COMPLEX NUMBERS AND QUADRATIC EQUATIONS

Chapter 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. Overview We know that the square of a real number is always non-negative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square root of a negative number? It is clear that a negative number can not have a real square root. So we need to extend the system of real numbers to a system in which we can find out the square roots of negative numbers. Euler (707-783) was the first mathematician to introduce the symbol i (iota) for positive square root of i.e., i. 5.. Imaginary numbers Square root of a negative number is called an imaginary number., for example, 5.. Integral powers of i 9 9 i3, 7 7i 7 i, i, i 3 i i i, i 4 (i ) ( ). To compute i n for n > 4, we divide n by 4 and write it in the form n 4m + r, where m is quotient and r is remainder (0 r 4) Hence i n i 4m+r (i 4 ) m. (i) r () m (i) r i r For example, (i) 39 i 4 9 + 3 (i 4 ) 9. (i) 3 i 3 i and (i) 435 i (4 08 + 3) (i) (4 08). (i) 3 (i) i. i 4 08 3 4 ( i ) () i () i If a and b are positive real numbers, then a b a b i a i b ab (ii) a. b ab if a and b are positive or at least one of them is negative or ero. However, a b ab if a and b, both are negative.

74 EXEMPLAR PROBLEMS MATHEMATICS 5..3 Complex numbers (a) A number which can be written in the form a + ib, where a, b are real numbers and i is called a complex number. (b) (c) (d) If a + ib is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as Re () a, Im () b. Order relations greater than and less than are not defined for complex numbers. If the imaginary part of a complex number is ero, then the complex number is known as purely real number and if real part is ero, then it is called purely imaginary number, for example, is a purely real number because its imaginary part is ero and 3i is a purely imaginary number because its real part is ero. 5..4 Algebra of complex numbers (a) (b) Two complex numbers a + ib and c + id are said to be equal if a c and b d. Let a + ib and c + id be two complex numbers then + (a + c) + i (b + d). 5..5 Addition of complex numbers satisfies the following properties. As the sum of two complex numbers is again a complex number, the set of complex numbers is closed with respect to addition.. Addition of complex numbers is commutative, i.e., + + 3. Addition of complex numbers is associative, i.e., ( + ) + 3 + ( + 3 ) 4. For any complex number x + i y, there exist 0, i.e., (0 + 0i) complex number such that + 0 0 +, known as identity element for addition. 5. For any complex number x + iy, there always exists a number a ib such that + ( ) ( ) + 0 and is known as the additive inverse of. 5..6 Multiplication of complex numbers Let a + ib and c + id, be two complex numbers. Then. (a + ib) (c + id) (ac bd) + i (ad + bc). As the product of two complex numbers is a complex number, the set of complex numbers is closed with respect to multiplication.. Multiplication of complex numbers is commutative, i.e.,.. 3. Multiplication of complex numbers is associative, i.e., (. ). 3. (. 3 )

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 75 4. For any complex number x + iy, there exists a complex number, i.e., ( + 0i) such that.., known as identity element for multiplication. 5. For any non ero complex number x + i y, there exists a complex number a ib such that, i.e., multiplicative inverse of a + ib a+ ib a + b. 6. For any three complex numbers, and 3,. ( + 3 ). +. 3 and ( + ). 3. 3 +. 3 i.e., for complex numbers multiplication is distributive over addition. 5..7 Let a + ib and ( 0) c + id. Then a+ ib c+ id ( ac + bd ) ( bc ad + i ) c + d c + d 5..8 Conjugate of a complex number Let a + ib be a complex number. Then a complex number obtained by changing the sign of imaginary part of the complex number is called the conjugate of and it is denoted by, i.e., a ib. Note that additive inverse of is a ib but conjugate of is a ib. We have :. ( ). + Re (), i Im() 3., if is purely real. 4. + 0 is purely imaginary 5.. {Re ()} + {Im ()}. 6. ( + ) +, ( ) 7. ( ) (. ) ( ) ( ), ( 0) ( ) 5..9 Modulus of a complex number Let a + ib be a complex number. Then the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute value) of and it is denoted by i.e., a + b

