To evaluate the given expression, we need to simplify the terms by expressing the numbers as powers of their prime factors. Then, we will apply the rules of exponents and follow the order of operations (BODMAS/PEMDAS).
The given expression is \( (243)^{2} \div (27)^{2} \times 6 \div 18 \).
We notice that 243 and 27 are powers of 3.
Substitute these into the expression:
\[ (3^5)^{2} \div (3^3)^{2} \times 6 \div 18 \]First, \(81 \times 6\):
\[ 81 \times 6 = 486 \]B) 27 is the correct answer. The step-by-step simplification of the expression leads to the value 27.
The question asks to identify a plant with 'hidden reproductive organs'. This term refers to plants where the reproductive structures are not visible externally, which is characteristic of cryptogams. We need to classify the given options into cryptogams or phanerogams (plants with visible reproductive organs).
A) Marsilea. Marsilea is a pteridophyte (fern), and pteridophytes are cryptogams, meaning their reproductive organs are hidden. They reproduce via spores, and their sexual reproduction involves inconspicuous gametophytes.
The problem involves finding an angle within a triangle given relationships between other angles. We will use algebraic substitution and the angle sum property of a triangle to solve this problem. The key is to express all angles in terms of a few variables and then use the given conditions.
Let's denote the angles:
From the above, we can find the angles of \( \triangle ABC \):
We are given \( \angle BAC = 60^\circ \). In \( \triangle ABC \), the sum of angles is \( 180^\circ \):
\[ \angle BAC + \angle ABC + \angle ACB = 180^\circ \]\[ 60^\circ + 3x + 3y = 180^\circ \]\[ 3(x + y) = 180^\circ - 60^\circ \]\[ 3(x + y) = 120^\circ \]\[ x + y = 40^\circ \]Now, consider \( \triangle BOC \). The sum of angles in \( \triangle BOC \) is \( 180^\circ \):
\[ \angle BOC + \angle OBC + \angle OCB = 180^\circ \]\[ \angle BOC + 2x + 2y = 180^\circ \]\[ \angle BOC + 2(x + y) = 180^\circ \]Substitute the value of \( (x + y) \) from Step 3 into the equation from Step 4:
\[ \angle BOC + 2(40^\circ) = 180^\circ \]\[ \angle BOC + 80^\circ = 180^\circ \]\[ \angle BOC = 180^\circ - 80^\circ \]\[ \angle BOC = 100^\circ \]D) The measure of \( \angle BOC \) is \( 100^\circ \).
The question asks for the full form of W3C in the context of computers and the Internet. This is a direct knowledge-based question related to common acronyms in the field of web technology.
Correct Option: A) World Wide Web Consortium is the correct full form of W3C. It is an international community that develops open standards to ensure the long-term growth of the Web.
To divide a decimal number by another decimal number, the easiest method is to convert the divisor into a whole number. This is done by multiplying both the dividend and the divisor by a power of 10 (10, 100, 1000, etc.) until the divisor becomes an integer. Then, perform the division as usual.
Thus, \(10.53 \div 0.09 = 117\).
Correct Option: A) 117
This problem involves finding the distance between the centers of two circles given their radii and the length of their transverse common tangent. We will use the formula for the length of a transverse common tangent, which relates the radii of the circles, the distance between their centers, and the length of the tangent.
Let the radii of the two circles be \(r_1\) and \(r_2\). Given:
Let the length of the transverse common tangent be \(L\). Given:
Let the distance between the centers of the two circles be \(d\). We need to find \(d\).
The formula for the length of a transverse common tangent between two circles is:
\[ L = \sqrt{d^2 - (r_1 + r_2)^2} \]Substitute the given values into the formula:
\[ 6 = \sqrt{d^2 - (4.5 + 3.5)^2} \]Calculate the sum of the radii:
\[ r_1 + r_2 = 4.5 + 3.5 = 8 \text{ cm} \]Substitute this value back into the equation:
\[ 6 = \sqrt{d^2 - (8)^2} \]Square both sides of the equation to eliminate the square root:
\[ 6^2 = d^2 - 8^2 \]\[ 36 = d^2 - 64 \]Solve for \(d^2\):
\[ d^2 = 36 + 64 \]\[ d^2 = 100 \]Take the square root of both sides to find \(d\):
\[ d = \sqrt{100} \]\[ d = 10 \text{ cm} \]Correct Option: B) 10 cm
The question asks to identify the biological process that converts sugar into alcohol. We need to recall the definitions of the given options and select the one that matches this description.
Correct Option: C) Fermentation is the biological process where microorganisms like yeast convert sugars into alcohol and carbon dioxide in the absence of oxygen.
The question asks for the establishment year of the National Informatics Centre (NIC). This is a factual recall question related to Indian government organizations and their history.
Correct Option: C) 1976 — The National Informatics Centre (NIC) was established in 1976 by the Government of India. It has been instrumental in driving e-governance and digital transformation initiatives across various government departments and ministries.
The problem is a coding-decoding question. We need to identify the pattern or mapping between letters and numbers from the given coded words and then apply that mapping to decode the new word.
A) 563201 is the correct code derived from the established letter-to-number mapping.
The given expression resembles the algebraic identity for the square of a sum: \((a+b)^2 = a^2 + 2ab + b^2\). By identifying the values of 'a' and 'b' from the expression, we can simplify the calculation significantly.
Identify the given expression:
\[ 394 \times 394 + 2 \times 394 \times 106 + 106 \times 106 \]Recognize the algebraic identity:
The expression is in the form \(a^2 + 2ab + b^2\).
Compare the given expression with the identity to find 'a' and 'b':
Substitute 'a' and 'b' into the identity \((a+b)^2\):
\[ (394 + 106)^2 \]Perform the addition inside the parenthesis:
\[ (500)^2 \]Calculate the square:
\[ 500 \times 500 = 250000 \]B) 250000 is the correct value of the expression.