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After 1st Lecture Q.1 The sequence S = i + 2i2 + 3i3 + upto 100 terms simplifies to where i = 1 : (A) 50 (1 i) (B) 25i (C) 25 (1 + i) (D) 100 (1 i) Q.2 If z + z3 = 0 then which of the following must be true on the complex plane? (A) Re(z) < 0 (B) Re(z) = 0 (C) Im(z) = 0 (D) z4 = 1 Q.3 What is the real part of (1 + i)50? (A) 0 (B) 225 (C) – 225 (D) – 250 Q.4 Given i = , the value of the sum 3 + 1+ i + 3 1 i + 3 1+ i + 3 1 i + .... 1+ i 1 i 1+ i 1 i , is (A) 2n2 + 2n (B) 2 i n2 + 2 i n (C) (1 + i)n2 (D) none of these Q.5 Let i = 1 . The product of the real part of the roots of z2 – z = 5 – 5i is (A) – 25 (B) – 6 (C) – 5 (D) 25 Q.6 The number of solutions of the equation z2 + z = 0 where z is a complex number, is : (A) 4 (B) 3 (C) 2 (D) 1 Q.7 In the quadratic equation x2 + (p + iq) x + 3i = 0 , p & q are real . If the sum of the squares of the roots is 8 then : (A) p = 3, q = 1 (B) p = –3, q = –1 (C) p = ± 3, q = ± 1 (D) p = 3, q = 1 Q.8 If a point P denoting the complex number z moves on the complex plane such that, Re z + Im z = 1 then the locus of z is : (A) a square (B) a circle (C) two intersecting lines (D) a line Q.9 The figure formed by four points 1 + 0 i ; 1 + 0 i ; 3 + 4 i & 25 – 3 4 i on the argand plane is : (A) a parallelogram but not a rectangle (B) a trapezium which is not equilateral (C) a cyclic quadrilateral (D) none of these 1 4 1 Q.10 Square root of x2 + x2 – i x x – 6 where x R is equal to : x 1 + 2i x 1 2i x + 1 + 2i x + 1 2i (A) ± x (B) ± x (C) ± x (D) ± x Q.11 If z = (3 + 7i) (p + iq) where p, q I – {0}, is purely imaginary then minimum value of | z |2 is 3364 (A) 0 (B) 58 (C) 3 (D) 3364 Q.12 Number of values of x (real or complex) simultaneously satisfying the system of equations 1 + z + z2 + z3 + .......... + z17 = 0 and 1 + z + z2 + z3 + .......... + z13 = 0 is (A) 1 (B) 2 (C) 3 (D) 4 Q.13 If x 3 3 + i + y 3 3 i = i where x, y R then (A) x = 2 & y = – 8 (B) x = – 2 & y = 8 (C) x = – 2 & y = – 6 (D) x = 2 & y = 8 Q.14 Number of complex numbers z satisfying z3 = z is (A) 1 (B) 2 (C) 4 (D) 5 Q.15 If x = 91/3 91/9 91/27 ......ad inf y = 41/3 4–1/9 41/27 ...... ad inf and z = (1 + i) – r then , the argument of the complex number w = x + yz is r=1 2 2 2 (A) 0 (B) – tan–1 3 (C) – tan–1 3 (D) – tan–1 3 DPP-2 After 2nd Lecture Q.1 The digram shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of F, which is (A) A (B) B (C) C (D) D Q.2 If z = x + iy & = 1 iz z i then = 1 implies that, in the complex plane : (A) z lies on the imaginary axis (B) z lies on the real axis (C) z lies on the unit circle (D) none Q.3 On the complex plane locus of a point z s

STUDY INNOVATIONSSeguir

