Algebra
- If 55x + 5 = 1, then x equals
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55x + 5 = 1
⇒ 55x × 55 = 1⇒ 55x = 1 55
⇒ 55x = 5−5 ⇒ 5x = – 5
⇒ x = – 1
Method 2 :
55x + 5 = 1
⇒ 55x + 5 = 5°
⇒ 5x + 5 = 0 ⇒ x = –1Correct Option: B
55x + 5 = 1
⇒ 55x × 55 = 1⇒ 55x = 1 55
⇒ 55x = 5−5 ⇒ 5x = – 5
⇒ x = – 1
Method 2 :
55x + 5 = 1
⇒ 55x + 5 = 5°
⇒ 5x + 5 = 0 ⇒ x = –1
- If x = y = 333 and z = 334, then the value of x3 + y3 + z3 – 3xyz is
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Using Rule 22,
x3 + y3 + z3 - 3xyz = 1 (x + y + z)[ (x - y)2 + (y - z)2 + (z - x)2 ] 2
Here , x = y = 333 and z = 334⇒ x3 + y3 + z3 - 3xyz = 1 (333 + 334 + 334)[ 02 + (-1)2 + 12 ] 2 ⇒ x3 + y3 + z3 - 3xyz = 1 × 1000[ 0 + 1 + 1 ] = 1000 2 Correct Option: C
Using Rule 22,
x3 + y3 + z3 - 3xyz = 1 (x + y + z)[ (x - y)2 + (y - z)2 + (z - x)2 ] 2
Here , x = y = 333 and z = 334⇒ x3 + y3 + z3 - 3xyz = 1 (333 + 334 + 334)[ 02 + (-1)2 + 12 ] 2 ⇒ x3 + y3 + z3 - 3xyz = 1 × 1000[ 0 + 1 + 1 ] = 1000 2
- If x = √5 + √3 and y = √5 - √3 then the value of (x4 – y4) is
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x = √5 + √3
x2 = ( √5 + √3 )2
x2 = 5 + 3 + 2√15 = 8 + 2√15
and y = √5 - √3
y2 = ( √5 - √3 )2
y2 = 8 - 2√15
∴ x4 – y4 = (x2 + y2)(x + y)(x - y)
⇒ x4 – y4 = (8 + 2√15 + 8 - 2√15) × (√5 + √3 + √5 - √3) × (√5 + √3 - √5 + √3)
⇒ x4 – y4 = 16 × 2√5 × 2√3 = 64√15Correct Option: A
x = √5 + √3
x2 = ( √5 + √3 )2
x2 = 5 + 3 + 2√15 = 8 + 2√15
and y = √5 - √3
y2 = ( √5 - √3 )2
y2 = 8 - 2√15
∴ x4 – y4 = (x2 + y2)(x + y)(x - y)
⇒ x4 – y4 = (8 + 2√15 + 8 - 2√15) × (√5 + √3 + √5 - √3) × (√5 + √3 - √5 + √3)
⇒ x4 – y4 = 16 × 2√5 × 2√3 = 64√15
- If x = ³√5 + 2 then the value of x³ – 6x² + 12x – 13 is
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x = ³√5 + 2
⇒ x - 2 = ³√5
Cubing on both sides ,
⇒ x3 - 3x2 × 2 + 3x.(-2)2 - 23 = 5
⇒ x3 - 6x2 + 12x - 8 = 5
⇒ x3 - 6x2 + 12x - 8 - 5 = 0
⇒ x3 - 6x2 + 12x - 13 = 0Correct Option: D
x = ³√5 + 2
⇒ x - 2 = ³√5
Cubing on both sides ,
⇒ x3 - 3x2 × 2 + 3x.(-2)2 - 23 = 5
⇒ x3 - 6x2 + 12x - 8 = 5
⇒ x3 - 6x2 + 12x - 8 - 5 = 0
⇒ x3 - 6x2 + 12x - 13 = 0
- If x = √5 + √3 and y = √5 - √3 then the value of (x4 – y4) is
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View Hint View Answer Discuss in Forum
x = √5 + √3
x2 = ( √5 + √3 )2
x2 = 5 + 3 + 2√15 = 8 + 2√15
and y = √5 - √3
y2 = ( √5 - √3 )2
y2 = 8 - 2√15
∴ x4 – y4 = (x2 + y2)(x + y)(x - y)
⇒ x4 – y4 = (8 + 2√15 + 8 - 2√15) × (√5 + √3 + √5 - √3) × (√5 + √3 - √5 + √3)
⇒ x4 – y4 = 16 × 2√5 × 2√3 = 64√15Correct Option: A
x = √5 + √3
x2 = ( √5 + √3 )2
x2 = 5 + 3 + 2√15 = 8 + 2√15
and y = √5 - √3
y2 = ( √5 - √3 )2
y2 = 8 - 2√15
∴ x4 – y4 = (x2 + y2)(x + y)(x - y)
⇒ x4 – y4 = (8 + 2√15 + 8 - 2√15) × (√5 + √3 + √5 - √3) × (√5 + √3 - √5 + √3)
⇒ x4 – y4 = 16 × 2√5 × 2√3 = 64√15
