The question asks to identify the specific Padma award conferred upon Ram Chet Chaudhary in the field of Science and Engineering in 2024. This requires knowledge of the recent Padma Awards recipients.
Correct Option: B) Padma Shri
Ram Chet Chaudhary was conferred with the Padma Shri in 2024 for his contributions to Science and Engineering, particularly in the field of agricultural science and plant breeding.
To find the average of prime numbers between 30 and 50, we first need to identify all the prime numbers in that range. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Once identified, we sum them up and divide by the count of these prime numbers.
The prime numbers between 30 and 50 are 31, 37, 41, 43, and 47.
There are 5 prime numbers in this range.
Correct Option: D) 39.8
If 4 sin²θ = 1 and θ is an acute angle, then the value of cos²θ + tan²θ = _____.
\( \frac{13}{12} \)
\( \frac{5}{13} \)
1
\( \frac{1}{4} \)
We are given an equation involving \(\sin^2\theta\) and need to find the value of an expression involving \(\cos^2\theta\) and \(\tan^2\theta\). Since \(\theta\) is an acute angle, we can find the value of \(\sin\theta\), then determine \(\theta\), or directly find \(\cos\theta\) and \(\tan\theta\) using trigonometric identities.
Find \(\sin\theta\):
Given the equation:
\[ 4 \sin^2\theta = 1 \]Divide by 4:
\[ \sin^2\theta = \frac{1}{4} \]Take the square root of both sides:
\[ \sin\theta = \sqrt{\frac{1}{4}} = \frac{1}{2} \]Since \(\theta\) is an acute angle (between \(0^\circ\) and \(90^\circ\)), \(\sin\theta\) must be positive.
Find \(\cos^2\theta\):
We use the fundamental trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\).
\[ \cos^2\theta = 1 - \sin^2\theta \]Substitute the value of \(\sin^2\theta\):
\[ \cos^2\theta = 1 - \frac{1}{4} = \frac{4-1}{4} = \frac{3}{4} \]Find \(\tan^2\theta\):
We use the identity \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), so \(\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta}\).
\[ \tan^2\theta = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{4} \times \frac{4}{3} = \frac{1}{3} \]Calculate \(\cos^2\theta + \tan^2\theta\):
Substitute the values found in steps 2 and 3:
\[ \cos^2\theta + \tan^2\theta = \frac{3}{4} + \frac{1}{3} \]To add these fractions, find a common denominator, which is 12:
\[ \cos^2\theta + \tan^2\theta = \frac{3 \times 3}{4 \times 3} + \frac{1 \times 4}{3 \times 4} = \frac{9}{12} + \frac{4}{12} = \frac{9+4}{12} = \frac{13}{12} \]Correct Option: A) \( \frac{13}{12} \)
The question asks about the occurrence of the division of labour in organisms. Division of labour refers to the specialization of different parts or components of an organism to perform specific functions. We need to consider both unicellular and multicellular organisms to determine where this principle applies.
D) both unicellular and multicellular organisms
Division of labour is a fundamental biological principle. In unicellular organisms, it occurs at the organelle level, where different organelles perform specific tasks. In multicellular organisms, it is highly developed, with cells, tissues, organs, and organ systems specializing in distinct functions.
The value of M in \( \left(\frac{p}{q}\right)^{2M+2} = \left(\frac{q}{p}\right)^{9-M} \) is:
To find the value of \(M\), we need to solve the given exponential equation. The key is to make the bases on both sides of the equation the same. We can use the property that \[\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\].