Educator em Study Innovations- After 1st Lecture Q.1 The sequence S = i + 2i2 + 3i3 + ........ upto 100 terms simplifies to where i = 1 : (A) 50 (1 i) (B) 25i (C) 25 (1 +i) (D) 100 (1 i) Q.2 If z + z3 =0 then which ofthe followingmust be true on the complex plane? (A) Re(z) < 0 (B) Re(z) = 0 (C) Im(z) = 0 (D) z4 = 1 Q.3 What is the real part of (1 + i)50? (A) 0 (B) 225 (C) – 225 (D) – 250 Q.4 Given i = 1 , the value of the sum i i i i i i i i 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 + i i i i i i i i 1 n 1 n 1 n 1 n ........ 1 3 1 3 1 3 1 3 , is (A) 2n2 + 2n (B) 2 i n2 + 2 i n (C) (1 + i)n2 (D) none of these Q.5 Let i = 1 . The product of the real part of the roots of z2 – z = 5 – 5i is (A) – 25 (B) – 6 (C) – 5 (D) 25 Q.6 The number of solutions of the equation z2 + z = 0 where z is a complex number, is : (A) 4 (B) 3 (C) 2 (D) 1 Q.7 In the quadratic equation x2 + (p + iq)x + 3i = 0, p & q are real . If the sum of the squares of the roots is 8 then : (A) p = 3, q = 1 (B) p = –3, q = –1 (C) p = ± 3, q = ± 1 (D) p = 3, q = 1 Q.8 If apoint P denotingthecomplex number z movesonthecomplex planesuch that, Re z + Im z = 1 then the locus of z is : (A) a square (B) a circle (C) twointersectinglines (D) a line Q.9 The figure formed by four points 1 + 0 i ; 1 + 0 i ; 3 + 4 i & 25 3 4 i on the argand plane is : (A) a parallelogram but not a rectangle (B) atrapezium which isnotequilateral (C) a cyclic quadrilateral (D) none of these Q.10 Square root of x2 + 2 x 1 – i 4 x 1 x – 6 where x R is equal to : (A) ± i 2 x 1 x (B) ± i 2 x 1 x (C) ± i 2 x 1 x (D) ± i 2 x 1 x Q.11 If z = (3 + 7i) (p + iq) where p, q I – {0}, is purelyimaginarythen minimum value of |z |2 is (A) 0 (B) 58 (C) 3364 3 (D) 3364 CLASS : XII (ABCD) Dpp on Complex Number DPP. NO.- 1
- Q.12 Numberofvalues of x (real orcomplex)simultaneouslysatisfying thesystem ofequations 1 + z + z2 + z3 + .......... + z17 = 0 and 1 + z + z2 + z3 + .......... + z13 = 0 is (A) 1 (B) 2 (C) 3 (D) 4 Q.13 If i 3 3 x + i 3 3 y = i where x, y R then (A) x = 2 & y = – 8 (B) x = – 2 & y = 8 (C) x = – 2 & y = – 6 (D) x = 2 & y = 8 Q.14 Numberofcomplexnumberszsatisfying z z3 is (A) 1 (B) 2 (C) 4 (D) 5 Q.15 If x = 91/3 91/9 91/27 ......ad inf y = 41/3 4–1/9 41/27 ...... ad inf and z = 1 r (1 + i) – r then , the argument of the complex number w = x + yz is (A) 0 (B) – tan–1 3 2 (C) – tan–1 3 2 (D) – tan–1 3 2 DPP-2 After 2nd Lecture Q.1 Thedigramshows several numbers inthe complex plane.Thecircleis theunitcirclecenteredattheorigin.Oneofthesenumbersisthereciprocal of F,which is (A)A (B) B (C) C (D) D Q.2 If z = x + iy& = 1 iz z i then = 1 implies that, in the complex plane : (A) z lies ontheimaginaryaxis (B) z lies on the real axis (C) z lies on theunit circle (D) none Q.3 Onthecomplex planelocus ofapointz satisfyingtheinequality 2 | z – 1 | < 3 denotes (A) region between the concentric circles of radii 3 and 1 centered at (1, 0) (B)regionbetweentheconcentriccirclesofradii3 and2 centeredat (1,0)excludingtheinnerand outer boundaries. (C)regionbetweentheconcentriccirclesofradii 3 and2 centered at (1,0)includingtheinnerand outer boundaries. (D)regionbetweentheconcentriccircles ofradii3 and2centeredat (1, 0)includingtheinner boundary andexcludingtheouterboundary. Q.4 The complex number z satisfies z + | z |= 2 + 8i. The value of |z | is (A) 10 (B) 13 (C) 17 (D) 23 Q.5 If S is the set of points in the complex plane such that z(3 + 4i) is a real number then S denotes a (A)circle (B) hyperbola (C)line (D) parabola Q.6 The locus of z, for arg z = – 3 is (A) same as the locus of z for arg z = 3 2 (B) same as the locus of z for arg z = 3 (C) the part of the straight line y x 3 = 0 with (y< 0, x > 0) (D) the part of the straight line y x 3 = 0 with (y> 0, x < 0)
