Questions asked by Rajat V.

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• Consider the linear operator T : C 3 → C 3 , defined by T (z1,z2,z3) = (z1 −iz2,iz1 +2z2 +iz3,−iz2 +z3). i) Compute T ∗ and check whether T is self-adjoint. ii) Check whether T is unitary.?
  
• Define T : R 3 → R 3 by T(x, y, x) = (x+y, y,2x−2y+2z). Check that T satisfies the polynomial (x−1) 2 (x−2). Find the minimal polynomial of T. ?
  
• Let V = R 3 , A = {(x, y,z)|y = 0} and B = {(x, y,z)|x = y = z}. Check whether R 3 = A⊕B ?
  
• Consider the basis #e_1 = (−2,4,−1)#, #e_2 = (−1,3,−1)# and #e_3 = (1,−2,1)# of #R^3# over #R#. Find the dual basis of #{e_1, e_2, e_3}#.?
  
• Let T : R³→ R³ be defined by T (x1, x2, x3) = (x1 −x3, x2 −x3, x1). Is T invertible? If yes, find a rule for T–¹ like the one which defines T ?
  
• Find the radius and the center of the circular section of the sphere |r| = 26 cut off by the plane r ·(2i+6j+3k) = 70 ?
  
• Complete the set S = {x³+x²+1,x²+x+1,x+1} to get a basis of P3 ?
  
• Check whether each of the following subsets of R³is linearly independent?
  
• #{α_1,α_2,...,α_n}# is a set only if all the #α_i#s follow a given rule. This statement is true/false? Please give reasons for your answer ?
  
• Every subring of a non-commutative ring is non-commutative.This statement is true/false?Please give reasons for your answer ?
  
• If #(R,*_1,*_2)# is a ring, then #ψ:R×R->R:ψ(r_1,r_2)=r_1*_2r_1# is a binary operation. This statement is true/false? Please give reasons your answer ?
  
• If (D,+,.) is an integral domain such that 1∈D,then D is a field.This statement is true/false?Please give reasons for your answer ?
  
• Which of following statements are true or false?Give reasons for your answers.(i)The function f, defined by f(x)=x^3-6x^2+16x-15,is increasing in RR.
  
• This statement is true or false?Please give reasons for your answer. Rolle's Theorem is applicable for the function f,defined by f(x)=1+x^(2/3) in the interval [-1,1] ?
  
• If # y=e^(mtan^(-1)x) # show that # (1+x^2)y_(n+1)+(2nx-m)y_n+n(n-1)y_(n-1)=0 #?
  
• Find the domain and range of the function #f# defined by #f(x,y)=(3x^2y^2)/(x^2+y^4)# ?
  
• Prove that #lim_(x→0)xsin(2/x)=0# ?
  
• Find #f@g# and #g@f# ,if they exist,for the functions #f(t)=4t,t∈RR,g(x,y)=x+y,x,y∈RR #?
  
• Find the mass of an object which is in the form of a cuboid#[0,1]×[2,4]×[1,3]#.The density at any point #(x,y,z)# on the cuboid is given by #delta(x,y,z)=x^2+y^2+z^2# ?
  
• Evaluate the following integral:(i)∫dx/√x-x² from 1/4 to 1/2?
  
• Solve the following differential equation? #(2x+y+3)dy/dx=x+2y+1#
  
• Verify that f(x)=x/8 can serve as the probability density function of a continuous random variable which can take on any value in the interval from 0 to 4 ?