Complex numbers 370 kho tài liệu bách khoa

KEY CONCEPTS DEFINITION : Complex numbers are definited as expressions of the form a + ib where a, b R & i = It is denoted by z i.e z = a + ib ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z) EVERY COMPLEX NUMBER CAN BE REGARDED AS Purely real if b = Purely imaginary if a = Imaginary if b Note : (a) The set R of real numbers is a proper subset of the Complex Numbers Hence the Complete Number system is N W I Q R C (b) Zero is both purely real as well as purely imaginary but not imaginary (c) i= (d) a is called the imaginary unit Also i² = l ; i3 = i ; i4 = etc b = a b only if atleast one of either a or b is non-negative CONJUGATE COMPLEX : If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z i.e z = a ib Note that : (i) z + z = Re(z) (ii) z z = 2i Im(z) (iii) z z = a² + b² which is real st th (iv) If z lies in the quadrant then z lies in the quadrant and z lies in the 2nd quadrant ALGEBRAIC OPERATIONS : The algebraic operations on complex numbers are similiar to those on real numbers treating i as a polynomial Inequalities in complex numbers are not defined There is no validity if we say that complex number is positive or negative e.g z > 0, + 2i < + i are meaningless However in real numbers if a2 + b2 = then a = = b but in complex numbers, z12 + z22 = does not imply z1 = z2 = EQUALITY IN COMPLEX NUMBER : Two complex numbers z1 = a1 + ib1 & z2 = a2 + ib2 are equal if and only if their real & imaginary parts coincide REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS : (a) Cartesian Form (Geometric Representation) : Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair (x, y) length OP is called modulus of the complex number denoted by z & is called the argument or amplitude eg z = x = tan y x y2 & (angle made by OP with positive x axis) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) NOTE : (i) z z is always non negative Unlike real numbers z = if z z if z is not correct (ii) Argument of a complex number is a many valued function If is the argument of a complex number then n + ; n I will also be the argument of that complex number Any two arguments of a complex number differ by 2n (iii) The unique value of (iv) Unless otherwise stated, amp z implies principal value of the argument (v) By specifying the modulus & argument a complex number is defined completely For the complex number + i the argument is not defined and this is the only complex number which is given by its modulus (vi) There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers (b) Trignometric / Polar Representation : z = r (cos + i sin ) where | z | = r ; arg z = Note: cos + i sin is also written as CiS such that – < is called the principal value of the argument ; z = r (cos i sin ) eix (c) e ix e ix e ix & sin x = are known as Euler's identities Also cos x = 2 Exponential Representation : ; z = re i z = rei ; | z | = r ; arg z = IMPORTANT PROPERTIES OF CONJUGATE / MODULI / AMPLITUDE : If z , z1 , z2 C then ; (a) z + z = Re (z) ; z (b) z = i Im (z) z1 z2 ; z1 z = z1 z z1 z = z1 + z ; ; z1 z2 |z| Re (z) ; | z | Im (z) ; | z | = | z | = | – z | ; z z = | z | ; ; |z| z1 ; z2 | z1 + z2 |2 + | z1 – z2 |2 = [| z1 |2 z1 (7) (z ) = z z1 z = z1 z1 z = z | z (c) ; = | z1 | , z2 | z2 | = z2 ; z2 0 , | zn | = | z |n ; | z |2 ] (i) z2 z1 + z z + z2 amp (z1 z2) = amp z1 + amp z2 + k (ii) amp (iii) amp(zn) = n amp(z) + 2k where proper value of k must be chosen so that RHS lies in ( z1 z2 = amp z1 amp z2 + k ; [ TRIANGLE INEQUALITY ] k I k I , ] VECTORIAL REPRESENTATION OF A COMPLEX : Every complex