The function \( f(x) = [x] + |x - 2| \) consists of two components:
1. The greatest integer function, \( [x] \), which has discontinuities at integer values of \( x \).
2. The absolute value function, \( |x - 2| \), which has a critical point at \( x = 2 \).
Now, consider the interval \( -2<x<3 \). The points where \( f(x) \) is not continuous or differentiable are determined by:
- Discontinuities in \( [x] \), which happen at \( x = -1, 0, 1, 2 \).
- A critical point in \( |x - 2| \) at \( x = 2 \).
So, the points where \( f(x) \) is not continuous are \( x = -1, 0, 1, 2 \), which gives us \( m = 4 \) discontinuities. The points where \( f(x) \) is not differentiable are due to the change in the slope at these points. Specifically, the function is not differentiable at \( x = 2 \), so \( n = 1 \).
Thus, \( m + n = 4 + 3 = 7 \).
Final Answer: \( m + n = 7 \).
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
For the circuit shown above, the equivalent gate is:
The expression given below shows the variation of velocity \( v \) with time \( t \): \[ v = \frac{At^2 + Bt}{C + t} \] The dimension of \( A \), \( B \), and \( C \) is:
Given below are two statements: one is labelled as Assertion (A) and the other one is labelled as Reason (R).
Assertion (A): Emission of electrons in the photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photoemissive substance.
Reason (R): A negative electric potential, which stops the emission of electrons from the surface of a photoemissive substance, varies linearly with the frequency of incident radiation.
In light of the above statements, choose the most appropriate answer from the options given below: