To find the smallest positive number that must be added to the greatest 4-digit number to make it exactly divisible by 307, we first identify the greatest 4-digit number. Then, we divide this number by 307 to find the remainder. The difference between the divisor (307) and the remainder will be the smallest number that needs to be added.
Identify the greatest 4-digit number: The greatest 4-digit number is 9999.
Divide the greatest 4-digit number by 307: We perform the division of 9999 by 307.
\[ 9999 \div 307 \] Let's perform the long division: \[ \begin{array}{r} 32 \\ 307\overline{)9999} \\ -921 \downarrow \\ \hline 789 \\ -614 \\ \hline 175 \\ \end{array} \] So, when 9999 is divided by 307, the quotient is 32 and the remainder is 175. This can be written as: \[ 9999 = (307 \times 32) + 175 \]Determine the number to be added: To make 9999 exactly divisible by 307, we need to add a number such that the sum becomes a multiple of 307. The current remainder is 175. This means 9999 is 175 more than a multiple of 307 (which is \(307 \times 32\)). To reach the next multiple of 307 (which is \(307 \times 33\)), we need to add the difference between the divisor and the remainder.
\[ \text{Number to be added} = \text{Divisor} - \text{Remainder} \] \[ \text{Number to be added} = 307 - 175 \] \[ \text{Number to be added} = 132 \] If we add 132 to 9999, the sum will be \(9999 + 132 = 10131\). Let's check if 10131 is divisible by 307: \[ 10131 \div 307 = 33 \] Since \(307 \times 33 = 10131\), the sum is exactly divisible by 307.C) 132
If \( \cos^{4}\theta - \sin^{4}\theta = \frac{2}{3} \), then the value of \( 1 - 2\sin^{2}\theta \) is:
The problem requires us to find the value of an expression involving trigonometric functions, given an equation relating them. We will use algebraic identities and fundamental trigonometric identities to simplify the given equation and then find the required value.
Start with the given equation:
\[ \cos^{4}\theta - \sin^{4}\theta = \frac{2}{3} \]Recognize that the left side of the equation is in the form of \(a^2 - b^2\), where \(a = \cos^2\theta\) and \(b = \sin^2\theta\). Apply the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\):
\[ (\cos^{2}\theta - \sin^{2}\theta)(\cos^{2}\theta + \sin^{2}\theta) = \frac{2}{3} \]Use the fundamental trigonometric identity \( \cos^{2}\theta + \sin^{2}\theta = 1 \):
\[ (\cos^{2}\theta - \sin^{2}\theta)(1) = \frac{2}{3} \]\[ \cos^{2}\theta - \sin^{2}\theta = \frac{2}{3} \]The expression we need to find is \( 1 - 2\sin^{2}\theta \). We know that \( \cos^{2}\theta = 1 - \sin^{2}\theta \). Substitute this into the simplified equation:
\[ (1 - \sin^{2}\theta) - \sin^{2}\theta = \frac{2}{3} \]Simplify the expression:
\[ 1 - 2\sin^{2}\theta = \frac{2}{3} \]D) The value of \( 1 - 2\sin^{2}\theta \) is \( \frac{2}{3} \), which matches option D.
The question asks to identify a gas that is light and non-inflammable, making it suitable for blowing balloons. We need to evaluate the properties of each given option.
D) Helium is light and non-inflammable, making it the safest and most effective gas for blowing balloons that need to float.
The question asks for the correct full form of the acronym IPBES. This requires knowledge of international environmental organizations and their official names.
C) Intergovernmental Science Policy Platform on Biodiversity and Ecosystem Services is the correct and official full form of IPBES. IPBES is an independent intergovernmental body established to strengthen the science-policy interface for biodiversity and ecosystem services for the conservation and sustainable use of biodiversity, long-term human well-being, and sustainable development.
We are given the total cost price of two Samsung and three Mi phones, and the total profit earned after selling them at specific profit percentages. We need to find the cost price of one Mi phone. We can set up a system of two linear equations based on the given information and solve for the unknown cost prices.
Correct Option: C) ₹5,400
To arrange the given items in the most meaningful order, we need to consider their relative sizes and hierarchical relationships, typically from the largest to the smallest or vice versa. In this case, the most logical order would be from the largest cosmic entity to the smallest, which is a human being.
Therefore, arranging them from largest to smallest gives us: Cosmos → Milky Way → Earth → Human. This corresponds to the sequence 1, 3, 2, 4.
Correct Option: C) 1, 3, 2, 4
To find the smallest perfect square number that must be added to 12519 to get another perfect square, we first need to find the square root of 12519. This will help us identify the nearest perfect squares. We will then look for the smallest perfect square difference that makes the sum a perfect square.
Find the square root of 12519:
We can estimate the square root of 12519. We know that:
Since 12519 is between 12100 and 14400, its square root will be between 110 and 120.
Let's try values close to 110:
So, \(111^2 = 12321\) and \(112^2 = 12544\).
Identify the next perfect square:
The number 12519 is not a perfect square. The next perfect square greater than 12519 is \(112^2 = 12544\).
Calculate the difference:
To make 12519 a perfect square (specifically, \(112^2\)), we need to add the difference between \(112^2\) and 12519.
\[ \text{Difference} = 112^2 - 12519 \] \[ \text{Difference} = 12544 - 12519 \] \[ \text{Difference} = 25 \]Check if the difference is a perfect square:
The difference we found is 25. We need to check if 25 is a perfect square. Yes, \(5^2 = 25\).
Therefore, 25 is the smallest perfect square number that must be added to 12519 to get a perfect square number (12544).
D) 25 — As calculated, adding 25 to 12519 results in 12544, which is \(112^2\). Since 25 itself is a perfect square (\(5^2\)), this is the correct answer.
This problem involves compound interest. The key property of compound interest is that if a sum of money multiplies by a certain factor in a given period, it will multiply by the same factor again in the same period. We can use the compound interest formula or a simpler conceptual approach based on powers to solve this problem.
Let the original principal amount be \(P\).
According to the problem, the sum doubles itself in 15 years. This means after 15 years, the amount \(A_1 = 2P\).
We want to find out when the amount becomes 8 times the original amount, i.e., \(A_n = 8P\).
We can express 8 as a power of 2: \(8 = 2^3\).
This implies that the money needs to double three times (2 times 2 times 2).
Since the money doubles every 15 years:
Alternatively, using the concept of powers:
If the amount becomes \(x\) times in \(T\) years, then it will become \(x^n\) times in \(n \times T\) years.
Here, \(x=2\) (doubles) and \(T=15\) years.
We want the amount to become 8 times. Since \(8 = 2^3\), we have \(n=3\).
Therefore, the total time will be \(n \times T = 3 \times 15 = 45\) years.
B) 45 years
The question asks for the year when the Anandpur Sahib Resolution was passed. This requires knowledge of significant historical events related to Indian political history, specifically concerning the state of Punjab and Sikh politics.
Correct Option: C) 1973
The Anandpur Sahib Resolution was passed by the Shiromani Akali Dal (SAD) in 1973 at Anandpur Sahib. It sought to redefine the center-state relationship in India, advocating for greater autonomy for Punjab and emphasizing the distinct religious and cultural identity of Sikhs. It became a foundational document for the Akali Dal's political agenda in the subsequent decades.
The question asks to identify the US missile defense systems offered to India as alternatives to the Russian S-400. This requires knowledge of current geopolitical defense deals and the specific missile systems offered by the US.
Correct Option: B) THAAD and PAC-3 are the US missile defense systems offered to India as alternatives to the Russian S-400s. These systems represent advanced capabilities in ballistic missile defense and air defense, respectively.