Identify the correct statement(s) from among the following.
(a) Resistivity of a material is dependent on the nature of the material.
(b) Resistivity of a material is independent of the area of cross section.
(c) Resistivity of a pure metal is more than its alloy.
Only (a)
Only (b)
Both (a) and (b)
Both (b) and (c)
This question tests the understanding of resistivity, its definition, and factors affecting it. We need to evaluate each statement based on the fundamental principles of electrical resistance and resistivity.
Resistivity (\(\rho\)) is an intrinsic property of a material. It quantifies how strongly a given material opposes the flow of electric current. Different materials (e.g., copper, iron, glass) have different atomic structures and electron configurations, which directly affect their ability to conduct electricity. Therefore, resistivity is indeed dependent on the nature of the material. This statement is correct.
The resistance (\(R\)) of a conductor is given by the formula:
\[ R = \rho \frac{L}{A} \]where \(\rho\) is resistivity, \(L\) is the length of the conductor, and \(A\) is its area of cross-section. From this formula, we can see that resistance \(R\) depends on the area of cross-section \(A\), but resistivity \(\rho\) itself is a material constant (at a given temperature) and does not depend on the dimensions (length or area of cross-section) of the object made from that material. This statement is correct.
Alloys are generally formed by mixing two or more metals or a metal with a non-metal. The presence of impurity atoms (in the case of alloys) disrupts the regular lattice structure of the pure metal. This disruption leads to more scattering of electrons, which in turn increases the resistivity. For example, nichrome (an alloy of nickel, chromium, and iron) has a much higher resistivity than its constituent pure metals. Therefore, the resistivity of an alloy is generally more than that of its constituent pure metals, not less. This statement is incorrect.
Based on the analysis, statements (a) and (b) are correct, while statement (c) is incorrect.
Correct Option: C) Both (a) and (b)
To find the coefficient of \(x^4y^3\) in the given expansion, we first expand each factor separately and then multiply the resulting terms. We only need to identify combinations of terms from each expansion that multiply to produce \(x^4y^3\).
We need to find terms from \((9x^2 - 12xy + 4y^2)\) that, when multiplied by terms from \((x^4 + 2x^2y^3 + y^6)\), result in \(x^4y^3\).
Only one combination yields \(x^4y^3\). The coefficient is \(18\).
Correct Option: B) 18
The problem asks us to identify numbers in a given series that are immediately preceded by an odd number and immediately followed by an odd number. We need to scan the series, identify odd and even numbers, and then check for the specified pattern.
B) 3
The laws of reflection hold true for which of the following mirror(s)?
(A) Concave mirror
(B) Convex mirror
(C) Plane mirror
The question asks about the applicability of the laws of reflection to different types of mirrors. We need to recall the fundamental principles of reflection and how they apply to various mirror surfaces.
B) A, B and C — The laws of reflection are fundamental and apply to all types of mirrors, including concave, convex, and plane mirrors. The curvature of the mirror changes how the reflected rays converge or diverge, but the basic laws of reflection (\( \angle i = \angle r \) and coplanarity) still hold true at every point on the mirror's surface.
To solve this problem, we need to compare the original position of each digit in the given number with its position after the digits are rearranged in descending order. We will list the original number, then sort its digits in descending order to form a new number, and finally compare the positions.
A) one — As determined in the step-by-step analysis, only one digit (the digit '4') retains its original position after the rearrangement.
The problem asks us to calculate the selling price required to achieve a 10% profit on a fan whose cost price is given. We need to first identify the cost price and then apply the profit percentage to find the desired selling price.
The information about the fan being "sold for Rs 800 less than the cost price" is extraneous and meant to distract. It describes a past transaction (a loss) and is not relevant to the current objective of finding the selling price for a 10% profit.
Correct Option: A) Rs 5,280
The problem provides two ratio equations involving variables \(x\) and \(y\). We need to solve these simultaneous equations to find the values of \(x\) and \(y\).
The first ratio is \(x : 6 = 6 : y\). This can be written as a proportion:
\[ \frac{x}{6} = \frac{6}{y} \]Cross-multiply to get:
\[ xy = 6 \times 6 \]\[ xy = 36 \quad \text{(Equation 1)} \]The second ratio is \(x : y = 6 : 6\). This can be written as:
\[ \frac{x}{y} = \frac{6}{6} \]The right side simplifies to 1:
\[ \frac{x}{y} = 1 \]\[ x = y \quad \text{(Equation 2)} \]Since \(x = y\), we can replace \(y\) with \(x\) in Equation 1:
\[ x \times x = 36 \]\[ x^2 = 36 \]Take the square root of both sides:
\[ x = \sqrt{36} \]\[ x = 6 \]Since \(x = y\) and \(x = 6\), then:
\[ y = 6 \]Check if \(x=6\) and \(y=6\) satisfy both original ratio conditions:
Both conditions are satisfied.
Correct Option: A) 6 and 6
In this question, a statement is followed by two conclusions. Which of the two conclusions is/are true with respect to the statement?
Statement:
F = E > D = A > Z < S = O
Conclusions:
I. F > D
II. D ≤ O
Only conclusion I is true
Neither conclusion I nor II is true
Both conclusions I and II are true
Only conclusion II is true
This question involves analyzing a statement of inequalities and determining the truthfulness of two given conclusions. We need to trace the relationships between the variables in the statement to verify each conclusion.
This question is based on the five, three-digit numbers given below:
(Left) 448 455 275 563 215 (Right)
What will be the resultant if the third digit of the largest number is added to the second digit of the smallest number?
To solve this problem, we need to identify the largest and smallest numbers from the given set of three-digit numbers. Once identified, we will extract the third digit of the largest number and the second digit of the smallest number and then add them together to find the resultant value.
B) 4 is the correct answer, as the sum of the third digit of the largest number (3) and the second digit of the smallest number (1) is 4.
The policy norms for Mission Vatsalya scheme implemented by the Ministry of Women and Child Development have been applicable from the 1st of which month in 2022?
The question asks about the implementation date of the policy norms for the Mission Vatsalya scheme. This requires knowledge of recent government scheme updates and their effective dates.
A) April — The policy norms for the Mission Vatsalya scheme, implemented by the Ministry of Women and Child Development, became applicable from April 1, 2022. This date marks the official commencement of the new guidelines for the scheme.