This problem can be solved by using a fundamental algebraic identity that relates the sum of two variables, their product, and the sum of their squares. The identity is \((x+y)^2 = x^2 + y^2 + 2xy\).
A) 45 is the correct value for \(x^2 + y^2\).
To find the maximum number of pieces that can be cut from a rod, we need to divide the total length of the rod by the length of each piece. Since we are looking for the maximum number of *whole* pieces, we will take the integer part of the division result, ignoring any fractional part (which represents the leftover length that is not enough to form another full piece).
To find the number of pieces, we perform the division:
\[ \text{Number of pieces} = \frac{\text{Total length of rod}}{\text{Length of each piece}} \]\[ \text{Number of pieces} = \frac{658}{34.6} \]When we divide 658 by 34.6, we get:
\[ \frac{658}{34.6} \approx 19.0173 \]Since we can only cut whole pieces, we take the integer part of the result. The integer part of \(19.0173\) is \(19\).
This means we can cut 19 full pieces, and there will be a small remaining length that is less than \(34.6 \text{ cm}\).
Correct Option: B) 19
The calculation shows that \(658 \div 34.6 \approx 19.0173\). Since only whole pieces can be cut, the maximum number of pieces is the integer part of this value, which is 19.
The question asks to identify the name of a specific robot introduced by the Chennai traffic police for traffic management in January 2019. This requires knowledge of current affairs and specific initiatives related to technology in policing.
D) Roadeo — The robot introduced by Chennai traffic police for traffic management in January 2019 was named Roadeo. It was deployed as a pilot project to assist in traffic regulation, particularly at busy intersections.
The problem asks us to evaluate two conclusions based on a given statement. We need to determine which conclusion(s) logically follow from the statement. The statement highlights a potential future scarcity of fuel due to rapid consumption of petrol and diesel.
Correct Option: B) Only II follows.
The question asks to identify which of the given options is NOT a part of the human excretory system. We need to recall the components of the excretory system and the respiratory system to differentiate.
Correct Option: C) Bronchiole is a part of the respiratory system, specifically the lungs, and is involved in air transport, not waste excretion.
The value of 12 + [(25 ÷ \( \frac{1}{5} \)) - (14 × 8)] - 9 × 4 is:
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we perform operations in the following order: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
First, evaluate the expressions inside the brackets:
The given expression is: \[12 + \left[\left(25 \div \frac{1}{5}\right) - \left(14 \times 8\right)\right] - 9 \times 4\]
Inside the first parenthesis: \(25 \div \frac{1}{5}\)
Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[ 25 \div \frac{1}{5} = 25 \times 5 = 125 \]Inside the second parenthesis: \(14 \times 8\)
\[ 14 \times 8 = 112 \]Substitute these values back into the main bracket:
\[ \left[\left(25 \div \frac{1}{5}\right) - \left(14 \times 8\right)\right] = [125 - 112] = 13 \]Now, substitute this result back into the original expression:
\[ 12 + 13 - 9 \times 4 \]Next, perform the multiplication outside the bracket:
\[ 9 \times 4 = 36 \]Substitute this back into the expression:
\[ 12 + 13 - 36 \]Finally, perform the addition and subtraction from left to right:
\[ 12 + 13 = 25 \] \[ 25 - 36 = -11 \]Thus, the value of the expression is -11.
B) -11 is the correct answer, as derived from the step-by-step calculation following the order of operations.
Two cards are drawn from a pack of 52 cards. The probability that out of 2 cards, one card is red and one card is black is:
\( \frac{25}{51} \)
\( \frac{13}{25} \)
\( \frac{26}{51} \)
\( \frac{1}{2} \)
To find the probability of drawing one red card and one black card from a pack of 52 cards, we need to determine the total number of ways to draw two cards and the number of ways to draw one red and one black card. We will use combinations for this calculation.
This is given by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \).
\[ \text{Total ways} = C(52, 2) = \frac{52!}{2!(52-2)!} = \frac{52!}{2!50!} = \frac{52 \times 51}{2 \times 1} = 26 \times 51 = 1326 \]There are 26 red cards, and we want to choose 1.
\[ \text{Ways to choose 1 red card} = C(26, 1) = \frac{26!}{1!(26-1)!} = \frac{26!}{1!25!} = 26 \]There are 26 black cards, and we want to choose 1.
\[ \text{Ways to choose 1 black card} = C(26, 1) = \frac{26!}{1!(26-1)!} = \frac{26!}{1!25!} = 26 \]Since these events are independent, we multiply the number of ways to choose one red card by the number of ways to choose one black card.
\[ \text{Favorable ways} = C(26, 1) \times C(26, 1) = 26 \times 26 = 676 \]The probability is the ratio of favorable ways to the total number of ways.
\[ P(\text{one red and one black}) = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{676}{1326} \]Simplify the fraction:
\[ \frac{676}{1326} = \frac{338}{663} = \frac{26 \times 13}{51 \times 13} = \frac{26}{51} \]C) \( \frac{26}{51} \) is the correct probability, as calculated by dividing the number of ways to draw one red and one black card by the total number of ways to draw two cards.
The problem asks us to find a number based on a relationship between two percentages of that number. We can set up an algebraic equation to represent the given condition and then solve for the unknown number.
Correct Option: A) 60
The question asks about the origin region of Cyclone Pabuk. To answer this, one needs to recall or know the geographical origin of this specific cyclone.
Correct Option: B) Gulf of Thailand
The question asks which international organization designated Rio de Janeiro as the World Capital of Architecture for 2020. This is a factual question related to international organizations and their cultural designations.
Correct Option: D) UNESCO