This question asks us to evaluate two arguments in relation to a given statement. We need to determine if each argument strengthens or weakens the statement. The statement is: "Schools should encourage self-discipline among children of all ages."
A) II weakens while I strengthens the statement.
The value of \(\sqrt{1225}\) + \(\sqrt{6.25}\) + \(\sqrt{12.25}\) – 8 is ________.
To solve this problem, we need to evaluate each square root term individually and then perform the addition and subtraction operations in the given expression. The key is to recognize perfect squares or convert decimal square roots into fractions to simplify calculations.
We need to find the number whose square is 1225. We know that \(30^2 = 900\) and \(40^2 = 1600\). Since 1225 ends in 5, its square root must end in 5. Thus, the square root is 35.
\[ \sqrt{1225} = 35 \]We can write 6.25 as a fraction:
\[ 6.25 = \frac{625}{100} \]\[ \sqrt{6.25} = \sqrt{\frac{625}{100}} = \frac{\sqrt{625}}{\sqrt{100}} \]\[ \sqrt{625} = 25 \quad \text{and} \quad \sqrt{100} = 10 \]\[ \sqrt{6.25} = \frac{25}{10} = 2.5 \]We can write 12.25 as a fraction:
\[ 12.25 = \frac{1225}{100} \]\[ \sqrt{12.25} = \sqrt{\frac{1225}{100}} = \frac{\sqrt{1225}}{\sqrt{100}} \]\[ \sqrt{1225} = 35 \quad \text{and} \quad \sqrt{100} = 10 \]\[ \sqrt{12.25} = \frac{35}{10} = 3.5 \]The given expression is \(\sqrt{1225} + \sqrt{6.25} + \sqrt{12.25} - 8\).
Substitute the calculated values:
\[ 35 + 2.5 + 3.5 - 8 \]First, add the positive terms:
\[ 35 + 2.5 + 3.5 = 35 + 6.0 = 41 \]Then, subtract 8:\[ 41 - 8 = 33 \]A) 33 is the correct answer, as derived from the step-by-step calculation.
The question asks to identify the incorrect statement regarding Mendeleev's Periodic Table. We need to recall the fundamental principles and characteristics of Mendeleev's classification of elements.
B) The properties of elements are the periodic functions of their atomic numbers.
This statement describes the Modern Periodic Law, proposed by Moseley, not Mendeleev's Periodic Law. Mendeleev's Periodic Law states that the properties of elements are a periodic function of their atomic masses. Therefore, this statement is incorrect in the context of Mendeleev's Periodic Table.
The value of \(\left(\frac{y^{b+c}}{y^{2a}}\right)\left(\frac{y^{c+a}}{y^{2b}}\right)\left(\frac{y^{a+b}}{y^{2c}}\right)\) is ________
The problem requires simplifying an expression involving exponents. We will use the laws of exponents, specifically the rules for division of powers with the same base \(\left(\frac{a^m}{a^n} = a^{m-n}\right)\) and multiplication of powers with the same base \(\left(a^m \cdot a^n = a^{m+n}\right)\).
First, simplify each term in the given expression using the rule \(\frac{a^m}{a^n} = a^{m-n}\):
\[\left(\frac{y^{b+c}}{y^{2a}}\right) = y^{(b+c) - 2a} = y^{b+c-2a}\]\[\left(\frac{y^{c+a}}{y^{2b}}\right) = y^{(c+a) - 2b} = y^{c+a-2b}\]\[\left(\frac{y^{a+b}}{y^{2c}}\right) = y^{(a+b) - 2c} = y^{a+b-2c}\]Now, multiply these simplified terms. When multiplying powers with the same base, we add their exponents:
\[y^{b+c-2a} \cdot y^{c+a-2b} \cdot y^{a+b-2c} = y^{(b+c-2a) + (c+a-2b) + (a+b-2c)}\]Next, sum the exponents:
\[(b+c-2a) + (c+a-2b) + (a+b-2c)\]Combine like terms:
So, the sum of the exponents is \(0 + 0 + 0 = 0\).
Substitute the sum of the exponents back into the expression:
\[y^0\]Recall that any non-zero number raised to the power of 0 is 1 (\(a^0 = 1\), where \(a \neq 0\)). Assuming \(y \neq 0\), the value of the expression is 1.
Correct Option: D) 1
The resistance of a wire depends on its resistivity, length, and cross-sectional area. Since the wires are identical in dimension (same length and cross-sectional area), their resistance will be directly proportional to their resistivity. We need to compare the resistivities of tungsten, manganin, and hard rubber to determine which has the least or highest resistance.
