Questions asked by Rohit J.

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• Give an example with justification of a function f:R1-->R2,where(R1,+,.) and (R2,+,.) are rings and such that f:(R1,+)-->(R2,+) is a group homomorphism but f is not a ring homomorphism.?
  
• Prove that if G is not equal to {e} and G has no proper non-trivial subgroup,then G is finite and o(G) is a prime number.?
  
• Let p be a prime and a∈N such that p|a^50.Show that p^50|a^50.?
  
• Let G be a group and H≤G.Prove that the only right coset of H in G that is a subring of G is H itself.?
  
• Give an example,with justification,of a group G with subgroups H and K such that HK is not a subgroup of G.?
  
• Which of the following statements are true/false? Give reasons for your answers. 1.The characteristic of a field containing (50-1) elements is 50.
  
• Give an example of a function which is not integrable on [0,π].Justify your answer.?
  
• Find the area enclosed by the curve r=a(1-cosθ)?
  
• Verify that f,defined by f(x)=x^3+4x+1 on [1,5], satisfies Lagrange's mean value theorem.?
  
• Give an example, with justification, of a periodic function which is not even.?
  
• Find the maxima and minima, if any, for the function f given by f(x)=x^4-14x^2+24x+6. Also find the values of the function at those points.?
  
• Find the length of the curve y=ln e^x-1/e^x+1 from x=1 to x=2?
  
• Prove that for every x>0,x/1+x^2<tan^-1<x?
  
• Let V=R³ and W={(x,y,z)|x+y+z=0} be a subspace of V.Which of the following pairs of vectors are in the same coset of W in V? (i)(1,3,2)and(2,2,2).(ii)(1,1,1)and(3,3,3).
  
• Which of the following statements are true/false? Justify your answer
  
• This statement is true/false? Justify your answer. If #A^k=0# for a square matrix #A#, then all the eigenvalues of #A# are zero.
  
• If z=e^(xy^2),x=tcost,y=tsint,compute dz/dt at t=π/2?
  
• Examine whether lim_"x→0"e^(1/x)/e^(1/x)+1 exits or not?
  
• For which value(s) of k,is the function f,defined as below,continuous at x=2?f(x)={3-kx,1≤x<2 and x²/4-3, x≥2.Further,at which other points in [1,infinity[ is f continuous,and why?
  
• Let #bbx=bbe_1+bbe_2-2bbe_3# and #bby=2bbe_1-bbe_2+bbe_3#,where #bbe_1,bbe_2,bbe_3# are unit vectors. Find #|bbx+2bby|# and #|bbx+bby#|?
  
• Show that the functions #f(x,y)=ln x-ln y # and #g(x,y)=(x^2+2y^2)/(2xy)# are functionally dependent?
  
• Find two level curves of the function #f(x,y)=(x+y)/(x-y),x≠y# and sketch them?
  
• Find the second Taylor's expansion of the function f(x,y)=xy^2+cos xy about the point (1,π/2)?
  
• Evaluate the integral of f(x,y,z)=x+z-3 over the cylinder bounded by x²+y²=1,z=0 and z=1?
  
• This statement is true or false?Please justify your answer. #f(x,y)=sin((x²y)/(x³+y³))/(ln(x+y/x))# is a homogeneous function of degree 2.
  
• These statements are true or false?Please justify your answer.(i)The function #f(x,y)=(x³y+1,x²+y²)# is locally invertible at#(1,2)#.(ii)The function #f(x,y)=x³+y³# is integrable on #[1,2]×[1,3].#
  
• These statements are true or false?Please justify your answer.(i)#lim_(x→0)(1/(x²)-1/(sin²x))# is in #(0/0)# form.(ii)Domain of #f(x,y)=(xy)/(x⁴+y⁴)# is #R²#.
  
• Let #f(x,y)={(x²y)/(x⁴+y²), " if " x⁴+y²≠0, {0, " if " x=y=0#. Check whether #lim_((x,y)→(0,0))f(x,y) # exists or not?
  
• If #f(x,y)=x^2tan^(-1)(y/x)-y^2tan^(-1)(x/y),x ne 0,y ne 0# and if #f(x,y)=0, x=0=y#, prove that #(partial^2f)/(partialx partialy)=(x^2-y^2)/(x^2+y^2)#?
  
• Check whether the function #f(x,y)={(6xy)/sqrt(x^2+y^2) " if " (x,y)≠(0,0), {0 if (x,y)=(0,0)# is continuous at the origin?
  
• Vectors a and b are given by a=3i-2j+k and b=-2i+2j+4k.Find (i)magniture of a and b and the angle between them. (ii)projection the vector (a+1/2b) onto a. (iii)which of the following vectors are perpendicular to a?c=-i-4j+2k,d=-3i+k,e=2i+2j-2k.?
  
• If p is the length of the perpendicular from the origin to the line #x/a+y/b=1#,prove that #1/(p²)=1/(a²)+1/(b²)#?
  
• For what values of m will the equation x²-2x(1+3m)+7(3+2m)=0 have equal roots?
  
• Determine the value of the constant a if the function #f(x)# defined below is continuous at #x=2#. #f(x)={(ax^2+7x; x≤2),(3x^2+3a; x>2):}# ?
  
• Suppose that a person selects a card at random from a deck of 52 cards and tells us that the selected card is red.Find the probability that the card is the kind of hearts given that it is red?
  
• For a population of size N=120 with a mean of 7.5 and a standard deviation of 1.5, find the estimated standard error of the mean for the following sample sizes:(i)n=9 (ii)n=25 ?
  
• Suppose that the temperature is normally distributed with expectation 50℃ and variance 4℃.What is the probability that the temperature T will be between 48℃ and 53℃?What is the probability that T≥52℃?
  
• A continuous random variable X has the p.d.f. F(x)={Ax(6-x)² if 0<x<6 ,{0 if otherwise Calculate the mean and standard deviation of X.Construct the distribution function F(x) and hence evaluate P(X>1/4).?
  
• Sir, please help me for getting the solve below question..?
  
• Sir/madam, could you help me to solve below question? i need to solve it..
  
• The following distribution gives the top speeds (in kilometers per hour) at which thirty races were clocked in auto race. Top Speed 145 150 160 170 No. of racers 9 8 11 2 Find the mean and variance of the distribution of the top speeds?
  
• Calculate the least square regression line where annual savings is the dependent variable and annual income is the independent variable.?