Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
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1 MCN COMPLEX NUMBER C The complex number Complex number is denoted by ie = a + ib, where a is called as real part of (denoted by Re and b is called as imaginary part of (denoted by Im Here i =, also i =, i = i; i = etc 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 Practice Problems : If n is natural number then the value of i n + i n + + i n + + i n + is (a (b 0 (c i (d i The value of (i 00 + (i 99 + (i + will be (a 0 (b (c i (d i [Answers : ( b ( a] C Algebraic Operations on Complex Number : Addition(a + bi + (c + di = a + bi + c + di = (a + c + (b + d i Subtraction (a + bi (c + di = a + bi c di = (a c + (b d i Multiplication (a + ib (c + id = ac + adi + bci + bdi = (ac bd + (ad + bci a bi a bi c di Division c di c di c di ac bd bc ad i c d c d 5 Inequalities in complex numbers are not defined 6 In real numbers if a + b = 0 then a = 0 = b however in complex numbers, + = 0 does not imply = = 0 Equality In Complex Number : If = Re( = Re( and I m ( = I m ( Practice Problems : i The value of i i i is equal to (a i (b i (c i (d i m i If then the least integral value of m is i (a (b (c 8 (d 0 ( ix i ( iy i If i, then the real value of x and y are given by i i (a x =, y = (b x =, y = (c x =, y = (d x =, y = [Answers : ( d ( b ( b] C Modulus of a Complex Number : If = a + ib, then it s modulus is denoted and defined by a b origin Infact is the distance of from Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
2 Properties of modulus MCN (i = (ii (provide 0 (iii + + (iv (Equality in (iii and (iv holds if and only if origin, and are collinear with and on the same side of origin C Representation of a Complex Number : (a Cartesian Form (Geometric Representation : Every complex number = 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 = = x y and = y tan x is called the argument or amplitude If is the argument of a complex number then n + ; n I will also be the argument of that complex number The unique value of such that < is called the principal value of the argument Unless otherwise stated, amp implies principal value of the argument The argument of =,, +,, = tan quadrant y, according as = x + iy lies in I, II, III or IVth x Properties of Argument of a Complex Number : (i arg ( = arg ( + arg ( + m for some integer m (ii arg ( / = arg ( arg ( + m for some integer m (iii arg ( = arg ( + m for some integer m (iv arg ( = 0 is real, for any complex number 0 (v arg ( = ± / is purely imaginary, for any complex number 0 (vi arg ( = angle of the line segment joining the point ( and point ( (b Trignometric/Polar Representation : = r (cos + i sin where = r; arg = ; = r(cos i sin cos + i sin is also written as CiS or e i (c Euler s Representation : re i ; r;arg ; re ix ix i ix ix e e e e Also cosx and sin x are known as Euler s identities (d Vectorial Representation : Every complex number can be considered as if it is the position vector of a point If the point P represents the complex number then, OP & OP Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
3 Practice Problems : MCN If = and arg = 5 then equals to 6 (a i (b i (c i (d i If x r = r r then x x x is (a (b (c (d 0 The amplitude or argument of ( i( i ( i will be (a 0 (b (c (d 6 [Answers : ( c ( c ( c] C5 Conjugate of a complex Number Conjugate of a complex number = a + ib is denoted and defined by Properties of conjugate a ib (i = (ii (iii ( ( ( ( (iv ( ( ( (v ( ( ( (vi ( 0 ( (vii ( ( (viii ( (ix arg( arg( 0 C6 Demoivere s Theorem : If n is any integer then (i (cos + i sin n = cos n + i sin n (ii (cos + i sin (cos + i sin (cos + i sin (cos + i sin (cos n + i sin n = cos ( n + i sin ( n Practice Problems : Simplify the following : (cos i sin (cos isin (i (cos5 i sin5 (cos i sin 5 5 i (sin i cos (ii (cos i sin If x = cos + i sin and c c nc, show that n ccos ( nx n x Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