76 EXEMPLAR PROBLEMS MATHEMATICS In the set of complex numbers > or < are meaningless but > or < are meaningful because and are real numbers. 5..0 Properties of modulus of a complex number. 0 0 i.e., Re () 0 and Im () 0. 3. Re () and Im () 4., 5. 6.., ( 0) + + +Re ( ) 7. + Re ( ) 8. + + 9. 0. + + + + a b b a ( a b )( ) In particular: + + + ( ). As stated earlier multiplicative inverse (reciprocal) of a complex number a + ib ( 0) is 5. Argand Plane a ib a + b A complex number a + ib can be represented by a unique point P (a, b) in the cartesian plane referred to a pair of rectangular axes. The complex number 0 + 0i represent the origin 0 ( 0, 0). A purely real number a, i.e., (a + 0i) is represented by the point (a, 0) on x - axis. Therefore, x-axis is called real axis. A purely imaginary number

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 77 ib, i.e., (0 + ib) is represented by the point (0, b) on y-axis. Therefore, y-axis is called imaginary axis. Similarly, the representation of complex numbers as points in the plane is known as Argand diagram. The plane representing complex numbers as points is called complex plane or Argand plane or Gaussian plane. If two complex numbers and be represented by the points P and Q in the complex plane, then PQ 5.. Polar form of a complex number Let P be a point representing a non-ero complex number a + ib in the Argand plane. If OP makes an angle θ with the positive direction of x-axis, then r (cosθ + isinθ) is called the polar form of the complex number, where r a + b and tanθ b. Here θ is called argument or amplitude of and we a write it as arg () θ. The unique value of θ such that θ is called the principal argument. arg (. ) arg ( ) + arg ( ) 5.. Solution of a quadratic equation arg arg ( ) arg ( ) The equations ax + bx + c 0, where a, b and c are numbers (real or complex, a 0) is called the general quadratic equation in variable x. The values of the variable satisfying the given equation are called roots of the equation. The quadratic equation ax + bx + c 0 with real coefficients has two roots given by b + D b and D, where D b 4ac, called the discriminant of the equation. a a Notes. When D 0, roots of the quadratic equation are real and equal. When D > 0, roots are real and unequal. Further, if a, b, c Q and D is a perfect square, then the roots of the equation are rational and unequal, and if a, b, c Q and D is not a perfect square, then the roots are irrational and occur in pair.

78 EXEMPLAR PROBLEMS MATHEMATICS When D < 0, roots of the quadratic equation are non real (or complex).. Let α, β be the roots of the quadratic equation ax + bx + c 0, then sum of the roots b c (α + β) and the product of the roots ( α. β) a a. 3. Let S and P be the sum of roots and product of roots, respectively, of a quadratic equation. Then the quadratic equation is given by x Sx + P 0. 5. Solved Exmaples Short Answer Type Example Evaluate : ( + i) 6 + ( i) 3 Solution ( + i) 6 {( + i) } 3 ( + i + i) 3 ( + i) 3 8 i 3 8i and ( i) 3 i 3 3i + 3i + i 3i 3 i Therefore, ( + i) 6 + ( i) 3 8i i 0i Example If x y ( x + iy ) 3 a + ib, where x, y, a, b R, show that a b (a + b ) Solution ( x + iy ) 3 a + ib x + iy (a + ib) 3 i.e., x + iy a 3 + i 3 b 3 + 3iab (a + ib) a 3 ib 3 + i3a b 3ab a 3 3ab + i (3a b b 3 ) x a 3 3ab and y 3a b b 3 Thus So, x a a 3b and y b 3a b x y a a b 3b 3a + b a b (a + b ). Example 3 Solve the equation, where x + iy Solution x y + ixy x iy Therefore, x y x... () and xy y... ()

