To find the number of factors of 1296 that are perfect squares, we first need to find the prime factorization of 1296. Then, we identify the factors that are perfect squares. A number is a perfect square if all the exponents in its prime factorization are even.
Prime Factorization of 1296:
We start by finding the prime factors of 1296:
\[ 1296 = 2 \times 648 \] \[ 648 = 2 \times 324 \] \[ 324 = 2 \times 162 \] \[ 162 = 2 \times 81 \] \[ 81 = 3 \times 27 \] \[ 27 = 3 \times 9 \] \[ 9 = 3 \times 3 \]So, the prime factorization of 1296 is \(2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3\), which can be written as:
\[ 1296 = 2^4 \times 3^4 \]Identify Perfect Square Factors:
A factor of \(1296 = 2^4 \times 3^4\) will be of the form \(2^a \times 3^b\), where \(0 \le a \le 4\) and \(0 \le b \le 4\).
For a factor to be a perfect square, the exponents \(a\) and \(b\) must both be even numbers.
Determine possible even exponents for each prime factor:
Calculate the total number of perfect square factors:
The total number of perfect square factors is the product of the number of choices for each exponent.
\[ \text{Number of perfect square factors} = (\text{Number of choices for } a) \times (\text{Number of choices for } b) \] \[ = 3 \times 3 \] \[ = 9 \]These factors are: \(2^0 3^0 = 1\), \(2^0 3^2 = 9\), \(2^0 3^4 = 81\), \(2^2 3^0 = 4\), \(2^2 3^2 = 36\), \(2^2 3^4 = 324\), \(2^4 3^0 = 16\), \(2^4 3^2 = 144\), \(2^4 3^4 = 1296\).
B) 9
The question asks to identify the organization responsible for establishing the Govardhan Eco Village in Maharashtra. This requires knowledge of prominent spiritual and social organizations in India and their projects.
C) ISKCON — The International Society for Krishna Consciousness (ISKCON) established the Govardhan Eco Village. It is an award-winning project located in Wada, Palghar district, Maharashtra, dedicated to rural development, environmental sustainability, and spiritual education based on Vedic principles.
We need to find the values of \(k\) for which the polynomial \(P(x) = x^2 + 5kx + k^2 + 5\) satisfies two conditions: it is exactly divisible by \(x + 2\) but not divisible by \(x + 3\). We will use the Remainder Theorem, which states that a polynomial \(P(x)\) is divisible by \(x - a\) if and only if \(P(a) = 0\).
Condition 1: \(P(x)\) is exactly divisible by \(x + 2\).
According to the Remainder Theorem, if \(P(x)\) is divisible by \(x + 2\), then \(P(-2) = 0\).
Substitute \(x = -2\) into the polynomial:
Solve the quadratic equation for \(k\).
We can factor the quadratic equation \(k^2 - 10k + 9 = 0\):
Condition 2: \(P(x)\) is not divisible by \(x + 3\).
According to the Remainder Theorem, if \(P(x)\) is not divisible by \(x + 3\), then \(P(-3) \neq 0\).
Substitute \(x = -3\) into the polynomial:
Check the values of \(k\) obtained from Condition 1 against Condition 2.
We have two possible values for \(k\): \(k = 1\) and \(k = 9\). We need to ensure that for these values, \(P(-3) \neq 0\).
Case 1: Let \(k = 1\).
Substitute \(k = 1\) into the expression for \(P(-3)\):
Case 2: Let \(k = 9\).
Substitute \(k = 9\) into the expression for \(P(-3)\):
Conclusion: The only value of \(k\) that satisfies both conditions is \(k = 9\).
B) 9 — As determined in the step-by-step solution, \(k=9\) is the only value that makes the polynomial divisible by \(x+2\) but not by \(x+3\).
Which of the given elements A, B, C, D and E with atomic numbers 2, 3, 7, 10 and 30 respectively belong to the same period?
To determine which elements belong to the same period, we need to find their electronic configurations. The period number of an element in the periodic table corresponds to the number of electron shells (principal energy levels) occupied by its electrons. Elements in the same period have the same number of occupied electron shells.
A) B, C, D — Elements B (Lithium), C (Nitrogen), and D (Neon) all have two occupied electron shells, placing them in the 2nd period of the periodic table.
The question asks for the commercial unit of energy. This requires understanding the standard units of power and energy and how they are used in practical, commercial applications, particularly for electricity billing.
Correct Option: C) Kilowatt hour is the commercial unit of energy. It is commonly used for measuring electricity consumption in homes and industries for billing purposes.
The problem is a coding-decoding question. We need to identify the pattern used to code "MAP" as "KYN" and then apply the same pattern to decode "REDUCE".
B) PCBSAC is the correct code for "REDUCE" based on the identified pattern.
This question tests knowledge of IUPAC nomenclature for organic compounds, specifically the prefixes used to denote the number of carbon atoms in a hydrocarbon chain. A standard set of prefixes is used for chains of different lengths.
Correct Option: D) But
This is a syllogism problem. We need to evaluate the given conclusions based on the provided statements. We can use Venn diagrams or logical deduction to determine which conclusions logically follow.
Imagine three concentric circles. The innermost circle represents "mud." The next larger circle, completely enclosing "mud," represents "vessels." The outermost circle, completely enclosing "vessels," represents "containers."
From our visualization or logical deduction, we know that all mud is vessels, and all vessels are containers. This implies that all mud is also containers. However, it does not mean that all containers are mud. There can be containers that are not vessels, and thus not mud. For example, a plastic box could be a container but not a vessel (in the context of this statement) and certainly not mud. Therefore, Conclusion I does not logically follow.
Since "All mud are vessels," it inherently means that the set of "mud" is a subset of "vessels." If all elements of "mud" are also elements of "vessels," then it must be true that some elements of "vessels" are "mud." This is a direct implication of the first statement. Therefore, Conclusion II logically follows.
B) Only conclusion 2 follows.
In a library, there are 10 different sections and the librarian categorises the books over 20 days. Which section has the least number of books?
Books Categorised
To determine which section has the least number of books, we need to analyze the graph provided. The graph shows the number of books categorized in each section.
Correct Option: A) Section 7
The question asks to identify the winner of the Golden Boot Award in the 2018 FIFA World Cup. This requires knowledge of the top goal scorer from that specific tournament.
Correct Option: A) Harry Kane won the Golden Boot Award in the 2018 FIFA World Cup by scoring 6 goals.