**Match the statements of column A and B.**

Column A |
Column B |

(a) The value of 1 + i^{2} + i^{4} + i^{6} + ... i^{20} is |
(i) purely imaginary complex number |

(b) The value of `i^(-1097)` is | (ii) purely real complex number |

(c) Conjugate of 1 + i lies in | (iii) second quadrant |

(d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |

(e) If a, b, c ∈ R and b^{2} – 4ac < 0, then the roots of the equation ax ^{2} + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |

(f) If a, b, c ∈ R and b^{2} – 4ac > 0, and b ^{2} – 4ac is a perfect square, then the roots of the equation ax ^{2} + bx + c = 0 |
(vi) may occur in conjugate pairs |

#### Solution

Column A |
Answers |

(a) The value of 1+ i^{2} + i^{4} + i^{6} + ... i^{20} is |
(ii) purely real complex number |

(b) The value of `i^(-1097)` is | (i) purely imaginary complex number |

(c) Conjugate of 1 + i lies in | (iv) Fourth quadrant |

(d) `(1 + 2i)/(1 - i)` lies in | (iii) second quadrant |

(e) If a, b, c ∈ R and b^{2} – 4ac < 0, then the roots of the equation ax ^{2} + bx + c = 0 are non real (complex) and |
(vi) may occur in conjugate pairs |

(f) If a, b, c ∈ R and b^{2} – 4ac > 0, and b ^{2} – 4ac is a perfect square, then the roots of the equation ax ^{2} + bx + c = 0 |
(v) may not occur in conjugate pairs |

**Explanation:**

**(a)** Because 1 + i^{2} + i^{4} + i^{6} + ... i^{20}

= 1 – 1 + 1 – 1 + ... + 1 = 1 ......(Which is purely a real complex number.)

**(b)** Because `i^(-1097)` = `1/((i)^1097)`

= `1/(i^(4 xx 274 + 1)`

= `1/((i^4)^274i)`

= `1/i`

= `i/i^2`

= –i

Which is purely imaginary complex number.

**(c)** Conjugate of 1 + i is 1 – i which is represented by the point (1, –1) in the fourth quadrant.

**(d)** Because `(1 + 2i)/(1 - i) = (1 + 2i)/(1 - i) xx (1 + i)/(1 + i)`

= `(-1 + 3i)/2`

= `-1/2 + 3/2 i`

Which is represented by the point `(- 1/2, 3/2)` in the second quadrant.

**(e)** If b^{2} – 4ac < 0 = D < 0 i.e., square root of D is a imaginary number.

Therefore, roots are x = `(-b +- "Imaginary Number")/(2a)`

i.e., roots are in conjugate pairs.

**(f)** Consider the equation `x^2 - (5 + sqrt(2)) x + 5 sqrt(2)` = 0, Where a = 1, b = `-(5 + sqrt(2))`, c = `5 sqrt(2)`, Clearly a, b, c ∈ R.

Now D = b^{2} – 4ac = `{- (5 + sqrt(2))}^2 - 4.1.5 sqrt(2) = (5 - sqrt(2))^2`.

Therefore x = `(5 + sqrt(2) +- 5 - sqrt(2))/2` = `5sqrt(2)` which do not form a conjugate pair.