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 79 From (), we have y 0 or x When y 0, from (), we get x x 0, i.e., x 0 or x. When x, from (), we get y + or y 4 3 4, i.e., y 3 ±. Hence, the solutions of the given equation are 0 + i0, + i0, + i 3, 3 i. Example 4 If the imaginary part of + is, then show that the locus of the point i + representing in the argand plane is a straight line. Solution Let x + iy. Then + ( x + iy ) + ( + ) + x i y i + i( x+ iy) + ( y) + ix {( x + ) + i y } {( ) } y ix {( y) + ix} {( y) ix} (x+ y) + i(y y x x) + y y+ x Thus + y y x x Im i + + y y+ x But + Im i + (Given) So y y x x + y y+ x y y x x y + 4y x i.e., x + y 0, which is the equation of a line. Example 5 If +, then show that lies on imaginary axis. Solution Let x + iy. Then +

80 EXEMPLAR PROBLEMS MATHEMATICS x y + ixy x+ iy + (x y ) + 4x y (x + y + ) 4x 0 i.e., x 0 Hence lies on y-axis. Example 6 Let and be two complex numbers such that + i 0 and arg ( ). Then find arg ( ). Solution Given that + i 0 i, i.e., i Thus arg ( ) arg + arg ( i ) arg ( i ) arg ( i ) + arg ( ) arg ( i ) + arg ( ) + arg ( ) arg ( ) 3 4 Example 7 Let and be two complex numbers such that + +. Then show that arg ( ) arg ( ) 0. Solution Let r (cosθ + i sin θ ) and r (cosθ + i sin θ ) where r, arg ( ) θ, r, arg ( ) θ. We have, + + r (cosθ+ cos θ ) + r (cosθ + sin θ ) r + r + + cos( θ θ ) ( + ) r r rr r r cos (θ θ ) θ θ i.e. arg arg Example 8 If,, 3 are complex numbers such that 3 + +, then find the value of + + 3. 3 Solution 3

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 8 3 3 3,, 3 3 Given that + + 3 + + 3, i.e., + + 3 + + 3 Example 9 If a complex number lies in the interior or on the boundary of a circle of radius 3 units and centre ( 4, 0), find the greatest and least values of +. Solution Distance of the point representing from the centre of the circle is ( 4+ i0) + 4. According to given condition + 4 3. Now + + 4 3 + 4 + 3 3+ 3 6 Therefore, greatest value of + is 6. Since least value of the modulus of a complex number is ero, the least value of + 0. Example 0 Locate the points for which 3< < 4 Solution < 4 x + y < 6 which is the interior of circle with centre at origin and radius 4 units, and > 3 x + y > 9 which is exterior of circle with centre at origin and radius 3 units. Hence 3 < < 4 is the portion between two circles x + y 9 and x + y 6. Example Find the value of x 4 + 5x 3 + 7x x + 4, when x 3 i Solution x + 3 i x + 4x + 7 0 Therefore x 4 + 5x 3 + 7x x + 4 (x + 4x + 7) (x 3x + 5) + 6 0 (x 3x + 5) + 6 6.

8 EXEMPLAR PROBLEMS MATHEMATICS Example Find the value of P such that the difference of the roots of the equation x Px + 8 0 is. Solution Let α, β be the roots of the equation x Px + 8 0 Therefore α + β P and α. β 8. Now α β ± ( α + β) 4αβ Therefore ± P 3 P 3 4, i.e., P ± 6. Example 3 Find the value of a such that the sum of the squares of the roots of the equation x (a ) x (a + ) 0 is least. Solution Let α, β be the roots of the equation Therefore, α + β a and αβ ( a + ) Now α + β (α + β) αβ (a ) + (a + ) (a ) + 5 Therefore, α + β will be minimum if (a ) 0, i.e., a. Long Answer Type Example 4 Find the value of k if for the complex numbers and, Solution L.H.S. k( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + ( ) ( ) R.H.S. k ( k )( )

