This problem involves calculating the work done by pipes (filling a cistern) over time. We'll determine the filling rate of each pipe, then use the total time the cistern was filled and the fact that one pipe was closed prematurely to find the duration for which both pipes operated together.
A) 8 minutes. This is the duration for which pipe B was open, after which it was closed, allowing pipe A to complete the remaining work.
This question asks us to identify the correct statement regarding the redox reaction: \( \text{ZnO} + \text{C} \rightarrow \text{Zn} + \text{CO} \). We need to understand the definitions of oxidation and reduction in terms of oxygen transfer.
Correct Option: A) ZnO is reduced to Zn. This statement is correct because ZnO loses oxygen to become Zn.
This problem involves ratios and basic algebraic equations. We are given the ratio of incomes and expenditures of two friends, along with their monthly savings. We need to find the income of one of the friends. We will set up equations based on the given ratios and the relationship between income, expenditure, and savings (Income = Expenditure + Savings).
Let the monthly income of Anuj be \(5x\) and Mathew be \(7x\), based on the given ratio of 5:7.
Let the monthly expenditure of Anuj be \(2y\) and Mathew be \(3y\), based on the given ratio of 2:3.
We know that Income - Expenditure = Savings. Both friends save Rs 10,000 every month.
For Anuj:
\[ 5x - 2y = 10000 \quad \text{(Equation 1)} \]For Mathew:
\[ 7x - 3y = 10000 \quad \text{(Equation 2)} \]We have a system of two linear equations with two variables \(x\) and \(y\). We can solve this system. Multiply Equation 1 by 3 and Equation 2 by 2 to eliminate \(y\):
Multiply Equation 1 by 3:
\[ 3 \times (5x - 2y) = 3 \times 10000 \]\[ 15x - 6y = 30000 \quad \text{(Equation 3)} \]Multiply Equation 2 by 2:
\[ 2 \times (7x - 3y) = 2 \times 10000 \]\[ 14x - 6y = 20000 \quad \text{(Equation 4)} \]Subtract Equation 4 from Equation 3:
\[ (15x - 6y) - (14x - 6y) = 30000 - 20000 \]\[ 15x - 14x - 6y + 6y = 10000 \]\[ x = 10000 \]Now that we have the value of \(x\), we can find the income of Anuj. The income of Anuj is \(5x\).
\[ \text{Anuj's Income} = 5 \times 10000 = 50000 \]C) Rs 50,000 is the correct income of Anuj, calculated using the given ratios and savings.
The equation x2−6x+9=0 can be written as:
The problem asks us to rewrite the given quadratic equation \(x^2 - 6x + 9 = 0\) in a factored form. We can achieve this by recognizing the pattern of a perfect square trinomial or by factoring the quadratic expression.
In our equation, \(x^2 - 6x + 9\):
Since \(x^2 - 6x + 9\) matches the form \(a^2 - 2ab + b^2\) with \(a=x\) and \(b=3\), it can be written as \((x - 3)^2\).
D) \((x - 3)^2 = 0\) is the correct representation of the given equation because \((x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9\).
When white light passes through a glass prism, it splits into its constituent colors, a phenomenon known as dispersion. This happens because different colors of light travel at different speeds in the prism material, leading to different refractive indices and thus different angles of deviation. The extent of deviation depends on the wavelength of the light.
Correct Option: C) violet
The question asks about the aim of the Digital Government Mission launched by the Ministry of Electronics and Information Technology in January 2022. To answer this, one needs to recall the core objectives and features of this specific mission.
B) To have services proactively anticipate citizens’ needs before they approach the services. This aligns perfectly with the stated objectives of the Digital Government Mission, which emphasizes a proactive, citizen-centric approach to governance through digital means.
This problem involves concepts of boats and streams. We need to find the speed of the boat in still water given the time taken for downstream and upstream journeys and the speed of the water. We will use the formulas for downstream speed and upstream speed, and the fact that the distance traveled in both directions is the same.
Let the speed of the boat in still water be \(v_b\) km/h.
Given the speed of the water (current) is \(v_c = 3\) km/h.
When the boat goes downstream (from P to Q), its speed is the sum of the boat's speed in still water and the speed of the current:
\[ v_{\text{downstream}} = v_b + v_c = (v_b + 3) \text{ km/h} \]The time taken to go downstream from P to Q is \(t_{\text{downstream}} = 3\) hours.
When the boat returns upstream (from Q to P), its speed is the difference between the boat's speed in still water and the speed of the current:
\[ v_{\text{upstream}} = v_b - v_c = (v_b - 3) \text{ km/h} \]The time taken to return upstream from Q to P is \(t_{\text{upstream}} = 4\) hours.
The distance from P to Q is the same as the distance from Q to P. We know that \(\text{Distance} = \text{Speed} \times \text{Time}\).
So, distance downstream = distance upstream:
\[ v_{\text{downstream}} \times t_{\text{downstream}} = v_{\text{upstream}} \times t_{\text{upstream}} \]Substitute the values into the equation:
\[ (v_b + 3) \times 3 = (v_b - 3) \times 4 \]Now, solve for \(v_b\):
\[ 3v_b + 9 = 4v_b - 12 \] \[ 9 + 12 = 4v_b - 3v_b \] \[ 21 = v_b \]Therefore, the speed of the boat in still water is 21 km/h.
