Question

please solve these two questions with full steps.

Answer 1

121)

Given , C = 45°
so,
A + B + C = 180°
A + B + 45° = 180°
A + B = 135°
take both sides tan

tan( A + B) = tan( 135°)

( tanA + tanB )/( 1- tanA. tanB) = tan(180°-45°) = -tan45° = -1

tanA + tanB = -( 1- tanA.tanB)

tanA + tanB = - 1 + tanA.tanB

1/cotA + 1/cotB = -1 + 1/cotA.cotB

( cotA + cotB)/( cotA.cotB) = { 1 - cotA.cotB)/cotA.cotB

cotA + cotB + cotA.cotB = 1

cotA + cotB + cotA.cotB +1 = 1 + 1 = 2

cotA + cotB ( 1+ cotA) +1 = 2

( 1+ cotA)( 1 + cotB) = 2

hence option (c) is answer .

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122 we know ,
1 , w , w² are the roots of x³ +1 = 0
where ,

now,

(x -2)³ + 27 = 0

( x -2)³ +(3)³ = 0

( x -2+ 3){ (x -2)² + (3)² - 3(x -2) } = 0

(x +1)( x² -4x + 4 +9 - 3x +6) = 0

(x +1)( x² -7x +19) = 0

x + 1= 0 and x² - 7x + 19 = 0

x = -1 and x = { 7±√27i)/2

x = 4/2 + ( 3 ±3√3i)/2

= 2 + 3( 1± √3i)/2

hence,
x = -1 , 2 +3 w , 2 +3 w²


option ( B) is answer