- Q.7 If z1 & 1 z represent adjacent vertices of a regular polygon of n sides with centre at the origin & if 1 2 z Re z Im 1 1 then the value of n is equal to : (A) 8 (B) 12 (C) 16 (D) 24 Q.8 If z1, z2 are two complex numbers & a, b are two real numbers then, az bz bz az 1 2 2 1 2 2 = (A) ( ) a b z z 2 1 2 2 2 (B) ( ) a b z z 1 2 2 2 (C) a b z z 2 2 1 2 2 2 (D) a b z z 2 2 1 2 2 2 Q.9 If x sin 2 1 x sin 2 3 i i is purelyimaginarythen x = (A) n ± 6 (B) n ± 3 (C) 2n ± 3 (D) 2n ± 6 Q.10 The complex number z = x + iy which satisfy the equation 1 5 z 5 z i i lie on : (A) the x-axis (B) the straight line y=5 (C) acirclepassingthroughtheorigin (D) the y-axis Q.11 All real numbers x which satisfythe inequality x 2 i 4 1 5 where i = 1 , x R are (A) [ 2 , ) (B) (– , 2] (C) [0, ) (D) [–2, 0] Q.12 Let z1, z2, z3 be three distinct complex numbers satisfying z1 1 = z2 1 = z3 1. If z1 + z2 + z3 = 3 then z1, z2, z3 must represent the vertices of : (A) anequilateraltriangle (B) an isosceles trianglewhichis notequilateral (C) a right triangle (D) nothing definite can besaid . Q.13 Let z = 1 sin + i cos where (0, /2), then the modulus and the principal value of the argument ofzarerespectively: (A) 2 1sin , 2 4 (B) 2 1sin , 2 4 (C) 2 1sin , 2 4 (D) 2 1sin , 2 4 Q.14 Number of real solution of the equation, z3 + iz 1 = 0 is : (A) zero (B) one (C) two (D) three Q.15 Apoint 'z'movesonthecurve z43i=2 inanargandplane.Themaximumandminimumvalues of z are : (A) 2, 1 (B) 6, 5 (C) 4, 3 (D) 7, 3
- CLASS : XII (ABCD) Dpp on Complex Number DPP. NO.- 3 After 3rd Lecture Q.1 If z1 & z2 are two non-zerocomplex numbers such that z1 + z2 =z1+z2, then Argz1 Arg z2 isequal to: (A) (B) /2 (C) 0 (D) /2 Q.2 Let Zbeacomplex numbersatisfyingthe equation (Z3 + 3)2 = – 16 then | Z | has the value equal to (A) 51/2 (B) 51/3 (C) 52/3 (D) 5 Q.3 Let i = 1 . Define a sequence of complex number byz1 = 0, zn + 1 = 2 n z + i for n 1. In the complex plane, howfar from theoriginis z111? (A) 1 (B) 2 (C) 3 (D) 110 Q.4 The points representing the complex number z for which |z + 5 |2 – |z – 5 |2= 10 lie on (A) astraight line (B) a circle (C) a parabola (D) the bisector of the line joining (5,0) & (5,0) Q.5 For Z1 = 6 3 i 1 i 1 ; Z2 = 6 i 3 i 1 ; Z3 = 6 i 3 i 1 which of the following holds good? (A) 2 3 | Z | 2 1 (B) | Z1 |4 + | Z2 |4 = | Z3 |–8 (C) 6 3 3 2 3 1 | Z | | Z | | Z | (D) 8 3 4 2 4 1 | Z | | Z | | Z | Q.6 Consider two complex numbers and as = 2 bi a bi a + 2 bi a bi a , where a, b R and = 1 z 1 z , where | z | = 1, then (A) Both and are purelyreal (B) Both and arepurelyimaginary (C) is purelyreal and is purelyimaginary (D) is purelyreal and ispurelyimaginary Q.7 Let Zis