Let 𝑓: ℝ ℝ be a function such that 𝑓(𝑥) = max{𝑥, 𝑥3}, 𝑥 ℝ, where ℝ is the set of all real numbers. The set of all points where 𝑓(𝑥) is NOT differentiable is
Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ⋯ , 𝑛} for some 𝑛 4. Let 𝑋 be the event that 1 occurs before 2 in the permutation, and 𝑌 the event that 3 occurs before 4. Which one of the following statements is TRUE?
Let A=1234412334122341 and B=3412412312342341 Let det(A) and det(B) denote the determinants of the matrices A and B, respectively. Which one of the options given below is TRUE?
Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}. Let 1, 2, 3, 4, and 5 be the five eigenvalues of A. Note that these eigenvalues need not be distinct. The value of 1 + 2 + 3 + 4 + 5 = ______.
Consider the functions I.e-x II.x2-sinx III.x3+1 Which of the above functions is/are increasing everywhere in [0,1]?
Let A and B be two n x n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements. I. rank(AB) = rank(A) rank(B) II. det(AB) = det(A) det(B) III. rank(A+B) rank(A) + rank(B) IV. det(A+B) det(A) + det(B) Which of the above statements is TRUE?
The number of permutations of the characters in LILAC so that no character appears in its original position, if the two Ls are indistinguishable, is ______.
For n2, let a {0, 1}n be a non-zero vector. Suppose that x is chosen uniformly at random from {0, 1}n. Then, the probability thati=1naixi is an odd number is ______.