Numbers and Algebra - Numbers and Algebra Section 1

1. If x - y = 1, then x3 - y3 - 3xy equals

  • Option : B
  • Explanation : x- y3- 3xy = x3 - y3 - 3xy(x y)....as (x - y) = 1 = (x - y)3= 1.
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2. If 6 ≥ x ≥ -2 and 4 ≥ y ≥ - 4, then limits for

If 6 ≥  x ≥ -2  and 4 ≥ y ≥ - 4, then limits for

where x and y are non zero integers, is

  • Option : D
  • Explanation :

    y 4 3 2 1 -1 -3 -4

    x 6 5 4 3 1 -1 -2

    Hence, minimum value of

    Number & Algebra

    And maximum value of

    Number & Algebra

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3. If (a, n)! is defined as product of n consecutive numbers starting from a, where a and n are both natural numbers, and if H is the HCF of (a, n)! and n!, then what can be said about H?

  • Option : D
  • Explanation : (a. n)! = product of n consecutive natural numbers starting from 'a' which is atleast divisible by n!. (n)! = product of n consecutive natural numbers. For n = 2 : (a. n)! = a(a + 1) and n! = 2 a(a + 1) is divisible by 2!. For n = 3 : (a n)! = a(a + 1)(a + 2) and n! = 6. One of the factors of a(a + 1)(a + 2) is divisible by 3 and other by 2. Thus, proceeding in this manner, (a. n)! and n! have HCF = n! ∴ H = n!.
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5. If x = b + c, y = c a, z = a b, then
x2 + y2 + z2 - 2xy - 2xz + 2yz is equal to

  • Option : B
  • Explanation :

    x+ y2 + z2-2xy - 2xz + 2yz 
    = (x - y - z)2
    =(b+c-c+a-a+b)2= 4b2

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