If \(\sqrt{6}\)=2.45, then the value of \(\sqrt{6}\)+\(\sqrt{24}\)+ \(\sqrt{54}\) is:
The problem asks us to calculate the value of an expression involving square roots, given the approximate value of \(\sqrt{6}\). The key is to simplify the square root terms \(\sqrt{24}\) and \(\sqrt{54}\) so that they can be expressed in terms of \(\sqrt{6}\). Once all terms are in the form of a multiple of \(\sqrt{6}\), we can factor out \(\sqrt{6}\) and substitute its given value to find the final answer.
We need to find the largest perfect square factor of 24. We know that \(24 = 4 \times 6\). Since 4 is a perfect square (\(2^2\)), we can write:
\[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \]Similarly, for \(\sqrt{54}\), we find the largest perfect square factor of 54. We know that \(54 = 9 \times 6\). Since 9 is a perfect square (\(3^2\)), we can write:
\[ \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6} \]The given expression is \(\sqrt{6} + \sqrt{24} + \sqrt{54}\). Substituting the simplified forms:
\[ \sqrt{6} + 2\sqrt{6} + 3\sqrt{6} \]Since all terms now involve \(\sqrt{6}\), we can combine their coefficients:
\[ (1 + 2 + 3)\sqrt{6} = 6\sqrt{6} \]We are given that \(\sqrt{6} = 2.45\). Substitute this value into the combined expression:
\[ 6 \times 2.45 \]Perform the multiplication:
\[ 6 \times 2.45 = 14.70 \]Thus, the value of \(\sqrt{6} + \sqrt{24} + \sqrt{54}\) is 14.70.
D) 14.70 is the correct value calculated by simplifying the square roots and substituting the given approximation for \(\sqrt{6}\).
The question asks for the full form of the INSPIRE scheme, a flagship program of the Department of Science & Technology (DST), Government of India. This requires knowledge of the exact acronym as defined by the government.
D) Innovation in Science Pursuit for Inspired Research
This is the officially recognized full form of the INSPIRE scheme, which is a program of the Department of Science & Technology (DST), Government of India. The scheme aims to attract talented youth to study science and pursue research careers.
To find the energy consumed by the bulb, we need to use the relationship between power, energy, and time. Power is defined as the rate at which energy is consumed or transferred. The formula connecting these quantities is \( \text{Energy} = \text{Power} \times \text{Time} \).
The energy consumed (\(E\)) is the product of power (\(P\)) and time (\(t\)).
\[ E = P \times t \]Substitute \(P = 40 \text{ W}\) and \(t = 100 \text{ s}\) into the equation:
\[ E = 40 \text{ W} \times 100 \text{ s} \]Perform the multiplication:
\[ E = 4000 \text{ J} \]The SI unit for energy is Joules (J).
Correct Option: D) 4000 J
If the radius of a sphere is 7 cm, then the surface area (in cm²) of the sphere is ______. (Take π = \( \frac{22}{7} \))
To find the surface area of a sphere, we use the standard formula for the surface area of a sphere, which is \(4\pi r^2\), where \(r\) is the radius of the sphere. We are given the radius and the value of \(\pi\).
Correct Option: A) 616 is the calculated surface area of the sphere.
The question asks for the full form of the POCSO Act. This is a direct knowledge-based question related to Indian legislation concerning child protection. Recalling the exact name of the act is crucial.
Correct Option: C) The Protection of Children from Sexual Offences Act
This is the correct and officially recognized full name of the Indian legislation enacted in 2012 to address child sexual abuse.
In a queue, Manoj is 10th from the front end and Ajay is 12th from the rear end, and there is no one between them. How many people are there in the queue if Ajay is standing behind Manoj in the queue?
This problem involves determining the total number of people in a queue based on the positions of two individuals from opposite ends and the condition that there is no one between them, with one person standing behind the other.
C) 22 — As calculated, there are 22 people in the queue.
To solve this problem, we first need to calculate the sum of the squares of 15 and 16. Then, we find the nearest perfect square to this sum. The difference between the sum and the nearest perfect square will be the smallest natural number that needs to be added or subtracted to make it a perfect square.
Calculate the square of 15:
\[ 15^2 = 15 \times 15 = 225 \]Calculate the square of 16:
\[ 16^2 = 16 \times 16 = 256 \]Calculate the sum of the squares:
\[ \text{Sum} = 225 + 256 = 481 \]Find the nearest perfect square to 481:
We need to find integers whose squares are close to 481.
Let's try \(20^2 = 400\)
Let's try \(21^2 = 441\)
Let's try \(22^2 = 484\)
Let's try \(23^2 = 529\)
The perfect squares closest to 481 are 441 and 484.
Determine the smallest number to add or subtract:
Difference from 441: \(481 - 441 = 40\). This means we would subtract 40.
Difference from 484: \(484 - 481 = 3\). This means we would add 3.
The question asks for the smallest natural number to be added or subtracted. Comparing 40 and 3, the smallest is 3.
B) 3 is the smallest natural number that needs to be added to 481 to make it a perfect square (484).
A) 81 — This is the square of 9. It is not directly related to the calculation of making 481 a perfect square.
C) 40 — This is the number that would need to be subtracted from 481 to get 441 (\(481 - 40 = 441\)). While it makes 481 a perfect square, it is not the smallest natural number compared to 3.
D) 19 — This number is not relevant to making 481 a perfect square by addition or subtraction.
In a certain code language, ‘BOAT’ is written as ‘DQCV’ and ‘PLAN’ is written as ‘RNCP’. How will ‘PHONE’ be written in that code language?
The problem is a coding-decoding question. We need to identify the pattern used to transform the given words and then apply the same pattern to the target word.
The pattern observed here is that each letter in the word is shifted two positions forward in the English alphabet.
The pattern holds true for the second example as well. Each letter is shifted two positions forward.
So, 'PHONE' will be coded as 'RJQPG'.
Correct Option: B) RJQPG
The question asks about the risk coverage for accidental death under the Pradhan Mantri Suraksha Bima Yojana (PMSBY) as of July 2022. To answer this, one needs to recall the key features and benefits of this government-backed insurance scheme.
Correct Option: D) Rs 2 lakh is the correct risk coverage for accidental death under the Pradhan Mantri Suraksha Bima Yojana (PMSBY).
The question asks to identify the true statement about non-biodegradable substances. We need to recall the definition and characteristics of non-biodegradable materials to evaluate each option.
Correct Option: D) Non-biodegradable substances persist in the environment.