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Find all the 8th roots of i3 − 3. Also show any one of them in an Argand diagram.
Which of the following statements are true or false? Give reasons for your answers.
(i) If P(A)= 0.4, P(A∪B)=0.7 and A and B are independent sets, then P(B)= 0.55.
(ii) By Chebyschev's inequality, P{|x-μ|≥2σ} ≤ 0.2
(i) If P(A)= 0.4, P(A∪B)=0.7 and A and B are independent sets, then P(B)= 0.55.
(ii) By Chebyschev's inequality, P{|x-μ|≥2σ} ≤ 0.2
Math
Find the values of a ∈R for which ia is a solution of 0 z z2 z7 z4 10 4 3 2
− + − + = .
Also find all the roots of this equation.
− + − + = .
Also find all the roots of this equation.
1. Which of the following statements are true? Justify your answers. (This means that if you
think a statement is false, give a short proof or an example that shows it is false. If it is
true, give a short proof for saying so.) (20)
i) f(n) = n −1 "nÎN, where f is the Euler-phi function.
ii) If 1 G and 2 G are groups, and 1 2 f :G ®G is a group homomorphism, then
o(G ) o(G ) 1 2 = .
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and HDG , then | G: H | = 2 .
v) Every element of n S has order at most n .
vi) If R is a ring and I is an ideal of R , then xr = rx " xÎI and rÎR .
vii) If S (n 3) n sÎ ³ is a product of an even number of disjoint cycles, then sign (s) =1.
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from [0, 1] to R form a ring with respect to pointwise
addition and multiplication.
think a statement is false, give a short proof or an example that shows it is false. If it is
true, give a short proof for saying so.) (20)
i) f(n) = n −1 "nÎN, where f is the Euler-phi function.
ii) If 1 G and 2 G are groups, and 1 2 f :G ®G is a group homomorphism, then
o(G ) o(G ) 1 2 = .
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and HDG , then | G: H | = 2 .
v) Every element of n S has order at most n .
vi) If R is a ring and I is an ideal of R , then xr = rx " xÎI and rÎR .
vii) If S (n 3) n sÎ ³ is a product of an even number of disjoint cycles, then sign (s) =1.
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from [0, 1] to R form a ring with respect to pointwise
addition and multiplication.
Prove that x * x * x *K* x (n times) = (1+ x) −1 " nÎN n and xÎX .
In a sequence of 4 trials with probability of success p=1/3,what is the probability that there will be exactly two successes?
A cylindrical tank with radius 3 m is being filled with water at a rate of 4 m3/min. How fast is the height of the water increasing?
Consider the linear operator T : C
4 → C
4
, defined by
T (z1,z2,z3,z4) = (−iz2,iz1,−iz4,z3).
i) Compute T
∗
and check whether T is self-adjoint.
ii) Check whether T is unitary.
4 → C
4
, defined by
T (z1,z2,z3,z4) = (−iz2,iz1,−iz4,z3).
i) Compute T
∗
and check whether T is self-adjoint.
ii) Check whether T is unitary.
question by finding the adjoint as well as
using Cayley-Hamiltion theorem
using Cayley-Hamiltion theorem
Reduce the conic x
2 −6xy+y
2 −4 = 0 to standard form. Hence the given conic.
2 −6xy+y
2 −4 = 0 to standard form. Hence the given conic.
