The question asks for the definition of an atomic number. We need to recall the basic structure of an atom and the role of its subatomic particles in defining an element's identity.
Correct Option: D) protons
The atomic number of an element is defined as the number of protons in the nucleus of each atom of that element. This number uniquely identifies an element.
When individuals work together to complete a task and their daily wage is the same, their share of the total earnings is proportional to the amount of work they do. The amount of work done by each person is inversely proportional to the number of days they take to complete the task alone. Therefore, we first find the work rate of each person, then the ratio of their work rates, and finally distribute the total earnings according to this ratio.
Since they work together and the daily wage is the same, their share of earnings will be in the ratio of their daily work rates (or the inverse ratio of days taken).
Ratio of work done by Imran : Tahir : Amla \[= \frac{1}{11} : \frac{1}{15} : \frac{1}{13}\]
To convert this ratio to whole numbers, find the Least Common Multiple (LCM) of 11, 15, and 13. Since 11, 15, and 13 are pairwise coprime, their LCM is their product:
\[ \text{LCM}(11, 15, 13) = 11 \times 15 \times 13 = 165 \times 13 = 2145 \]Multiply each fraction in the ratio by the LCM:
\[ \text{Imran} : \text{Tahir} : \text{Amla} = \left(\frac{1}{11} \times 2145\right) : \left(\frac{1}{15} \times 2145\right) : \left(\frac{1}{13} \times 2145\right) \] \[ \text{Imran} : \text{Tahir} : \text{Amla} = 195 : 143 : 165 \]The total earnings are ₹2,012.
\[ \text{Imran's share} = \left(\frac{\text{Imran's ratio part}}{\text{Total ratio parts}}\right) \times \text{Total earnings} \] \[ \text{Imran's share} = \left(\frac{195}{503}\right) \times 2012 \]Notice that \(2012 = 4 \times 503\).
\[ \text{Imran's share} = 195 \times 4 = \text{₹}780 \]Correct Option: A) ₹780
If x⁴ + \( \frac{1}{x^4} \) = 322 and x > 1, then what is the value of x - \( \frac{1}{x} \)?
The problem requires us to find the value of \(x - \frac{1}{x}\) given \(x^4 + \frac{1}{x^4} = 322\) and \(x > 1\). We can use algebraic identities to work backward from \(x^4 + \frac{1}{x^4}\) to \(x - \frac{1}{x}\).
We are given the equation:
\[ x^4 + \frac{1}{x^4} = 322 \]Recall the identity \((a^2 + b^2)^2 = a^4 + b^4 + 2a^2b^2\). Let \(a = x\) and \(b = \frac{1}{x}\). Then \(a^2 = x^2\) and \(b^2 = \frac{1}{x^2}\). So, \(a^2b^2 = x^2 \cdot \frac{1}{x^2} = 1\).
Applying this, we have:
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + \frac{1}{x^4} + 2 \cdot x^2 \cdot \frac{1}{x^2} \]\[ \left(x^2 + \frac{1}{x^2}\right)^2 = x^4 + \frac{1}{x^4} + 2 \]Substitute the given value of \(x^4 + \frac{1}{x^4}\) into the equation:
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 322 + 2 \]\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 324 \]Take the square root of both sides:
\[ x^2 + \frac{1}{x^2} = \pm\sqrt{324} \]\[ x^2 + \frac{1}{x^2} = \pm 18 \]Since \(x > 1\), \(x^2\) is positive and \( \frac{1}{x^2}\) is positive. Therefore, \(x^2 + \frac{1}{x^2}\) must be positive.
\[ x^2 + \frac{1}{x^2} = 18 \]Now, we need to find \(x - \frac{1}{x}\). Recall the identity \((a - b)^2 = a^2 + b^2 - 2ab\). Let \(a = x\) and \(b = \frac{1}{x}\). Then \(ab = x \cdot \frac{1}{x} = 1\).
