This problem involves simple interest. We are given the rate of interest and two different time periods, along with the difference in interest earned for these periods. We can use the simple interest formula to set up an equation and solve for the principal amount (initial sum invested).
B) Rs. 5,000 is the correct answer because the calculations show that the principal amount is Rs. 5,000.
This is a statement and conclusion type of question. We need to analyze the given statement and determine which of the conclusions logically and definitely follow from the information provided in the statement. We should not use any external knowledge beyond what is explicitly stated.
B) Only conclusion 1 follows.
The question asks to identify the Indian Chief Minister (as of 2018) whose father was a 'Maharaja'. This requires knowledge of the family backgrounds of prominent Indian political figures, particularly those who were Chief Ministers around 2018.
Correct Option: D) Amarinder Singh
Captain Amarinder Singh served as the Chief Minister of Punjab from 2002 to 2007 and again from 2017 to 2021. His father was Maharaja Yadavindra Singh, the last ruling Maharaja of the princely state of Patiala. Therefore, Amarinder Singh's father was indeed a 'Maharaja'.
The surface areas of three faces of a cuboid sharing a vertex are 20 m², 32 m² and 40 m². What is the volume of the cuboid?
To find the volume of a cuboid, we need its length, width, and height. The problem provides the surface areas of three faces that share a common vertex. We can use these areas to determine the dimensions and then calculate the volume.
Let the dimensions of the cuboid be length \(l\), width \(w\), and height \(h\).
The three faces sharing a vertex have areas given by \(l \times w\), \(w \times h\), and \(h \times l\). Let's assign the given areas to these products:
To find the volume \(V\) of the cuboid, which is \(V = lwh\), we can multiply the three equations:
\[ (lw) \times (wh) \times (hl) = 20 \times 32 \times 40 \]\[ l^2 w^2 h^2 = 25600 \]Recognize that \(l^2 w^2 h^2\) is the square of the volume \((lwh)^2\):
\[ (lwh)^2 = 25600 \]Take the square root of both sides to find the volume \(V = lwh\):
\[ lwh = \sqrt{25600} \]\[ lwh = 160 \]Therefore, the volume of the cuboid is \(160 \text{ m}^3\).
D) \(160 \text{ m}^3\) is the correct volume calculated from the given face areas.
This question tests fundamental knowledge of chemical reactions, specifically the reaction between metals and acids. Understanding the general reaction pattern for these substances is key to identifying the correct products.
Here, zinc (a metal) reacts with hydrochloric acid to produce zinc chloride (a salt) and hydrogen gas.
A) A salt and Hydrogen. This is the characteristic product of the reaction between most metals and acids. The metal displaces hydrogen from the acid, forming a salt and releasing hydrogen gas.
The problem asks us to find the missing term in a letter series. We need to identify the pattern in the given terms by analyzing the position of each letter in the alphabet.
B (2), D (4), ?, H (8)
The sequence of positions is 2, 4, ?, 8. This is an arithmetic progression with a common difference of 2. So, the missing number is \(4 + 2 = 6\). The 6th letter of the alphabet is F.
G (7), I (9), ?, M (13)
The sequence of positions is 7, 9, ?, 13. This is an arithmetic progression with a common difference of 2. So, the missing number is \(9 + 2 = 11\). The 11th letter of the alphabet is K.
L (12), N (14), ?, R (18)
The sequence of positions is 12, 14, ?, 18. This is an arithmetic progression with a common difference of 2. So, the missing number is \(14 + 2 = 16\). The 16th letter of the alphabet is P.
The missing letters are F, K, P. Therefore, the missing term is FKP.
C) FKP — As determined by analyzing the pattern of each letter's position in the alphabet, the first, second, and third letters each follow an arithmetic progression with a common difference of 2.
The question asks about the use of chlorine gas in the manufacturing process of common chemical compounds. We need to identify which of the given options utilizes chlorine gas in its production.
A) Bleaching powder — Chlorine gas is a key reactant in the manufacture of bleaching powder (\(\text{CaOCl}_2\)) from dry slaked lime (\(\text{Ca(OH)}_2\)).
Which of the following is a reducible fraction?
\(\frac{105}{112}\)
\(\frac{91}{15}\)
\(\frac{41}{17}\)
\(\frac{79}{26}\)
A reducible fraction is one where the numerator and the denominator share a common factor greater than 1. To identify a reducible fraction, we need to find the Greatest Common Divisor (GCD) of the numerator and the denominator. If the GCD is greater than 1, the fraction is reducible.
A) \(\frac{105}{112}\) is a reducible fraction because both \(105\) and \(112\) are divisible by \(7\).
This problem involves an algebraic identity. We are given the condition \(a + b + c = 0\) and need to find the value of \((a^3 + b^3 + c^3)^2\). We will use the special case of the sum of cubes identity when \(a + b + c = 0\).
Recall the algebraic identity for the sum of cubes:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \]We are given the condition \(a + b + c = 0\). Substitute this into the identity:
\[ a^3 + b^3 + c^3 - 3abc = (0)(a^2 + b^2 + c^2 - ab - bc - ca) \] \[ a^3 + b^3 + c^3 - 3abc = 0 \]From the above equation, we can express \(a^3 + b^3 + c^3\) as:
\[ a^3 + b^3 + c^3 = 3abc \]The problem asks for the value of \((a^3 + b^3 + c^3)^2\). Substitute the expression from step 3 into this:
\[ (a^3 + b^3 + c^3)^2 = (3abc)^2 \]Simplify the expression:
\[ (3abc)^2 = 3^2 \cdot a^2 \cdot b^2 \cdot c^2 = 9a^2b^2c^2 \]Correct Option: D) 9a²b²c²
The problem asks us to find the value of an expression involving cosecants, given an equation involving cotangents. We can use the fundamental trigonometric identity relating cotangent and cosecant, which is \(1 + \cot^2\theta = \text{cosec}^2\theta\), to transform the given equation and the expression to be evaluated into terms of the same trigonometric function.
We are given the equation:
\[ \cot^4\theta + \cot^2\theta = 3 \]We know the trigonometric identity:
\[ 1 + \cot^2\theta = \text{cosec}^2\theta \]From the identity, we can express \(\cot^2\theta\) in terms of \(\text{cosec}^2\theta\):
\[ \cot^2\theta = \text{cosec}^2\theta - 1 \]Now, substitute this expression for \(\cot^2\theta\) into the given equation. First, let's rewrite \(\cot^4\theta\) as \((\cot^2\theta)^2\):
\[ (\cot^2\theta)^2 + \cot^2\theta = 3 \]Substitute \(\cot^2\theta = \text{cosec}^2\theta - 1\) into the equation:
\[ (\text{cosec}^2\theta - 1)^2 + (\text{cosec}^2\theta - 1) = 3 \]Let's expand the terms:
\[ (\text{cosec}^4\theta - 2\text{cosec}^2\theta + 1) + (\text{cosec}^2\theta - 1) = 3 \]Combine like terms:
\[ \text{cosec}^4\theta - 2\text{cosec}^2\theta + \text{cosec}^2\theta + 1 - 1 = 3 \]\[ \text{cosec}^4\theta - \text{cosec}^2\theta = 3 \]The expression we need to find is \(\text{cosec}^4\theta - \text{cosec}^2\theta\), which we have just calculated to be 3.
D) 3 is the correct answer. By using the identity \(1 + \cot^2\theta = \text{cosec}^2\theta\) and substituting \(\cot^2\theta = \text{cosec}^2\theta - 1\) into the given equation, we directly obtain the value of the required expression.