The problem provides an equation relating \(a^2+b^2+c^2\) and \(ab+bc+ca\). We need to use this relationship to simplify the expression \((a+b+c)^2\). The key insight is to recognize that the given condition implies a specific relationship between \(a\), \(b\), and \(c\).
Analyze the given condition: We are given the equation \( a^{2}+b^{2}+c^{2}= ab+bc+ca \).
Rearrange the equation: Multiply the entire equation by 2:
\[ 2a^{2}+2b^{2}+2c^{2}= 2ab+2bc+2ca \]Move all terms to one side:
\[ 2a^{2}+2b^{2}+2c^{2} - 2ab - 2bc - 2ca = 0 \]Group terms to form squares: Rearrange the terms to form perfect squares:
\[ (a^{2} - 2ab + b^{2}) + (b^{2} - 2bc + c^{2}) + (c^{2} - 2ca + a^{2}) = 0 \]This simplifies to:
\[ (a-b)^{2} + (b-c)^{2} + (c-a)^{2} = 0 \]Deduce the relationship between a, b, and c: Since \(a, b, c\) are real numbers, the square of any real number is non-negative. The sum of non-negative terms can only be zero if each term is zero. Therefore:
\[ (a-b)^{2} = 0 \implies a-b = 0 \implies a = b \] \[ (b-c)^{2} = 0 \implies b-c = 0 \implies b = c \] \[ (c-a)^{2} = 0 \implies c-a = 0 \implies c = a \]This implies that \( a = b = c \).
Substitute this relationship into the expression to be found: We need to find the value of \( (a+b+c)^{2} \). Since \( a=b=c \), we can substitute \(b=a\) and \(c=a\) into the expression:
\[ (a+a+a)^{2} \] \[ (3a)^{2} \] \[ 9a^{2} \]A) \( 9a^{2} \)
This is an analogy question where we need to find the relationship between the first pair of words and apply the same relationship to the second pair to find the missing word.
Correct Option: A) EVENTS. History is the study of past events. Just as numbers are central to mathematics, events are central to history.
The question asks for the founding year of Facebook. This is a factual question requiring knowledge of the history of major technology companies.
Correct Option: B) Facebook was launched on February 4, 2004, by Mark Zuckerberg and his Harvard roommates.
The question asks for the chemical formula of Ozone. To answer this, one needs to recall the basic chemical composition of Ozone, which is a well-known allotrope of oxygen.
C) O₃ is the correct chemical formula for Ozone, indicating that it is a molecule composed of three oxygen atoms.
The question asks us to evaluate an assertion and a reason regarding earthworms and their impact on agriculture. We need to determine the truthfulness of both statements and whether the reason correctly explains the assertion.
B) A is false but R is true.
This question tests general knowledge about animal sounds. The task is to identify the specific sound associated with a seal from the given options.
Seals are known to make a variety of sounds, but one of their most commonly recognized vocalizations, especially when communicating or asserting territory, is a barking sound.
A) Bark — Seals are well-known for making barking sounds, which are often heard in their natural habitats or in captivity. This is their characteristic vocalization.
This is a coding-decoding problem. We need to identify the pattern used to code the word "STUDENT" as 103 and then apply the same pattern to code the word "SCHOOL". The common pattern in such problems involves assigning numerical values to letters, usually based on their alphabetical position.
Sum \( = 19 + 20 + 21 + 4 + 5 + 14 + 20 = 103 \).
This matches the given code for STUDENT, confirming our pattern.
Sum \( = 19 + 3 + 8 + 15 + 15 + 12 = 72 \).
A) The sum of the numerical positions of the letters in "SCHOOL" is 72, which matches option A.
If 'P' means 'divided by', 'T' means 'add', 'M' means 'subtract' and 'D' means 'multiplied by', then what will be the value of the following expression?
12 M 12 D 28 P 7 T 15
This problem involves substituting given mathematical operations for letters and then evaluating the resulting expression using the order of operations (BODMAS/PEMDAS).
The given expression is \(12 \text{ M } 12 \text{ D } 28 \text{ P } 7 \text{ T } 15\).
Replacing the letters with their corresponding mathematical operations, we get:
\[ 12 - 12 \times 28 \div 7 + 15 \]BODMAS/PEMDAS stands for Brackets/Parentheses, Orders/Exponents, Division/Multiplication (from left to right), Addition/Subtraction (from left to right).
The value of the expression is \(-21\).
Correct Option: A) -21 — This is the correct value obtained by following the given substitutions and the order of operations.
The question asks to identify the dentist responsible for killing Cecil the Lion in 2015. This is a factual recall question related to a widely reported international incident. Candidates should recall the name associated with this event.
Correct Option: C) Walter Palmer was the American dentist who killed Cecil the Lion during a hunting trip in Zimbabwe in July 2015. The incident sparked international outrage and drew significant attention to trophy hunting and wildlife conservation.