The question asks to identify a specific campaign launched in January 2018 by the Union Minister for Environment, Forest and Climate Change, which focuses on recognizing small positive actions for environmental protection. We need to recall the major environmental initiatives launched around that time by the Ministry.
Correct Option: C) Green Good Deeds
This question is based on clock directions and spatial reasoning. Rajni's watch shows quarter to 6, which means 5:45. We need to determine the direction of the minute hand when the hour hand points towards the south. To solve this, we identify the positions of both hands and use their relative directions on the clock.
Quarter to 6 means 15 minutes before 6 o'clock.
Therefore, the time is:
5:45
At 5:45, the hour hand lies between 5 and 6, closer to 6.
It has completed three-quarters of its movement from 5 towards 6.
Therefore, the hour hand is approximately pointing towards the number 6.
At 5:45, the minute hand points exactly towards the number 9.
Each number on the clock represents five minutes.
\(9 \times 5 = 45\)
Therefore, the minute hand points towards 9.
The question states that the hour hand points towards the south.
Since the hour hand is approximately at 6, the number 6 is taken to represent south.
Accordingly, the clock directions are:
The minute hand points towards the number 9.
When 6 points south, 9 points west.
Therefore, the minute hand points towards west.
At exactly 5:45, the hour hand is positioned at:
\(5 \times 30^\circ + 45 \times 0.5^\circ = 172.5^\circ\)
The minute hand is positioned at:
\(45 \times 6^\circ = 270^\circ\)
The angle between the two hands is:
\(270^\circ - 172.5^\circ = 97.5^\circ\)
Therefore, the minute hand is slightly north of west when the hour hand points exactly south. Among the four cardinal directions provided, west is the correct choice.
Correct Option: B) West
At quarter to 6, the minute hand points towards 9, while the hour hand is close to 6. Since the hour hand points south, the minute hand points approximately west. Therefore, option B is correct.
We need to find the number of two-digit numbers (excluding 14) that, when their digits are reversed, result in a new number that is 27 greater than the original number. We can represent a two-digit number algebraically and set up an equation based on the given condition.
Let the two-digit number be \(10t + u\), where \(t\) is the tens digit and \(u\) is the units digit.
When the digits are reversed, the new number is \(10u + t\).
According to the problem, the new number is 27 greater than the original number. So, we can write the equation:
\[ (10u + t) - (10t + u) = 27 \]Simplify the equation:
\[ 10u + t - 10t - u = 27 \]\[ 9u - 9t = 27 \]Divide the entire equation by 9:
\[ u - t = 3 \]This equation tells us that the units digit must be 3 greater than the tens digit.
Now, we need to find pairs of digits \((t, u)\) that satisfy this condition, keeping in mind the constraints for \(t\) and \(u\).
So, the numbers that satisfy the condition are 14, 25, 36, 47, 58, and 69.
The question asks for "how many other two-digit numbers" increase by 27. The number 14 is given in the question as an example. Therefore, we need to exclude 14 from our count.
The numbers other than 14 are 25, 36, 47, 58, and 69. There are 5 such numbers.
D) 5 — There are 6 numbers in total (14, 25, 36, 47, 58, 69) that satisfy the condition. Since the question asks for "other" numbers, we exclude 14, leaving 5 numbers.
The question asks about the type of user interface that MS-DOS, a well-known operating system, utilizes. We need to recall the fundamental characteristics of MS-DOS to identify its interface type.
C:\>) to perform tasks like opening files, running programs, or managing directories. There were no icons, windows, or mouse-based interactions. This is the defining characteristic of a Command Line Interface.Correct Option: A) MS-DOS is an operating system that has a Command Line Interface. Users interact with it by typing commands at a prompt.
The question asks for the year in which A.R. Rahman won two Academy Awards (Oscars). This is a factual recall question related to significant achievements in Indian cinema on an international platform.
Correct Option: C) 2009. A.R. Rahman won two Oscars at the 81st Academy Awards ceremony held in February 2009 for his work on the film Slumdog Millionaire.
This problem can be solved using the concept of work and time. We will calculate the individual work rates of A and B and then use them to find the time A alone takes to complete the task.
