We need to find the least number that satisfies two conditions: first, it leaves a remainder of 3 when divided by 8, 12, and 16; and second, it leaves no remainder (i.e., is perfectly divisible) when divided by 7. This problem involves finding the Least Common Multiple (LCM) and then applying the remainder conditions.
First, find the prime factorization of each number:
The LCM is the product of the highest powers of all prime factors involved:
\[ \text{LCM}(8, 12, 16) = 2^4 \times 3^1 = 16 \times 3 = 48 \]The number, when divided by 8, 12, and 16, leaves a remainder of 3. This means the number can be expressed in the form:
\[ N = \text{LCM}(8, 12, 16) \times k + 3 \]\[ N = 48k + 3 \]where \(k\) is a positive integer.The number \(N\) must be perfectly divisible by 7. So, \(48k + 3\) must be divisible by 7.
We can test values of \(k\) starting from 1:
Since we are looking for the least number, \(k=3\) gives us the smallest such number.
The number is 147.
All conditions are met.
C) 147 is the least number that satisfies both conditions.
The question asks for the launch year of the Nano Mission (Mission on Nano Science and Technology) in India. This requires knowledge of specific government initiatives and their timelines.
Correct Option: B) 2007
The Nano Mission was launched by the Government of India in May 2007. Its primary objective was to support basic research, infrastructure development, human resource development, and international collaboration in the field of Nanoscience and Nanotechnology.
The question asks about the development of India's first beam weapon, KALI-5000. To answer this, one needs to know which Indian scientific organizations were involved in its creation.
Correct Option: A) DRDO & BARC. The KALI-5000 project, India's first beam weapon, was a collaborative effort between the Defence Research and Development Organisation (DRDO) and the Bhabha Atomic Research Centre (BARC).
To find the area of a rhombus, we need the lengths of both its diagonals. We are given the side length and one diagonal. We can use the property that the diagonals of a rhombus bisect each other at right angles to find the length of the second diagonal using the Pythagorean theorem. Once both diagonals are known, the area can be calculated using the formula: Area \( = \frac{1}{2} \times d_1 \times d_2 \).
Let the side of the rhombus be \(a = 17\) cm.
Let one diagonal be \(d_1 = 16\) cm.
The diagonals of a rhombus bisect each other at right angles. This means that half of each diagonal forms the legs of a right-angled triangle, with the side of the rhombus as the hypotenuse.
Half of the given diagonal \(d_1\) is \[\frac{d_1}{2} = \frac{16}{2} = 8\] cm.
Let the other diagonal be \(d_2\). Half of the other diagonal is \( \frac{d_2}{2} \).
Using the Pythagorean theorem in one of the right-angled triangles formed by the diagonals and a side:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \]Substitute the known values:
\[ (8)^2 + \left(\frac{d_2}{2}\right)^2 = (17)^2 \]\[ 64 + \left(\frac{d_2}{2}\right)^2 = 289 \]Solve for \( \left(\frac{d_2}{2}\right)^2 \):
\[ \left(\frac{d_2}{2}\right)^2 = 289 - 64 \]\[ \left(\frac{d_2}{2}\right)^2 = 225 \]Take the square root of both sides to find \( \frac{d_2}{2} \):
\[ \frac{d_2}{2} = \sqrt{225} \]\[ \frac{d_2}{2} = 15 \text{ cm} \]Now, find the full length of the second diagonal \(d_2\):
\[ d_2 = 2 \times 15 = 30 \text{ cm} \]Calculate the area of the rhombus using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]\[ \text{Area} = \frac{1}{2} \times 16 \times 30 \]\[ \text{Area} = 8 \times 30 \]\[ \text{Area} = 240 \text{ cm}^2 \]B) 240 cm² is the correct area of the rhombus.
We are given the ratio of Laxman's and Gopal's salaries for the last year. We are also given the ratio of individual salaries from last year to the present year for both Laxman and Gopal. Using these ratios, we can find the ratio of their present salaries. Finally, given their total present salaries, we can calculate Laxman's present salary.
