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A particle P moves on the curve with polar equation r = 1/ (2 - sinx) . Given that at any instant t, during the motion, r^2 (dx/dt) = 4,
(i) write an expression for r(dx/dt) in terms of x.
(ii) Show that dr/dt = 4cosx and 1/3 <=r<=1.
(iii) Find the speed of P when x = 0.
(iv) Prove that the force acting on P is directed towards the pole.

1.   Prove that a cycle of length l is odd if l is even.

Prove that the product of an even permutation and an odd permutation is odd.

— If m1, m2 belongs to the same orbit then St(m1) and St(m2) are conjugate to each other if m2=p(g)m1 then St(m2)=gSt(m1) g ...

Find a maximal ideal of R [x] containing

the ideal <x²— 1, x³— 1>

### If m1, m2 belongs to the same orbit then St(m1) and St(m2) are conjugate to each other if m2=p(g)m1 then St(m2)=gSt(m1)g-1

Let s= {p/q €Q|(q,7) =1} Define a relation ~ on s by p/q ~ a/b iff 7| (bp-aq). check whether or not ~ is an equivalence relation on s?

Which of the following statements are true, and which short proof or a counter-example.

i)There is no non-abelian group of 12

ii)If in a group G every element is of finite order, then G is a finite order.

iii)The homomorphic image of a non-cyclic group is non cyclic.

iv) If a is an integral domain, then R /I is an integral domain for every non-zero ideal I of R .

v)If I and J are ideals of a ring R, then so is I U J

explain the definition of a localization of ring
explain the examples in localization of a ring
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