Algebra
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If 2x – 3(4 – 2x) < 4x – 5 < 4x + 2x , 3
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2x – 3(4 – 2x) < 4x – 5 < 4x + 2x 3 ⇒ 2x – 12 + 6x < 4x – 5 < 12x + 2x 3 ⇒ 8x – 12 < 4x – 5 < 14x 3
⇒ 24x – 36 < 12x – 15 < 14x
When x = 0,
–36 < –15 < 0Correct Option: C
2x – 3(4 – 2x) < 4x – 5 < 4x + 2x 3 ⇒ 2x – 12 + 6x < 4x – 5 < 12x + 2x 3 ⇒ 8x – 12 < 4x – 5 < 14x 3
⇒ 24x – 36 < 12x – 15 < 14x
When x = 0,
–36 < –15 < 0
- Which of the following equations has equal roots?
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The roots of quadratic equation ax2+ bx + c = 0 will be equal
if b2 – 4ac = 0
Option (1),
3x2 – 6x + 2 = 0
a = 3, b = –6, c = 2
∴ b2 – 4ac = (–6)2 – 4 × 3 × 2
= 36 – 24 = 12 ≠ 0
Option (2),
3x2 – 6x + 3 = 0
a = 3, b = –6, c = 3
∴ b2 – 4ac = (–6)2 – 4 × 3 × 3
= 36 – 36 = 0
Option (3),
x2 – 8x + 8 = 0
∴ b2 – 4ac = (–8)2 – 4 × 8
= 64 – 32 = 32 ≠ 0
Option (4),
4x2 – 8x + 2 = 0
∴ b2 – 4ac = (–8)2 – 4 × 4 × 2
= 64 – 32
= 32 ≠ 0Correct Option: B
The roots of quadratic equation ax2+ bx + c = 0 will be equal
if b2 – 4ac = 0
Option (1),
3x2 – 6x + 2 = 0
a = 3, b = –6, c = 2
∴ b2 – 4ac = (–6)2 – 4 × 3 × 2
= 36 – 24 = 12 ≠ 0
Option (2),
3x2 – 6x + 3 = 0
a = 3, b = –6, c = 3
∴ b2 – 4ac = (–6)2 – 4 × 3 × 3
= 36 – 36 = 0
Option (3),
x2 – 8x + 8 = 0
∴ b2 – 4ac = (–8)2 – 4 × 8
= 64 – 32 = 32 ≠ 0
Option (4),
4x2 – 8x + 2 = 0
∴ b2 – 4ac = (–8)2 – 4 × 4 × 2
= 64 – 32
= 32 ≠ 0
- If 5x – 40 = 3x, then the numerical value of (2x – 11) is
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5x – 40 = 3x
⇒ 5x – 3x = 40⇒ 2x = 40 ⇒ x = 40 = 20 2
∴ 2x – 11 = 2 × 20 – 11
= 40 – 11 = 29Correct Option: A
5x – 40 = 3x
⇒ 5x – 3x = 40⇒ 2x = 40 ⇒ x = 40 = 20 2
∴ 2x – 11 = 2 × 20 – 11
= 40 – 11 = 29
- If 4 (2x + 3) > 5 – x and 5x –3 (2x – 7) > 3x – 1, then x can take which of the following values?
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4 (2x + 3) > 5 – x
⇒ 8x + 12 > 5 – x
⇒ 8x + x > 5 – 12
⇒ 9x > –7⇒ x > –7 9
Again,
5x – 3 (2x – 7) > 3x – 1
⇒ 5x – 6x + 21 > 3x – 1
⇒ –x + 21 > 3x – 1
⇒ –x – 3x > – 21 – 1
⇒ –4x > –22
⇒ 4x < 22⇒ x < 22 i.e., x < 5.5 4
∴ Required value of x = 5Correct Option: C
4 (2x + 3) > 5 – x
⇒ 8x + 12 > 5 – x
⇒ 8x + x > 5 – 12
⇒ 9x > –7⇒ x > –7 9
Again,
5x – 3 (2x – 7) > 3x – 1
⇒ 5x – 6x + 21 > 3x – 1
⇒ –x + 21 > 3x – 1
⇒ –x – 3x > – 21 – 1
⇒ –4x > –22
⇒ 4x < 22⇒ x < 22 i.e., x < 5.5 4
∴ Required value of x = 5
- What should be added to 8 (3x – 4y) to obtain (18x – 18y) ?
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Required answer
= (18x – 18y) – 8 (3x – 4y)
= 18x – 18y – 24x + 32y
= 14y – 6xCorrect Option: C
Required answer
= (18x – 18y) – 8 (3x – 4y)
= 18x – 18y – 24x + 32y
= 14y – 6x