4 If x + (/x = cos and y + (/y = cos etc, then prove that MCN x y (i xy cos( (ii cos( xy y x m n x y (iv cos(m n m n n m x y y x m n (iii x y cos(m n [Answers : (i (ii ] C Cube Root of Unity : (i i The cube root of unity are, i, (ii If is one of the imaginary cube roots of unity then + + = 0 In general + r + r = 0; where r I but is not the multiple of (iii In polar form the cube roots of unity are : cos0 i sin 0;cos i sin,cos i sin (iv The three cube roots of unity when plotted on the argand plane constitute the varties of an equilateral triangle (v The following factorisation should be remembered : Practice Problems : (a, b, c R and is the cube root of unity a b = (a b (a b (a b ; a + a + = (a (a a + b = (a + b (a + b (a + b ; a + ab + b = (a b (a b a + b + c abc = (a + b + c (a + b + c (a + b + c If,, are the cube roots of unity, prove that (i ( + ( + = (ii ( + ( + ( + ( + 8 = (iii ( = ( = 9 (iv ( + ( + ( + 8 to n factors = n (v (vi n n 0,, when n is a multipleof when n is not a multipleof (a + + (a + + (a to n factors = (a n a b c c a b (vii a b c b c a = If and are complex cube roots of unity, prove that (i x + y = (x + y (x + y ( x + y (ii x y = (x y (x y ( x y If is an imaginary cube root of unity, prove that 0 Given + + = A, + + = B, + + = C where is cube root of unity (i express,, in terms of A, B, C (ii prove that A + B + C = ( + + Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
5 MCN 5 [Answers : ( A B C, A B C, A B C ] C8 nth Roots of Unity : If,,, n are the n, nth root of unity then : (i (ii (iii (iv They are in GP with common ratio e i(/n p p p p = 0, if p is not an integral multiple of n n = n if p is an integral multiple of n Practice Problems : ( ( ( n = n ( + ( + ( + n = 0 if n is even and if n is odd n = or according as n is odd or even If,,,, n are the nth roots of unity and n is an odd natural number then find the value of ( + ( + ( + ( + n C9 (i Rotation theorem If P( and Q( are two complex numbers such that =, then = e i where = POQ (ii If P(, Q( and R( are three complex numbers and PQR =, then e i Practice Problems : If,, are vertices of an equilateral having its circumcentre at origin such that = + i then find and Show that the triangle whose vertices are the points represented by the complex numbers,, on the Argand plane is equilateral if and only if 0, that is if and only if + + = + + If,, be the affixes of the vertices A, B and C respectively of a triangle ABC having centroid at G such that = 0 is the mid-point of AG, then prove that + + = 0 (a Complex numbers,, are the vertices A,B,C respectively of an isosceles right angled triangle with right angle at C Show that ( = ( ( (b If + cos = 0 then the origin, form vertices of an isosceles triangle with vertical angle Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
6 MCN 6 5 Show that the triangle whose vertices are,, and are directly similar if ' ' ',, ' ' ' 0 C0 Logarithm of a Complex Quantity : (i Loge( i Log e ( i n tan where n I n (ii i i represents a set of positive real numbers given by e,ni C Geometrical Properties : (i (ii Distance Formula : If and are affixies of the two points P and Q respectively then distance between P and Q is given by Section Formula : If and are affixes of the two points P and Q respectively and point C divides the line joining P and Q internally in the ratio m : n then affix to C is given by m n m n m n If C divides PQ in the ratio m : n externally then m n (iii Condition of collinearty : If a, b, c are three real numbers such that a + b + c = 0; where a + b + c = 0 and a, b, c are not all simultaneoulsy ero, then the complex numbers, and are collinear Important Results : ( If the vertices A, B, C of a triangle represents the complex numbers,, respectively and a, b, c are the length of sides then, (i Centroid of the ABC = (ii Orthocenter of the ABC = ( asec A (bsecb (c secc tan A tanb tanc or asec A bsecb csecc tan A tanb tanc (iii Incentre of the ABC = (a + b + c / (sin A + sin B + sin C ( amp ( = is a ray emanating from the origin inclined at an angle to the x-axis ( = is the perpendicular bisector of the line joining to ( The equation of a line joining and is given by, = + t( where t is a real parameter (5 = ( + it where t is a real parameter is a line through the point & perpendicular to the line joining to the origin (6 The equation of a line passing through and can be expressed in the determinant form as 0 This is also the condition for three complex numbers to be collinear The Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
