Algebra
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If 
a + 1 
= -2 , then the value of a1000 + a-1000 is a
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a + 1 = -2 a
⇒ a2 + 1 = –2a
⇒ a2 + 2a + 1 = 0
⇒ (a + 1)2 = 0 ⇒ a = –1
∴ (a)1000 + (a)-1000 = (-1)1000 + (-1)-1000 = 1 + 1 = 2Correct Option: A
a + 1 = -2 a
⇒ a2 + 1 = –2a
⇒ a2 + 2a + 1 = 0
⇒ (a + 1)2 = 0 ⇒ a = –1
∴ (a)1000 + (a)-1000 = (-1)1000 + (-1)-1000 = 1 + 1 = 2
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If a2 = b + c , b2 = a + c , c2 = b + a, then what will be the value of 1 + 1 + 1 = ? a + 1 b + 1 c + 1
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a2 = b + c
⇒ a2 + a = a + b + c
⇒ a (a + 1) = a + b + c⇒ 1 = a a + 1 a + b + c
Similarly , b2 = a + c⇒ 1 = b b + 1 a + b + c
and c2 = b + a⇒ 1 = c c + 1 a + b + c ∴ 1 + 1 + 1 = a + b + c = a + b + c = 1 a + 1 b + 1 c + 1 a + b + c a + b + c a + b + c a + b + c Correct Option: C
a2 = b + c
⇒ a2 + a = a + b + c
⇒ a (a + 1) = a + b + c⇒ 1 = a a + 1 a + b + c
Similarly , b2 = a + c⇒ 1 = b b + 1 a + b + c
and c2 = b + a⇒ 1 = c c + 1 a + b + c ∴ 1 + 1 + 1 = a + b + c = a + b + c = 1 a + 1 b + 1 c + 1 a + b + c a + b + c a + b + c a + b + c
- If a, b, c and d satisfy the equations
a + 7b + 3c + 5d = 0,
8a + 4b + 6c + 2d = –4
2a + 6b + 4c + 8d = 4,
5a + 3b + 7c + d = –4,then (a + d) = ? (b + c)
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8a + 4b + 6c + 2d = –4
2a + 6b + 4c + 8d = 4
On adding,
10a + 10b + 10c + 10d = 0
⇒ a + b + c + d = 0
⇒ a + d = –(b + c)
⇒ a + d = -1 b + c Correct Option: C
8a + 4b + 6c + 2d = –4
2a + 6b + 4c + 8d = 4
On adding,
10a + 10b + 10c + 10d = 0
⇒ a + b + c + d = 0
⇒ a + d = –(b + c)
⇒ a + d = -1 b + c
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If x = y = z then (x + y + z) is (b - c)(b + c - 2a) (c - a)(c + a - 2b) (a - b)(a + b - 2c)
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x = y = z = k (b - c)(b + c - 2a) (c - a)(c + a - 2b) (a - b)(a + b - 2c)
∴ x = k (b – c) (b + c – 2a) = k{b2 – c2 – 2a(b – c)}
y = k (c – a) (c + a – 2b) = k{c2 – a2 – 2b(c – a)}
z = k (a – b) (a + b – 2c) = k{a2 – b2 – 2c(a – b)}
∴ x + y + z = k[ b2 – c2 + c2 – a2 + a2 – b2 – 2 {a(b – c) + b (c – a) + c (a – b)} ]
x + y + z = 0 – 2(ab – ac + bc – ab + ac – bc) = 0Correct Option: B
x = y = z = k (b - c)(b + c - 2a) (c - a)(c + a - 2b) (a - b)(a + b - 2c)
∴ x = k (b – c) (b + c – 2a) = k{b2 – c2 – 2a(b – c)}
y = k (c – a) (c + a – 2b) = k{c2 – a2 – 2b(c – a)}
z = k (a – b) (a + b – 2c) = k{a2 – b2 – 2c(a – b)}
∴ x + y + z = k[ b2 – c2 + c2 – a2 + a2 – b2 – 2 {a(b – c) + b (c – a) + c (a – b)} ]
x + y + z = 0 – 2(ab – ac + bc – ab + ac – bc) = 0
- The simplified value of following is :
3 a5 b6 c3 × 3 a b5 c4 15 9 10 a2 b c3 27
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Expression = 3 a5 b6 c3 × 3 a b5 c4 15 9 10 a2 b c3 27 Expression = 3 × 5 × 27 × a6 b11 c7 15 9 10 a2 b c3 Expression = 3 a6 - 2 b11 - 1 c7 - 3 10 Expression = 3 a4 b10 c4 10 ∴ am × an = am + n
am ÷ an = am - n
Correct Option: C
Expression = 3 a5 b6 c3 × 3 a b5 c4 15 9 10 a2 b c3 27 Expression = 3 × 5 × 27 × a6 b11 c7 15 9 10 a2 b c3 Expression = 3 a6 - 2 b11 - 1 c7 - 3 10 Expression = 3 a4 b10 c4 10 ∴ am × an = am + n
am ÷ an = am - n