If A and B together can complete a task in 20 days, their combined work rate is \(\frac{1}{20}\) of the task per day.
\[ \text{Work rate of (A + B)} = \frac{1}{20} \text{ task/day} \]If B alone can complete the same task in 60 days, B's work rate is \(\frac{1}{60}\) of the task per day.
\[ \text{Work rate of B} = \frac{1}{60} \text{ task/day} \]The work rate of A alone can be found by subtracting B's work rate from the combined work rate of A and B.
\[ \text{Work rate of A} = \text{Work rate of (A + B)} - \text{Work rate of B} \]\[ \text{Work rate of A} = \frac{1}{20} - \frac{1}{60} \]To subtract these fractions, find a common denominator, which is 60.
\[ \text{Work rate of A} = \frac{3}{60} - \frac{1}{60} \]\[ \text{Work rate of A} = \frac{3 - 1}{60} \]\[ \text{Work rate of A} = \frac{2}{60} \]\[ \text{Work rate of A} = \frac{1}{30} \text{ task/day} \]If A's work rate is \(\frac{1}{30}\) of the task per day, then A alone will take the reciprocal of this rate to complete the entire task.
\[ \text{Time taken by A} = \frac{1}{\text{Work rate of A}} \]\[ \text{Time taken by A} = \frac{1}{\frac{1}{30}} \]\[ \text{Time taken by A} = 30 \text{ days} \]Correct Option: C) 30 days
To find the value of \(x^2 - y^2\), we will substitute the given expressions for \(x\) and \(y\) into the expression and then simplify using algebraic identities and trigonometric identities.
We are given:
\[ x = a \text{cosec}\theta + b \text{cot}\theta \]\[ y = b \text{cosec}\theta + a \text{cot}\theta \]We need to find \(x^2 - y^2\). This can be factored as \((x-y)(x+y)\) or calculated directly as \(x^2 - y^2\).
Let's calculate \(x^2\) and \(y^2\) first.
\[ x^2 = (a \text{cosec}\theta + b \text{cot}\theta)^2 \]\[ x^2 = a^2 \text{cosec}^2\theta + b^2 \text{cot}^2\theta + 2ab \text{cosec}\theta \text{cot}\theta \quad \text{(Equation 1)} \]\[ y^2 = (b \text{cosec}\theta + a \text{cot}\theta)^2 \]\[ y^2 = b^2 \text{cosec}^2\theta + a^2 \text{cot}^2\theta + 2ab \text{cosec}\theta \text{cot}\theta \quad \text{(Equation 2)} \]Now, subtract Equation 2 from Equation 1:
\[ x^2 - y^2 = (a^2 \text{cosec}^2\theta + b^2 \text{cot}^2\theta + 2ab \text{cosec}\theta \text{cot}\theta) - (b^2 \text{cosec}^2\theta + a^2 \text{cot}^2\theta + 2ab \text{cosec}\theta \text{cot}\theta) \]Distribute the negative sign:
\[ x^2 - y^2 = a^2 \text{cosec}^2\theta + b^2 \text{cot}^2\theta + 2ab \text{cosec}\theta \text{cot}\theta - b^2 \text{cosec}^2\theta - a^2 \text{cot}^2\theta - 2ab \text{cosec}\theta \text{cot}\theta \]The term \(2ab \text{cosec}\theta \text{cot}\theta\) cancels out:
\[ x^2 - y^2 = a^2 \text{cosec}^2\theta + b^2 \text{cot}^2\theta - b^2 \text{cosec}^2\theta - a^2 \text{cot}^2\theta \]Recall the fundamental trigonometric identity: \(\text{cosec}^2\theta - \text{cot}^2\theta = 1\).
Substitute this into the expression:
\[ x^2 - y^2 = a^2 (1) - b^2 (1) \]\[ x^2 - y^2 = a^2 - b^2 \]A) \( a^{2} - b^{2} \)
The step-by-step calculation clearly shows that \(x^2 - y^2\) simplifies to \(a^2 - b^2\) using algebraic expansion and the trigonometric identity \(\text{cosec}^2\theta - \text{cot}^2\theta = 1\).
The question asks to identify the odd one out from a list of memorials. To do this, we need to understand the significance of each memorial and find a common theme among three of them, with one being distinct.
B) Smriti Sthal is the odd one out because it is a collective memorial complex for multiple leaders, whereas Shantivan, Kisan Ghat, and Raj Ghat are individual memorial sites dedicated to specific national leaders.
The question asks to identify the river known by the names Tsangpo and Dihang. This requires knowledge of the geographical names of major rivers in India and its neighboring regions.
Correct Option: D) Brahmaputra
The question asks about the land revenue system developed by Holt Mackenzie. We need to identify the specific system associated with his name and distinguish it from other land revenue systems introduced by the British in India.
Correct Option: B) Mahalwari system — Holt Mackenzie developed the Mahalwari system. This system was introduced in the North-Western Provinces, parts of Central India, and Punjab. Under this system, the land revenue was collected from the 'Mahal' (village or a group of villages) and the settlement was made with the village headman or a group of elders, who were collectively responsible for paying the revenue to the British.