If a force 'F' acts on a body such that body gets displaced through the distance 'x' in the direction of the force, then the work done by the force is _______.
The question asks for the work done by a force when it causes a displacement in its own direction. The fundamental definition of work done by a constant force is the product of the magnitude of the force and the distance displaced in the direction of the force.
D) Fx is the correct expression for the work done when a force \(F\) causes a displacement \(x\) in its own direction.
The problem describes a relationship between two sets of letters based on their positions in the English alphabet. We need to identify this pattern and apply it to a new set of letters to find the corresponding option.
D) OMKHF is the correct option as it follows the established pattern of shifting each letter forward by 4 positions in the English alphabet.
The question asks to identify the statement that is NOT a property of solids. We need to recall the general characteristics of solids and evaluate each option against these properties.
Correct Option: A) It is not difficult to change their shape.
This is a coding-decoding problem where we need to find the code for a specific word by comparing common words and their corresponding codes in given statements. We will identify the common word 'day' in the provided phrases and then find the common code associated with it.
D) mn is the correct code for 'day' because it is the common code found in both given statements corresponding to the common word 'day'.
To solve this problem, we first need to calculate the dimensions (length and breadth) of the first carpet. Then, we will determine the dimensions of the second carpet based on the given percentage increases. After finding the dimensions of the second carpet, we will calculate its area and finally multiply the area by the cost per square meter to find the total cost.
Correct Option: C) Rs. 4082.40
To find the simple interest, we use the standard formula for simple interest: \(SI = \frac{P \times R \times T}{100}\), where \(P\) is the principal amount, \(R\) is the rate of interest per annum, and \(T\) is the time in years.
Correct Option: A) βΉ192 is the calculated simple interest.
The question asks us to simplify the trigonometric expression \((1-\sec^2\theta)(1-\sin\theta)(1+\sin\theta)(1+\cot^2\theta)\). We can solve this by applying standard trigonometric identities and simplifying the resulting expression.
Identify the given expression:
\[ (1-\sec^2\theta)(1-\sin\theta)(1+\sin\theta)(1+\cot^2\theta) \]We need to simplify this expression using trigonometric identities.
Apply the identity for secant:
We know that:
\[ 1+\tan^2\theta=\sec^2\theta \]Rearranging:
\[ 1-\sec^2\theta=-\tan^2\theta \]Apply the difference of squares identity:
Using \((a-b)(a+b)=a^2-b^2\):
\[ (1-\sin\theta)(1+\sin\theta)=1-\sin^2\theta \]Since:
\[ 1-\sin^2\theta=\cos^2\theta \]Therefore:
\[ (1-\sin\theta)(1+\sin\theta)=\cos^2\theta \]Apply the identity for cotangent:
We know that:
\[ 1+\cot^2\theta=\csc^2\theta \]Substitute the identities into the expression:
\[ =(-\tan^2\theta)(\cos^2\theta)(\csc^2\theta) \]Expressing tangent and cosecant in terms of sine and cosine:
\[ =-\frac{\sin^2\theta}{\cos^2\theta} \times\cos^2\theta \times\frac{1}{\sin^2\theta} \]Cancel the common factors:
\[ =-1 \] \[ \boxed{-1} \]Therefore, the simplified value of the expression is -1.
Correct Option: B) -1 : Using the identities \(1-\sec^2\theta=-\tan^2\theta\), \((1-\sin\theta)(1+\sin\theta)=\cos^2\theta\), and \(1+\cot^2\theta=\csc^2\theta\), the expression simplifies to \(-\tan^2\theta\cos^2\theta\csc^2\theta=-1\). Hence, Option B is the correct answer.
A) 2: This is incorrect because applying the trigonometric identities gives -1, not 2.
C) 0: This is incorrect because the expression simplifies to -1. The factors do not produce zero for any angle at which the original expression is defined.
D) 1: This is incorrect because the identity \(1-\sec^2\theta=-\tan^2\theta\) introduces a negative sign. Therefore, the simplified result is -1 rather than 1.
The question asks for the specific title of a book authored by Parimal Nathwani that was presented to Prime Minister Modi in 2024. This is a factual recall question related to current affairs and notable publications.
Correct Option: C) Call of the Gir
To find the combined ratio \(a : b : c\), we need to make the common term, \(b\), equal in both given ratios. We will find the least common multiple (LCM) of the values of \(b\) in the given ratios and then adjust the ratios accordingly.
Identify the given ratios:
Identify the common term and its values:
The common term is \(b\). Its value in the first ratio is \(2\), and in the second ratio, it is \(4\).
Find the LCM of the values of the common term:
The values of \(b\) are \(2\) and \(4\). The LCM of \(2\) and \(4\) is \(4\).
Adjust the first ratio:
To make the value of \(b\) in the first ratio equal to the LCM (\(4\)), we need to multiply both parts of the ratio \(a : b\) by a factor. Since \(2 \times 2 = 4\), we multiply both parts of \(3 : 2\) by \(2\).
Adjust the second ratio:
The value of \(b\) in the second ratio is already \(4\), which is equal to the LCM. So, this ratio remains unchanged.
Combine the adjusted ratios:
Now that the value of \(b\) is the same in both ratios (\(4\)), we can combine them to get \(a : b : c\).
B) 6 : 4 : 5 is the correct ratio obtained by making the common term 'b' equal in both given ratios.