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An athlete is running in four races and in each race she has 60% chance of winning. What is the probability that she will win atl east two races?
Find the area between = -2 & 3
What ia the total area tocthe left of z = 2.18?
What ia the total area tocthe left of z = 2.18?
Obtain the Fourier series for the following periodic function which has a period of 2π:
f(x) = x^2 for -π ≤x ≤π
f(x) = x^2 for -π ≤x ≤π
Show that there are infinitely many values of a for which x7+15x2-30x+a is irreducible in Q[x].
Let R= Z[root 2] and M={a+b root2€ R|5|a and 5|b}.
i) Show that M is an ideal of R
ii) Show that if 5 |a or 5|b , then 5|(a2 + b2), for a,b € Z
iii) Hence show that if N is an ideal of R properly containing M, then N= R
iv) Show that R/M is a field , and give two distinct non- zero elements of this field
i) Show that M is an ideal of R
ii) Show that if 5 |a or 5|b , then 5|(a2 + b2), for a,b € Z
iii) Hence show that if N is an ideal of R properly containing M, then N= R
iv) Show that R/M is a field , and give two distinct non- zero elements of this field
Let D={f(x,y)+g(x,y) i |f , g€Z[x,y]}subset to C[x,y]. Check whether D is a UFD or not
Math
Which of the following statements are true or false? Give reasons for your answers.
i) Union of any two convex sets is convex.
ii) The number of decision variables in the dual of a LPP is the same as the number of
decision variables in the primal.
iii) The cost vector in a LPP becomes an activity vector in its dual problem.
iv) In a transportation problem, if all the source availabilities ai and all the requirements
bj are integers, then the optimal solution consists of integers only.
v) A pay-off matrix can have more than one saddle point.
i) Union of any two convex sets is convex.
ii) The number of decision variables in the dual of a LPP is the same as the number of
decision variables in the primal.
iii) The cost vector in a LPP becomes an activity vector in its dual problem.
iv) In a transportation problem, if all the source availabilities ai and all the requirements
bj are integers, then the optimal solution consists of integers only.
v) A pay-off matrix can have more than one saddle point.
Define a relation R on Z by R={(n,n+3k)| k€Z}.
Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on Z.
Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on Z.
a) Check whether or not A={z€C*||z|€Q} is a subgroup of
i) (C*, .)
ii) (C ,+)
b) Let (G, .) be a finite abelian group and m€N. Prove that
S={g€G |(o(g),m)=1}<_G
i) (C*, .)
ii) (C ,+)
b) Let (G, .) be a finite abelian group and m€N. Prove that
S={g€G |(o(g),m)=1}<_G
Consider the set X=R\{-1}. Define * on X by
x1 * x2= x1+ x2+ x1x2 ¥ x1,x2 € X.
i) check whether (X,*) is a group or not.
ii) Prove that x*x*x*....*x (n times)=(1+x)n -1¥n€N and x€X
x1 * x2= x1+ x2+ x1x2 ¥ x1,x2 € X.
i) check whether (X,*) is a group or not.
ii) Prove that x*x*x*....*x (n times)=(1+x)n -1¥n€N and x€X
