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Mathematics Test 76
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Mathematics Test 76

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  • Question 1/10
    4 / -1

    If eq. x2 + bx + ca = 0 & x2 + cx + ab = 0 have a common root then eq. containing their other roots as a roots :-

    Solutions

    x2 + bx + ca = 0           α β

    x2 + cx + ab = 0           α, γ

    –          –       –
                             
    (b–c) x + (a) (c–b) = 0

    (b–c)  x = (b – c) a

    (x = a = α)

    αβ = ac             αγ = ab

    aβ = ac             aγ = ab

    β = c                γ = b

    eq. is x2 – (b + c)x + bc = 0

     

  • Question 2/10
    4 / -1

    If x = 2 + 21/3 + 22/3 then value of x3 – 6x2 + 6x is:-

    Solutions

    x – 2 = 21/3 + 22/3

    cutting both side

    x3 – 8 – 3x · 2 (x–2) = (21/3 + 22/3)3

    x3 – 8 – 6x2 + 12x = 2 + 22 + 3.2 (x–2)

    x3 – 8 – 6x2 + 12x = 6 + 6x – 12

    x3 – 6x2 + 6x = 2

     

  • Question 3/10
    4 / -1

    All the values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0 are greater than –2 but less than 4, lie in the interval-

    Solutions

    x2 + px + q = 0

    tan30° + tan15° = –p

    tan30° tan15° = q

     1 =

    –p = 1 – q

    q – p = 1

    2 + q – p = 3

     

     

  • Question 4/10
    4 / -1

     (wherexi∈R –{0}) be the roots of x3 – (2a + 1)x2 + bx – 27 = 0, a,b ∈ R. The value of 'a' cannot be-

  • Question 5/10
    4 / -1

    If the graph of |y| = f(x), where f(x) = ax2 + bx + c, a,b,c∈R, a≠0, has maximum vertical height is 4, then :-

    Solutions

    Graph of |y| = f(x)

    When a > 0           When a < 0

    hence a < 0

     

  • Question 6/10
    4 / -1

    If α, β are the roots of then equation ax2 + bx + c = 0 then the roots of the equation a(x + 3)2 + b(x + 3)  (x + 2) + c(x + 2)2 = 0 are :-

    Solutions

     

  • Question 7/10
    4 / -1

    If α1 < α2 < α3 < α4 < α5 < α6, then the equation

    (x–α1)(x – α3)(x – α5) + 3(x – α2)(x – α4)(x – α6)=0 has :-

    Solutions

     

  • Question 8/10
    4 / -1

    The number of real solution of equation (3/2)x = - x2 + 5x - 10 :-

    Solutions

    Let f(x) = –x2 + 5x – 10 and g(x) =  (3/2)x

    f(x)max = -D/4a = -15/4 and g(x) > 0

    so f(x) = g(x) has no Real solutions.

     

  • Question 9/10
    4 / -1

    Consider the equation x2+x– n = 0, where n∈N, and n∈[5,100]. Then total Number of different values of n, so that the given equation has Integral roots is :-

    Solutions

    x2 + x – n = 0, n∈[5,100]

    is Integer so D should be odd & perfect square

    Let 1 + 4n = (2λ+1)2; (λ∈I)

    ⇒ n = λ (λ+1) so λ = 2,3,4, ....... 9

    so n has 8 values

     

  • Question 10/10
    4 / -1

    Roots of the equation

    a(b–c)x2 + b(c–a)x + c(a–b) = 0 are :-

    Solutions

    Sum of coefficient = 0 so one root is 1

    and other root is 1, c(a-b)/a(b-c) 

     

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