The given equation is:
\[ \left(\frac{p}{q}\right)^{2M+2} = \left(\frac{q}{p}\right)^{9-M} \]To make the bases the same, we can rewrite the right side of the equation. We know that \[\left(\frac{q}{p}\right) = \left(\frac{p}{q}\right)^{-1}\]. So, substitute this into the equation:
\[ \left(\frac{p}{q}\right)^{2M+2} = \left(\left(\frac{p}{q}\right)^{-1}\right)^{9-M} \]Apply the exponent rule \( (a^m)^n = a^{mn} \):
\[ \left(\frac{p}{q}\right)^{2M+2} = \left(\frac{p}{q}\right)^{-(9-M)} \]\[ \left(\frac{p}{q}\right)^{2M+2} = \left(\frac{p}{q}\right)^{M-9} \]Now that the bases are the same, we can equate the exponents:
\[ 2M+2 = M-9 \]Solve for \(M\). Subtract \(M\) from both sides:
\[ 2M - M + 2 = -9 \]\[ M + 2 = -9 \]Subtract 2 from both sides:
\[ M = -9 - 2 \]\[ M = -11 \]Correct Option: B) −11
To find the missing number in the series, we need to identify the pattern or the common difference between consecutive terms. We will calculate the difference between each adjacent pair of numbers to determine if there is an arithmetic progression or another type of sequence.
Therefore, the missing number in the series is 296.
Correct Option: C) 296
The question states that a man sold an article for ₹247.50 and earned a profit of 12.5%. We need to determine the cost price of the article. Since the profit percentage is calculated on the cost price, we can use the relationship between selling price, cost price and profit percentage.
Identify the given information:
Apply the formula for selling price:
When an article is sold at a profit:
\[ SP=CP\times\frac{100+\text{Profit \%}}{100} \]Substituting the given values:
\[ 247.50=CP\times\frac{100+12.5}{100} \] \[ 247.50=CP\times\frac{112.5}{100} \]Rearrange the formula to calculate the cost price:
\[ CP=\frac{247.50\times100}{112.5} \] \[ CP=\frac{24750}{112.5} \] \[ CP=220 \]Understand the result:
The cost price of the article is ₹220.
At a profit of 12.5%, the profit amount is:
\[ \frac{12.5}{100}\times220=27.50 \]Adding the profit to the cost price gives:
\[ 220+27.50=247.50 \]Therefore, the cost price of the article is ₹220.
Correct Option: A) ₹220 — The selling price is ₹247.50, and the profit percentage is 12.5%. Using the formula \(CP=\frac{SP\times100}{100+\text{Profit \%}}\), we get \(CP=\frac{247.50\times100}{112.5}=220\). Therefore, the cost price of the article is ₹220. Hence, Option A is the correct answer.
B) ₹225: This is incorrect because a cost price of ₹225 with a profit of 12.5% would give a selling price of \(225\times1.125=253.125\) rupees, not ₹247.50.
C) ₹210: This is incorrect because a cost price of ₹210 with a profit of 12.5% would give a selling price of \(210\times1.125=236.25\) rupees, not ₹247.50.
D) ₹224: This is incorrect because a cost price of ₹224 with a profit of 12.5% would give a selling price of \(224\times1.125=252\) rupees, not ₹247.50.
The problem requires us to perform two main operations on the word "TYPING": first, change each letter to its preceding letter in the English alphabet, and second, rearrange the new set of letters in alphabetical order. Finally, we need to identify the letter that is first to the right in this newly rearranged group.
The letters are S, X, O, H, M, F. Arranging them alphabetically gives:
F, H, M, O, S, X
In the rearranged group F, H, M, O, S, X, the letters are ordered from left to right. The letter "first to the right" means the second letter from the left.
The first letter is F.
The second letter (first to the right of the first letter) is H.
B) X is the rightmost letter in the alphabetically rearranged sequence F, H, M, O, S, X. This interpretation aligns with one of the given options.
This problem involves analyzing logical statements and drawing conclusions based on them. We will use Venn diagrams to represent the given statements and then evaluate each conclusion to see if it logically follows from the diagrams.
Correct Option: A) Only conclusion (I) follows
The question asks about Thomson's model of an atom, specifically which sub-atomic particles are embedded in a sphere of positive charge. This requires recalling the key features of J.J. Thomson's atomic model, often called the "plum pudding" or "watermelon" model.
Correct Option: A) Only electrons are embedded in the sphere of positive charge in Thomson's model. The positive charge itself was considered to be uniformly spread throughout the sphere, not concentrated in distinct particles like protons.