Important Questions for Class 11 Chapter 5 - Complex number and Quadratic Equations

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Important questions based on NCERT syllabus for Class 11 Chapter 5-Complex number and Quadratic Equations:

Question 1: let 2x^2 + 3x + 8 = 0. Without computing the roots find
i) 1/α + 1/β
ii) α^2 + β^2
iii) α^3 + β^3

Solution:

Let α and β be the roots of the equation
sum of the roots α + β = -b/a = -3/2
product of the roots αβ = c/a = 8/2 = 4

i) 1/α + 1/β = α + β/αβ [ by taking LCM]
Substitute vales of α + β and αβ in the above we get
-3/2/4 = -3/2 x 1/4= -3/8

ii) α^2 + β^2 = (α + β)^2 - 2αβ
= (-3/2)^2 - 2(4)
= 9/4 - 8
= (9-32)/4
= -23/4

iii) α^3 + β^3 = (α + β)^3 - 3αβ(α + β)
= (-3/2)^3 -3(4)(-3/2)
= 27/8 - 18
= 27-144/8
= 117/8

Question 2: Find (i+2)^2
Solution: We know that (a+b)^2 = a^2 + b^2 +2ab
i^2 + 2^2 + 2(i)2 [i^2 = 1]
= 1 + 4 + 4i
= 4i + 5

Question 3: Find the sum of the roots of the equation x^2 + x+ 1 = 0

Solution:

Given x^2 + x+ 1 = 0
we apply the formula to find the roots of the equation.
x = -b ±√ (b^2 - 4ac)/2ab
here b = 1 a = 1 c = 1
Substitute to get the roots
x = -1± √(1 - 4(1)1))/2(1) = (-1 ± √(1 - 4))/2 = -1 ±√-3)/2 = -1±3i/2
Hence the roots of equation -1+3i/2, -1-3i/2