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 83 Hence, equating LHS and RHS, we get k. Example 5 If and both satisfy + arg ( ), then find 4 Im ( + ). Solution Let x + iy, x + iy and x + iy. Then + (x + iy) + (x iy) x + iy x + y... () Since and both satisfy (), we have x + y... and x + y (x x ) (y + y ) (y y ) y y (y + y ) x x Again (x x ) + i (y y ) y y Therefore, tan θ x x, where θ arg ( ) y y tan 4 x x i.e., y y x x since θ 4 From (), we get y + y, i.e., Im ( + )... () Objective Type Questions Example 6 Fill in the blanks: (i) The real value of a for which 3i 3 ai + ( a)i + 5 is real is. (ii) If and arg (), then. 4 (iii) The locus of satisfying arg () 3 is. 4 n 3 (iv) The value of ( ), where n N, is.

84 EXEMPLAR PROBLEMS MATHEMATICS (v) The conjugate of the complex number i is. + i (vi) If a complex number lies in the third quadrant, then its conjugate lies in the. (vii) If ( + i) ( + i) ( + 3i)... ( + ni) x + iy, then 5.8.3... (4 + n ). Solution (i) 3i 3 ai + ( a)i + 5 3i + a + 5 + ( a)i a + 5 + ( a ) i, which is real if a 0 i.e. a. (ii) cos + sin + ( + ) 4 4 i i i (iii) Let x + iy. Then its polar form is r (cos θ + i sin θ), where tan θ y x and (iv) (v) θ is arg (). Given that θ. Thus. 3 y tan y 3x, where x > 0, y > 0. 3 x Hence, locus of is the part of y 3x in the first quadrant except origin. 4 n 3 4n 3 4n 3 Here ( ) ( i) ( i) ( i) ( i) i i 3 i i i i i i + i i i i + i + i i i + Hence, conjugate of i is i. + i (vi) Conjugate of a complex number is the image of the complex number about the x-axis. Therefore, if a number lies in the third quadrant, then its image lies in the second quadrant. (vii) Given that ( + i) ( + i) ( + 3i)... ( + ni) x + iy... () ( + i) (+ i) (+ 3 i)...( + ni) x+ iy ( x iy) ( ) i.e., ( i) ( i) ( 3i)... ( ni) x iy... () 3

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 85 Multiplying () and (), we get 5.8.3... (4 + n ) x + y. Example 7 State true or false for the following: (i) Multiplication of a non-ero complex number by i rotates it through a right angle in the anti- clockwise direction. (ii) The complex number cosθ + i sinθ can be ero for some θ. (iii) If a complex number coincides with its conjugate, then the number must lie on imaginary axis. (iv) The argument of the complex number ( +i 3 ) ( + i) (cos θ + i sin θ) is 7 + θ (v) The points representing the complex number for which + < lies in the interior of a circle. (vi) If three complex numbers, and 3 are in A.P., then they lie on a circle in the complex plane. (vii) If n is a positive integer, then the value of i n + (i) n+ + (i) n+ + (i) n+3 is 0. Solution (i) True. Let + 3i be complex number represented by OP. Then i 3 + i, represented by OQ, where if OP is rotated in the anticlockwise direction through a right angle, it coincides with OQ. (ii) False. Because cosθ + isinθ 0 cosθ 0 and sinθ 0. But there is no value of θ for which cosθ and sinθ both are ero. (iii) False, because x + iy x iy y 0 number lies on x-axis. (iv) True, arg () arg ( + i 3 ) + arg ( + i) + arg (cosθ + isinθ) 7 + +θ + θ 3 4 (v) False, because x + iy + < x + iy (x + ) + y < (x ) + y which gives 4x < 0. + 3 (vi) False, because if, and 3 are in A.P., then is the midpoint of and 3, which implies that the points,, 3 are collinear. (vii) True, because i n + (i) n+ + (i) n+ + (i) n+3 i n ( + i + i + i 3 ) i n ( + i i) i n (0) 0