7 MCN above equation on manipulating, takes the form r 0 where r is real and is a non ero complex constant ( The equation of circle having centre 0 and radius r is : 0 = r General equation of the circle a a b 0 where a is a complex number and b is real number Centre of the circle is a and radius is a b (8 The equation of the circle described on the line segment joining & as diameter is arg or ( ( ( ( 0 (9 Condition for four given points,, & to be concyclic is the number should be real Hence the equation of a circle through non collinear points, & can be taken as ( ( ( ( is real ( ( ( ( ( ( ( ( (0 Arg represent (i a line segment if = (ii Pair of ray if = 0 (iii a part of circle, if 0 < < ( Area of triangle formed by the points, & is i ( Perpendicular distance of a point 0 from the line r 0 is 0 0 r ( (i Complex slope of a line r 0 is (ii (iii Complex slope of a line joining by the points & is Complex slope of a line makine angle with real axis = e i ( & are the complex slopes of two lines (i If lines are parallel then = (ii If lines are perpendicular then + = 0 (5 If + = K > then locus of is an ellipse whose focii are & (6 If 0 = r then locus of is parabola whose focus is 0 and directrix is the line 0 0 r 0 Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
8 ( If k where k 0 or then locus of is circle MCN 8 (8 If = K < then locus of is a hyperbola, whose focii are & C (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 & will be the reflection points for the straight line r 0 if and only if; r 0, where r is real and is non ero complex constant (b Inverse points wrt a circle : Two points P & Q are said to be inverse wrt a circle with centre O and radius, if : (i the point O, P, Q are collinear and P, Q are on the same side of O (ii OP OQ = The two points & will be the inverse points wrt the circle r 0 if and only if r 0 Practice Problems : Find the radius and centre of the circle ( i ( i 0 Determine the value of k for which equation ( i ( i k 0 represent a circle Show that the points representing the complex numbers ( + i, ( i and i are collinear Find the perpendicular bisector of + i and 5 + 6i 5 If,, are the affixes of the vertices of a triangle having its circumcentres at the origin If is the affix of its orthocentre then prove that + + = 0 6 Find the locus of a complex number in the Argand plane, satisfying ( + i = 5 Show that the locus of a complex number satisfying arg is a circle Find the equation of the circle in cartesian coordinates 8 Locate the points in the Argand plane representing the complex numbers = x + iy for which (i + + < (ii arg( i = 6 (iii + + = (iv arg ( + i arg ( i = / i 9 Find the locus of the complex number in the Argand plane if i i 0 If = x + iy and, =, then find the locus of i [Answers : ( (,, ( k 5 ( ( 8 i (8 i 6 0 (6 Circle ( x + y = (8 (i Interior of the ellipse having foci at (, 0 and (, 0 and major axis of length units (ii A straight line passing through (, and making an angle of /6 with x-axis (iii Ellipse with foci at + 0i and + 0i and centre at origin (iv Locus of point is a circle with diameter AB and centre at origin with radius (0 lies on the real axis] C 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 products of lengths of the two pairs of its opposite sides ie = + Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
9 MCN 9 SINGLE CORRECT CHOICE TYPE If lies on the circle i =, then the value of arg is equal to (a (c 6 (b (d The area of the triangle on the Argand diagram formed by the complex numbers, i and + i is (a (c (b (d none If arg ( / = arg( /, then the value of is (a (b ½ (c (d none If (a + + (b + + (c + + (d + =, (a + + (b + + (c + + (d + = ( where and are the imaginary cube roots of unity, the value of is (a + + (b + + (c + + (d + (a (b (c (d a ib 5 The value of tan i loge a ib is ab (a a b ab (c a b ab (b b a (d none 6 ABCD is a rhombus It s diagonals AC and BD intersect at P and satisfy BD = AC If D + i, P i, then the complex number representing the point A can be (a 6 + i (b 6 i (c + i (d + i The location of the complex number = x + iy for which log / {log / ( + + } < 0 (a (b (c (d interiof of the parabola inside a circle empty set none 8 The value of the expression ( + ( + + ( + ( + + ( + ( (n + (n + (n + is where is the cube root of unity (a (b (c (d n (n n(n n(n n n none of these 9 If a, b, c and u, v, w are the complex numbers representing the vertices of two triangles such that c = ( r a + rb and w = ( ru + rv, where r is a complex number, then the two triangles : (a (b (c (d have the same area are similar are congurent none of these 0 If ( is a cube root of unity then i i i i = (a 0 (b (c i (d Center of the arc represented by i arg is i (a (5i 5 (b (5i 5 (c (9i 5 (d (9i 5 Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