number can be considered as if it is the position vector of that point If the point P represents the complex number z then, OP = z & OP = z ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) NOTE : (i) If OP = z = r ei then OQ = z1 = r ei ( of unequal magnitude then OQ (ii) (iii) (i) (ii) (iii) (iv) (v) 10 (b) z 12 + z 22 + z 23 = z 20 Q The i i , 2 If w is one of the imaginary cube roots of unity then + w + w² = In general + wr + w2r = ; where r I but is not the multiple of In polar form the cube roots of unity are : 2 4 cos + i sin ; cos + i sin , cos + i sin 3 3 The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral triangle The following factorisation should be remembered : (a, b, c R & is the cube root of unity) 2) ; a3 b3 = (a b) (a b) (a ²b) ; x2 + x + = (x ) (x a3 + b3 = (a + b) (a + b) (a + 2b) ; a3 + b3 + c3 3abc = (a + b + c) (a + b + ²c) (a + ²b + c) nth ROOTS OF UNITY : If , , , n are the n , nth root of unity then : (i) They are in G.P with common ratio ei(2 /n) & The cube roots of unity are , (iv) (ii) z1 z2 z2 z3 z3 z1 = CUBE ROOT OF UNITY : (iii) (i) OP e i DEMOIVRE’S THEOREM : Statement : cos n + i sin n is the value or one of the values of (cos + i sin )n ¥ n theorem is very useful in determining the roots of any complex quantity Note : Continued product of the roots of a complex quantity should be determined using theory of equations (ii) 11 = z e i If OP and OQ are If A, B, C & D are four points representing the complex numbers z1, z2 , z3 & z4 then z z AB CD if is purely real ; z z1 z z3 AB CD if z z is purely imaginary ] If z1, z2, z3 are the vertices of an equilateral triangle where z0 is its circumcentre then (a) z 12 + z 22 + z 23 + ) 1p + p + p + + p n = if p is not an integral multiple of n = n if p is an integral multiple of n (1 & 1) (1 2) (1 n 1) = n (1 + 1) (1 + 2) (1 + n 1) = if n is even and if n is odd n = or according as n is odd or even THE SUM OF THE FOLLOWING SERIES SHOULD BE REMEMBERED : sin n n cos + cos + cos + + cos n = cos sin 2 sin n n sin sin 2 = (2 /n) then the sum of the above series vanishes sin + sin + sin + + sin n = Note : If ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) 12 STRAIGHT LINES & CIRCLES IN TERMS OF COMPLEX NUMBERS : nz mz If z1 & z2 are two complex numbers then the complex number z = divides the joins of z1 m n & z2 in the ratio m : n (A) Note: (i) If a , b , c are three real numbers such that az1 + bz2 + cz3 = ; where a + b + c = and a,b,c are not all simultaneously zero, then the complex numbers z1 , z2 & z3 are collinear (a) Centroid of the (b) Orthocentre of the a sec A z1 ABC = b sec B z c sec C z OR z1 tan A z tan B z tan C z z1 z z1z z1z = 0, which on manipulating takes the form as z z r=0 where r is real and is a non zero complex constant The equation of circle having centre z0 & radius " is : z z0 = " or z z z0 z z z + z z0 "² = which is of the form z z r = , r is real centre & radius r Circle will be real if r The equation of the circle described on the line segment joining z1 & z2 as diameter is : (i) arg (J) : Complex equation of a straight line through two given points z & z2 can be written as zz (I) z z1 = This is also the condition for three complex numbers to be collinear z2 z z1 z (H) z3 amp(z) = is a ray emanating from the origin inclined at an angle to the x axis z a = z b is the perpendicular bisector of the line joining a to b The equation of a line joining z1 & z2 is given by ; z = z1 + t (z1 z2) where t is a perameter z = z1 (1 + it) where t is a real parameter is a line through the point z1 & perpendicular to oz1 The