Find the value of n in the given equation.
271.5 × 91.25 ÷ 35 = 3n
To find the value of \(n\), we need to simplify the given equation by expressing all terms with the same base. In this case, the base will be 3, as 27 and 9 are powers of 3. Then, we will use the laws of exponents to combine the terms and solve for \(n\).
Express all numbers as powers of 3:
We know that \(27 = 3^3\) and \(9 = 3^2\).
Substitute these into the equation:
\[ (3^3)^{1.5} \times (3^2)^{1.25} \div 3^5 = 3^n \]Apply the power of a power rule \((a^m)^p = a^{m \times p}\):
For the first term:
\[ (3^3)^{1.5} = 3^{3 \times 1.5} = 3^{4.5} \]For the second term:
\[ (3^2)^{1.25} = 3^{2 \times 1.25} = 3^{2.5} \]Substitute these back into the equation:
\[ 3^{4.5} \times 3^{2.5} \div 3^5 = 3^n \]Apply the product rule \(a^m \times a^p = a^{m+p}\) and the quotient rule \(a^m \div a^p = a^{m-p}\):
First, combine the multiplication:
\[ 3^{4.5} \times 3^{2.5} = 3^{4.5 + 2.5} = 3^7 \]Now, apply the division:
\[ 3^7 \div 3^5 = 3^{7-5} = 3^2 \]Equate the result to \(3^n\):
So, we have:
\[ 3^2 = 3^n \]Solve for \(n\):
Since the bases are the same, the exponents must be equal.
\[ n = 2 \]D) 2 β By converting all terms to the base 3 and applying the laws of exponents, we found that \(n=2\).
The question asks for the year when the demand for Pakistan was officially made at the Lahore Session of the Muslim League. This is a direct factual question related to Indian history, specifically the Pakistan Movement.
The demand for Pakistan was officially made at the Lahore Session of the Muslim League in 1940 through the adoption of the Lahore Resolution.
To find the ratio of the areas, we first need to establish a relationship between the side of the square and the radius of the circle using the given condition that their perimeters are equal. Then, we will calculate the areas of both shapes and find their ratio.
Let the side of the square be \(s\).
Let the radius of the circle be \(r\).
The perimeter of the square is \(P_{\text{square}} = 4s\).
The circumference of the circle is \(C_{\text{circle}} = 2\pi r\).
Given that the perimeter of the square and the circumference of the circle are the same:
\[ 4s = 2\pi r \]From the equality, we can express \(s\) in terms of \(r\):
\[ s = \frac{2\pi r}{4} = \frac{\pi r}{2} \]The area of the square is \(A_{\text{square}} = s^2\).
The area of the circle is \(A_{\text{circle}} = \pi r^2\).
Substitute the expression for \(s\) into the area of the square formula:
\[ A_{\text{square}} = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4} \]The ratio is \(\frac{A_{\text{square}}}{A_{\text{circle}}}\):
\[ \frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{\frac{\pi^2 r^2}{4}}{\pi r^2} \]\[ \frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{\pi^2 r^2}{4} \times \frac{1}{\pi r^2} \]\[ \frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{\pi^2}{4\pi} \]\[ \frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{\pi}{4} \]Therefore, the ratio of the area of the square to that of the circle is \(\pi : 4\).
Correct Option: A) \(\pi : 4\)
The question asks to identify the technology that provides high-speed internet access over traditional modem lines. We need to evaluate each option based on its function and how it relates to internet access over existing telephone infrastructure.
Correct Option: A) ADSL (Asymmetric Digital Subscriber Line) is a technology that enables high-speed data transmission over existing copper telephone lines. It allows for simultaneous voice communication and internet access by using different frequency bands for each, offering significantly faster speeds than traditional dial-up modems.
The question asks to identify the largest producer of crude oil and natural gas in India as of 2021. This requires knowledge of major public sector undertakings (PSUs) in India's energy sector.
B) ONGC β Oil and Natural Gas Corporation (ONGC) is a Maharatna Public Sector Undertaking (PSU) of the Government of India. It is India's largest crude oil and natural gas company, contributing around 71% of domestic production. It is engaged in exploration, development, and production of crude oil and natural gas, both in India and internationally.
The question asks about the individual who presides over the very first meeting of a newly constituted Lok Sabha. This involves understanding the initial procedures and roles of various constitutional functionaries in the Indian parliamentary system.
Correct Option: C) Protem Speaker β The Protem Speaker is appointed by the President specifically to administer the oath to the newly elected members and to preside over the first sitting of the Lok Sabha until a permanent Speaker is elected.
The standard equation of a circle centered at the origin is given by \( x^2 + y^2 = r^2 \), where \( r \) is the radius. We are given the equation of a circle and a point through which it passes. To find the value of 'a' (which represents the radius in this case), we can substitute the coordinates of the given point into the circle's equation.
\( (0)^{2} + (4)^{2} = a^{2} \)
\( 0 + 16 = a^{2} \)
\( 16 = a^{2} \)
\( a = \sqrt{16} \)
\( a = 4 \)
Since 'a' represents the radius of the circle, it must be a positive value.B) The value of 'a' is 4.
The question asks to identify one of the founders of the Red Cross society. This requires knowledge of the history and key figures associated with humanitarian organizations.
A) Jean Henry Dunant is widely recognized as the principal founder of the Red Cross movement and the recipient of the first Nobel Peace Prize along with FrΓ©dΓ©ric Passy in 1901 for his role in founding the ICRC.
The question asks to identify the scientist responsible for introducing the system of scientific naming of organisms. This refers to the binomial nomenclature system, a fundamental concept in biology and taxonomy.
A) Carolus Linnaeus
Carolus Linnaeus, a Swedish botanist, zoologist, and physician, is widely regarded as the "father of modern taxonomy." He formalized binomial nomenclature, the modern system of naming organisms, in his work Species Plantarum (1753) and Systema Naturae (10th edition, 1758). This system assigns each species a unique two-part name, consisting of the genus and species epithet, which is universally recognized and helps in unambiguous communication among scientists worldwide.
The value of \(\frac{1+\tan^2\theta}{1+\cot^2\theta}\) is:
\( \tan ^{2}\theta \)
\( \cot ^{2}\theta \)
\( \sec ^{2}\theta \)
\( \operatorname{cosec} ^{2}\theta \)
To simplify the given trigonometric expression, we will use fundamental trigonometric identities that relate tangent, cotangent, secant, and cosecant functions.
Correct Option: A) \( \tan ^{2}\theta \)