Applying this, we have:
\[ \left(x - \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} - 2 \cdot x \cdot \frac{1}{x} \]\[ \left(x - \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} - 2 \]Substitute the value of \(x^2 + \frac{1}{x^2} = 18\):
\[ \left(x - \frac{1}{x}\right)^2 = 18 - 2 \]\[ \left(x - \frac{1}{x}\right)^2 = 16 \]Take the square root of both sides:
\[ x - \frac{1}{x} = \pm\sqrt{16} \]\[ x - \frac{1}{x} = \pm 4 \]We are given that \(x > 1\). If \(x > 1\), then \( \frac{1}{x}\) will be less than 1 (and positive). Thus, \(x - \frac{1}{x}\) must be positive.
For example, if \(x=2\), then \[x - \frac{1}{x} = 2 - \frac{1}{2} = 1.5\], which is positive. If \(x=0.5\), then \(x - \frac{1}{x} = 0.5 - 2 = -1.5\), which is negative. Since \(x > 1\), \(x - \frac{1}{x}\) must be positive.
Therefore,
\[ x - \frac{1}{x} = 4 \]Correct Option: C) 4
The question asks about the standard storage capacity of a 3.5-inch floppy disk. This is a common piece of general knowledge related to computer hardware history, often tested in competitive exams.
Correct Option: B) 1.44 MB is the standard storage capacity of a high-density 3.5-inch floppy disk. While earlier double-density (DD) versions existed with lower capacities (e.g., 720 KB), the 1.44 MB HD disk was the most common and is generally what is referred to when speaking of a "3.5-inch floppy disk" without further qualification.
The question asks to identify which of the given animals belongs to the class Amphibia. To answer this, one needs to recall the characteristics of amphibians and classify each option accordingly.
D) Toad — Toads are classic examples of amphibians, belonging to the order Anura within the class Amphibia. They undergo metamorphosis from an aquatic larval stage to a terrestrial adult stage.
The question asks to identify the name of India's research station established in Antarctica in 2012. This is a factual question related to India's scientific missions in polar regions.
Correct Option: B) Bharati is India's third permanent research station in Antarctica, commissioned in 2012. It is located between the Thala Fjord and Quilty Bay, east of Stornes in the Larsemann Hills, Antarctica.
The question asks about the primary objective of the "MOBILISE YOUR CITY" (MYC) agreement signed between India and France. To answer this, one needs to recall the core focus of the MYC initiative.
C) Green House Gas (GHG) emissions. The MYC initiative, supported by France, aims to reduce Green House Gas (GHG) emissions from urban transport by promoting sustainable mobility solutions in partner cities.
The question asks for the geographical location where the Western Ghats and Eastern Ghats mountain ranges converge. This is a direct factual question related to the physical geography of India.
Correct Option: A) Nilgiri
The Nilgiri Hills, also known as the Nilgiri Mountains, are located at the trijunction of Karnataka, Kerala, and Tamil Nadu. They form a critical geographical junction where the Western Ghats and the Eastern Ghats meet, creating a complex of mountains and plateaus.
To find the length of the longest side of the quadrilateral, we first need to determine the actual lengths of its sides. We are given the ratio of the sides and the total perimeter. We can use a common multiplier for the ratio to represent the side lengths and then set up an equation using the perimeter.
Let the four sides of the quadrilateral be \(4x\), \(5x\), \(6x\), and \(7x\), where \(x\) is a common multiplier.
The perimeter of the quadrilateral is the sum of the lengths of its four sides. We are given that the perimeter is \(176 \text{ cm}\).
Set up the equation for the perimeter:
\[ 4x + 5x + 6x + 7x = 176 \]Combine the terms on the left side:
\[ (4 + 5 + 6 + 7)x = 176 \]\[ 22x = 176 \]Solve for \(x\):
\[ x = \frac{176}{22} \]\[ x = 8 \]Now that we have the value of \(x\), we can find the length of each side:
Identify the longest side. Comparing the lengths \(32 \text{ cm}\), \(40 \text{ cm}\), \(48 \text{ cm}\), and \(56 \text{ cm}\), the longest side is \(56 \text{ cm}\).
D) 56 cm — The longest side is \(7x\), which calculates to \(7 \times 8 = 56 \text{ cm}\).
The question asks to identify the individual who is NOT a co-founder of Apple Inc. This requires knowledge of the founding members of Apple Inc.
Correct Option: B) Mark Zuckerberg is not related to Apple Inc. as its co-founder. He is famously known as the co-founder of Facebook (now Meta Platforms).