equation of a line passing through z1 & z2 can be expressed in the determinant form as z z1 z2 (G) z1 z Circumcentre of the ABC = : (Z1 sin 2A + Z2 sin 2B + Z3 sin 2C) ! (sin 2A + sin 2B + sin 2C) (d) (B) (C) (D) ABC = a sec A b sec B c sec C tan A tan B tan C Incentre of the ABC = (az1 + bz2 + cz3) ! (a + b + c) (c) (E) (F) represent the complex nos z1, z2, z3 respectively, then : If the vertices A, B, C of a (ii) z z2 z z1 = ± or (z z1) ( z z 2) + (z z2 ) ( z z 1) = Condition for four given points z1 , z2 , z3 & z4 to be concyclic is, the number z z1 z z is real Hence the equation of a circle through 3non collinear points z1, z2 & z3 can be z z z z1 taken as z z z z1 z z1 z z is real # z z z z1 z z1 z z = z z z3 z1 z z1 z3 z ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) 13.(a) Reflection points for a straight line : Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ Note that the two points denoted by the complex numbers z1 & z2 will be the reflection points for the straight line z z r if and only if ; z1 z r , where r is real and is non zero complex constant (b) Inverse points w.r.t a circle : Two points P & Q are said to be inverse w.r.t a circle with centre 'O' and radius ", if : (i) the point O, P, Q are collinear and on the same side of O (ii) OP OQ = "2 Note that the two points z1 & z2 will be the inverse points w.r.t the circle z z z z r if and only if z1 z z1 z r 14 PTOLEMY’S THEOREM : It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides i.e z1 z3 z2 z4 = z1 z2 z3 z4 + z1 z4 z2 z3 15 LOGARITHM OF A COMPLEX QUANTITY : $ Loge ( + i $) = Loge ( ² + $²) + i 2n tan (i) where n 2n ii represents a set of positive real numbers given by e (ii) I ,n I VERY ELEMENTARY EXERCISE Q.1 Simplify and express the result in the form of a + bi 2i (a) i (b) i (9 + i) (2 i) 4i i (c) 2i (d) 2i 5i i 2i (e) i 5i 2 i i (f) A square P1P2P3P4 is drawn in the complex plane with P1 at (1, 0) and P3 at (3, 0) Let Pn denotes the point (xn, yn) n = 1, 2, 3, Find the numerical value of the product of complex numbers (x1 + i y1)(x2 + i y2)(x3 + i y3)(x4 + i y4) Q.2 Given that x , y R, solve : (a) (x + 2y) + i (2x 3y) = 4i (b) (x + iy) + (7 5i) = + 4i (c) x² y² i (2x + y) = 2i (d) (2 + 3i) x² (3 2i) y = 2x 3y + 5i Q.3 Find the square root of : Q.4 (a) (b) Q.5 Among the complex numbers z satisfying the condition z least positive argument Q.6 Solve the following equations over C and express the result in the form a + ib, a, b (a) ix2 3x 2i = (b) (1 + i) x2 (2 i) x i = Q.7 Locate the points representing the complex number z on the Argand plane: (a) (a) + 40 i (b) 11 60 i (c) 50 i If f (x) = x4 + 9x3 + 35x2 x + 4, find f ( – + 4i) If g (x) = x4 x3 + x2 + 3x 5, find g(2 + 3i) z + 2i = ; (b) z z = ; (c) z z 3i , find the number having the = ; (d) R z = z Q.8 If a & b are real numbers between & such that the points z1 = a + i, z2 = + bi & z3 = form an equilateral triangle, then find the values of 'a' and 'b' Q.9 Let z1 = + i and z2 = – – i Find z3 Q.10 For what real values of x & y are the numbers C such that triangle z1, z2, z3 is equilaterial + ix2 y & x2 + y + 4i conjugate complex? ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.11 Find the modulus, argument and the principal argument of the complex numbers (i) (cos 310° i sin 310°) Q.12 (ii) (cos 30° + i sin 30°) If (x + iy)1/3 = a + bi ; prove that (a2 b2) = x a (iii) 4i i (1 i) y b z z2 c\R and z z2 Q.13 Let z be a complex number such that z Q.14 Prove the identity, | z1z |2 | z1 z |2 | z1 | | z | Q.15 Prove the identity, | z1 z | | z1 z |2 | z1 |2 | z |2 Q.16 For any two complex numbers, prove that z1 z 2 z1 R, then prove that | z | =1 z2 = z1 z2 Also give the geometrical interpretation of this identity Find all non zero complex numbers Z satisfying Z = i Z² If the complex numbers z1, z2, zn lie on the unit circle |z| = then show that |z1 + z2 + +zn| = |z1–1+ z2–1+ +zn–1| Q.17 (a) (b) Q.18 Find the Cartesian equation of the locus of 'z' in the complex plane satisfying, | z – | + z + | = 16 Q.19 Let z = (0, 1) n C Express % zk in terms of the positive integer n k z i where z = x + iy, where x, y 2z Q.20 If the complex number w is purely imaginary then locus of z is (A) a straight line Consider a complex number w = (B) a circle with centre R 1 , and radius 4 1 , and passing through origin (D) neither a circle nor a straight line (C) a circle with centre Q.21 If the complex number w is purely real then locus of z is (A) a straight line passing through origin (B) a straight line with gradient and y intercept (–1) (C) a straight line with gradient and y intercept (D) none Q.22 If | w | = then the locus of P is (A) a point circle (C) a real circle (B) an imaginary circle (D) not a circle ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) EXERCISE–I Q.1 Simplify and express the result in the form of a + bi : (a) i (9 + i) (2 i) 4i i (b) 2i 1 (c) 2i 5i 2i 5i i2 i2 (e) i i i i Find the modulus , argument and the principal argument of the complex numbers (d) Q.2 (i) z = + cos (iii) z = Q.3 10 12i 12i Given that x, y + i sin 10 (ii) (tan1 – i)2 12i 12i (iv) i i cos 5 sin R, solve : x y 2i 2i (c) x² y² i (2x + y) = 2i (d) (2 + 3i) x² (3 (e) 4x² + 3xy + (2xy 3x²)i = 4y² (x2/2) + (3xy 2y²)i Q.4(a) Let Z is complex satisfying the equation, z2 – (3 + i)z + m + 2i = 0, Suppose the equation has a real root, then find the value of m (a) (x + 2y) + i (2x 3y) = 4i (b) 6i 8i 2i) y = 2x 3y + 5i where m R (b) a, b, c are real numbers in the polynomial, P(Z) = 2Z4 + aZ3 + bZ2 + cZ + If two roots of the equation P(Z) = are and i, then find the value of 'a' Q.5(a) Find the real values of x & y for which z1 = 9y2 10 i x and z2 = 8y2 20 i are conjugate complex of each other (b) Find the value of x4 x3 + x2 + 3x if x = + 3i Q.6 Solve the following for z : z2 – (3 – i)z = (5i – 5) Q.7(a) If i Z3 + Z2 Z + i = 0, then show that | Z | = (b) Let z1 and z2 be two complex numbers such that z1 2z 2 z1z = and | z2 | 1, find | z1 | (c) Let z1 = 10 + 6i & z2 = + 6i If z is any complex number such that the argument of, Q.8 z z2 is , then prove that z 9i = Show that the product, i ( i 1 & ' Q.9 z z1 2( i &1 &' 22 ( & 1 i & ' 2n ( & & is equal to ' 22 n (1+ i) where n Let z1, z2 be complex numbers with | z1 | = | z2 | = 1, prove that | z1 + | + | z2 + | + | z1z2 + | ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.10 Interpret the following locii in z Q.11 C (a) < z 2i < (c) Arg (z + i) Arg (z i) = /2 Let A = {a (b) z 2i (z 2i) iz (d) Arg (z a) = /3 where a = + 4i Re R | the equation (1 + 2i)x3 – 2(3 + i)x2 + (5 – 4i)x + 2a2 = 0} has at least one real root Find the value of % a2 a A Q.12 P is a point on the Aragand diagram On the circle with OP as diameter two points Q & R are taken such that ) POQ = ) QOR = If ‘O’ is the origin & P, Q & R are represented by the complex numbers Z1 , Z2 & Z3 respectively, show that : Z22 cos = Z1 Z3 cos² Q.13 Let z1, z2, z3 are three pair wise distinct complex numbers and t1, t2, t3 are non-negative real numbers such that t1 + t2 + t3 = Prove that the complex number z = t1z1 + t2z2 + t3z3 lies inside a triangle with vertices z1, z2, z3 or on its boundry Q.14 Let A * z1 ; B * z2; C * z3 