86 EXEMPLAR PROBLEMS MATHEMATICS Example 8 Match the statements of column A and B. Column A Column B (a) The value of +i + i 4 + i 6 +... i 0 is (i) purely imaginary complex number (b) The value of i 097 is (ii) purely real complex number (c) Conjugate of +i lies in (iii) second quadrant (d) + i i lies in (iv) Fourth quadrant (e) If a, b, c R and b 4ac < 0, (v) may not occur in conjugate pairs then the roots of the equation ax + bx + c 0 are non real (complex) and (f) If a, b, c R and b 4ac > 0, (vi) may occur in conjugate pairs and b 4ac is a perfect square, then the roots of the equation ax + bx + c 0 Solution (a) (ii), because + i + i 4 + i 6 +... + i 0 + +... + (which is purely a real complex number) (b) (i), because i 097 097 4 74+ () i i which is purely imaginary complex number. {( ) } ( ) i i i i i i 4 74 (c) (iv), conjugate of + i is i, which is represented by the point (, ) in the fourth quadrant. (d) (iii), because + i + i + i + 3 i + 3 i, which is i i + i 3 represented by the point, in the second quadrant. (e) (vi), If b 4ac < 0 D < 0, i.e., square root of D is a imaginary number, therefore, roots are conjugate pairs. ± x b Imaginary Number, i.e., roots are in a

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 87 (f) (v), Consider the equation x (5 + ) x + 5 0, where a, b (5 + ), c 5, clearly a, b, c R. Now D b 4ac { (5 + )} 4..5 (5 ). Therefore 5+ ± 5 x 5, which do not form a conjugate pair. Example 9 What is the value of Solution i, because i 4n+ 4n i 4 n + 4 n 4 n 4 n i i i i i i i i i i i i i Example 0 What is the smallest positive integer n, for which ( + i) n ( i) n? + i Solution n, because ( + i) n ( i) n i (i) n which is possible if n ( i 4 ) Example What is the reciprocal of 3 + 7 i Solution Reciprocal of Therefore, reciprocal of 3 + 7 i 3 7 i 3 7 i 6 6 6 Example If 3 + i 3 and 3 + i, then find the quadrant in which lies. Solution 3 + i 3 3+ 3 3 3 + 3 + 4 4 i i which is represented by a point in first quadrant.? n

88 EXEMPLAR PROBLEMS MATHEMATICS Example 3 What is the conjugate of Solution Let 5+ i + 5 i 5+ i 5 i 5+ i + 5 i 5+ i + 5 i 5+ i 5 i 5+ i + 5 i 5+ i+ 5 i+ 5+ 44 5+ i 5+ i 3 i 3 i 0 3 i Therefore, the conjugate of 0 + 3 i Example 4 What is the principal value of amplitude of i? Solution Let θ be the principle value of amplitude of i. Since tan θ tan θ tan θ 4 4 Example 5 What is the polar form of the complex number (i 5 ) 3? Solution (i 5 ) 3 (i) 75 i 4 8+3 (i4 ) 8 (i) 3 i 3 i 0 i Polar form of r (cos θ + i sinθ) cos + isin cos i sin Example 6 What is the locus of, if amplitude of 3i is? 4 Solution Let x + iy. Then 3i (x ) + i (y 3) 3 Let θ be the amplitude of 3i. Then tan θ y x 3 tan y since θ 4 x 4

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 89 y 3 i.e. x y + 0 x Hence, the locus of is a straight line. Example 7 If i, is a root of the equation x + ax + b 0, where a, b R, then find the values of a and b. a Solution Sum of roots ( i) + ( + i) a. (since non real complex roots occur in conjugate pairs) b Product of roots, ( ) ( ) i + i b Choose the correct options out of given four options in each of the Examples from 8 to 33 (M.C.Q.). Example 8 + i + i 4 + i 6 +... + i n is (A) positive (B) negative (C) 0 (D) can not be evaluated Solution (D), + i + i 4 + i 6 +... + i n + +... ( ) n which can not be evaluated unless n is known. Example 9 If the complex number x + iy satisfies the condition +, then lies on (A) x-axis (B) circle with centre (, 0) and radius (C) circle with centre (, 0) and radius (D) y-axis Solution (C), + ( x+ ) + iy (x +) + y which is a circle with centre (, 0) and radius. Example 30 The area of the triangle on the complex plane formed by the complex numbers, i and + i is: (A) (B) (C) (D) none of these

