1. It is given that at least one of a,b,c∈R is nonnegative. Find the equation of surface of rotation when the straight line ax−1=by−1=cz−1 rotates about the z-axis.2. Let B⊂Rn(n≥2) be a unit open ball and function u,v is continuous on b and twice continuously differentiable in B. It satisfies the condition
(i) −Δu−(1−u2−v2)u=0
(ii) −Δv−(1−u2−v2)v=0
(iii) u(x)=v(x)=0, where x∈∂B.
Also, x=(x1,x2,⋯,xn), Δu=∂x12∂2u+∂x22∂2u+⋯+∂xn2∂2u.
(∂B is the boundary of B.)
Prove that u2(x)+v2(x)≤1(∀x∈B).3. Let f(x)=x2021+a2020x2020+a2019x2019+⋯+a2x2+a1x+a0 be a polynomial with integer coefficients and a0=0. For any 0≤k≤2020, there exists ∣ak∣≤40. Prove that it is not possible for all roots of f(x)=0 to be a real number.4. Let P be an unitary and symmetric matrix. Prove that there exists an inverse complex matrix Q such that P=QQ−1.5. Let α be an integer greater than 1. Prove that
(1) ∫0+∞dx∫0+∞e−tαxsinxdt=∫0+∞dt∫0+∞e−tαxsinxdx.
(2) Evaluate ∫0+∞sinx3dx⋅∫0+∞sinx23dx.6. It satisfies: ∀x∈R, f′(x)≥6+f(x)−f2(x) and g′(x)≤6+g(x)−g2(x). Prove that
(1) ∀x∈(0,1) and x∈R, there exists ξ∈(−∞,x) such that f(ξ)≥3−ε.
(2) ∀x∈R, f(x)≥3.
(3) ∀x∈R, there exists η∈(−∞,x) such that g(η)≤3.
(4) ∀x∈R, g(x)≤3.