To find the smallest number that is divisible by 10, 14, and 28 and is a perfect square, we first need to find the Least Common Multiple (LCM) of these numbers. Then, we will adjust the prime factorization of the LCM to make it a perfect square.
Find the prime factorization of each number:
Calculate the LCM of 10, 14, and 28:
The LCM is found by taking the highest power of each prime factor present in the numbers.
So, \[ \text{LCM}(10, 14, 28) = 2^2 \times 5^1 \times 7^1 = 4 \times 5 \times 7 = 140 \] The LCM is 140.
Make the LCM a perfect square:
For a number to be a perfect square, all the exponents in its prime factorization must be even. The prime factorization of the LCM is \(2^2 \times 5^1 \times 7^1\).
Here, the exponents of 5 and 7 are 1 (odd). To make them even, we need to multiply the LCM by \(5^1\) and \(7^1\).
The smallest number that is a perfect square and divisible by 10, 14, and 28 will be:
\[ \text{Smallest perfect square} = (2^2 \times 5^1 \times 7^1) \times (5^1 \times 7^1) \] \[ = 2^2 \times 5^2 \times 7^2 \] \[ = (2 \times 5 \times 7)^2 \] \[ = (70)^2 \] \[ = 4900 \]Correct Option: C) 4900
The number 4900 is divisible by 10 (4900/10 = 490), 14 (4900/14 = 350), and 28 (4900/28 = 175). Also, 4900 is a perfect square (\(70^2\)).
The question asks to identify M's father. We need to analyze each statement individually and then combined to see if they provide enough information to determine the relationship. We are looking for a definitive answer about M's father.
Since neither statement alone nor both statements combined provide enough information to identify M's father, both statements are not sufficient.
Correct Option: B) Statement I and II both are not sufficient.
The question asks to identify the folk music form with which the world-famous Chhattisgarhi folk singer Teejan Bai is associated. This requires knowledge of prominent Indian folk artists and their respective art forms.
Teejan Bai is a renowned exponent of Pandavani, a traditional performing art form from Chhattisgarh, India. In Pandavani, artists narrate tales from the Mahabharata, often with musical accompaniment and dramatic gestures. Teejan Bai is particularly famous for her unique style of narrating these epic stories, earning her national and international recognition, including the Padma Vibhushan, India's second-highest civilian award.
A) Pandavani — Teejan Bai is indeed a world-renowned exponent of Pandavani, a folk music and theatrical performance style from Chhattisgarh that involves narrating tales from the Mahabharata.
The problem defines two new operations, '0' and 'Ø', with specific rules. We need to evaluate a given expression by applying these rules step-by-step, following the order of operations (parentheses first).
Using the definition of 'Ø', where \(a=8\) and \(b=6\):
\[ 8Ø6 = \frac{1}{2} \times 8 \times 6 \] \[ 8Ø6 = \frac{1}{2} \times 48 \] \[ 8Ø6 = 24 \]Using the definition of 'Ø', where \(a=6\) and \(b=4\):
\[ 6Ø4 = \frac{1}{2} \times 6 \times 4 \] \[ 6Ø4 = \frac{1}{2} \times 24 \] \[ 6Ø4 = 12 \]The expression becomes \((24)0(12)\).
Using the definition of '0', where \(a=24\) and \(b=12\):
\[ 24012 = 2(24 - 12) \] \[ 24012 = 2(12) \] \[ 24012 = 24 \]D) 24 — The step-by-step calculation shows that the value of \((8Ø6)0(6Ø4)\) is 24.
This problem involves work and time. We need to determine the individual work rates of A and B, and then combine them to find their combined work rate. Finally, we can calculate the time taken for A and B to complete the task together.