are three complex numbers denoting the vertices of an acute angled triangle If the origin ‘O’ is the orthocentre of the triangle, then prove that z1 z + z1 z2 = z2 z + z z3 = z3 z1 + z z1 hence show that the ABC is a right angled triangle + z1 z + z1 z2 = z2 z + z z3 = z3 z1 + z z1 = Q.15 Let + i$; , $ R, be a root of the equation x3 + qx + r = 0; q, r independent of & $, whose one root is Q.16 Find the sum of the series 1(2 – )(2 – one of the imaginary cube root of unity 2) + 2(3 – ) (3 – R Find a real cubic equation, 2) (n – 1)(n – )(n – 2) where Q.17 If A, B and C are the angles of a triangle e iA iC D= e e iB e iC e 2iB eiA e iB eiA e iC where i = then find the value of D Q.18 If w is an imaginary cube root of unity then prove that : (a) (1 w + w2) (1 w2 + w4) (1 w4 + w8) to 2n factors = 22n (b) If w is a complex cube root of unity, find the value of (1 + w) (1 + w2) (1 + w4) (1 + w8) to n factors Q.19 Prove that sin sin i cos i cos n = cos sin Q.20 i cos n n + i sin n n Hence deduce that =0 i cos 5 If cos ( $) + cos ($ -) + cos () = 3/2 then prove that: (a) cos = = sin (b) sin ( + $) = = cos ( + $) 2 (c) sin = cos = 3/2 (d) sin = sin ( + $ + -) (e) cos = cos ( + $ + -) (f) cos3 ( + ) + cos3 ( + $) + cos3 ( + -) = cos ( + ) cos ( + $) cos ( + -) where + i sin R ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) is Q.21 Resolve Z5 + into linear & quadratic factors with real coefficients Deduce that : 4·sin 10 ·cos = Q.22 If x = 1+ i ; y = i & z = , then prove that xp + yp = zp for every prime p > Q.23 Dividing f(z) by z i, we get the remainder i and dividing it by z + i, we get the remainder + i Find the remainder upon the division of f(z) by z² + Q.24(a) Let z = x + iy be a complex number, where x and y are real numbers Let A and B be the sets defined by A = {z | | z | 2} and B = {z | (1 – i)z + (1 + i) z 4} Find the area of the region A / B (b) For all real numbers x, let the mapping f (x) = , where i = If there exist real number x i a, b, c and d for which f (a), f (b), f (c) and f (d) form a square on the complex plane Find the area of the square Q.25 Column-I Column-II (A) Let w be a non real cube root of unity then the number of distinct elements (P) in the set (1 w w w n ) m | m, n N is Let 1, w, w2 be the cube root of unity The least possible (Q) degree of a polynomial with real coefficients having roots 2w, (2 + 3w), (2 + 3w2), (2 – w – w2), is = + 4i and $ = (2 + 4i) are two complex numbers on the complex plane (R) (B) (C) z z $ A complex number z satisfying amp moves on the major (S) segment of a circle whose radius is EXERCISE–II p q Q.1 r If q r p r 0; where p , q , r are the moduli of non zero complex numbers u, v, w respectively,, p q w w u prove that, arg = arg v v u Q.2 Let Z = 18 + 26i where Z0 = x0 + iy0 (x0, y0 Find the value of x0y0(x0 + y0) Q.3 Show that the locus formed by z in the equation z3 + iz = never crosses the co-ordinate axes in the Im(z) Re( z) Im( z) Argand’s plane Further show that |z| = x5 = 10x2 + 10x + If Q.5 Prove that , with regard to the quadratic equation z2 + (p + ip0) z + q + iq0 = where p , p0, q , q0 are all real (i) if the equation has one real root then q 02 pp q + qp 02 = (ii) if the equation has two equal roots then p2 p02 = 4q & pp = 2q State whether these equal roots are real or complex If the equation (z + 1)7 + z7 = has roots z1, z2, z7, find the value of (a) % Re( Zr ) r + 2, prove that Q.4 Q.6 is the fifth root of and x = R) is the cube root of Z having least positive argument and (b) % Im(Zr ) r ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 10 