10 The roots of the equation = 0 are represented by the vertices of (a (b (c (d The value of p (p a square an equilateral triangle a rhombus none of these 0 q q q sin icos is (a ( i (b ( + i (c 8( + i (d 8( i If = + icot, (a (b (c (d cosec cosec < < 0 then is equal to cosec or cosec none of these 5 If is a nonreal complex number then which of the following may not be real? (a + (b + (c ( (d + 6 The area of the region in the Argand plane in which can belong, satisfying the inequalities and +, is 5 (a 5 (b (c (d none of these A( and B( are the points on the circles = and = respectively, then (a max = (b min = (c max = (d min = 8 If the centroid of an equilateral triangle ABC in the Argand plane be at =, the vertex C is at i and A, B, C are in the anticlockwise sense then the vertex A is at (a = (b = + (c = 0 (d = 9 Let, be complex numbers such that i = 0 and arg( = Then arg equals (a (b p MCN 0 0 If is one of the roots of the equation : a 0 n + a n + + a n + a n =, where a i < for i = 0,, n, then (a (c (b (d If = + then lies on (a (b (c (d the real axis the imaginary axis a circle an ellipse If, and are cube roots of unity represented on the complex plane by the points A, B and C respectively If P, Q and R are the centroinds of AOB, BOC and AOC respectively then the area of PQR is (a (c 8 (b (d 0 is one of the roots of the equation n cos 0 + n cos + cos n + cos n = where i R, then (a (c 6 0 (b 0 0 (d none of these The point of intersection of the curves arg ( i + = and arg ( + i = is 6 given by (a ( + i (b i (c + i (d no intersection points is possible 5 If tangents drawn to circle = at A( and B( meet at P( p, then (a (c p p (b p ( (d p = (c (d 5 Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
11 MCN ANSWERS (SINGLE CORRECT CHOICE TYPE b 6 a d 6 b b b c b c a a 8 b d 8 b b b 9 b b 9 c d 5 c 0 a 5 d 0 a 5 c EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE Comprehension- Consider the following equation : x + ax + bx + c = 0 which has roots,, are the points A, B, C on the argard plane (Gaussian plane and a, b, c are complex number The centroid of the triangle is the point represented by (a a (b b (c a (d none The triangle ABC will be equilateral if (a b = a (b a = b (c a = b (d a = b The triangle ABC will be equilateral such that = = then the value of a is (a 0 (b (c ½ (d none The triangle ABC will be equilateral then its circumcentre is given by (a 9a 6b (b 6a 9b 6a 9b (c (d none Comprehension- Consider = + t + t < i t, where t is real and 5 The modulus of is directly proportional to t n then the value of n is (a 0 (b (c (d ½ 6 The modulus of is (a (b (c (d depends on t The locus of for any value of t ( 0 is (a a straight line (b a circle (c a parabola (d none Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
12 MATRIX-MATCH TYPE Matching- Match the value of column A with suitable given values of column B at the given x Column - A Column - B (A The value of (p 66 x + x + 6x + x + 9 at x = (B The value of (q 0 x + x x + 5 at x (C The value of (r x 6 + x + x + at i x (D The value of (s x + x x + 5 at x Matching- Match the column A with suitable values of column B Column - A (A ( i / (p Column - B 5/ (B ( + i /5 (q (C 6 / (r / 5 5 cos i sin cos i sin 5 cos i sin 5 5 (D ( + i 5 (s none Matching- Match the column A with suitable values of x given in column B Column - A (A x 5 = 0 (p Column - B 6 6 cos i sin 5 5 (B x 6 + = 0 (q (C x 5 + x x = 0 (r (D x + x + x + x + = 0 (s Matching- MCN cos i sin cos i sin 6 6 cos i sin 5 5 If is variable and and are two fix complex numbers Match with locus of given in column B according to given conditions given in column A Column - A Column - B (A If = then (p locus of is a circle (B If = k where (q locus of is k then perpendicular bisector of line joining and (C If + where (r locus of is k then ellipse (D If = k (s locus of is where k then hyperbola (A (B (C Matching-5 If is variable and and are two fix complex numbers If Amp then match with locus of given in column B according to given conditions of given in column A Column - A Column - B If then (p locus of is circle If then (q locus of is straight line If then (r locus of is major arc of circle (D If = 0 or then (s locus of is minor arc of circle Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