90 EXEMPLAR PROBLEMS MATHEMATICS Solution (C), Let x + iy. Then i y ix. Therefore, + i (x y) + i (x + y) Required area of the triangle ( + ) x y Example 3 The equation + i + i represents a (A) straight line (B) circle (C) parabola (D) hyperbola Solution (A), + i + i + ( i) ( i) PA PB, where A denotes the point (, ), B denotes the point (, ) and P denotes the point (x, y) lies on the perpendicular bisector of the line joining A and B and perpendicular bisector is a straight line. Example 3 Number of solutions of the equation + 0 is (A) (B) (C) 3 (D) infinitely many Solution (D), + 0, 0 x y + ixy + x + y 0 x + ixy 0 x (x + iy) 0 x 0 or x + iy 0 (not possible) Therefore, x 0 and 0 So y can have any real value. Hence infinitely many solutions. Example 33 The amplitude of sin + i( cos ) is 5 5 (A) 5 (B) 5 Solution (D), Here r cos θ sin (C) 5 5 and r sin θ cos 5 (D) 0

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 9 cos sin 5 0 Therefore, tan θ sin sin.cos 5 0 0 tan θ tan i.e., θ 0 0 5.3 EXERCISE Short Answer Type. For a positive integer n, find the value of ( i) n. Evaluate 3 n n n + ( i + i ), where n N. 3 3 + i i 3. If x + iy, then find (x, y). -i + i 4. If ( + i) i x + iy, then find the value of x + y. i n 00 5. i If + i a + ib, then find (a, b). 6. If a cos θ + i sinθ, find the value of + a a 7. If ( + i) ( i), then show that i. 8. If x + iy, then show that + ( + ) + b 0, where b R, represents a circle. + 9. If the real part of is 4, then show that the locus of the point representing in the complex plane is a circle. 0. Show that the complex number, satisfying the condition arg on a circle.. Solve the equation + + i. lies + 4

9 EXEMPLAR PROBLEMS MATHEMATICS Long Answer Type. If + + ( + i), then find. 3. If arg ( ) arg ( + 3i), then find x : y. where x + iy 4. Show that 3 represents a circle. Find its centre and radius. 5. If + is a purely imaginary number ( ), then find the value of. 6. and are two complex numbers such that and arg ( ) + arg ( ), then show that. 7. If ( ) and, then show that the real part of + is ero. 8. If, and 3, 4 are two pairs of conjugate complex numbers, then find arg + arg. 4 3 9. If... n, then show that + + 3+... + n + + +... +. 3 n 0. If for complex numbers and, arg ( ) arg ( ) 0, then show that. Solve the system of equations Re ( ) 0,.. Find the complex number satisfying the equation + ( + ) + i 0. i 3. Write the complex number in polar form. cos + isin 3 3 4. If and w are two complex numbers such that w and arg () arg (w), then show that w i.

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 93 Objective Type Questions 5. Fill in the blanks of the following (i) For any two complex numbers, and any real numbers a, b, + + a b b a... (ii) The value of 5 9 is... 3 (iii) ( i) The number 3 i is equal to... (iv) The sum of the series i + i + i 3 +... upto 000 terms is... (v) Multiplicative inverse of + i is... (vi) If and are complex numbers such that + is a real number, then... (vii) arg () + arg ( 0) is... (viii) If + 4 3, then the greatest and least values of + are... and... (ix) If, then the locus of is... + 6 (x) If 4 and arg () 5, then... 6 6. State True or False for the following : (i) The order relation is defined on the set of complex numbers. (ii) Multiplication of a non ero complex number by i rotates the point about origin through a right angle in the anti-clockwise direction. (iii) For any complex number the minimum value of + is. (iv) The locus represented by i is a line perpendicular to the join of (, 0) and (0, ). (v) If is a complex number such that 0 and Re () 0, then Im ( ) 0. (vi) The inequality 4 < represents the region given by x > 3.