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.7 Find the roots of the equation Zn = (Z + 1)n and show that the points which represent them are collinear on the complex plane Hence show that these roots are also the roots of the equation sin m n Z + sin m n Z + = Q.8 If the expression z5 – 32 can be factorised into linear and quadratic factors over real coefficients as (z5 – 32) = (z – 2)(z2 – pz + 4)(z2 – qz + 4) then find the value of (p2 + 2p) Q.9 Let z1 & z2 be any two arbitrary complex numbers then prove that : z1 z2 | z1 | | z | | z1 | | z | z1 + z2 Q.10 If Zr, r = 1, 2, 3, 2m, m N are the roots of the equation 2m Z2m + Z2m-1 + Z2m-2 + + Z + = then prove that r%1 Z = m r Q.11(i) Let Cr's denotes the combinatorial coefficients in the expansion of (1 + x)n, n an = C0 + C3 + C6 + C9 + bn = C1 + C4 + C7 + C10 + and cn = C2 + C5 + C8 + C11 + , then prove that (a) a 3n b3n c 3n – 3anbncn = 2n, (b) N If the integers (an – bn)2 + (bn – cn)2 + (cn – an)2 = (ii) Prove the identity: (C0 – C2 + C4 – C6 + .)2 + (C1 – C3 + C5 – C7 + .)2 = 2n Q.12 Let z1 , z2 , z3 , z4 be the vertices A , B , C , D respectively of a square on the Argand diagram taken in anticlockwise direction then prove that : & (ii) 2z4 = (1 i) z1 + (1 + i) z3 (i) 2z2 = (1 + i) z1 + (1 i)z3 Q.13 Show that all the roots of the equation i x Q.14 Prove that: n ix 1 ia ia a R are real and distinct (a) cos x + nC1 cos 2x + nC2 cos 3x + + nCn cos (n + 1) x = 2n cosn (b) sin x + nC1 sin 2x + nC2 sin 3x + + nCn sin (n + 1) x = 2n cosn (c) cos Q.15 2n + cos 2n + cos 2n + + cos x x cos sin 2n = 2n 1 n n 2 2 x When n N Show that all roots of the equation a0zn + a1zn – + + an – 1z + an = n, where | | 1, i = 0, 1, 2, , n lie outside the circle with centre at the origin and radius Q.16 x n n The points A, B, C depict the complex numbers z1 , z2 , z3 respectively on a complex plane & the angle B & C of the triangle ABC are each equal to ( ) Show that (z2 z3)² = (z3 z1) (z1 z2) sin2 ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 11 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) 32 Q.17 Evaluate: % p 10 (3 p 2) p 2q sin 11 % q 2q i cos 11 a Q.19 Let , $ be fixed complex numbers and z is a variable complex number such that, z + z c = c Let a, b, c be distinct complex numbers such that b = b Q.18 a = k Find the value of k $ = k Find out the limits for 'k' such that the locus of z is a circle Find also the centre and radius of the circle R is defined by f (z) = | z3 – z + 2| Find the maximum value of f (z) Q.20 C is the complex number f : C if | z | = Q.21 Let f (x) = logcos 3x (cos i x) if x and f (0) = K (where i = ) is continuous at x = then find the value of K 20 = Q.22 If of and f(x) = A0 + % k Ak xk, then find the value of, f (x) + f ( x) + + f( 6x) independent Q.23 Find the set of points on the argand plane for which the real part of the complex number (1 + i)z is positive where z = x + iy , x, y R and i = Q.24 If a and b are positive integer such that N = (a + ib)3 – 107i is a positive integer Find N Q.25 If the biquadratic x4 + ax3 + bx2 + cx + d = (a, b, c, d R) has non real roots, two with sum + 4i and the other two with product 13 + i Find the value of 'b' EXERCISE–III Q.1(a) If z1 , z2 , z3 are complex numbers such that z1 + z2 + z3 is : (A) equal to z1 = z2 = z3 = (B) less than (C) greater than (b) If arg (z) < , then arg ( z) (A) z1 z2 = 1, then z3 (D) equal to arg (z) = (B) (C) (D) [ JEE 2000 (Screening) + out of 35 ] 2 Q.2 Given , z = cos n + i sin , 'n' a positive integer, find the equation whose roots are, 2n = z + z3 + + z2n & $ = z2 + z4 + + z2n [ REE 2000 (Mains) out of 100 ] Q.3 Find all those roots of the equation z12 – 56z6 – 512 = whose imaginary part is positive [ REE 2000, out of 100 ] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 12 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.4(a) The complex numbers z1, z2 and z3 satisfying (A) of area zero (C) equilateral z1 z i are the vertices of a triangle which is z2 z3 (B) right-angled isosceles (D) obtuse – angled isosceles (b) Let z1 and z2 be nth roots of unity which subtend a right angle at the origin Then n must be of the form (A) 4k + (B) 4k + (C) 4k + (D) 4k [ JEE 2001 (Scr) + out of 35 ] Q.5(a) Let i Then the value of the determinant 1 (A) (B) ( – 1) 1 2 (C) is (D) (1 – ) (b) For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 – – 4i| = 5, the minimum value of |z1 – z2| is (A) (B) (C) (D) 17 [JEE 2002 (Scr) 3+3] (c) Let a complex number , 1, be a root of the equation p+q p q z – z – z + = where p, q are distinct primes Show that either + + + + p–1 = or + + + + q–1 = , but not both together [JEE 2002, (5) ] Q.6(a) If z1 and z2 are two complex numbers such that | z1 | < < | z2 | then prove that (b) Prove that there exists no complex number z such that | z | < and z1 z2 z1 z n % a r zr = where | ar | < r [JEE-03, + out of 60] Q.7(a) is an imaginary cube root of unity If (1 + (A) (B) 2)m = (1 + 4)m , then least positive integral value of m is (C) (D) [JEE 2004 (Scr)] (z ) k, (z $) [JEE 2004, out of 60 ] (b) Find centre and radius of the circle determined by all complex numbers z = x + i y satisfying where i 2, $ $1 i$ are fixed complex and k Q.8(a) The locus of z which lies in shaded region is best represented by (A) z : |z + 1| > 2, |arg(z + 1)| < /4 (B) z : |z - 1| > 2, |arg(z – 1)| < /4 (C) z : |z + 1| < 2, |arg(z + 1)| < /2 (D) z : |z - 1| < 2, |arg(z - 1)| < /2 (b) If a, b, c are integers not all equal and w is a cube root of unity (w 1), then the minimum value of |a + bw + cw2| is (A) (B) (C) 2 [JEE 2005 (Scr), + 3] (D) (c) If one of the vertices of the square circumscribing the circle |z – 1| = is i Find the other vertices of square [JEE 2005 (Mains), 4] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 13 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.9 If w = w wz is purely real, then the set of z + i$ where $ and z 1, satisfies the condition that values of z is (A) {z : | z | = 1} (B) {z : z = z ) (D) {z : | z | = 1, z 1} [JEE 2006, 3] Q.10(a) A man walks a distance of units from the origin towards the North-East (N 45° E) direction From there, he walks a distance of units towards the North-West (N 45° W) direction to reach a point P Then the position of P in the Argand plane is (C) (4 3i)ei z (b) If | z | = and z ± 1, then all the values of lie on z2 (A) 3ei + 4i (B) (3 4i)e i (C) {z : z 1} (A) a line not passing through the origin (C) the x-axis (D) (3 4i )e i (B) | z | = (D) the y-axis [JEE 2007, 3+3] Q.11(a)A particle P starts from the point z0 = + 2i, where i = It moves first horizontally away from origin by units and then vertically away from origin by units to reach a point z1 From z1 the particle moves units in the direction of the vector ˆi ˆj and then it moves through an angle in anticlockwise direction on a circle with centre at origin, to reach a point z2 The point z2 is given by (A) + 7i (B) – + 6i (C) + 6i (D) – + 7i (b) Comprehension (3 questions together) Let A, B, C be three sets of complex numbers as defined below A = z : Im z B = z :| z i | (i) C = z : Re((1 i )z ) The number of elements in the set A / B / C is (A) (B) (C) (ii) Let z be any point