Convert the following Cartesian coordinates to the corresponding polar coordinates using positive `r` and negative `rdot` (i) `( i i ) (iii)(( i v ) (v)-1,1( v i ))( v i i )` (viii) (ii) `( i x ) (x)(( x i ) (xii)2,-3( x i i i ))( x i v )` (xv)

Convert the following Cartesian coordinates to the cooresponding polar coordinates using positive r. (i) (1,-1) (ii) (-3,-4)

Convert the following points from polar coordinates to the corresponding Cartesian coordinates. (i) ( i i ) (iii)(( i v ) (v)2, (vi)pi/( v i i )3( v i i i ) (ix) (x))( x i ) (xii) (ii) ( x i i i ) (xiv)(( x v ) (xvi)0, (xvii)pi/( x v i i i )2( x i x ) (xx) (xxi))( x x i i ) (xxiii) (iii) ( x x i v ) (xxv)(( x x v i ) (xxvii)-sqrt(( x x v i i i )2( x x i x ))( x x x ) , (xxxi)pi/( x x x i i )4( x x x i i i ) (xxxiv) (xxxv))( x x x v i ) (xxxvii)

Convert the following polar coordinates to its equivalent Cartesian coordinates. (i) ( i i ) (iii)2,pi)( i v ) (v) (ii) ( v i ) (vii)(( v i i i ) (ix)sqrt(( x )2( x i ))( x i i ) , (xiii)pi/( x i v )6( x v ) (xvi) (xvii))( x v i i i ) (xix) (iii) ( x x ) (xxi)(( x x i i ) (xxiii)-3,-( x x i v )pi/( x x v )6( x x v i ) (xxvii) (xxviii))( x x i x ) (xxx)

Number of four-digit numbers of the form N=a b c d which satisfy following three conditions (i) ( i i ) (iii) 4000lt=N<6000 (iv) (v) (vi) ( v i i ) N (viii) (ix) is a multiple of 5 (x) ( x i ) (xii)3lt=b

If ((1+i)/(1-i))^(x)=1 AA n in N is (i) x = 2n+1 (ii) x =4n (iii) x=2n (iv) x=4n+1

Given points P(2,3),Q(4,-2),a n dR(alpha,0) (i) Find the value of ( i i )alpha( i i i ) (iv) if ( v ) (vi) P R+r Q (vii) (viii) is minimum (ix) Find the value of ( x )alpha( x i ) (xii) if ( x i i i ) (xiv)|( x v ) P R-R Q|( x v i ) (xvii) is maximum

Convert the following in the polar form : (i) (1+7i)/((2-i)^2) (ii) (1+3i)/(1-2i)

Convert the following in the form of (a+ib) : (i) (1+i)^(4) (ii) (-3+(1)/(2)i)^(3) (iii) (1-i)(3+4i) (iv) (1+i)(1+ 2i)(1+ 3i) (v) (3+5i)/(6-i) (vi) ((2+3i)^(2))/(2+i) (vii) ((1+ i)(2+i))/((3+i)) (viii) (2-i)^(-3)

Simplify the following : (i) i^(97) (ii) i^(8) (iii) (1)/(i^(3)) (iv) (-i)^(14) (v) i^(-22) (vi) i^(-63)

If i= sqrt(-1) , prove that following (x+1+ i) (x + 1- i) (x-1 + i) (x-1-i)= x^(4) + 4