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**Complex Numbers One Mark Questions PREPARED BY:**

R.RAJENDRAN. M.A., M. Sc., M. Ed., K.C.SANKARALINGA NADAR HR. SEC. SCHOOL, CHENNAI-21

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1. The value of (a) (b) 0 (c) – (d) 1 2. The modulus and amplitude of the complex number [e3 – i /4]3 are respectively (a) (b) (c) (d)

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3. If (m – 5) + i(n + 4) is the complex conjugate of (2m + 3) + i(3n – 2), then (n, m) are (a) (– ½, –8) (b) (– ½, 8) (c) (½, –8) (d) (½, 8) 4. If x2 + y2 = 1 then the value of (a) x – iy (b) 2x (c) – 2iy (d) x + iy

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5. The modulus of the complex number 2 + i3 is (a) 3 (b) 13 (c) 7 (d) 7 6. If A + iB = (a1 + ib1) (a2 + ib2) (a3 + ib3), then A2 + B2 is (a) a12 + b12 + a22 + b22 + a32 + b32 (b) (a1 + a2 + a3)2 + (b1 + b2 + b3)2 (c) (a12 + b12)(a22 + b22)(a32 + b32) (d) (a12 + a22 + a32)(b12 + b22 + b32)

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7. The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram in order if and only if (a) z1+ z4 = z2 + z3 (b) z1+ z3 = z2 + z4 (c) z1+ z2 = z3 + z3 (d) z1 – z2 = z3 – z4 8. If a = 3 + i and z = 2 – i, then the points on the Argand diagram representing az, 3az and – az are (a) vertices of a right angled triangle (b) vertices of an equilateral triangle (c) vertices of an isosceles triangle (d) collinear

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9. If z represents a complex number then arg (z) + arg ( ) is (a) /4 (b) /2 (c) 0 (d) /3 10. If the amplitude of a complex number is /2 then the number is (a) purely imaginary (b) purely real (c) 0 (d) neither real nor imaginary

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11. If the point represented by the complex number iz is rotated about the origin through the angle /2 in the counter clockwise direction then the complex number representing the new position is (a) iz (b) – iz (c) –z (d) z 12. The polar form of the complex number (i25)3 is (a) cos /2 + isin /2 (b) cos + isin (c) cos – isin (d) cos /2 – isin /2

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13. If P represents the variable complex number z and if |2z – 1| = 2|z| then the locus of P is (a) the straight line x = ¼ (b) the straight line y = ¼ (c) the straight line z = ½ (d) the circle x2 + y2 – 4x – 1 = 0 14. The value of is (a) cos + isin (b) cos – isin (c) sin – icos (d) sin – icos

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15. If – lies in the third quadrant then z lies in the (a) First quadrant (b) second quadrant (c) third quadrant (d) fourth quadrant 16. If zn = cos isin then z1z2z3…..z6 is (a) (b) –1 (c) i (d) – i

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17. If x = cos + isin, then the value of is (a) 2cos n (b) 2isin n (c) 2sin n (d) 2i cos n 18. If a = cos – isin, b = cos – isin , c = cos – isin then (a2c2 – b2)/abc is (a) cos2( – + ) + isin2( – + ) (b) –2cos( – + ) (c) – 2isin( – + ) (d) 2cos( – + )

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19. If z1 = 4 + 5i, z2 = – 3 + 2i then the value of is (a) (b) (c) (d) 20. The value of i + i22 + i23 + i24 + i25 is (a) i (b) –i (c) (d) – 1

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21. The conjugate of i13 + i14 + i15 + i16 is (a) (b) – 1 (c) (d) – i 22. If – i + 2 is one root of the equation ax2 – bx + c = 0, then the other root is (a) – i – (b) i – 2 (c) 2 + i (d) 2i + 1

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23. The quadratic equation whose roots are i7 is (a) x2 + 7 = 0 (b) x2 – 7 = 0 (c) x2 + x + 7 = 0 (d) x2 – x – 7 = 0 24. The equation having 4 – 3i and 4 + 3i as roots is (a) x2 + 8x + 25 = 0 (b) x2 + 8x – 25 = 0 (c) x2 – 8 x + 25 = 0 (d) x2 – 8x – 25 = 0