13 Matching-6 If a complex number = x + iy is represented by a point p on xy plane Then match the column A with suitable amplitude given in column B Column - A (A Amp (R + (p Column - B (B Amp (0 (q (C Amp (R (r 0 (D Amp (iy (s not defined (A (B (C (D Matching- Column - A Consider the equation + = 0 Column - A 5 cos cos cos cos sin tan cos sin tan 5 cos 5 sin 5 tan Column - B Column - B (p (q (r (s MULTIPLE CORRECT CHOICE TYPE If + i then (a the greatest value of is 5 + (b the least value of is 5 (c the greatest value of is + (d the least value of is Among the complex numbers which satisfies 5i 5, then (a (b (c (d The complex number has least positive argument equals to tan The complex number has maximum possible argument equals to tan The least modulus of the complex number is 0 The maximum modulus of the complex number is 0 If sin + sin + sin = 0 and cos + cos + cos = 0; then (a cos + 8cos + cos = 0 MCN (b sin ( + + sin ( + + sin ( + = 0 (c cos + 8cos + cos 0 (d sin ( + + sin ( + + sin ( + 0 If is real and is a complex number and u and v be the real and imaginary parts of ( (cos i sin + ( (cos + i sin, then the locus of the points on the Argand plane representing the complex number such that v = 0 is (a (b (c (d a circle of unit radius with centre at the point (, 0 a straight line through the centre of the circle a real axis an imaginary axis 5 ABCD is a rhombus in the argand plane, the affixes of whose vertices are,, and respectively and are in anticlockwise sense If CBA, then (a [( i ( i ] (b [( i ( i ] (c [( i ( i ] (d [( i ( i ] 6 The complex numbers satisfying are (a 0 (b (c (d If = a + ib and = c + id are complex numbers such that = = and R( 0, then the pair of complex numbers = a + ic and = b + id satisfy (a = (b = (c ( 0 (d R Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
14 8 If is any non-ero complex number, then (a arg (b + arg (c (d both (a and (b are incorrect none of these 9 If,, n are n, nth roots of unity, then (a (5 (5 (5 n = (b n = 0 (c (d 0 If amp ( ( ( n = n none of these 5 n, where then the points representing, and are (a (b (c (d concyclic collinear vertices of a right-angled triangle vertices of an isosceles right-angled triangle Assertion-Reason Type Each question contains STATEMENT- (Assertion and STATEMENT- (Reason Each question has choices (A, (B, (C and (D out of which ONLY ONE is correct (A Statement- is True, Statement- is True; Statement- is a correct explanation for Statement- (B (C (D MCN Statement- is True, Statement- is True; Statement- is NOT a correct explanation for Statement- Statement- is True, Statement- is False Statement- is False, Statement- is True STATEMENT- : If,, are the affixes of the vertices of a triangle having its circumcentre at the origin If is the affix of its orthocentre then + + = 0 STATEMENT- : For an equilateral triangle the orthocentre will coincide with the centroid of the triangle STATEMENT- : The following equation ( i ( i 0 represents a circle STATEMENT- : The radius of the circle is unit STATEMENT- : If, and are three complex numbers, then Im( Im( Im( 0 STATEMENT- : Im i { } is a unimodular complex number STATEMENT- : arg( arg STATEMENT- : cos(arg isin(arg i e i e 5 let r and r, where r >, r > STATEMENT- : STATEMENT- : (Answers EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE a d a a 5 a 6 c b MATRIX-MATCH TYPE [A-r; B-p; C-q; D-p] [A-s; B-r; C-q; D-p] [A-p, s; B-r; C-q; D-p, s] [A-q; B-p; C-r; D-s] 5 [A-r; B-s; C-p; D-q] 6 [A-r; B-s; C-q; D-p] [A-p; B-q; C-r; D-s] MULTIPLE CORRECT CHOICE TYPE a, b a, c, d a, b a, b 5 a, b 6 a, b, c, d a, b, c, d 8 a, b 9 a, b, c 0 a, c ASSERTION-REASON TYPE B C A D 5 B Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
15 MCN 5 INITIAL STEP EXERCISE (SUBJECTIVE Solve for x, y R : (x xi (x yi ( yi 5i Let be a non-real complex number lying on the circle = then prove that arg i tan arg i tan If,, are roots of x x + x + = 0 (and is imaginary cube root of unity, then find the value of If,,,, be the roots of x 5 = 0 then prove that where is a non real complex cube root of unity 5 Let = 0 + 6i, = + 6i If is a complex number such that the argument of ( /( is / then prove that 9i = 6 Let and be roots of the equation + p + q = 0, where the coefficients p and q may be complex numbers Let A and B represents and in the complex plane If AOB = 0 and OA = OB, when O is the origin, prove that p = qcos / and are two complex numbers such