The question asks about the number of elements discovered when Newlands' Law of Octaves was proposed. This requires recalling a specific historical fact related to the development of the periodic table.
B) 56. John Newlands proposed his Law of Octaves in 1865, and at that time, 56 elements were known.
The problem asks us to identify the relationship between the numbers in the given sets and then find an option that follows the same relationship. We need to analyze the given pairs \((4, 24)\) and \((11, 143)\) to deduce the rule.
Let the first number be \(x\) and the second number be \(y\).
Here, \(x = 4\) and \(y = 24\).
We can observe that \(24\) is a multiple of \(4\). Specifically, \(24 = 4 \times 6\).
We can also express \(6\) in terms of \(x\). \(6 = 4 + 2\), or \(6 = x + 2\).
So, the relationship could be \(y = x \times (x + 2)\).
Here, \(x = 11\) and \(y = 143\).
Using the derived relationship \(y = x \times (x + 2)\):
\[ 11 \times (11 + 2) = 11 \times 13 = 143 \]The relationship holds true for both given sets.
We need to find an option \((x, y)\) where \(y = x \times (x + 2)\).
Here \(x = 9\). According to the rule, \(y\) should be \(9 \times (9 + 2) = 9 \times 11 = 99\). Since \(102 \neq 99\), this option is incorrect.
Here \(x = 7\). According to the rule, \(y\) should be \(7 \times (7 + 2) = 7 \times 9 = 63\). Since \(42 \neq 63\), this option is incorrect.
Here \(x = 5\). According to the rule, \(y\) should be \(5 \times (5 + 2) = 5 \times 7 = 35\). Since \(40 \neq 35\), this option is incorrect.
Here \(x = 2\). According to the rule, \(y\) should be \(2 \times (2 + 2) = 2 \times 4 = 8\). This matches the given second number in the option.
Correct Option: D) (2, 8)
This is a syllogism problem that can be solved using Venn diagrams or by analyzing the relationships between the categories. We will draw Venn diagrams for the given statements and then check if the conclusions logically follow.
C) Both conclusions I and II follow.
The problem asks us to find a relationship between pairs of numbers and apply that same relationship to a new number. We are given two pairs: 13 : 195 and 2 : 8. We need to find the number that relates to 12 in the same way. The rule states that operations should be performed on the whole numbers, not on their constituent digits.
Analyze the first pair: 13 : 195
Let's try to find a mathematical operation that transforms 13 into 195. A common pattern in such problems involves squaring the number, cubing it, or multiplying it by a related number.
Analyze the second pair: 2 : 8
Now, let's apply the same relationship \(n^2 + 2n\) to the number 2 to see if it results in 8.
The established pattern is \(n : n^2 + 2n\).
Apply the pattern to the third number: 12 : ?
We need to find the number that corresponds to 12 using the pattern \(n^2 + 2n\).
B) 168 is the correct answer, as it follows the established pattern \(n^2 + 2n\).
If \( \left(\frac{12}{7}\right)^{5} \times \left(\frac{7}{12}\right)^{8} \times \left(\frac{12}{7}\right)^{16} = \left(\frac{7}{12}\right)^{2m+10} \), then m =?
-1
\(\frac{23}{2}\)
\(\frac{19}{23}\)
\(-\frac{23}{2}\)
To solve this problem, we need to simplify the given equation using the laws of exponents. The key is to express all terms with the same base. We will use the property that \[\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\] and \( a^m \times a^n = a^{m+n} \).
The given equation is:
\[ \left(\frac{12}{7}\right)^{5} \times \left(\frac{7}{12}\right)^{8} \times \left(\frac{12}{7}\right)^{16} = \left(\frac{7}{12}\right)^{2m+10} \]We can convert \( \left(\frac{7}{12}\right) \) to \( \left(\frac{12}{7}\right)^{-1} \) or vice versa. Let's aim to have all terms with the base \( \left(\frac{12}{7}\right) \) on the left side and \( \left(\frac{7}{12}\right) \) on the right side.