94 EXEMPLAR PROBLEMS MATHEMATICS (vii) Let and be two complex numbers such that + +, then arg ( ) 0. (viii) is not a complex number. 7. Match the statements of Column A and Column B. Column A Column B (a) The polar form of i + 3 is (i) Perpendicular bisector of segment joining (, 0) and (, 0) (b) The amplitude of + 3 is (ii) On or outside the circle having centre at (0, 4) and radius 3. (c) If +, then (iii) locus of is (d) If + i i, then (iv) Perpendicular bisector of segment locus of is joining (0, ) and (0, ). (e) Region represented by (v) cos + isin 6 6 + 4i 3 is (f) Region represented by (vi) On or inside the circle having centre 3 + 4 3 is ( 4, 0) and radius 3 units. (g) Conjugate of + i lies in (vii) First quadrant i (h) Reciprocal of i lies in (viii) Third quadrant i 8. What is the conjugate of? ( i) 9. If, is it necessary that? 30. If ( a + ) x + iy, what is the value of x a i + y?

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 95 3. Find if 4 and arg () 5. 6 3. Find ( + i) ( + i) (3 + i ) 33. Find principal argument of ( + i 3 ). 5i 34. Where does lie, if. + 5i Choose the correct answer from the given four options indicated against each of the Exercises from 35 to 50 (M.C.Q) 35. sinx + i cos x and cos x i sin x are conjugate to each other for: (A) x n (B) x n+ (C) x 0 (D) No value of x 36. The real value of α for which the expression i sin α is purely real is : + isinα (A) ( n +) (B) (n + ) (C) n (D) None of these, where n N 37. If x + iy lies in the third quadrant, then also lies in the third quadrant if (A) x > y > 0 (B) x < y < 0 (C) y < x < 0 (D) y > x > 0 38. The value of ( + 3) ( + 3) is equivalent to (A) +3 (B) 3 (C) + 3 (D) None of these x 39. + i If i, then (A) x n+ (B) x 4n (C) x n (D) x 4n +, where n N

96 EXEMPLAR PROBLEMS MATHEMATICS 3 4ix 40. A real value of x satisfies the equation α iβ ( αβ R, ) 3+ 4ix if α + β (A) (B) (C) (D) 4. Which of the following is correct for any two complex numbers and? (A) (B) arg ( ) arg ( ). arg ( ) (C) + + (D) + 4. The point represented by the complex number i is rotated about origin through an angle in the clockwise direction, the new position of point is: (A) + i (B) i (C) + i (D) + i 43. Let x, y R, then x + iy is a non real complex number if: (A) x 0 (B) y 0 (C) x 0 (D) y 0 44. If a + ib c + id, then (A) a + c 0 (B) b + c 0 (C) b + d 0 (D) a + b c + d i+ 45. The complex number which satisfies the condition lies on i (A) circle x + y (B) the x-axis (C) the y-axis (D) the line x + y. 46. If is a complex number, then (A) > (B) (C) < (D) 47. + + is possible if (A) (B) (C) arg ( ) arg ( ) (D)

COMPLEX NUMBERS AND QUADRATIC EQUATIONS 97 48. The real value of θ for which the expression + i cos θ i cosθ is a real number is: (A) n + (B) n+ ( ) n 4 4 (C) n ± (D) none of these. 49. The value of arg (x) when x < 0 is: (A) 0 (B) (C) (D) none of these 50. 7 If f (), where + i, then f ( ) (A) (B) (C) (D) none of these.