complex satisfyingtheequation, z2 – (3+ i)z +m + 2i= 0, wherem R. Supposethe equation has a real root. The additiveinverse of non real root, is (A) 1 – i (B) 1 + i (C) – 1 – i (D) –2 Q.8 The minimum value of 1+z+ 1zwhere z is a complex number is : (A) 2 (B) 3/2 (C) 1 (D) 0 Q.9 Thecomplexnumberswhoserealandimaginarypartsareintegersandsatisfytherelationz z3 + z3 z = 350 forms arectangle on theArgandplane, the length ofwhosediagonal is (A) 5 (B) 10 (C) 15 (D) 25 Q.10 If | z – 5 + 12 i | 1 and the least and greatest values of | z | are m and n and if l be the least positive value of x 1 x 24 x2 (x > 0), then l is (A) 2 n m (B) m + n (C) m (D) n Q.11 The system ofequations z i z 1 2 1 Re where z is a complex number has : (A) no solution (B)exactlyonesolution(C) two solutions (D) infinitesolution
- Q.12 Let C1 and C2 are concentric circles of radius 1 and 8/3 respectively having centre at (3, 0) on the argand plane. If the complex number z satisfies theinequality, log1/3 2 | 3 z | 11 2 | 3 z | 2 >1 then: (A) z lies outside C1 but inside C2 (B) z lies inside of both C1 and C2 (C) z lies outside both of C1 and C2 (D) none of these Q.13 Identifytheincorrect statement. (A) no nonzero complex number z satisfies the equation, z = 4z (B) z = z implies that z is purelyreal (C) z = z implies that z is purelyimaginary (D) if z1, z2 are the roots of the quadratic equation az2 + bz + c = 0 such that Im(z1 z2) 0 then a, b, c must bereal numbers. Q.14 The equation ofthe radical axis ofthe two circles represented bythe equations, z 2 = 3 and z 2 3i = 4 on the complex plane is : (A) 3y + 1 = 0 (B) 3y 1 = 0 (C) 2y 1 = 0 (D) none Q.15 If z1 = 3+ 5i ; z2 = – 5 – 3i andz is a complex number lying on theline segment joining z1 & z2 then arg z can be : (A) 3 4 (B) 4 (C) 6 (D) 5 6 Q.16 Givenz =f(x)+i g(x)wheref, g: (0, 1) (0,1)arerealvaluedfunctions then, whichofthefollowing holds good? (A) z = 1 1 ix + i 1 1 ix (B) z = 1 1 ix + i 1 1 ix (C) z = 1 1 ix + i 1 1 ix (D) z = 1 1 ix + i 1 1 ix Q.17 z1 = i 1 a ; z2 = i 2 b ; z3 = a – bi for a, b R if z1 – z2 = 1 then the centroid of the triangle formed bythe points z1 , z2 , z3 in the argand’s plane is givenby (A) 9 1 (1 + 7i) (B) 3 1 ( 1 + 7i) (C) 3 1 (1 – 3i) (D) 9 1 (1 – 3i) Q.18 If the equation, z3 +(3 + i)z2 –3z – (m + i) = 0 where m R has at least one real root then mcan have thevalueequal to (A) 1 or 3 (B) 2 or 5 (C) 3 or 5 (D) 1 or 5 Q.19 Theregionrepresentedbyinequalities ArgZ< 3 ;|Z|<2;Im(z)>1intheArganddiagramisgivenby (A) (B) (C) (D) Q.20 Number of complex numbers z such that | z | = 1 and z z z z = 1 is (A) 4 (B) 6 (C) 8 (D) more than 8
