This problem can be solved by using the algebraic identity for the square of the sum of three terms: \((a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\). We are given the values of \((a+b+c)\) and \((a^2 + b^2 + c^2)\), and we need to find the value of \((ab + bc + ca)\).
The fundamental identity relating the sum of terms and the sum of their pairwise products is:
\[ (a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]We are given:
Substitute these values into the identity:
\[ (6)^2 = 14 + 2(ab + bc + ca) \]Calculate the square of 6:
\[ 36 = 14 + 2(ab + bc + ca) \]Subtract 14 from both sides of the equation:
\[ 36 - 14 = 2(ab + bc + ca) \] \[ 22 = 2(ab + bc + ca) \]Divide both sides by 2:
\[ \frac{22}{2} = ab + bc + ca \] \[ 11 = ab + bc + ca \]Thus, the value of \((ab + bc + ca)\) is 11.
B) 11
The question asks to identify which of the given options is classified as a dwarf planet. This requires knowledge of the classification of celestial bodies in our solar system, specifically the definition of a dwarf planet as established by the International Astronomical Union (IAU).
Correct Option: D) Pluto
Biogas is a mixture of gases produced by the anaerobic decomposition of organic matter. Understanding the composition of biogas is key to identifying its major component.
Correct Option: C) Methane is the major component of biogas, typically ranging from 50% to 75% of its volume, making it the primary combustible gas.
The question asks about the company that introduced Mac OS. This is a factual question related to the history of operating systems and technology companies. Recall the major players in the personal computing industry and their flagship products.
Correct Option: C) Apple
Apple Inc. is the company that developed and introduced Mac OS. The first version was released in 1984 with the original Macintosh computer.
The problem presents a series of letter groups, and we need to find the missing group. We will analyze the pattern within each group and the pattern between consecutive groups to determine the missing term.
So, the pattern within each group is: (2nd letter), (4th letter), (1st letter), (3rd letter), (5th letter) of a consecutive alphabetical sequence of five letters.
Correct Option: C) LNKMO
Find the least number required to be added to 3105 so that it is exactly divisible by 3, 4, 5 and 6.
To find the least number that needs to be added to 3105 so that it is exactly divisible by 3, 4, 5, and 6, we first need to find the Least Common Multiple (LCM) of 3, 4, 5, and 6. This LCM will be the smallest number that is divisible by all these numbers. Then, we find the remainder when 3105 is divided by this LCM. The difference between the LCM and this remainder will be the least number to be added.
Correct Option: D) 15
To find the new ratio of milk to water when two mixtures are combined, we need to calculate the total amount of milk and the total amount of water from both vessels. Since the vessels have equal volume, we can assume a convenient total volume for each vessel to simplify calculations.
Correct Option: A) 11:13
The question asks to identify the substance that extinguishes fire. Fire requires three elements to burn: fuel, oxygen, and heat (this is known as the "fire triangle"). To extinguish a fire, one or more of these elements must be removed or reduced. We need to find a gas that can effectively cut off the oxygen supply or cool down the burning material.
Correct Option: B) COâ‚‚ extinguishes fire because it is a non-combustible gas that is denser than air. When sprayed on a fire, it forms a blanket over the burning material, cutting off the supply of oxygen, which is necessary for combustion. It also helps to cool down the burning substance.
The question asks to identify the odd one out from four pairs of words. This is a classification-based reasoning question where we need to find the relationship between the words in each pair and identify the pair that does not follow the same relationship as the others.
Correct Option: C) Milky Way : Constellation — The Milky Way is a galaxy, not a constellation. This pair does not follow the correct classification relationship established by the other options.
The question asks for the name of the main branch of the Ganges River after it enters Bangladesh. This requires knowledge of the geographical names of rivers as they flow through different regions, specifically the Ganges in India and Bangladesh.
Correct Option: D) After entering Bangladesh, the main branch of the Ganges is known as the Padma river. This is the correct geographical designation for the Ganges in Bangladesh.