Define variables for last year's salaries:
Let Laxman's salary last year be \(L_1\) and Gopal's salary last year be \(G_1\).
Given, the ratio of Laxman's salary to Gopal's salary last year was 3 : 5.
\[ \frac{L_1}{G_1} = \frac{3}{5} \]Express last year's salaries in terms of a common variable:
From the ratio, we can write:
\[ L_1 = 3x \]\[ G_1 = 5x \]Define variables for present year's salaries:
Let Laxman's salary this year be \(L_2\) and Gopal's salary this year be \(G_2\).
Use the individual salary ratios to find present salaries:
For Laxman, the ratio of last year's salary to present year's salary is 2 : 3.
\[ \frac{L_1}{L_2} = \frac{2}{3} \]\[ L_2 = \frac{3}{2} L_1 \]Substitute \(L_1 = 3x\) into the equation for \(L_2\):
\[ L_2 = \frac{3}{2} (3x) = \frac{9x}{2} \]For Gopal, the ratio of last year's salary to present year's salary is 4 : 5.
\[ \frac{G_1}{G_2} = \frac{4}{5} \]\[ G_2 = \frac{5}{4} G_1 \]Substitute \(G_1 = 5x\) into the equation for \(G_2\):
\[ G_2 = \frac{5}{4} (5x) = \frac{25x}{4} \]Find the ratio of their present salaries:
\[ \frac{L_2}{G_2} = \frac{\frac{9x}{2}}{\frac{25x}{4}} = \frac{9x}{2} \times \frac{4}{25x} = \frac{9 \times 2}{25} = \frac{18}{25} \]Calculate the total present salary:
Given that their total salaries for the present year are ₹4,300.
\[ L_2 + G_2 = 4300 \]From the ratio of present salaries, we can write:
\[ L_2 = 18k \]\[ G_2 = 25k \]Substitute these into the total salary equation:
\[ 18k + 25k = 4300 \]\[ 43k = 4300 \]\[ k = \frac{4300}{43} = 100 \]Find Laxman's present salary:
Laxman's present salary \(L_2 = 18k\).
\[ L_2 = 18 \times 100 = 1800 \]Thus, Laxman's present salary is ₹1,800.
Correct Option: A) ₹1,800
The question asks about the eligibility criteria for the Chairman of the National Human Rights Commission (NHRC) of India. This requires knowledge of the Protection of Human Rights Act, 1993, which governs the establishment and functioning of the NHRC.
Correct Option: D) a retired Chief Justice of India
As of 31st October 2020, The International Monetary Fund (IMF) has _____ member countries.
The question asks for the number of member countries in the International Monetary Fund (IMF) as of a specific date, October 31, 2020. This requires knowledge of the IMF's membership status around that time.
C) 190 — As of October 31, 2020, the International Monetary Fund (IMF) had 190 member countries. Andorra joined as the 190th member on October 16, 2020.
The question asks to identify which of the given options is NOT a part of the United Nations (UN). To answer this, one needs to know the principal organs of the UN.
Correct Option: A) The State Council is not a part of the United Nations. It is a governmental body found in certain countries, such as China, but not an organ of the UN.
The question asks to identify the first female jawan in the Indian Army who joined the 969 Railway Engineer Regiment of the Territorial Army (TA) in 2011. This requires knowledge of significant milestones and personalities in the Indian armed forces.
Sapper Shanti Tigga was indeed the first female jawan in the Indian Army. She joined the 969 Railway Engineer Regiment of the Territorial Army (TA) in 2011. Her achievement marked a significant step for women in the Indian armed forces, as she was the first woman to be recruited as a jawan, a rank traditionally reserved for men.
B) Sapper Shanti Tigga is the correct answer. She was the first female jawan in the Indian Army and joined the 969 Railway Engineer Regiment of the Territorial Army (TA) in 2011.
The question asks for the full form of NCSM, an autonomous organization under the Ministry of Culture. This requires knowledge of common acronyms related to Indian government bodies and cultural institutions.
Correct Option: D) National Council of Science Museums