that Find is unimodular, while is not unimodular 8 Find all circles which are orthogonal to = and = 9 Interpret the following equations geometrically on the Argand plane : (i < i < (ii / < arg ( < / 0 Prove that the roots of the equation 5 8x x x + = 0 are cos,cos and cos Hence, obtain the equation whose roots are (i (ii sec tan,sec,sec 5,tan,tan 5 Also evaluate the value of 5 sec sec sec For complex numbers and, prove that = if and only if = or If and, then show that ( + (arg arg If P, A, B represent the complex numbers x + iy, 6i and respectively and P moves in such a manner that PA = PB then prove that : ( i ( i Also prove that the locus of the point P is a circle whose centre is the point ( i and radius 0 i i i If A ib, principal values only being considered, prove that (i (ii tan A A + B = e B B / A 5 If is represented on the argand plane by a point on the circle =, prove that i tan (arg (iii logcos / (iv 8i( (v log 5i Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
16 MCN 6 FINAL STEP EXERCISE (SUBJECTIVE If = a where is a complex number and a is a real number, find the least and the greatest values of Also find a for which the greatest and the least value of are equal If A, B, C represent the complex numbers,, respectively on the complex plane and the angles at B and C are each equal to (, then prove that ( ( ( sin (a Let A, A,A n be vertices of an n sided regular polygon such that (b A A A A A A value of n Find the Assume that A i (i =,,, n are the vertices of a regular polygon inscribed in a circle of radius unity Find the value of (i A A + A A + + A A n (ii A A A A A A n Show that the equation of a straight line in the argand plane can be put in the form b b c, where b is a non-ero complex constant and c is real 5 If = e i/ and f(x = A 0 + A 0 k x k k, then find the value of f(x + f(x + + f( 6 x, independent of 6 If and both satisfy the relation and arg ( =, then find the imaginary part of ( + If a is a complex number such that a = Find the values of a, so that equation a + + = 0 has one purely imaginary root 8 Find the equation of two lines making an angle of 5 0 with the line ( i + ( + i + = 0 and passing through (, 9 A, B, C are the points representing the complex numbers,, respectively on the complex plane and the circumcentre of the triangle ABC lies at the origin If the altitude of the triangle through the vertex A meets the circumcircle again at P, then prove that P represents the complex number / 0 Let,, be three non ero complex numbers and If 0 then prove that (i (ii,, lie on a circle with the centre at origin arg arg If and are two complex numbers such that i, prove that k, where k is a real number Find the angle between the lines from the origin to the points + and in terms of k Let a complex number,, be a root of the equation Z p + q Z p Z q + = 0, where p, q are distinct primes Show that either p = 0 or q = 0, but not both together Find the equation of the circle in complex form which touches the line i i 0 and the lines ( i = ( + i and ( + i + (i i = 0 are the normals of the circle For every real a 0, find all the complex numbers that satisfy the equation a + + ia = 0 5 Find a R if the complex number is to satisfy =, {a( + i i} and + a (a + i > simultaneously for at least one 6 A cubic equation f(x = 0 has one real root a and two complex roots ± i Points A, B, C represent roots a, + i and i respectively on the argand diagram Show that the roots of the derived equation f (x 0 are complex if A falls inside one of the two equilateral triangles described on base BC Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
17 MCN ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE x =, y = ; x =, y = = 8 ib (8 b 9 (i A circle with center at (, and radius r (, (ii Region between the lines y = x and y = x (iii Exterior of a circle of radius 0 with center at (, 0 (iv Interior of the parabola x = y (v The region included between two concentric circles having centre at 0 + 5i and radii and units respectively ANSWERS SUBJECTIVE (FINAL STEP EXERCISE max a a, min a a, a = 0 (a (b (i n (ii n 5 (A 0 + A x + A x 6 a = cos + i sin, where cos 5 8 k ( i ( i 0 tan k ( i ( i 0 i For 0 a, there is no solution for a, a a (a i 5 a or a 5 5 Einstein Classes, Unit No 0, 0, Vardhman Ring Road Plaa, Vikas Puri Extn, Outer Ring Road New Delhi 0 08, Ph : 96905, 85
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