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25. If is a root of the equation ax2 + bx + 1 = 0 where a, b are real then (a, b) is (a) (1, 1) (b) (1, –1) (c) (0, 1) (d) (1, 0) 26. If –i + 3 is a root of x2 – 6x + k = 0 then k is (a) (b) 5 (c) (d) 10

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27. If is a cube root of unity, then the value of (1 – + 2)4 + (1 + – 2)4 is (a) (b) 32 (c) – (d) – 32 28. If is a cube root of unity, then the value of (1 – )(1 – 2)(1 – 4)(1 – 8) is (a) (b) – 9 (c) (d) 32

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29. Which one of the following is incorrect? (a) |z1z2| = |z1| |z2| (b) |z1 + z2| |z1| + |z2| (c) |z1 – z2| > |z1| – |z2| (d) |z1 + z2|2 =(z1 + z2)( ) 30. If is nth root of unity, then (a) 1 + 2 + 4 + …… = + 3 + 5+ ……. (b) n = 0 (c) n = (d) = n – 1

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The complex number form of is (a) (b) – (c) (d) 35i The complex number form of 3 – is (a) (b) (c) 3 – i7 (d) 3 + i7

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Real and imaginary parts of are (a) 4, 3 (b) 4, – 3 (c) – 3, (d) 3, 4 The complex conjugate of is (a) (b) (c) (d) Real and imaginary parts of 3/2 i are (a) 0, 3/2 (b) 3/2, 0 (c) 2, (d) 3, 2

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The complex conjugate of – 4 – 9i (a) – 4 + 9i (b) 4 + 9i (c) 4 – 9i (d) – 4 – 9i The complex conjugate of 5 is (a) 5 (b) –5 (c) i5 (d) – i5 The standard form (a + ib) of 3 + 2i + (– 7 – i) is (a) 4 – i (b) – 4 + i (c) 4 + i (d) 4 + 4i

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If a + ib = (8 – 6i) – (2i – 7) then the values of a and b are (a) 8, – (b) 8, (c) 15, (d) 15, – 8 If p + iq = (2 – 3i)(4 + 2i) then q is (a) (b) – (c) – (d) 8 The conjugate of (2 + i)(3 – 2i) is (a) 8 – i (b) – 8 – i (c) – 8 + i (d) 8 + i

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The real and imaginary parts of (2 + i)(3 – 2i) are (a) – 1, 8 (b) – 8, 1 (c) 8, – (d) – 8, – 1 The modulus values of – 2 + 2i and 2 – 3i are (a) 5, (b) 25, 13 (c) 22, 13 (d) – 4, 1 The modulus values of – 3 – 2i and 4 + 3i are (a) 5, (b) 5, 7 (c) 6, (d) 13, 5

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The cube roots of unity are (a) in GP with common ratio (b) in GP with common difference 2 (c) in AP with common difference (d) in AP with common difference 2 The arguments of nth roots of a complex number differ by (a) (b) (c) (d)

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Which of the following statements is correct? (a) negative complex numbers exist (b) order relation does not exist in real numbers (c) order relation exist in complex numbers (d) (1 + i) > (3 – 2i) is meaningless If | z – z1 | = | z – z2 | then the locus of z is (a) a circle with center at the origin (b) a circle with center at z1 (c) a straight line passing through the origin (d) is a perpendicular bisector of the line joining z1 and z2

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Which of the following statement are correct? (1) Re (z) z (2) Im (z) z (3) = z (4) | | =( )n (a) 1 and 2 (b) 2 and 3 (c) 2, 3 and 4 (d) 1, 3 and 4 The value of is (a) 2Re(z) (b) Re(z) (c) Im(z) (d) 2Im(z)

