To solve this problem, we need to determine the cardinal number of the given set \( A = \{-2, -1, 0, 1, 2\} \).
1. Understanding Cardinal Number:
The cardinal number of a set refers to the number of distinct elements present in that set.
2. Counting the Elements:
The set \( A \) contains the elements: -2, -1, 0, 1, 2
So, total elements = 5
3. Evaluating the Options:
(1) 5 – Correct
(2) 4 – Incorrect
(3) -2 – Incorrect (cardinal number is never negative)
(4) 2 – Incorrect
Final Answer:
The correct option is (A) 5.
For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:
\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \).
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let
\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).
If the given figure shows the graph of polynomial \( y = ax^2 + bx + c \), then: