G.M. of the product equals the product of the individual G.M.s.
Correct Option: B) G1·G2·…·Gr — Because the nth root of a product equals the product of the nth roots, the geometric mean of the product of r sets is G₁ · G₂ · ... · Gᵣ. Hence, Option B is the correct answer.
A) log G1 + log G2 + … + log Gr: This is incorrect because this sum equals log G, not G itself.
C) G1 + G2 + … + Gr: This is incorrect because geometric means combine by multiplication for a product of sets, not by addition.
D) log G1 × log G2 × … × log Gr: This is incorrect because neither G nor log G is a product of logarithms; log G is the sum of the logarithms.
The given symbol represents ________ element?
The diagram provided represents the symbol for an element. Recognizing the symbol is key to identifying the correct element.
Correct Option: B) Hydrogen
The question asks to identify the author of the book 'One Indian Girl'. This is a direct factual recall question related to Indian literature and authors, common in general knowledge sections of competitive exams.
Correct Option: A) Chetan Bhagat
Chetan Bhagat is a popular Indian author, columnist, and screenwriter. He is known for his English-language novels, many of which have been adapted into Bollywood films. 'One Indian Girl' is one of his well-known novels, published in 2016. It tells the story of Radhika Mehta, an investment banker, and her struggles with love, career, and societal expectations.
The question asks to identify the recipients of the Whitley Awards 2017, also known as the Green Oscar Awards. This is a factual question related to current affairs and environmental conservation awards.
Correct Option: C) Sanjay Gubbi and Purnima Devi Barman were the Indian recipients of the Whitley Awards in 2017 for their significant contributions to wildlife conservation in India.
The question asks to identify the group in the Modern Periodic Table that contains elements with a completely filled valence shell and are chemically inactive. This describes the characteristics of noble gases.
Correct Option: B) 18 — Group 18 elements (noble gases) have a completely filled valence shell (duplet for He, octet for others) and are chemically inactive due to their stable electron configuration.
To find the speed of the stream, we first need to calculate the speed of the boat upstream. We are given the distance covered upstream and the time taken. Once we have the upstream speed, we can use the formula relating upstream speed, speed of the boat in still water, and speed of the stream.
Identify the given values:
Calculate the speed of the boat upstream (\(v_u\)):
The formula for speed is:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]So, the upstream speed is:
\[ v_u = \frac{D}{T} = \frac{38\ \text{km}}{4\ \text{h}} = 9.5\ \text{km/h} \]Relate upstream speed to boat speed and stream speed:
Let the speed of the stream be \(v_s\). The upstream speed is the difference between the speed of the boat in still water and the speed of the stream:
\[ v_u = v_b - v_s \]Solve for the speed of the stream (\(v_s\)):
Rearrange the formula to find \(v_s\):
\[ v_s = v_b - v_u \]Substitute the known values:
\[ v_s = 12\ \text{km/h} - 9.5\ \text{km/h} = 2.5\ \text{km/h} \]Correct Option: C) 2.5
This question tests the understanding of how the speed of light changes when it passes from one medium to another, specifically from a rarer medium to a denser medium. This phenomenon is governed by the refractive index of the media.
Correct Option: C) speeds down
The question asks to identify the sport Gurmeet Singh is associated with. This requires knowledge of prominent Indian athletes and their respective sports.
Gurmeet Singh is a well-known Indian athlete who has represented the country in various international competitions.
He is particularly recognized for his achievements in race walking.
He was the first Indian to win a gold medal in the Asian Race Walking Championships in 2012.
A) Race walk - Gurmeet Singh is an accomplished Indian race walker. He has won medals at international events, including the Asian Race Walking Championships.
The problem asks us to solve a quadratic equation for \(x\), with the constraint that \(x\) must be a natural number (\(x \in N\)). We will use algebraic manipulation to isolate \(x\).
Start with the given equation:
\[ (x - 2)^2 - 36 = 0 \]Add 36 to both sides of the equation:
\[ (x - 2)^2 = 36 \]Take the square root of both sides. Remember to consider both positive and negative roots:
\[ x - 2 = \pm \sqrt{36} \]\[ x - 2 = \pm 6 \]This gives us two possible cases:
Case 1: \(x - 2 = 6\)
Add 2 to both sides:
\[ x = 6 + 2 \]\[ x = 8 \]Case 2: \(x - 2 = -6\)
Add 2 to both sides:
\[ x = -6 + 2 \]\[ x = -4 \]Now, consider the condition given in the problem: \(x \in N\). Natural numbers (\(N\)) are positive integers \(\{1, 2, 3, \dots\}\).
From Case 1, \(x = 8\). Since 8 is a positive integer, it is a natural number.
From Case 2, \(x = -4\). Since -4 is a negative integer, it is not a natural number.
Therefore, the only valid solution for \(x\) that satisfies the given condition is \(x = 8\).
Correct Option: D) 8
The value \(x=8\) satisfies the equation and the condition that \(x\) is a natural number.
The question asks to identify the type of tissue our bones are. This requires knowledge of the different types of animal tissues and their classifications.
Correct Option: D) Connective
Bones are a prime example of specialized connective tissue. Connective tissues are diverse and include bone, cartilage, blood, adipose tissue, and fibrous connective tissue. They all share the common characteristic of having cells embedded in an extracellular matrix. In the case of bone, this matrix is hard and calcified, providing strength and rigidity.