Rewrite \( \left(\frac{7}{12}\right)^{8} \) as \( \left(\frac{12}{7}\right)^{-8} \).
The equation becomes:
\[ \left(\frac{12}{7}\right)^{5} \times \left(\frac{12}{7}\right)^{-8} \times \left(\frac{12}{7}\right)^{16} = \left(\frac{7}{12}\right)^{2m+10} \]Using the rule \( a^m \times a^n = a^{m+n} \):
\[ \left(\frac{12}{7}\right)^{5 + (-8) + 16} = \left(\frac{7}{12}\right)^{2m+10} \] \[ \left(\frac{12}{7}\right)^{5 - 8 + 16} = \left(\frac{7}{12}\right)^{2m+10} \] \[ \left(\frac{12}{7}\right)^{13} = \left(\frac{7}{12}\right)^{2m+10} \]To compare the exponents, the bases must be the same. We can rewrite \( \left(\frac{12}{7}\right)^{13} \) as \( \left(\frac{7}{12}\right)^{-13} \).
\[ \left(\frac{7}{12}\right)^{-13} = \left(\frac{7}{12}\right)^{2m+10} \]Since the bases are now equal, we can equate the exponents:
\[ -13 = 2m + 10 \]Subtract 10 from both sides:
\[ -13 - 10 = 2m \] \[ -23 = 2m \]Divide by 2:
\[ m = -\frac{23}{2} \]Correct Option: D) \(-\frac{23}{2}\)
The problem requires us to modify a given number based on specific rules for odd and even digits, then identify two specific digits in the new number, and finally calculate their sum.
Correct Option: D) 8
This problem involves directions and distances, which can be solved by visualizing the path taken and using the Pythagorean theorem. We need to determine the unknown distance traveled in the north direction by setting up a coordinate system or drawing a diagram.
A) 4 km is the correct answer. Based on the step-by-step analysis, the distance \(x\) from A to B is 4 km.
Rajesh has 140 litres of Oil A and 220 litres of Oil B. He fills a number of identical containers with the two types of oil in a manner that each container has only one type of oil, and all containers are completely filled. What can be the maximum volume (in litres) of each container that Rajesh uses, so that all the oil that Rajesh has, of both types, can be poured into these containers?
To find the maximum volume of each container, we need to find the largest possible volume that can divide both the total volume of Oil A and the total volume of Oil B without leaving any remainder. This is equivalent to finding the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of the two given volumes.
Volume of Oil A = \(140\) litres
Volume of Oil B = \(220\) litres
Each container must have only one type of oil, and all containers must be completely filled. This means the volume of each container must be a factor of the total volume of Oil A and also a factor of the total volume of Oil B.
We need to find the maximum volume of each container. Therefore, we need to find the Highest Common Factor (HCF) of \(140\) and \(220\).
For Oil A:
\[ 140 = 2 \times 70 = 2 \times 2 \times 35 = 2^2 \times 5 \times 7 \]For Oil B:
\[ 220 = 2 \times 110 = 2 \times 2 \times 55 = 2^2 \times 5 \times 11 \]The HCF is found by taking the lowest power of all common prime factors.
Common prime factors are \(2\) and \(5\).
Lowest power of \(2\) is \(2^2\).
Lowest power of \(5\) is \(5^1\).
\[ \text{HCF}(140, 220) = 2^2 \times 5 = 4 \times 5 = 20 \]The maximum volume of each container that Rajesh can use is \(20\) litres.
Correct Option: B) 20
The question asks for an important reason contributing to the success of population control programs. This requires understanding that the effectiveness of any program ultimately depends on its acceptance and adoption by the target population.
Correct Option: C) Acceptance of contraceptive methods by people. The success of any population control program, regardless of the specific methods offered, fundamentally depends on the willingness of the target population to accept and utilize those methods. Without public acceptance, even the best-designed programs and most effective contraceptives will fail to achieve their objectives.
To determine the formula unit by which succeeding members of an alkane homologous series differ, we need to recall the general formula for alkanes and examine the chemical formulas of the first few members of the series.
Correct Option: D) \(\text{CH}_2\)