Recall the formula for resistance: The resistance \(R\) of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]where \(\rho\) is the resistivity of the material, \(L\) is the length of the wire, and \(A\) is its cross-sectional area.
Analyze the given information: The problem states that the three wires (tungsten, manganin, and hard rubber) are identical in dimension. This means their length \(L\) and cross-sectional area \(A\) are the same for all three.
Relate resistance to resistivity: Since \(L\) and \(A\) are constant, the resistance \(R\) is directly proportional to the resistivity \(\rho\):
\[ R \propto \rho \]Therefore, the material with the lowest resistivity will have the lowest resistance, and the material with the highest resistivity will have the highest resistance.
Compare the resistivities of the materials:
Order the resistivities: Based on the typical values:
\[ \rho_{\text{Tungsten}} < \rho_{\text{Manganin}} < \rho_{\text{Hard Rubber}} \]Numerically:
\[ 5.6 \times 10^{-8} \ \Omega \cdot \text{m} < 4.3 \times 10^{-7} \ \Omega \cdot \text{m} < \approx 10^{13} \ \Omega \cdot \text{m} \]Determine the resistance order: Since resistance is proportional to resistivity, the order of resistance will be the same:
\[ R_{\text{Tungsten}} < R_{\text{Manganin}} < R_{\text{Hard Rubber}} \]This means tungsten has the least resistance, and hard rubber has the highest resistance.
B) Tungsten is least — As determined in the step-by-step analysis, tungsten has the lowest resistivity among the three materials, and since the dimensions are identical, it will have the least resistance.
The question asks about the orientation of magnetic field lines around a straight current-carrying conductor. This can be determined using the Right-Hand Thumb Rule (also known as Maxwell's corkscrew rule).
B) perpendicular to the direction of current
According to the Right-Hand Thumb Rule, the magnetic field lines form concentric circles around the straight conductor, lying in planes perpendicular to the conductor's length. Since the current flows along the length of the conductor, the magnetic field lines are perpendicular to the direction of the current.
Which of the following states bagged the first prize in the 3rd National Water Awards presented by then President of India, Shri Ram Nath Kovind in March 2022?
The question asks about the winner of the first prize in the 3rd National Water Awards presented in March 2022. This is a factual recall question related to current affairs and government awards.
Correct Option: D) Uttar Pradesh bagged the first prize in the "Best State" category at the 3rd National Water Awards, presented in March 2022 by then President of India, Shri Ram Nath Kovind.
The question asks about the recommendations of the Swaran Singh Committee regarding the inclusion of certain features in the Constitution of India. This requires knowledge of Indian constitutional history, specifically the context of the Emergency period and the amendments made during that time.
Correct Option: A) Fundamental duties. The Swaran Singh Committee was set up in 1976 to recommend amendments to the Constitution. Its most significant recommendation was the inclusion of a separate chapter on Fundamental Duties in the Constitution. These duties were subsequently incorporated into the Constitution as Part IV-A (Article 51-A) by the 42nd Constitutional Amendment Act, 1976.
This problem involves solving a system of linear equations derived from age ratios. We will set up equations based on the given present age ratio and the ratio after 9 years, then solve for the unknown variable to find Keerthi's present age.
Define variables based on the present age ratio:
Let the present age of Kamal be \(3x\) years.
Let the present age of Keerthi be \(5x\) years.
This is derived from the given ratio of their present ages, 3:5.
Express their ages after 9 years:
After 9 years, Kamal's age will be \(3x + 9\) years.
After 9 years, Keerthi's age will be \(5x + 9\) years.
Formulate an equation based on the future age ratio:
We are given that after 9 years, the ratio of their ages will be 3:4. So, we can write the equation:
Solve the equation for \(x\):
Cross-multiply the terms:
Distribute the numbers:
\[ 12x + 36 = 15x + 27 \]Rearrange the terms to isolate \(x\):
\[ 36 - 27 = 15x - 12x \] \[ 9 = 3x \]Divide by 3:
\[ x = \frac{9}{3} \] \[ x = 3 \]Calculate Keerthi's present age:
Keerthi's present age was defined as \(5x\). Substitute the value of \(x = 3\):
A) 15 years is the present age of Keerthi, as calculated.
The question asks to identify a property that is NOT necessarily true for a rhombus. We need to examine each given option and determine if it is a defining characteristic or a derived property of all rhombuses, or if it is a property that only holds true for a specific type of rhombus (like a square).
Correct Option: C) Diagonals are equal.
This property is not necessarily true for all rhombuses. It is only true for a square, which is a special type of rhombus where all angles are right angles. In a general rhombus, the diagonals are usually of different lengths.