Find the value of \(\sqrt{0.5}\)
To find the value of \(\sqrt{0.5}\), we can convert the decimal to a fraction and then simplify the square root. Alternatively, we can perform long division for the square root, or approximate the value by considering squares of numbers close to 0.5.
First, convert \(0.5\) into a fraction:
\[ 0.5 = \frac{5}{10} = \frac{1}{2} \]Now, we need to find the square root of \(\frac{1}{2}\):
\[ \sqrt{0.5} = \sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]To simplify, rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{2}\):
\[ \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]We know that the approximate value of \(\sqrt{2}\) is \(1.414\).
\[ \frac{\sqrt{2}}{2} \approx \frac{1.414}{2} \]D) The calculated value of \(\sqrt{0.5}\) is approximately \(0.707\).
The question asks to identify which of the given options is NOT a Kharif crop. To answer this, we need to recall the classification of crops in India based on seasons: Kharif, Rabi, and Zaid. Kharif crops are typically sown with the onset of the monsoon (June-July) and harvested in September-October. Rabi crops are sown in winter (October-December) and harvested in spring (April-June).
Correct Option: A) Wheat is a Rabi crop, not a Kharif crop. It is sown in winter (October-December) and harvested in spring (April-June).
6 7 4 8 5 8 8 4 8 3 2 5 8 6 7 8 3 8
If the 1st term of the above sequence is interchanged with the 2nd term, the 3rd with the 4th, and so on, then how many 8s are there that are divisible by the number on their left as well as their right?
The problem requires us to first rearrange the given sequence of numbers according to the specified rule and then identify the number of '8's that satisfy a specific divisibility condition.
A) 0 — As determined in the step-by-step analysis, there are no '8's in the rearranged sequence that are divisible by both their left and right neighbors.
To find the year whose calendar will be the same as 2018, we need to calculate the number of odd days for each year starting from 2018 until the cumulative sum of odd days becomes a multiple of 7 (i.e., 0 odd days). A normal year has 1 odd day, and a leap year has 2 odd days.
Since the cumulative odd days become 7 (which is \(0 \pmod 7\)) at the end of 2023, the calendar for 2018 will repeat in the year immediately following this cycle, which is 2024. However, 2024 is not an option, and the question asks for a past year.
We need to find a year \(X\) such that the number of odd days between year \(X\) and 2018 (exclusive of 2018, inclusive of \(X\)) is a multiple of 7. Let's go backward from 2018.
Since the cumulative odd days from 2007 to 2017 (inclusive) is 14, which is a multiple of 7 (\(14 \div 7 = 2\), remainder 0), the calendar for 2007 will be the same as 2018. Both 2007 and 2018 are normal years, which is a necessary condition for their calendars to be identical.
Correct Option: A) 2007
Virat can complete a task in 20 days, whereas Anu can complete the same task in 15 days. How long will it take for Virat and Anu to complete the task if they worked together?
12\( \frac{1}{7} \) days
5 days
8\( \frac{4}{7} \) days
17\( \frac{1}{2} \) days
This problem can be solved by calculating the individual work rates of Virat and Anu and then adding them to find their combined work rate. The total time taken to complete the task together is the reciprocal of their combined work rate.
Calculate Virat's one-day work:
Virat can complete the task in 20 days. So, in one day, Virat completes:
Calculate Anu's one-day work:
Anu can complete the task in 15 days. So, in one day, Anu completes:
Calculate their combined one-day work:
When they work together, their one-day work rates add up:
To add these fractions, find a common denominator, which is the Least Common Multiple (LCM) of 20 and 15. The LCM of 20 and 15 is 60.
\[ \text{Combined one-day work} = \frac{3}{60} + \frac{4}{60} \] \[ \text{Combined one-day work} = \frac{3+4}{60} = \frac{7}{60} \text{ of the task} \]Calculate the total time to complete the task together:
If they complete \(\frac{7}{60}\) of the task in one day, then the total time taken to complete the entire task (which is 1 whole task) is the reciprocal of their combined one-day work:
Convert the improper fraction to a mixed number:
\[ \frac{60}{7} = 8 \frac{4}{7} \text{ days} \]C) \(8\frac{4}{7}\) days is the correct answer, as calculated above.
A) \(12\frac{1}{7}\) days: This would be incorrect. It might arise from an incorrect calculation of the combined work rate or a misinterpretation of the problem.
B) 5 days: This would imply a much faster completion time, possibly from averaging the days (\((20+15)/2 = 17.5\)) or simply guessing. It does not follow the logic of work and time problems.
D) \(17\frac{1}{2}\) days: This is the average of the individual times (\((20+15)/2 = 17.5\)). When people work together, the time taken is always less than the time taken by the faster individual, so this option is logically incorrect.
To answer this question, we need to recall the positions of metalloids in the Modern Periodic Table and count how many metalloids are present in each specified period.
Correct Option: A) 4, 5
Period 4 contains Germanium (Ge) and Arsenic (As). Period 5 contains Antimony (Sb) and Tellurium (Te). Both periods have two metalloids each.
We are given Manoj's initial expenditure on food as a percentage of his income. After an increment in income, his expenditure on food as a percentage of his new income changes, but the absolute amount spent on food remains the same. We can set up an equation based on the constant food expenditure to find his initial income.
D) 4,500 — This is the correct initial income calculated by equating the absolute food expenditure before and after the income increment.
The question asks about the type of lens used to correct myopia. Myopia, also known as nearsightedness, is a common vision defect where distant objects appear blurry. We need to identify which type of lens can diverge light rays before they reach the eye, effectively pushing the focal point back onto the retina.
B) Concave — A concave lens is a diverging lens that spreads out light rays before they enter the eye. This shifts the focal point backward, allowing the image to form correctly on the retina for myopic eyes.
If cot4θ + cot2θ = 4, then csc4θ − csc2θ =?
The question asks to identify the type of motion indicated by vibration. Vibration is a specific type of oscillatory motion. We need to understand the characteristics of vibration to choose the correct option.
D) Rapid to and fro. This option precisely describes the nature of vibration, which involves quick, repetitive movement back and forth from a central position.