Algebra
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If x = y = z , then b + c c + a a + b
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x = y b + c b + a = x - y = x - y ; y = z b + c - c -a b - a c + a a + b = y - z = y - z ; z = x c + a - a - b c - b a + b b + c = z - x = z - x a + b - b - c a - c ∴ x - y = y - z = z - x b - a c - b a - c Correct Option: A
x = y b + c b + a = x - y = x - y ; y = z b + c - c -a b - a c + a a + b = y - z = y - z ; z = x c + a - a - b c - b a + b b + c = z - x = z - x a + b - b - c a - c ∴ x - y = y - z = z - x b - a c - b a - c
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If x = 3 + 2√2 and xy = 1, then the value of x2 + 3xy + y2 is x2 − 3xy + y2
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x = 3 + 2√2
⇒ xy = 1⇒ y = 1 = 1 × 3 - 2√2 3 + 2√2 3 + 2√2 3 - 2√2 = 3 - 2√2 = 3 - 2√2 9 - 8
∴ x + y
= 3 + 2√2 + 3 - 2√2 = 6∴ x² + 3xy + y² = (x + y)² + xy x² - 3xy + y² (x - y)² - 5xy = 36 + 1 = 37 36 - 5 31 Correct Option: D
x = 3 + 2√2
⇒ xy = 1⇒ y = 1 = 1 × 3 - 2√2 3 + 2√2 3 + 2√2 3 - 2√2 = 3 - 2√2 = 3 - 2√2 9 - 8
∴ x + y
= 3 + 2√2 + 3 - 2√2 = 6∴ x² + 3xy + y² = (x + y)² + xy x² - 3xy + y² (x - y)² - 5xy = 36 + 1 = 37 36 - 5 31
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If p + q = 10 and pq = 5, then the numerical value of p + q will be q p
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p + q = p² + q² p q pq = (p + q) - 2pq pq = 100 - 2 × 5 = 90 = 18 5 5 Correct Option: D
p + q = p² + q² p q pq = (p + q) - 2pq pq = 100 - 2 × 5 = 90 = 18 5 5
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If n = 7 + 4 √3, then the value of √n + 1 is : √n
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n = 7 + 4√3 = 7 + 2 × 2 × √3
= 4 + 3 + 2 × 2 × √3
= (2 + √3)²
∴ √n = 2 + √3∴ 1 = 1 = 2 √n 2 + √3 = 1 × 2 - √3 = 2 - √3 2 + √3 2 - √3 ∴ √n + 1 = 2 + √3 + 2 - √3 = 4 √n Correct Option: B
n = 7 + 4√3 = 7 + 2 × 2 × √3
= 4 + 3 + 2 × 2 × √3
= (2 + √3)²
∴ √n = 2 + √3∴ 1 = 1 = 2 √n 2 + √3 = 1 × 2 - √3 = 2 - √3 2 + √3 2 - √3 ∴ √n + 1 = 2 + √3 + 2 - √3 = 4 √n
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If a + b + c = 0, then the value of a2 + b2 + c2 is a2 − bc
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a + b + c = 0
⇒ b + c = –a
On squaring both sides,
⇒ (b + c)2 = a2
⇒ b2 + c2 + 2bc = a2
⇒ a2 + b2 + c2 + 2bc = 2a2
⇒ a2 + b2 + c2 = 2a2 − 2bc = 2(a2 − bc)∴ a2 + b2 + c2 = 2(a2 − bc) = 2 a2 − bc a2 − bc Correct Option: C
a + b + c = 0
⇒ b + c = –a
On squaring both sides,
⇒ (b + c)2 = a2
⇒ b2 + c2 + 2bc = a2
⇒ a2 + b2 + c2 + 2bc = 2a2
⇒ a2 + b2 + c2 = 2a2 − 2bc = 2(a2 − bc)∴ a2 + b2 + c2 = 2(a2 − bc) = 2 a2 − bc a2 − bc