Calculate A's daily work rate:
A alone can finish the task in 30 days. So, A's one day's work is:
\[ \text{A's 1 day's work} = \frac{1}{30} \text{ of the task} \]Calculate the work done by A in 6 days:
A works for 6 days. So, the amount of work A completes is:
\[ \text{Work done by A in 6 days} = 6 \times \frac{1}{30} = \frac{6}{30} = \frac{1}{5} \text{ of the task} \]Calculate the remaining work:
The total work is 1 (or 100%). After A works for 6 days, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{5} = \frac{4}{5} \text{ of the task} \]Calculate B's daily work rate:
B finishes the remaining \(\frac{4}{5}\) of the task in 24 days. So, B's one day's work is:
\[ \text{B's 1 day's work} = \frac{\frac{4}{5}}{24} = \frac{4}{5 \times 24} = \frac{4}{120} = \frac{1}{30} \text{ of the task} \]Calculate the combined daily work rate of A and B:
A's 1 day's work is \(\frac{1}{30}\) and B's 1 day's work is \(\frac{1}{30}\). Their combined 1 day's work is:
\[ \text{(A + B)'s 1 day's work} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15} \text{ of the task} \]Calculate the total time taken by A and B together:
If A and B together complete \(\frac{1}{15}\) of the task in one day, they will complete the entire task in:
\[ \text{Time taken by (A + B) together} = \frac{1}{\frac{1}{15}} = 15 \text{ days} \]B) 15
The problem presents a series of number pairs and asks us to find the missing number in the last pair based on the relationship observed in the preceding pairs. We need to identify the pattern that connects the first number to the second number in each given pair.
A) 110 is the correct answer because it follows the established pattern where the second number is the first number multiplied by (first number - 1).
The question asks to identify the countries involved in the joint venture for the BrahMos supersonic missile. This requires knowledge of international defense collaborations, specifically regarding India's missile development programs.
Correct Option: B) India and Russia
The BrahMos supersonic cruise missile is a joint venture between India's Defence Research and Development Organisation (DRDO) and Russia's NPO Mashinostroyeniya. The name "BrahMos" is a combination of the names of two rivers: the Brahmaputra of India and the Moskva of Russia, signifying the collaborative nature of the project.
What is the place value of 5 in the number 56789214?
To find the place value of a digit in a number, we need to identify its position from the rightmost digit (units place). Each position corresponds to a power of 10. The place value is the digit multiplied by the power of 10 corresponding to its position.
So, the place value of \(5\) in \(56789214\) is \(5 \times 10^7\).
B) \(5 \times 10^7\) — This is the correct representation of the place value of 5, as it is in the ten millions position.
This is a blood relations question. We need to deduce the relationship between 'A' and 'my' (the speaker) based on the given statements. We will break down the statements and establish the connections step-by-step.
Correct Option: A) father
If A is my father, then B (A's son) is my brother. My son's paternal uncle would be my brother (B). This aligns perfectly with the given statements.
Find the value of \(\frac{(3.17+9.12)^2+(3.17-9.12)^2}{3.17\times3.17+9.12\times9.12}\).
The problem asks us to simplify a given algebraic expression involving specific numbers. We can simplify this expression by recognizing and applying a standard algebraic identity. The key identity to use here is \((a+b)^2 + (a-b)^2 = 2(a^2+b^2)\).
The given expression is:
\[ \frac{(3.17+9.12)^2+(3.17-9.12)^2}{3.17\times3.17+9.12\times9.12} \]Let \(a = 3.17\) and \(b = 9.12\). Substituting these values into the expression, we get:
\[ \frac{(a+b)^2+(a-b)^2}{a\times a+b\times b} = \frac{(a+b)^2+(a-b)^2}{a^2+b^2} \]Recall the algebraic identity: \((a+b)^2 + (a-b)^2 = (a^2+2ab+b^2) + (a^2-2ab+b^2) = 2a^2+2b^2 = 2(a^2+b^2)\).
Using the identity, the numerator \((a+b)^2+(a-b)^2\) becomes \(2(a^2+b^2)\). So, the expression transforms into:
\[ \frac{2(a^2+b^2)}{a^2+b^2} \]Since \(a^2+b^2\) is in both the numerator and the denominator, and \(a^2+b^2 \neq 0\) (as \(a\) and \(b\) are non-zero real numbers), we can cancel out the term \(a^2+b^2\):
\[ \frac{2\cancel{(a^2+b^2)}}{\cancel{(a^2+b^2)}} = 2 \]The value of the expression is 2.
Correct Option: A) 2 is the correct answer because applying the algebraic identity \((a+b)^2 + (a-b)^2 = 2(a^2+b^2)\) simplifies the given expression to 2.