- CLASS : XII (ABCD) Dpp on Complex Number DPP. NO.- 4 After 4th Lecture Q.1 If be a complex nth root of unity, then ( ) ar b r r n 1 1 is equal to : (A) n n a ( ) 1 2 (B) nb n 1 (C) na 1 (D) none Q.2 Let z beacomplex number havingtheargument , 0 < </2and satisfyingtheequalityz 3i=3. Then cot 6 z is equal to : (A) 1 (B) 1 (C) i (D) i Q.3 If the complex number z satisfies the condition z 3, then the least value of z z 1 is equal to : (A) 5/3 (B) 8/3 (C) 11/3 (D) none of these Q.4 Given that z satisfies z + z 1 = 2 cos 13°, find an angle B so that 0 < B < 2 and z2 + 2 z 1 = 2cosB. (A) 23° (B) 24° (C) 25° (D) 26° Q.5 Let A, B, C represent the complex numbers z1, z2, z3 respectively on the complex plane . If the circumcentre ofthetriangleABC lies at theorigin, thentheorthocentre is represented by the complex number : (A) z1 + z2 z3 (B) z2 + z3 z1 (C) z3 + z1 z2 (D) z1 + z2 + z3 Q.6 Given zp = cos P 2 + i sin P 2 , then n Lim (z1 z2 z3 ....zn) = (A) 1 (B) 1 (C) i (D) – i Q.7 The maximum & minimum values of z + 1 when z + 3 3 are : (A) (5 , 0) (B) (6 , 0) (C) (7 , 1) (D) (5 , 1) Q.8 If z3 + (3 + 2i) z + (–1 + ia) = 0 has one real root, then the value of 'a' lies in the interval (a R) (A) (– 2, – 1) (B) (– 1, 0) (C) (0, 1) (D) (1, 2) Q.9 If x = a + bi is a complex number such that x2 = 3 + 4i and x3 = 2 + 11i where i = 1 , then (a + b) equal to (A) 2 (B) 3 (C) 4 (D) 5 Q.10 If Arg (z + a) = 6 and Arg (z – a) = 3 2 ; R a , then (A) z is independent of a (B) | a | = | z + a | (C) z = a Cis 6 (D) z = a Cis 3 Q.11 If z1, z2, z3 are the vertices of the ABC on the complex plane & are also the roots of the equation, z3 3z2 + 3z + x = 0, then the condition for the ABC to be equilateral triangle is : (A) 2 = (B) = 2 (C) 2 = 3 (D) = 32
- Q.12 The locus represented bythe equation, z 1 + z + 1 = 2 is : (A) an ellipse with focii (1,0) ; (1,0) (B)one ofthefamilyof circles passingthrough the points ofintersection ofthe circles z 1=1 and z + 1 = 1 (C) the radical axis of the circles z 1 = 1 and z + 1 = 1 (D) the portion of the real axis between the points (1,0) ; (1,0) including both. Q.13 The points z1 =3 + 3 i and z2 = 2 3 + 6i aregiven on a complex plane. Thecomplex number lying on the bisector of the angle formed bythe vectors z1 and z2 is : (A) z = 3 2 3 2 3 2 2 i (B) z = 5 + 5i (C) z = 1 i (D) none Q.14 Let z1 & z2 be non zero complex numbers satisfying the equation, z1 2 2 z1z2 + 2 z2 2 = 0. The geometrical nature ofthetriangle whose vertices arethe origin and the points representing z1 & z2 is : (A)anisoscelesright angledtriangle (B)aright angledtrianglewhichisnot isosceles (C)anequilateraltriangle (D)anisosceles trianglewhich is notright angled . Q.15 Let P denotes a complex number z on the Argand's plane, and Q denotes a complex number 2 | z | 2 CiS 4 where = amp z. If 'O' is the origin, then the OPQ is : (A) isosceles but not right angled (B) right angled but not isosceles (C) right isosceles (D) equilateral . Q.16 OntheArgandplanepoint 'A'denotesacomplexnumberz1.Atriangle OBQis madedirectilysimiliar tothetriangle OAM, whereOM = 1 as showninthefigure.IfthepointBdenotesthecomplexnumberz2, then the complex number corresponding to the point 'Q'is (A) z1 z2 (B) z z 1 2 (C) z z 2 1 (D) z z z 1 2 2 Q.17 z1 & z2 are two distinct points in an argand plane. If a z1 = b z2 , (where a, b R) then the point a z bz 1 2 + bz a z 2 1 is a point on the : (A) line segment [2, 2] of the real axis (B) line segment [2, 2] of the