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Complex Numbers and Quadratic Equations

Short Answer Type

41.

Find the number of non-zero integral solutions of the equation.

$space space space space space space space space space space space space open vertical bar 1 minus straight i close vertical bar to the power of straight x space equals 2 to the power of straight x space end exponent$

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42.

If $space space straight alpha space and space straight beta$ are two different complex number with $space space space space space space space open vertical bar straight beta close vertical bar space equals space 1 space then space find space open vertical bar fraction numerator straight beta minus straight alpha over denominator 1 minus top enclose straight alpha straight beta end fraction close vertical bar$

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43.

If (a + ib) (c+id)(e+if)(g+ih) = A+iB, then show that : (a²+ b²) (c²+ d²) (e²+ f²) (g²h²) = A²+ B²

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44.

Convert the following in the polar form.

$space space space space space space space space left parenthesis straight i right parenthesis space space space space space fraction numerator 1 plus 7 straight i over denominator left parenthesis 2 minus straight i right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis ii right parenthesis space space space space space fraction numerator 1 plus 2 straight i over denominator 1 minus 3 straight i end fraction$

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45.

Find the values of x and y, if.

3x + (-2x + y) i = 6-2i

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46.

Find the values of x and y, if.

(2x + y - 5) + (3x - 2y - 4) i = 0

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47.
Find the values of x and y, if.

$space space space space space space space space space space fraction numerator open parentheses 1 space plus space straight i close parentheses space straight x space minus space 2 straight i over denominator 3 space plus space straight i end fraction plus fraction numerator left parenthesis 2 minus 3 straight i right parenthesis space straight y space plus space straight i over denominator 3 space minus space straight i end fraction space equals space straight i$

space space space space space space space space space space fraction numerator left parenthesis 1 plus straight i right parenthesis space straight x space minus space 2 straight i over denominator 3 space plus space straight i end fraction plus fraction numerator left parenthesis 2 minus 3 straight i right parenthesis space straight y space plus space straight i over denominator 3 space minus space straight i end fraction space equals space straight i

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= i

rightwards double arrow space space space space space space space space space space space space space space fraction numerator 3 straight x space plus space 3 ix space minus 6 straight i space minus space ix space minus straight i squared straight x space plus space 2 straight i squared space plus space 6 straight y space minus space 9 iy space plus space 3 straight i space plus space 2 iy space minus space 3 straight i squared straight y space plus space straight i squared over denominator 3 squared space minus space straight i squared end fraction space equals space straight i

rightwards double arrow space space space space space space space space space space space space space space space space fraction numerator 3 straight x space plus space 2 ix space minus space 3 straight i space plus space straight x space minus 2 space plus space 6 straight y space minus 9 iy space plus space 3 straight i space plus space 2 iy space minus space 3 straight i squared straight y space plus space straight i squared over denominator 9 space plus space 1 end fraction space equals space straight i

rightwards double arrow

(4x + 9y -3) + i (2x - 7y - 3) = 10i

Equating real and imaginary parts, we get

Re (z) : 4x + 9y - 3 = 0

rightwards double arrow

4x + 9y = 3 ....(i)
Im (z) : 2x - 7y - 3 = 10

rightwards double arrow

2x - 7y = 13 ...(ii)

Multipling (ii) by 2, we get
4x - 14y = 26

Subtracting (iii) from (i), we get

23y = -23

rightwards double arrow

y = -1

Using in (i), we get

4x + 9(-1) = 3

rightwards double arrow

4x = 12

rightwards double arrow

x =3

Hence, x - 3 and y = -1

144 Views

48.

Write the conjugates of the following complex numbers :

$space space space space space space space space space space space minus space 5 space plus 1 half straight i$

115 Views

49.

Write the conjugates of the following complex numbers :

$space space space space space space space space space 7 over 5$

123 Views

50.

Write the conjugates of the following complex numbers :

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Previous Year Papers

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Test Yourself

Short Answer Type

47. Find the values of x and y, if.