The question asks to identify the scientist who provided the basic theory about the nature of matter. This requires knowledge of fundamental contributions to atomic theory in chemistry.
Correct Option: D) John Dalton proposed the atomic theory, which described matter as being composed of indivisible particles called atoms. His theory was a cornerstone in understanding the nature of matter and chemical reactions.
The problem presents a coding pattern where groups of three letters are mapped to a single letter. We need to identify the rule governing this mapping and apply it to find the missing letter for the given group.
C) I — Based on the derived rule \((1\text{st letter} + 3\text{rd letter} - 2\text{nd letter}) - 3\), the calculation for KDE results in 9, which corresponds to the letter I.
Aman's great-grandfather is 105 years old. Aman's father Ram Singh's age is \( \frac{1}{3} \) of his grandfather. Five years ago, when Aman was born, Ram Singh's mother was double the age of Ram Singh. What is the difference of age between Aman and Ram Singh?
This problem involves calculating the ages of different family members at various points in time. We will work backward from the oldest known age, calculating each person's current age, and then determine the age difference between Aman and Ram Singh.
Given that Aman's great-grandfather is 105 years old.
Aman's father Ram Singh's age is \(\frac{1}{3}\) of his grandfather. This statement implies that Ram Singh's father (Aman's grandfather) is the person whose age is being referenced. Let's assume "his grandfather" refers to Aman's grandfather, who is the son of Aman's great-grandfather.
However, the phrasing "Aman's father Ram Singh's age is \(\frac{1}{3}\) of his grandfather" is ambiguous. It could mean Ram Singh's father (Aman's grandfather) or Ram Singh's grandfather (Aman's great-grandfather). Given the context of the great-grandfather's age being 105, it is more logical that Ram Singh's age is \(\frac{1}{3}\) of his *own* grandfather (Aman's great-grandfather).
So, Ram Singh's current age is:
\[ \text{Ram Singh's current age} = \frac{1}{3} \times \text{Aman's great-grandfather's age} \]\[ \text{Ram Singh's current age} = \frac{1}{3} \times 105 = 35 \text{ years} \]Five years ago, when Aman was born, Ram Singh's mother was double the age of Ram Singh.
Five years ago, Ram Singh's age was \(35 - 5 = 30\) years.
Five years ago, Ram Singh's mother's age was \(2 \times 30 = 60\) years.
So, Ram Singh's mother's current age is \(60 + 5 = 65\) years.
Aman was born 5 years ago. Therefore, Aman's current age is 5 years.
Difference = Ram Singh's current age - Aman's current age
\[ \text{Difference} = 35 - 5 = 30 \text{ years} \]B) 30 Years is the correct difference in age between Aman and Ram Singh.
\(\frac{3\sqrt{5}+\sqrt{125}}{\sqrt{80}+6\sqrt{5}}\)is:
a natural number
an irrational number
a rational number
an integer
The value of \(\sqrt{4}\) is:
The question asks for the value of \(\sqrt{4}\). This involves understanding the definition of the principal square root of a non-negative number.
Correct Option: B) only 2
The question asks about the number of times India reached the ICC Men's Cricket World Cup (ODI) finals up to 2021. It's important to recall India's history in the tournament finals.
Correct Option: D) Three
The question asks to identify the person who popularized the term "Megalopolis" in 1957. This is a factual recall question related to urban geography and the history of urban planning concepts.
Correct Option: D) Jean Gottman
To find the diagonal of a rectangle, we first need to determine its length and width. We are given the area and perimeter of the rectangle. We can set up a system of two equations with two variables (length and width) and solve for them. Once the length and width are known, the diagonal can be found using the Pythagorean theorem.
Let the length of the rectangle be \(l\) units and the width be \(w\) units.
The area of a rectangle is given by the formula: \(\text{Area} = l \times w\).
Given Area = 60 units, so we have:
\[ lw = 60 \quad \text{(Equation 1)} \]The perimeter of a rectangle is given by the formula: \(\text{Perimeter} = 2(l + w)\).
Given Perimeter = 34 units, so we have:
\[ 2(l + w) = 34 \]Divide the perimeter equation by 2:
\[ l + w = \frac{34}{2} \] \[ l + w = 17 \quad \text{(Equation 2)} \]Now we have a system of two equations:
1) \(lw = 60\)
2) \(l + w = 17\)
From Equation 2, we can express \(l\) in terms of \(w\):
\[ l = 17 - w \]Substitute this expression for \(l\) into Equation 1:
\[ (17 - w)w = 60 \] \[ 17w - w^2 = 60 \] \[ w^2 - 17w + 60 = 0 \]This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring the quadratic equation:
We need two numbers that multiply to 60 and add up to -17. These numbers are -12 and -5.
\[ (w - 12)(w - 5) = 0 \]So, \(w = 12\) or \(w = 5\).
If \(w = 12\), then from \(l = 17 - w\), we get \(l = 17 - 12 = 5\).
If \(w = 5\), then from \(l = 17 - w\), we get \(l = 17 - 5 = 12\).
So, the dimensions of the rectangle are 12 units by 5 units (length and width can be interchanged).
The diagonal of a rectangle (\(d\)) can be found using the Pythagorean theorem, as the diagonal forms the hypotenuse of a right-angled triangle with the length and width as its legs:
\[ d^2 = l^2 + w^2 \] \[ d^2 = 12^2 + 5^2 \] \[ d^2 = 144 + 25 \] \[ d^2 = 169 \] \[ d = \sqrt{169} \] \[ d = 13 \text{ units} \]Correct Option: B) 13 units
The question asks to identify the first Governor-General of independent India. This requires knowledge of the history of India's independence and the key figures involved in the transfer of power.
B) Lord Mountbatten was indeed the last Viceroy of British India and subsequently became the first Governor-General of independent India, serving from 1947 to 1948.
The problem asks us to find the fourth term in an analogy, where the relationship between the first and second terms must be applied to the third term to find the fourth. This is a letter-based analogy, so we need to identify the pattern or rule that transforms the first term (EIOR) into the second term (HFRO).
D) MITT — Following the established pattern of +3, -3, +3, -3 for the letter positions, JLQW transforms into MITT.