imaginaryaxis (C) unit circle z = 1 (D) the line with arg z = tan1 2 . Q.18 When the polynomial 5x3 + Mx + N is divided by x2 + x + 1 the remainder is 0. The value of (M + N) is equal to (A) – 3 (B) 5 (C) – 5 (D) 15 Q.19 1 1 1 ....... is equal to : (A) or 2 (B) or 2 (C) 1 + i or 1 i (D) 1 + i or 1 i where is the imaginarycube root of unityand i = 1 Q.20 If q1,q2,q3 arethe roots oftheequation, x3 +64= 0,then thevalueofthedeterminant 2 1 3 1 3 2 3 2 1 q q q q q q q q q is (A) 1 (B) 4 (C) 10 (D) zero
- CLASS : XII (ABCD) Dpp on Complex Number DPP. NO.- 5 After 5th Lecture Q.1 If = ei2/n, then (11 – ) (11 – 2) ...... (11 – n–1) = (A) 11n–1 (B) 10 1 11n (C) 10 1 11 1 n (D) 11 1 11 1 n Q.2 z is a complex number such that z + z 1 = 2 cos 3°, then the value of z2000 + 2000 z 1 + 1 is equal to (A) 0 (B) – 1 (C) 1 3 (D) 1 – 3 Q.3 If, betherootsof theequationu2 2u+2=0& if cot=x+1,then n n ) x ( ) x ( isequalto (A) n sin n sin (B) n cos n cos (C) n cos n sin (D) n sin n cos Q.4 Thecomplex numbersatisfyingtheequation3 =8i andlyingin thesecondquadrantonthecomplex planeis (A) – 3 + i (B) – 2 3 + 2 1 i (C) – 3 2 + i (D) – 3 + 2i Q.5 Consider the equation 10z2 – 3iz – k = 0, where z is a complex variable and i2 = – 1. Which of the followingstatementsisTrue? (A) For all real positive numbers k, both roots arepure imaginary. (B) For real negative real numbers k, both roots arepure imaginary. (C) For all pure imaginarynumbers k, both roots are real and irrational. (D) For all complex numbers k, neitherroot is real. Q.6 The complex number z satisfies the condition z z 25 =24 .Themaximum distancefrom the origin of co-ordinates to the point z is : (A) 25 (B) 30 (C) 32 (D) none of these Q.7 If the expression x2m + xm + 1 is divisible by x2 + x +1, then : (A) m is anyodd integer (B) m is divisible by3 (C) m is not divisible by3 (D) none of these Q.8 5 4 ) cos (sin ) sin (cos i i = (A) cosisin (B) cos9isin9 (C) sin9icos9 (D) sinicos Q.9 If p2 p + 1 = 0 then the value of p3n is (n I) : (A) 1, 1 (B) 1 (C) 1 (D) 0 Q.10 If z1 = 2 + 3 i , z2 = 3 – 2 i and z3 = – 1 – i 3 2 then which of the following is true? (A) arg 2 3 z z = arg 1 2 1 3 z z z z (B) arg 2 3 z z = arg 1 2 z z (C) arg 2 3 z z = 2 arg 1 2 1 3 z z z z (D) arg 2 3 z z = 2 1 arg 1 2 1 3 z z z z
- Q.11 If Ar (r = 1, 2, 3, ..... , n) are the vertices of a regular polygon inscribed in a circle of radius R, then (A1 A2)2 + (A1 A3)2 + (A1 A4)2 + ...... + (A1 An)2 = (A) nR2 2 (B) 2 nR2 (C) 4 R2 cot 2n (D) (2n 1) R2 Q.12 If D = b a c a c b c b a 2 2 2 ; D = b a c a c b c b a where is thenonreal cube root of unitythen which of the following does not hold good? (A) D = 0 if (a + b + c) = 0 and a ,b, c all distinct (B) D = 0 if a = b = c and (a + b + c) 0 (C) D = – D (D) D = D Q.13 If z is a complex number satisfying the equation, Z6 + Z3 + 1 = 0. If this equation has a root rei with 90° < < 180° then the value of '' is (A) 100° (B) 110° (C) 160° (D) 170° Q.14 If z1, z2, z3, z4 are the vertices of a square in that order, then which of the following do(es) not hold good? (A) z z z z 1 2 3 2 ispurelyimaginary (B) z z z z 1 3 2 4 is purelyimaginary (C) z z z z 1 2 3 4 is purelyimaginary (D) none of these Q.15 Given , respectivelythe fifth andthe fourth non-real roots of unity, then thevalue of (1 + ) (1 + ) (1 + 2) (1 + 2) (1 +3) (1 + 4) is (A) 0 (B) (1 + + 2) (1 – 2) (C) (1 + ) (1 + + 2) (D) 1 Q.16 Number of orderedpair(s) (z, ) of the complex numbers z and satisfying thesystem of equations, z3 + 7 = 0 and z5 . 