02 - COMPLEX NUMBERS Page 1 ( Answers at the end of all questions ) l w l = 1, then z lies on
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1 0 - COMPLEX NUMBERS Page ( ) If the cube roots of uity are,,, the the roots of the equatio ( x - ) + 8 = 0 are ( a ) -, - +, - - ( b ) -, -, -, ( c ) -, -, - ( d ) -, +, + [ AIEEE 005 ] ( ) If z ad z are two o-zero complex umbers such that l z + l = l z l + l z l, the arg z - arg z is equal to ( a ) ( ) If w = z z - i ( b ) - ( c ) 0 ( d ) - ad l w l =, the z lies o [ AIEEE 005 ] ( a ) a ellipse ( b ) a circle ( d ) a straig t lie ( d ) a parabola [ AIEEE 005 ] ( ) Let z, w be complex umbers such that z + i w = 0 ad arg zw =. The arg z equals ( a ) ( b ) ( c ) ( 5 ) If z = x - y ad z = p + iq, the ( d ) 5 x p p y + q + q is equal to [ AIEEE 00 ] ( a ) ( b ) - ( c ) ( d ) - [ AIEEE 00 ] 6 ) If l z - l = l z l +, the z lies o ( a ) the real axis ( b ) the imagiary axis ( c ) a circle ( c ) a ellipse [ AIEEE 00 ] ( 7 ) Let z ad z be two roots of the equatio z + az + b = 0, z beig complex. Further assume that the origi, z ad z form a equilateral triagle. The ( a ) a = b ( b ) a = b ( c ) a = b ( d ) a = b [ AIEEE 00 ]
2 0 - COMPLEX NUMBERS Page ( 8 ) If z ad w are two o-zero complex umbers such that l zw l = ad ( 9 ) If Arg ( z ) - Arg ( w ) =, the z w is equal to ( a ) ( b ) - ( c ) i ( d ) - i [ AIEEE 00 ] x + i - i =, the the value of smallest positive iteger is gi by ( a ) x = ( b ) x = ( c ) x = + ( d ) x = + [ AIEEE 00 ] ( 0 ) If,, are the cube roots of uity, the the alue of = ( ) If is ( a ) ( b ) 0 ( c ) ( d ) [ AIEEE 00 ] c + i = a ib, where a, b, c are real, the the value of a + b is c - i ( a ) ( b ) ( c ) c ( d ) - c [ AIEEE 00 ] ( ) If z = x + y, the l z - l = l z - l represets ( a ) x xis ( b ) y-axis ( c ) a circle ( d ) lie parallel to y-axis [ AIEEE 00 ] If the cube roots of uity are, ad, the the value of + ( a ) ( b ) - ( c ) ( d ) [ AIEEE 00 ] is ( ) If a = cos α + i si α ad b = cos β + i si β, the the value of ab + ab is ( a ) si ( α + β ) ( b ) cos ( α + β ) ( c ) si ( α - β ) ( d ) cos ( α - β ) [ AIEEE 00 ]
3 0 - COMPLEX NUMBERS Page ( 5 ) If α is cube root of uity, the for N, the value of α + + α + 5 is ( a ) - ( b ) 0 ( c ) ( d ) [ AIEEE 00 ] ( 6 ) Four poits P ( -, 0 ), Q (, 0 ), R ( -, ) ad S ( -, - ) are give o a complex plae, equatio of the locus of the shaded regio excludig the boudaries is give by ( a ) l z + l > ad l arg ( z + ) l < ( b ) l z + l > ad l arg ( z + ) l < ( c ) l z - l > ad l arg ( z - ) l < ( d ) l z - l > ad l arg ( z - ) l < [ IIT 005 ] ( 7 ) If is cube root of uity ( ), the the least value of where is a positive iteger such that ( + ) = ( + ) is ( a ) ( b ) ( c ) 5 ( d ) 6 [ IIT 00 ] ( 8 ) The complex umber z is such that l z l =, z - ad = of is ( a ) ( 9 ) Let = l z + l ( b ) - l z + l ( c ) l z + l z -, the real part z + ( d ) 0 [ IIT 00 ] - + i. The the value of the determiat - - is ( a ) ( b ) ( - ) ( c ) ( d ) ( - ) [ IIT 00 ] ( 0 ) For all complex umbers z, z satisfyig l z l = ad l z - - i l = 5, the miimum value of l z - z l is ( a ) 0 ( b ) ( c ) 7 ( d ) 7 [ IIT 00 ]
4 0 - COMPLEX NUMBERS Page ( ) The complex umbers z, z ad z satisfyig of a triagle which is z - z - i = are the vertices z - z ( a ) of area zero ( b ) right-agled isosceles ( c ) equilateral ( d ) obtuse-agled isosceles [ IIT 00 ] ( ) If z ad z be th roots of uity which subted a right gle at the origi, the must be of the form ( a ) k + ( b ) k + ( c ) k + ( d ) k [ IIT 00 ] ( ) If arg ( z ) < 0, the arg ( - z ) - arg ( z ) = ( a ) ( b ) - ( c ) - ( ) If z, z ad z are complex umbers such that l z l = l z l = l z l = + + =, the l z + z + z l is z z ( d ) [ IIT 000 ] ( a ) ( b ) < ( c ) > ( d ) [ IIT 000 ] 65 i ( 5 ) If i = - th + 5 i is equal to ( a ) - i ( b ) - + i ( c ) i ( d ) - i [ IIT 999 ] ( 6 ) If is a imagiary cube root of uity, the ( + - ) 7 equals ( a ) 8 ( b ) - 8 ( c ) 8 ( d ) - 8 [ IIT 998 ] ( 7 ) The value of the sum ( i + i + ), where i = -, equals = ( a ) i ( b ) i - ( c ) - i ( d ) 0 [ IIT 998 ]
