Which city hosted the 55th International Film Festival of India in November 2024?
The question asks for the host city of the 55th International Film Festival of India (IFFI) in November 2024. This is a factual question related to current events and general knowledge about major cultural festivals in India.
Correct Option: D) Panaji is the capital city of Goa and has been the permanent venue for the International Film Festival of India since 2004. Therefore, the 55th edition in November 2024 will also be held there.
To find the missing number in the series, we need to identify the pattern or rule governing the sequence of numbers. This often involves looking at the differences between consecutive terms, or differences of differences (second-order differences).
Correct Option: D) 74
To solve this problem, we need to compare the positions of digits in the original number with their positions after the digits are rearranged in ascending order. We will list the original number and then the rearranged number, aligning them to easily identify digits whose positions remain unchanged.
Correct Option: A) None
Find the value of \(\frac{(0.06)^2+(0.66)^2+(0.666)^2}{(0.006)^2+(0.066)^2+(0.0666)^2}\).
The problem asks us to simplify a fraction where both the numerator and the denominator consist of sums of squares of decimal numbers. The key to solving this efficiently is to observe the relationship between the numbers in the numerator and the denominator. We can factor out a common term from the denominator to relate it to the numerator.
D) 100
The question asks to identify the first notable king of the Pushyabhuti dynasty, along with his reign period. This requires knowledge of the early rulers of the Pushyabhuti dynasty and their chronological order.
B) Prabhakaravardhana — He is widely considered the first important and powerful ruler of the Pushyabhuti dynasty, whose reign (580-605 CE) marked the beginning of the dynasty's rise to prominence.
The problem involves finding the present age of Hitesh based on relationships between his age, his son's age, and his daughter's age, and their combined age three years ago. We will set up equations based on the given information and solve for the present ages.
Let Hitesh's present age be \(H\) years.
Let his son's present age be \(S\) years.
Let his daughter's present age be \(D\) years.
According to the problem:
3 years ago, their ages were:
The sum of their ages 3 years ago was 123 years:
\[(H - 3) + (S - 3) + (D - 3) = 123\]\[H + S + D - 9 = 123\]\[H + S + D = 123 + 9\]\[H + S + D = 132 \quad \text{(Equation 3)}\]Substitute \(H = 3S\) and \(D = S - 3\) into Equation 3:
\[3S + S + (S - 3) = 132\]\[5S - 3 = 132\]\[5S = 132 + 3\]\[5S = 135\]\[S = \frac{135}{5}\]\[S = 27\]Now that we have the son's present age (\(S = 27\) years), we can find Hitesh's present age using Equation 1:
\[H = 3S\]\[H = 3 \times 27\]\[H = 81\]So, Hitesh's present age is 81 years.
Daughter's present age \(D = S - 3 = 27 - 3 = 24\) years.
Ages 3 years ago: Hitesh \(81-3=78\), Son \(27-3=24\), Daughter \(24-3=21\).
Sum of ages 3 years ago: \(78 + 24 + 21 = 123\). This matches the given information.
Correct Option: A) 81
The question asks about the legal status and nature of proceedings in a Lok Adalat under Indian law. Understanding the provisions of the Legal Services Authorities Act, 1987, is crucial to answer this question.
C) Proceedings are judicial and Lok Adalat is deemed a civil court under CrPC. This option accurately reflects the legal provisions. Section 22(4) of the Legal Services Authorities Act, 1987, explicitly states that a Lok Adalat shall be deemed to be a Civil Court for the purpose of Section 195 and Chapter XXVI of the Code of Criminal Procedure, 1973, making its proceedings judicial.
The given equation is a general form of a circle. By completing the square for the terms involving \(x\) and \(y\), we can transform the equation into the standard form of a circle, \((x-h)^2 + (y-k)^2 = r^2\). This will allow us to find the values of \(x\) and \(y\) that satisfy the equation, or determine if it represents a single point.
Rearrange the given equation to group terms with \(x\) and terms with \(y\):
\[ x^{2} - 10x + y^{2} + 26y + 194 = 0 \]Complete the square for the \(x\) terms. To make \(x^2 - 10x\) a perfect square, we need to add \( \left( \frac{-10}{2} \right)^{2} = (-5)^2 = 25 \).
Complete the square for the \(y\) terms. To make \(y^2 + 26y\) a perfect square, we need to add \( \left( \frac{26}{2} \right)^{2} = (13)^2 = 169 \).
Add and subtract these values to the equation:
\[ (x^{2} - 10x + 25) + (y^{2} + 26y + 169) + 194 - 25 - 169 = 0 \]Rewrite the perfect squares and simplify the constant terms:
\[ (x - 5)^{2} + (y + 13)^{2} + 194 - 194 = 0 \] \[ (x - 5)^{2} + (y + 13)^{2} = 0 \]For the sum of two squares to be zero, each square must be zero. This is because squares of real numbers are always non-negative.
\[ (x - 5)^{2} = 0 \implies x - 5 = 0 \implies x = 5 \] \[ (y + 13)^{2} = 0 \implies y + 13 = 0 \implies y = -13 \]Now that we have the values of \(x\) and \(y\), substitute them into the expression \(x^2 + y^2\):
\[ x^{2} + y^{2} = (5)^{2} + (-13)^{2} \] \[ x^{2} + y^{2} = 25 + 169 \] \[ x^{2} + y^{2} = 194 \]Correct Option: B) The value of \(x^2 + y^2\) is 194.
The question asks about the individual or office responsible for launching the IndiaAI initiative. This is a knowledge-based question related to recent government initiatives and their official launches.
Correct Option: B) The IndiaAI initiative, a comprehensive national program for artificial intelligence, was officially launched by the Union Minister for Electronics and Information Technology (MeitY). This ministry is responsible for policy, promotion, and development in the IT sector, including emerging technologies like AI.
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then evaluate the conclusions based on these diagrams. We need to consider all possible valid Venn diagram configurations to ensure a conclusion logically follows.
B) Both conclusions (I) and (II) follow.