11 = 1 is : (A) 7 (B) 5 (C) 3 (D) 2 Q.17 If p = a + b+ c2; q = b + c+ a2 and r = c + a+ b2 where a, b, c 0 and is the complex cube root ofunity,then: (A) p + q + r = a + b + c (B) p2 + q2 + r2 = a2 + b2 + c2 (C) p2 + q2 + r2 = 2(pq + qr + rp) (D) none of these Q.18 IfAand B be two complex numbers satisfying A B B A = 1. Then the two points represented byAand Bandthe origin form thevertices of (A)anequilateraltriangle (B)an isosceles triangle which is not equilateral (C) anisosceles trianglewhich isnot right angled (D)aright angled triangle Q.19 If is an imaginarycube root of unity, then the value of, (p + q)3 + (p+ q2)3 + (p 2 + q )3 is (A) p3 + q3 (B) 3 (p3 + q3) (C) 3 (p3 + q3) p q (p + q) (D) 3 (p3 + q3) + p q (p + q) Q.20 OnthecomplexplanetrianglesOAP&OQRaresimiliarandl(OA) = 1. If the points P and Q denotes the complex numbers z1 & z2 then the complex number 'z' denoted bythe point R is given by: (A) z1 z2 (B) z z 1 2 (C) z z 2 1 (D) z z z 1 2 2
- CLASS : XII (ABCD) Dpp on Complex Number DPP. NO.- 6 After 6th Lecture Q.1 The expression 8 8 8 8 8 cos sin 1 cos sin 1 i i = (A) 1 (B) 1 (C) i (D) – i Q.2 If z2 – z + 1 = 0 then the value of 2 24 24 2 3 3 2 2 2 2 z 1 z ........ z 1 z z 1 z z 1 z is equal to (A) 24 (B) 32 (C) 48 (D) None Q.3 If the six solutions of x6 =– 64 are writtenin the form a+ bi, where a and b arereal, then theproduct of those solutions witha > 0, is (A) 4 (B) 8 (C) 16 (D) 64 Q.4 IftheequationoftheperpendicularbisectorofthelinejoiningtwocomplexnumbersP(z1)andQ(z2)are the complex plane is 0 r z z then and r are respectivelyare (A) z2 – z1 and | z1 |2 + | z2 |2 (B) z1 – z2 and | z1 |2 – | z2 |2 (C) 1 2 z z and | z1 |2 + | z2 |2 (D) z2 – z1 and | z1 |2 – | z2 |2 Q.5 If & are imaginarycube roots of unity then n + n is equal to : (A) 2 cos 2 3 n (B) cos 2 3 n (C) 2isin 2 3 n (D) isin 2 3 n Q.6 If z + 4 3, z C, then the greatest and least value of z + 1 are : (A) (7 , 1) (B) (6 , 1) (C) (6 , 0) (D) none Q.7 If z1 & z2 are two complex numbers & if arg z z z z 1 2 1 2 = 2 but z z 1 2 z z 1 2 then the figure formed bythe points represented by 0, z1, z2 & z1 + z2 is : (A) a parallelogram but not a rectangleor a rhombous (B) a rectangle but not a square (C) a rhombous but not a square (D) a square Q.8 If zn = cos ( ) ( ) 2 1 2 3 n n + i sin ( ) ( ) 2 1 2 3 n n , then Limit n (z1 . z2 . z3 . ...... zn) = (A) cos 3 + i sin 3 (B) cos 6 + i sin 6 (C) cos 12 + i sin 12 (D) none Q.9 The straight line (1 + 2i)z + (2i – 1) z = 10i on the complex plane, has intercept onthe imaginaryaxis equal to (A) 5 (B) 5/2 (C) – 5/2 (D) – 5 Q.10 If cos + i sin is a root of the equation xn + a1xn 1 + a2xn 2 +......+ an 1x+ an = 0 then the value of a r r r n cos 1 = (A) 0 (B) 1 (C) 1 (D) none
