To find the smallest natural number that must be multiplied by a given number to make it a perfect square, we need to perform prime factorization of the given number. A number is a perfect square if all the exponents in its prime factorization are even. We then identify the prime factors with odd exponents and multiply the original number by the missing factors to make all exponents even.
We start by finding the prime factors of 1083.
So, the prime factorization of 1083 is:
\[ 1083 = 3 \times 19 \times 19 = 3^1 \times 19^2 \]For a number to be a perfect square, all the exponents in its prime factorization must be even.
In the prime factorization of 1083, we have:
To make the exponent of 3 even, we need to multiply \(3^1\) by \(3^1\), which means we need to multiply the entire number by 3.
If we multiply 1083 by 3:
\[ 1083 \times 3 = (3^1 \times 19^2) \times 3^1 = 3^{1+1} \times 19^2 = 3^2 \times 19^2 = (3 \times 19)^2 = 57^2 \]Thus, multiplying by 3 makes 1083 a perfect square.
The smallest natural number required is 3.
Correct Option: A) 3
This question tests knowledge about the roles of specific hormones in human puberty, particularly in boys. We need to identify which hormone is responsible for the secondary sexual characteristics observed in males during puberty.
Correct Option: D) Statement A is true but B is false.
The question asks for the reason behind the continuous movement of particles of matter. This is a fundamental concept in physics and chemistry related to the states of matter and energy.
C) kinetic energy — Particles of matter are continuously moving because they possess kinetic energy. Kinetic energy is the energy an object possesses due to its motion. The formula for kinetic energy is \(KE = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is velocity. Since particles are always in motion (unless at absolute zero temperature), they always have kinetic energy.
To determine if a number is prime, we need to check if it has any divisors other than 1 and itself. A common method is to test divisibility by prime numbers up to the square root of the given number. If no such prime divides the number, then the number is prime.
Correct Option: B) 373 is a prime number because it has no divisors other than 1 and itself.
The problem asks us to identify the pair of letter clusters that does not follow the same pattern as the others. We need to analyze the relationship between the letters within each cluster and between the two clusters in each pair based on their positions in the English alphabet.
C) XB – YC is the pair that does NOT belong to the group. In this pair, the second cluster (YC) is formed by adding 1 to the position of each letter of the first cluster (XB) (X+1=Y, B+1=C). In the other options, the second cluster is formed by subtracting 1 from the position of each letter of the first cluster.
This is a coding-decoding problem where letters are assigned specific numerical codes. We need to identify the code for each common letter across the given words to deduce the code for 'R'.
Let's compare the words and their codes to find the numerical value for each letter.
Let's list the letters and their potential codes:
By comparing all three words, we can establish the unique code for each letter:
From our consolidated list, the code for 'R' is 9.
A) 9 is the correct code for 'R' as derived from the given coding scheme.
This problem involves calculating distance, time, and speed. We first determine the total distance to the destination using the initial speed and time. Then, we calculate the new time taken to cover the same distance with the increased speed. Finally, we find the difference between the original time and the new time to determine how much earlier the destination is reached.
Given:
The formula for distance is \(\text{Distance} = \text{Speed} \times \text{Time}\).
\[ \text{Distance} = S_1 \times T_1 = 10 \text{ km/h} \times 2 \text{ h} = 20 \text{ km} \]The speed is increased by 5 km/h.
\[ \text{New Speed } (S_2) = S_1 + 5 \text{ km/h} = 10 \text{ km/h} + 5 \text{ km/h} = 15 \text{ km/h} \]Using the same distance (20 km) and the new speed (15 km/h).
The formula for time is \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
\[ \text{New Time } (T_2) = \frac{\text{Distance}}{S_2} = \frac{20 \text{ km}}{15 \text{ km/h}} = \frac{4}{3} \text{ hours} \]\[ \frac{4}{3} \text{ hours} = 1 \frac{1}{3} \text{ hours} \]This means 1 hour and \(\frac{1}{3}\) of an hour.
To convert \(\frac{1}{3}\) hour to minutes:
\[ \frac{1}{3} \text{ hour} \times 60 \text{ minutes/hour} = 20 \text{ minutes} \]So, the new time \(T_2\) is 1 hour and 20 minutes.Original Time (\(T_1\)) = 2 hours
New Time (\(T_2\)) = 1 hour 20 minutes
Difference in time = Original Time - New Time
Difference = 2 hours - (1 hour 20 minutes)
Difference = 2 hours - 1 hour - 20 minutes
Difference = 1 hour - 20 minutes
Difference = 60 minutes - 20 minutes = 40 minutes
Correct Option: C) 40 min
To solve this problem, we need to perform algebraic subtraction. The phrase "subtract A from B" means we calculate \(B - A\). In this case, we need to subtract the expression \(x^2 - y^2\) from \(y^2 - x^2\).
Identify the two expressions: The first expression is \(x^2 - y^2\), and the second expression is \(y^2 - x^2\).
Set up the subtraction: We are subtracting \(x^2 - y^2\) from \(y^2 - x^2\). This can be written as:
\[ (y^2 - x^2) - (x^2 - y^2) \]Distribute the negative sign: When subtracting an expression, we change the sign of each term within the parentheses being subtracted.
\[ y^2 - x^2 - x^2 - (-y^2) \] \[ y^2 - x^2 - x^2 + y^2 \]Combine like terms: Group the \(y^2\) terms together and the \(x^2\) terms together.
\[ (y^2 + y^2) + (-x^2 - x^2) \] \[ 2y^2 - 2x^2 \]Factor out the common term: Both terms have a common factor of 2. We can also factor out -2 to match the options.
\[ 2(y^2 - x^2) \]Alternatively, factoring out \(-2\):
\[ -2(x^2 - y^2) \]The result of the subtraction is \(2y^2 - 2x^2\), which can be factored as \(2(y^2 - x^2)\). This is equivalent to \(-2(x^2 - y^2)\) because \(y^2 - x^2 = -(x^2 - y^2)\). Therefore, option A is correct.
B) \(-2(x^2 + y^2)\): This would imply that the terms \(x^2\) and \(y^2\) were added, or that their signs were both negative in the final expression, which is not the case. Our result is \(2y^2 - 2x^2\), not \(-2x^2 - 2y^2\).
C) \(2(x^2 + y^2)\): This option suggests that both \(x^2\) and \(y^2\) terms were positive and added together, which is incorrect. The \(x^2\) term is negative in our final expression.
D) \(2(x^2 - y^2)\): This option is the negative of our result \(2(y^2 - x^2)\). If we had subtracted \(y^2 - x^2\) from \(x^2 - y^2\), we would get this result.
The problem presents a proportion and asks for the value of an unknown variable \(x\). A proportion states that two ratios are equal. We can set up an equation based on the given proportion and solve for \(x\).
The given proportion is \(48 : x :: x : 75\). This can be written as an equation of two ratios:
\[ \frac{48}{x} = \frac{x}{75} \]To solve for \(x\), we can cross-multiply the terms:
\[ 48 \times 75 = x \times x \]Simplify both sides of the equation:
\[ 3600 = x^2 \]To find \(x\), take the square root of both sides:
\[ x = \sqrt{3600} \]Calculate the square root:
\[ x = 60 \]The problem states that \(x > 0\), so we take the positive square root. Thus, \(x = 60\).
Correct Option: D) 60