Correct Option: C) 21 km/h
Find the value of \(\frac{1}{(1+\cot^{2}x)^{2}}+\frac{\tan^{2}x}{(1+\tan^{2}x)^{2}}+\frac{1}{1+\tan^{2}x}\) is:
To solve this problem, we will use fundamental trigonometric identities to simplify the given expression. The key identities are:
Substitute the identities \(1 + \cot^2 x = \csc^2 x\) and \(1 + \tan^2 x = \sec^2 x\) into the given expression:
\[ E = \frac{1}{(1+\cot^{2}x)^{2}}+\frac{\tan^{2}x}{(1+\tan^{2}x)^{2}}+\frac{1}{1+\tan^{2}x} \] \[ E = \frac{1}{(\csc^{2}x)^{2}}+\frac{\tan^{2}x}{(\sec^{2}x)^{2}}+\frac{1}{\sec^{2}x} \] \[ E = \frac{1}{\csc^{4}x}+\frac{\tan^{2}x}{\sec^{4}x}+\frac{1}{\sec^{2}x} \]Rewrite the terms using \(\sin x = 1/\csc x\) and \(\cos x = 1/\sec x\):
\[ E = \sin^{4}x + \frac{\tan^{2}x}{\sec^{4}x} + \cos^{2}x \]Substitute \(\tan x = \frac{\sin x}{\cos x}\) into the second term:
\[ E = \sin^{4}x + \frac{\left(\frac{\sin x}{\cos x}\right)^{2}}{\sec^{4}x} + \cos^{2}x \] \[ E = \sin^{4}x + \frac{\frac{\sin^{2}x}{\cos^{2}x}}{\frac{1}{\cos^{4}x}} + \cos^{2}x \]Simplify the second term:
\[ \frac{\frac{\sin^{2}x}{\cos^{2}x}}{\frac{1}{\cos^{4}x}} = \frac{\sin^{2}x}{\cos^{2}x} \times \cos^{4}x = \sin^{2}x \cos^{2}x \]Substitute this back into the expression for \(E\):
\[ E = \sin^{4}x + \sin^{2}x \cos^{2}x + \cos^{2}x \]Factor out \(\sin^{2}x\) from the first two terms:
\[ E = \sin^{2}x(\sin^{2}x + \cos^{2}x) + \cos^{2}x \]Apply the Pythagorean identity \(\sin^{2}x + \cos^{2}x = 1\):
\[ E = \sin^{2}x(1) + \cos^{2}x \] \[ E = \sin^{2}x + \cos^{2}x \]Apply the Pythagorean identity again:
\[ E = 1 \]D) The simplified value of the expression is 1.
L, G, K, P, H and Z live on eight different floors of the same multistorey building, with 2 floors being vacant. The lowermost floor in the building is numbered 1, the floor above it is numbered 2 and so on till the topmost floor is numbered 8. H lives on floor No. 1, K on No. 2, and P on No. 8. One immediate floor above and one immediate floor below Z are vacant. G’s floor is immediately above the floor on which L lives and G’s floor is also immediately below P’s floor.
Who lives on floor No. 4?
This is a logical reasoning problem involving arrangement of people on different floors of a building. We need to deduce the floor assignments based on the given clues and identify who lives on floor No. 4.
List the floors: There are 8 floors, numbered 1 to 8.
Identify fixed positions:
Let's represent this in a table:
| Floor No. | Resident |
| 8 | P |
| 7 | |
| 6 | |
| 5 | |
| 4 | |
| 3 | |
| 2 | K |
| 1 | H |
Use the clue about G and L: "G's floor is immediately above the floor on which L lives and G's floor is also immediately below P's floor."
Update the table:
| Floor No. | Resident |
| 8 | P |
| 7 | G |
| 6 | L |
| 5 | |
| 4 | |
| 3 | |
| 2 | K |
| 1 | H |
Use the clue about Z and vacant floors: "One immediate floor above and one immediate floor below Z are vacant."
Update the table:
| Floor No. | Resident |
| 8 | P |
| 7 | G |
| 6 | L |
| 5 | Vacant |
| 4 | Z |
| 3 | Vacant |
| 2 | K |
| 1 | H |
Final check: All conditions are met. H (1), K (2), P (8). G (7) is below P (8), L (6) is below G (7). Z (4) has vacant floors (3 and 5) immediately above and below. Two floors are vacant (3 and 5). All 6 people are placed.
Answer the question: Who lives on floor No. 4? From our table, Z lives on floor No. 4.
The question asks for the name of the sacred text of Christianity, which is described as having two main parts: the Old Testament (Hebrew text from Jesus's time) and the New Testament (writings about Jesus Christ and the early church). We need to identify the correct term among the given options.
Correct Option: B) The Bible is the sacred text of Christianity, consisting of the Old Testament and the New Testament, as described in the question.