in A / B / C Then, | z + – i |2 + | z – – i |2 lies between (A) 25 and 29 (B) 30 and 34 (C) 35 and 39 (D) 40 and 44 (iii) Let z be any point in A / B / C and let w be any point satisfying | w – – i | < Then, | z | – | w | + lies between (A) –6 and (B) –3 and (C) –6 and (D) –3 and [JEE 2008, + + + 4] (D) 15 Q.12(a) Let z = cos + i sin Then the value of % Im(z 2m ) at = 2° is m 1 (A) sin 23 (b) (B) sin 23 (C) sin 23 (D) sin 23 Let z = x + iy be a complex number where x and y are integers Then the area of the rectangle whose vertices are the roots of the equation z z zz 350 is (A) 48 (B) 32 (C) 40 (D) 80 [JEE 2009, + 3] ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 14 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) VERY ELEMENTARY EXERCISE 25 21 12 24 i; (b) i; (c) + 4i; (d) 5 25 22 + 0i; (e) i; (f) 15 29 Q.1 (a) Q.2 (a) x =1, y = 2; (b) (2, 9); (c) ( , 2) or Q.3 (a) ± (5 + 4i) ; (b) ± (5 6i) (c) ± 5(1 + i) Q.5 – Q.7 (a) on a circle of radius with centre ( 1, 2) ; (b) on a unit circle with centre at origin (c) on a circle with centre ( 15/4, 0) & radius 9/4 ; (d) a straight line Q.8 a=b=2 Q.10 x = 1, y = or x = 1, y = Q.11 (i) Modulus = , Arg = k + 3; ; (d) (1 ,1) , Q.9 7(1, 0) 4(1,1) Q.19 6(0,1) 4(0, 0) i , i ,i for n 4k for n 4k for n 4k for n 4k Q.4 (a) 160 ; (b) Q.6 (a) z3 = tan (K x2 64 Q.18 Q.20 B (77 +108 i) i , 2i (b) (1 i) and z '3 5 (K I) , Principal Arg = (K 18 18 + , Principal Arg = 6 , Arg = k (iii) Modulus = (a) 3 i (ii) Modulus = , Arg = k Q.17 , 5i or i 3( i) I) I) , Principal Arg = tan 12 y2 48 Q.21 C Q.22 C EXERCISE–I Q.1 (a) 21 12 +0i 29 i (b) + i (c) Q.2 (i) Principal Arg z = ; z = cos (ii) Modulus = sec21 , Arg = n (iii) Principal value of Agr z = (iv) Modulus = (d) (2 – (a) 2, (b) – 11/2 Q.5 z = (2 + i) or (1 – 3i) (b) ; Arg z = k k I ) ; Principal value of Arg z = & z = 2 11 11 , Principal Arg = 20 20 Q.3(a) x = 1, y = 2; (b) x = & y = ; (c) ( , 2) or Q.4 Q.6 Q.7 i or i ) , Principal Arg = (2 – & z = cos ec , Arg z = n 22 i (e) + , 3 (a) [( 2, 2) ; ( 2, 2)] ; (d) (1 ,1) , (b) ; (e) x =K, y = 3K K R (77 +108 i) ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 15 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) Q.10 (a) The region between the co encentric circles with centre at (0 , 2) & radii & units 1 + 2i and radius 2 (c) semi circle (in the 1st & 4th quadrant) x² + y² = (d) a ray emanating from the point (b) region outside or on the circle with centre (3 + 4i) directed away from the origin & having equation x Q.11 18 Q.15 x3 + qx Q.17 – r=0 Q.16 n (n 1) ( n &' Q.18 (b) one if n is even ; Q.21 (Z + 1) (Z² 2Z cos 36° + 1) (Z² Q.24 (a) – ; (b) 1/2 y 3 n w² if n is odd Q.23 i z 2Z cos 108° + 1) 2 i Q.25 (A) R; (B) Q; (C) P EXERCISE–II Q.2 12 Q.18 – Q.6 (a) – or – , (b) zero Q.24 Q.19 k > $ Q.17 48(1 - i) Q.20 | f (z) | is maximum when z = , where is the cube root unity and | f (z) | = 13 Q.21 K = – Q.22 7A0 + 7A7 x7 + 7A14 x14 Q.23 required set is constituted by the angles without their boundaries, whose sides are the straight lines y = ( 1) x and y + ( 1) x = containing the x axis Q.24 198 Q.25 51 EXERCISE–III Q.1 (a) A (b) A Q.2 z2 + z + Q.3 +1 + i , i , sin n sin 2i = 0, where = 2n Q.4 (a) C, (b) D Q.5 (a) B ; (b) B Q.7 (a) D; (b) Centre * k 2$ k , Radius = Q.8 (a) A, (b) B, (c) z2 = – i ; z3 = Q.10 (a) D ; (b) D Q.11 (a) D ; (b) (i) B; (ii) C; (iii) D (k 1) | k $ |2 i ; z4 = k | $ |2 i | |2 k Q.9 D Q.12 (a) D; (b) A ETOOS Academy Pvt Ltd : F-106, Road No 2, Indraprastha Industrial Area, End of Evergreen Motors 16 (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005)