5 ( 8 ) If 6i 0 - i i - i 0 - COMPLEX NUMBERS Page 5 = x + iy, the ( a ) x =, y = ( b ) x =, y = ( c ) x = 0, y = ( d ) x = 0, y = 0 [ IIT 998 ] ( 9 ) For positive itegers,, the value of the expressio ( ) ( ) ( ) i i i ( i 7 ) , where i = - is a real umber if ad oly if ( a ) = + ( b ) = - ( c ) = ( d ) >, > 0 [ IIT 996 ] ( 0 ) If ( ) is a cube root of uity a ( + ) 7 = A + B, the A ad B are respectively the umbers ( a ) 0, ( b ), ( c ), 0 ( d ) -, [ IIT 995 ] ( ) If ( ) is a cube root of uity, the - i - i + i - i equals ( a ) 0 ( b ) ( c ) i ( d ) [ IIT 995 ] ( ) If z ad be two o-zero complex umbers such that l z l = l l ad Arg z + Arg =, the z equals a ) ( b ) - ( c ) ( d ) - [ IIT 995 ] ( ) If z ad w be two complex umbers such that l z l, l w l ad l z + iw l = l z - iw l =, the z equals ( a ) or i ( b ) i or - i ( c ) or - ( d ) i or - [ IIT 995 ] ( ) The complex umbers si x + i cos x ad cos x - i si x are cojugate to each other for ( a ) x = ( b ) x = 0 ( c ) x = ( + / ) ( d ) o value of x [ IIT 988 ]
6 0 - COMPLEX NUMBERS Page 6 ( 5 ) If z ad z are two o-zero complex umbers such that l z + z l = l z l + l z l, the arg z - arg z is equal to ( a ) - ( b ) - 6 k ( 6 ) The value of si 7 k = ( c ) 0 ( d ) k - i cos is 7 ( e ) [ II 987 ] ( a ) - ( b ) 0 ( c ) - i ( d ) i ( e ) oe of these [ IIT 987 ] ( 7 ) Let z ad z be complex umbers such that z z ad l z l = l z l. If z has z positive real part ad z has egative imagia y part, the + z may be z - z ( a ) zero ( b ) real ad positive ( c ) real ad egative ( d ) purely imagiary ( e ) oe of hese [ IIT 986 ] ( 8 ) If a, b, c ad u, v, w re complex umbers represetig the vertices of two triagles such that c = ( - r ) a + b ad w = ( - r ) u + rv, where r is a complex umber, the the two riagles ( a ) have the same area ( b ) are similar ( c ) are cogruet ( d ) oe of these [ IIT 985 ] ( 9 ) If z = a + b ad z = c + id are complex umbers such that l z l = l z l = ad Re ( z z ) = 0, the the pair of complex umbers w = a + ic ad w = b + id s tisfies ( a ) l w l = ( b ) l w l = ( c ) Re ( w w ) = 0 ( d ) oe of these [ IIT 985 ] ( 0 ) If z = x + iy ad w = z - - iz, the l w l = implies that, i the complex plae, i ( a ) z lies o the imagiary axis ( b ) z lies o the real axis ( c ) z lies o the uit circle ( d ) Noe of these [ IIT 98 ]
7 0 - COMPLEX NUMBERS Page 7 ( ) The poits z, z, z, z i the complex plae are the vertices of a parallelogram take i order if ad oly if ( a ) z + z = z + z ( b ) z + z = z + z ( c ) z + z = z + z ( d ) Noe of these [ IIT 98 ] ( ) The iequality l z - l < l z - l represets the regio give by ( ) If z = ( a ) Re ( z ) > 0 ( b ) Re ( z ) < 0 ( c ) Re ( z ) > ( d ) oe of these [ IIT 98 ] 5 5 i i + + -, the ( a ) Re ( z ) = 0 ( b ) Im ( z ) = 0 ( c ) Re ( z ) > 0, Im ( z ) > 0 ( d ) Re ( z ) > 0, Im ( z ) < 0 [ IIT 98 ] ( ) If the cube roots of uity are,, the the roots of the equatio ( x - ) + 8 = 0 are ( a ) -, +, ( b ) -, -, - ( c ) -, -, - ( d ) oe of these [ IIT 979 ] Aswers c c c c d b c d a b a d b b b a b d b b c d a a c d b d d b a b c d a,e d a,d b a,b,c b b d b b
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