- Q.11 LetA(z1)andB(z2)representtwocomplex numbers on thecomplex plane. Supposethecomplex slope of the line joiningAand B is defined as 2 1 2 1 z z z z . Then the lines l1 with complex slope 1 and l2 with complexslope 2 onthe complex plane will be perpendicular to eachother if (A) 1 + 2 = 0 (B) 1 – 2 = 0 (C) 12 = –1 (D) 1 2 = 1 Q.12 If the equation, z4 + a1z3 + a2z2 + a3z + a4 = 0, where a1, a2, a3, a4 are real coefficients different from zero has a pure imaginaryroot then the expression a a a 3 1 2 + a a 1 4 a a 2 3 has the value equal to: (A) 0 (B) 1 (C) 2 (D) 2 Q.13 P(z1),Q(z2),R(z3)and S(z4)arefourcomplex numbers representingthevertices of arhombus taken in order onthe complex plane, thenwhich oneofthefollowingis INCORRECT? (A) 3 2 4 1 z z z z is purelyreal (B) amp 4 2 4 1 z z z z amp 4 3 4 2 z z z z (C) 4 2 3 1 z z z z ispurelyimaginary (D) | z1 – z3 | | z2 – z4 | Q.14 SupposeAis a complex number & n N, such that An = (A+ 1)n = 1, then the least value of n is (A) 3 (B) 6 (C) 9 (D) 12 Q.15 Intercept made bythe circle z z + z + z + r = 0 on the real axis on complex plane, is (A) r ) ( (B) r 2 ) ( 2 (C) r ) ( 2 (D) r 4 ) ( 2 Q.16 If Zr ; r = 1, 2, 3,..., 50 are the roots of the equation r 0 50 (Z)r = 0, then the value of 1 1 1 50 Zr r is (A) 85 (B) 25 (C) 25 (D) 75 Q.17 All roots of the equation, (1 + z)6 + z6 = 0 : (A)lie onaunit circle withcentreat the origin (B) lie on a unit circle with centre at (- 1, 0) (C)lie onthevertices of aregularpolygon with centreat the origin (D)arecollinear Q.18 If z&w aretwocomplex numbers simultaneouslysatisfyingtheequations, z3 + w5 = 0 and z2 . w4 = 1 , then : (A) z and w both are purelyreal (B) z is purelyreal and w is purelyimaginary (C) wis purelyreal andzis purelyimaginarly (D) z and w both are imaginary. Q.19 If 1, z1, z2, z3, ...... , zn 1 be the nth roots of unityand be a non real complex cube root of unitythen the product r n 1 1 ( zr) can not be equal to : (A) 0 (B) 1 (C) 1 (D) 1 + Q.20 Whichofthefollowingrepresentsapointinanargands'plane,equidistant fromtherootsoftheequation (z + 1)4 = 16z4? (A) (0, 0) (B) 1 3 0 , (C) 1 3 0 , (D) 0 2 5 , ]
- ANSWER KEY DPP-1 Q.1 A Q.2 B Q.3 A Q.4 D Q.5 B Q.6 C Q.7 C Q.8 A Q.9 C Q.10 A Q.11 D Q.12 A Q.13 B Q.14 D Q.15 C DPP-2 Q.1 C Q.2 B Q.3 D Q.4 C Q.5 C Q.6 C Q.7 A Q.8 D Q.9 B Q.10 A Q.11 A Q . 12 A Q.13 A Q . 14 A Q.15 D DPP-3 Q.1 C Q.2 B Q.3 B Q.4 A Q.5 B Q.6 C Q.7 C Q.8 A Q.9 B Q.10 B Q.11 B Q.12 A Q.13 D Q.14 B Q.15 D Q.16 B Q.17 A Q.18 D Q.19 B Q.20 C DPP-4 Q.1 C Q.2 C Q.3 B Q.4 D Q.5 D Q.6 B Q.7 A Q.8 B Q.9 B Q.10 D Q.11 A Q.12 D Q.13 B Q.14 A Q.15 C Q.16 C Q.17 A Q.18 C Q.19 A Q.20 D DPP-5 Q.1 B Q.2 A Q.3 A Q.4 A Q.5 B Q.6 A Q.7 C Q.8 D Q.9 A Q.10 C Q.11 B Q.12 C Q.13 C Q.14 C Q.15 A Q.16 D Q.17 C Q.18 A Q.19 B Q.20 A DPP-6 Q.1 B Q.2 C Q.3 A Q.4 D Q.5 A Q.6 C Q.7 C Q.8 B Q.9 A Q.10 C Q.11 A Q.12 B Q.13 B Q.14 B Q.15 D Q.16 B Q.17 D Q.18 A Q.19 C Q . 20 C