This problem can be visualized as a right-angled triangle. The wall and the ground form the two perpendicular sides, and the ladder acts as the hypotenuse. We can use the Pythagorean theorem to find the length of the ladder.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
\[ c^2 = a^2 + b^2 \]C) 6.5 m — The calculation using the Pythagorean theorem yields 6.5 m as the length of the ladder.
How many different values can be stored by a 32 bit register?
This question asks about the number of unique values a 32-bit register can store. In digital systems, information is represented in binary (0s and 1s). The number of distinct values that can be represented by a fixed number of bits is determined by the formula \(2^n\), where \(n\) is the number of bits.
A) \(2^{32}\) — This is the correct formula for calculating the number of unique values that can be stored by a register of \(n\) bits. For a 32-bit register, it is \(2^{32}\).
The question asks to identify the rivers that define the Bari Doab region. Doabs are fertile lands located between two confluent rivers. The name of a doab often reflects the names of the rivers that bound it.
A) Beas and Ravi — The Bari Doab is indeed the region situated between the Beas and Ravi rivers. This is a well-known geographical term associated with the Punjab plains.
The question asks to identify the Indian scheme funded by a 0.5% cess levied on all taxable products since November 2015. This requires knowledge of specific government schemes and their funding mechanisms.
Correct Option: B) Swachh Bharat Abhiyan is the scheme that was funded by a 0.5% cess levied on all taxable products (specifically services initially, then broadened) since November 2015. This cess was known as the Swachh Bharat Cess.
This problem involves the concept of relative speed and distance. When a train crosses a man, the distance covered is the length of the train itself. When it crosses a platform, the distance covered is the sum of the train's length and the platform's length. We can use the formula \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\) to set up equations and solve for the unknown length of the train.
D) 260 m
The principal value of cot⁻¹\(\left(-\frac{1}{\sqrt{3}}\right)\) is:
\(\frac{\pi}{6}\)
\(\frac{2\pi}{3}\)
\(\frac{3\pi}{2}\)
\(\frac{\pi}{3}\)
To find the principal value of \(\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\), we need to recall the definition and range of the principal value branch for the inverse cotangent function. The principal value branch of \(\cot^{-1}x\) is \((0, \pi)\).
Let \(y = \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\).
By definition of the inverse cotangent function, this means \(\cot y = -\frac{1}{\sqrt{3}}\).
We know that \[\cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}}\].
Since \(\cot y\) is negative, \(y\) must lie in the second quadrant because the principal value range for \(\cot^{-1}x\) is \((0, \pi)\). In the second quadrant, cotangent is negative.
Using the identity \(\cot(\pi - \theta) = -\cot\theta\), we can write:
\[ \cot y = -\cot\left(\frac{\pi}{3}\right) \] \[ \cot y = \cot\left(\pi - \frac{\pi}{3}\right) \]Simplifying the expression inside the cotangent function:
\[ \pi - \frac{\pi}{3} = \frac{3\pi - \pi}{3} = \frac{2\pi}{3} \]So, we have \(\cot y = \cot\left(\frac{2\pi}{3}\right)\).
Therefore, \(y = \frac{2\pi}{3}\).
We must check if this value lies within the principal value range \((0, \pi)\). Since \(0 < \frac{2\pi}{3} < \pi\), this is the principal value.
Correct Option: B) \(\frac{2\pi}{3}\)
The question asks to identify the correct description of the 'taungya system'. We need to evaluate each option based on the known definition and characteristics of the taungya system.
C) It is the system of growing agricultural crops between rows of planted trees. This option accurately defines the taungya system, which is a method of agroforestry where short-term agricultural crops are cultivated between young forest trees during their initial establishment phase.
The question asks about properties that remain unchanged as one moves down a group in the periodic table. We need to analyze how each given property (number of shells, metallic character, valency, and atomic size) changes or stays constant within a group.
Based on this analysis, valency is the property that does not change as one goes down a group.
C) Valency — Elements in the same group have the same number of valence electrons, and thus exhibit the same valency. For instance, all alkali metals (Group 1) have a valency of 1, and all halogens (Group 17) have a valency of 1 (or other valencies depending on the compound, but their primary combining capacity remains consistent within the group).
The question asks to identify the name of the scheme launched by the Department of Posts in September 2020, aimed at ensuring universal coverage of flagship postal schemes in rural areas. This requires knowledge of recent government initiatives related to postal services.
B) Five Star Villages: This is the correct scheme launched by the Department of Posts in September 2020 to ensure universal coverage of flagship postal schemes in rural areas.
The question asks for the location of a specific hydrotherapy treatment centre for elephants. This is a factual recall question related to current affairs or general knowledge about wildlife conservation and animal welfare facilities in India.
Correct Option: A) Mathura is the correct answer. The Wildlife SOS Elephant Conservation and Care Centre (ECCC) in Mathura, Uttar Pradesh, has established a specialized hydrotherapy treatment centre for elephants. This facility utilizes hydrotherapy pools to provide relief and treatment for elephants suffering from chronic ailments such as arthritis, joint pain, and foot problems, which are common among captive or rescued elephants due to past abuse or neglect.