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• For a > b > 0, consider D = {(x,y,z) ∈ R^3 : x^2 + y^2 + z^2 ≤ a^2 and x^2 + y^2 ≥ b^2}. Then, the surface area of the boundary of the solid 𝐷 is
• Let 𝑓: ℝ → ℝ be defined by f(x) = (x^2 + 1)^2/(x^4 + x^2 + 1) for x ∈ ℝ. Then, which one of the following is TRUE?
• Consider the following two statements. P: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that 𝑓 is continuous at x = 1 and 𝑔 is discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1. Q: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that both 𝑓 and 𝑔 are discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1. Which one of the following holds?
• For p,q,r ∈ R, r ≠ 0 and n ∈ N, let a_n = p^n·n^q(n/(n+2))^(n^2) and b_n = ((n^n)/(n! r^n))(sqrt((n+2)/n)). Then, which one of the following statements is TRUE?
• Let 𝑃_7(𝑥) be the real vector space of polynomials, in the variable 𝑥 with real coefficients and having degree at most 7, together with the zero polynomial. Let 𝑇: 𝑃_7(𝑥) → 𝑃_7(𝑥) be the linear transformation defined by T(f(x)) = f(x) + df(x)/dx. Then, which one of the following is TRUE?
• For a matrix 𝑀, let Rowspace(𝑀) denote the linear span of the rows of 𝑀 and Colspace(𝑀) denote the linear span of the columns of 𝑀. Which of the following hold(s) for all 𝐴, 𝐵, 𝐶 ∈ 𝑀_{10}(ℝ) satisfying 𝐴 = 𝐵𝐶 ?
• Let 𝑓: ℝ → ℝ be a solution of the differential equation d^2y/dx^2 - 2dy/dx + y = 2e^x for x ∈ ℝ. Consider the following statements. P: If 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓′(𝑥) > 0 for all 𝑥 ∈ ℝ . Q: If 𝑓′(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ . Then, which one of the following holds?
• Consider G = {m + n·sqrt2 : m,n ∈ Z} as a subgroup of the additive group ℝ. Which of the following statements is/are TRUE?
• Let S = {f : R → R : f is polynomial and f(f(x)) = (f(x))^2024 for x ∈ R}. Then, the number of elements in S is _____________
• Let M = ([0, 0, 0, 0, -1], [2, 0, 0, 0, -4], [0, 2, 0, 0, 0], [0, 0, 2, 0, 3], [0, 0, 0, 2, 2]). If 𝑝(𝑥) is the characteristic polynomial of 𝑀, then 𝑝(2) − 1 equals _____________