Algebra
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If x is a prime number and –1 ≤ 2x − 7 ≤ 1 then the number of values of x is 5
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–1 ≤ 2x − 7 ≤ 1 5
⇒ –5 ≤ 2x – 7 ≤ 5
⇒ –5 + 7 ≤ 2x – 7 + 7 ≤ 5 + 7
⇒ 2 ≤ 2x ≤ 12
⇒ 1 ≤ x ≤ 6Correct Option: D
–1 ≤ 2x − 7 ≤ 1 5
⇒ –5 ≤ 2x – 7 ≤ 5
⇒ –5 + 7 ≤ 2x – 7 + 7 ≤ 5 + 7
⇒ 2 ≤ 2x ≤ 12
⇒ 1 ≤ x ≤ 6
- a2 + b2 + c2 = 2 (a – b – c) – 3, then the value of (a + b + c) is
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a2 + b2 + c2 = 2a – 2b – 2c – 3
⇒ (a2 – 2a + 1) + (b2 + 2b + 1) + (c2 + 2c + 1) = 0
⇒ (a – 1)2 + (b + 1)2 + (c + 1)2 = 0
⇒ a = 1, b = –1, c = –1
∴ a + b + c = 1 – 1 – 1 = –1Correct Option: C
a2 + b2 + c2 = 2a – 2b – 2c – 3
⇒ (a2 – 2a + 1) + (b2 + 2b + 1) + (c2 + 2c + 1) = 0
⇒ (a – 1)2 + (b + 1)2 + (c + 1)2 = 0
⇒ a = 1, b = –1, c = –1
∴ a + b + c = 1 – 1 – 1 = –1
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If x = 2 + √3 , then x2 + 1 is equal to x2
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x = 2 + √3
∴ 1 = 1 x 2 + √3 = 2 − √3 (2 + √3)(2 − √3) = 2 − √3 = 2 − √3 4 −3 ∴ x2 + 1 = 
x + 1 
2 − 2 x2 x
= (2 + √3 + 2 − √3)2 – 2
= 42 – 2 = 16 – 2 = 14Correct Option: D
x = 2 + √3
∴ 1 = 1 x 2 + √3 = 2 − √3 (2 + √3)(2 − √3) = 2 − √3 = 2 − √3 4 −3 ∴ x2 + 1 = x + 1 2 − 2 x2 x
= (2 + √3 + 2 − √3)2 – 2
= 42 – 2 = 16 – 2 = 14
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If a2 – 4a – 1 = 0, a ≠ 0, then the value of a2 + 3a + 1 − 3 is a2 a
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a2 – 4a – 1 = 0
⇒ a2 – 1 = 4a
On dividing both sides by a,⇒ a − 1 = 4 ...(i) a Expression = a2 + 3a + 1 − 3 a2 a = a2 + 1 + 3a − 3 a2 a = a − 1 2 + 2 + 3 a − 1 a a
= (4)2 + 2 + 3 × 4
= 16 + 2 + 12 = 30Correct Option: D
a2 – 4a – 1 = 0
⇒ a2 – 1 = 4a
On dividing both sides by a,⇒ a − 1 = 4 ...(i) a Expression = a2 + 3a + 1 − 3 a2 a = a2 + 1 + 3a − 3 a2 a = a − 1 2 + 2 + 3 a − 1 a a
= (4)2 + 2 + 3 × 4
= 16 + 2 + 12 = 30
- What is the slope between the lines y − √3 x− 5 = 0 and √3y − x + 6 = 0
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Angle between two lines is
tan θ = 1 √3 ∴ Slope = 1 √3 Correct Option: B
Angle between two lines is
tan θ = 1 √3 ∴ Slope = 1 √3