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The value of is (a) 2Im(z) (b) 2i Im(z) (c) Im(z) (d) i Im(z) The value of is (a) | z | (b) | z |2 (c) 2| z | (d) 2| z |2 The fourth roots of unity are (a) 1 i, – 1 i (b) i, 1 i (c) 1, i (d) 1, – 1

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If is a cube root of unity then (a) 2 = (b) 1 + = 0 (c) 1 + + 2 = 0 (d) 1 – + 2 = 0 The principal value of arg z lies in the interval (a) (b) (–, ] (c) [0, ] (d) (–, 0] The fourth roots of unity form the vertices of (a) an equilateral triangle (b) a square (c) a hexagon (d) a rectangle

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If z1 and z2 are any two complex numbers then which one of the following is false (a) Re(z1 + z2) = Re(z1) + Re(z2) (b) Im(z1 + z2) = Im(z1) + Im(z2) (c) arg(z1 + z2) = arg(z1) + arg(z2) (d) | z1 z2 | = | z1 | | z2 | Cube roots of unity are (a) 1, (b) i, (c) 1, (d) i,

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The number of values of (cos + isin)p/q where p and q are non-zero integers prime to each other, is (a) p (b) q (c) p + q (d) p – q The value of ei + e – i is (a) 2 cos (b) cos (c) 2sin (d) sin The value of ei – e – i is (a) sin (b) 2sin (c) isin (d) 2i sin

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Geometrical interpretation of is (a) reflection of z on real axis (b) reflection of z on imaginary axis (c) rotation of z about origin (d) rotation of z about origin through /2 in clockwise direction If z1 = a + ib, z2 = - a + ib then z1 – z2 lies on (a) real axis (b) imaginary axis (c) the line y = x (d) the line y = – x

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Which one of the following is incorrect? (a) (cos + isin )n = cos n + isin n (b) (cos – isin )n = cos n – isin n (c) (sin + icos )n = sin n + icos n (d) = cos – isin Polynomial equation P(x) = 0 admits conjugate pairs of roots only if the coefficients are (a) imaginary (b) complex (c) real (d) either real or complex

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Identify the correct statement (a) sum of the moduli of two complex numbers is equal to their modulus of the sum (b) modulus of the product of the complex numbers is equal to sum of their moduli (c) arguments of the product of two complex numbers is the product of their arguments (d) arguments of the product of two complex numbers is equal to sum of their arguments

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Which of the following is not true? (a) (b) (c) Re(z) = (d) Im(z) = If z1 and z2 are complex numbers then which of the following is meaningful? (a) z1 < z2 (b) z1 > z2 (c) z1 z2 (d) z1 z2

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Which of the following is incorrect? (a) Re (z) | z | (b) Im (z) | z | (c) z = | z |2 (d) Re (z) | z | Which of the following is incorrect? (a) | z1 + z2 | | z1 | + | z2| (b) | z1 – z2 | | z1 | + | z2| (c) | z1 – z2 | | z1| – | z2| (d) | z1 + z2 | | z1 | + | z2|

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Which of the following is incorrect? (a) is the mirror image of z on the real axis (b) the polar form of is (r, – ) (c) – z is the point symmetrical to z about the origin (d) the polar form of – z is(–r, – ) Which of the following is an incorrect regarding nth root of unity? (a) the number of distinct roots is n (b) the roots are in GP with common ratio cis (c) the arguments are in AP with common difference (d) product of the roots is 0 and the sum of the roots is 1

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Which of the following is incorrect? (a) multiplying a complex number by i is equivalent to rotating the number counter clockwise about the origin through an angle 90 (b) multiplying a complex number by – i is equivalent to rotating the number clockwise about the origin through an angle 90 (c) dividing a complex number by i is equivalent to rotating the number counter clockwise about the origin through an angle 90 (d)dividing a complex number by i is equivalent to rotating the number clockwise about the origin through an angle 90

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THE END

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Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9.

Warm-Up Use the quadratic formula to solve each equation. 6 minutes 1) x 2 + 12x + 35 = 02) x 2 + 81 = 18x 3) x 2 + 4x – 9 = 04) 2x 2 = 5x + 9.

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