Maths-
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Question

If alpha comma beta be that roots 4 x squared minus 16 x plus lambda equals 0 where lambda element of R,  such that 1 less than alpha less than 2 and 2 less than beta less than 3 then the number of integral solutions of λ is

  1. 5
  2. 6
  3. 2    
  4. 3

Hint:

The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. It also goes by the name quadratic equations. Here we have to find the number of integral solutions of λ. 

The correct answer is: 3


    A quadratic equation, or sometimes just quadratics, is a polynomial equation with a maximum degree of two. It takes the following form:
    ax² + bx + c = 0
    where a, b, and c are constant terms and x is the unknown variable.
    Now we have given alpha comma beta are the roots of equation 4 x squared minus 16 x plus lambda equals 0 where lambda element of R,  such that 1 less than alpha less than 2 and 2 less than beta less than 3.
    The equation is:

    4x216x+λ=0

    First finding the discriminant, we get:
    D=b2-4ac 
    Applying it, we get:
    D=16216λ>0
    λ<16...........................(i)
    Now, we have sum of roots as: alpha plus beta equals 16 over 4 equals 4
    we have product of the roots alpha cross times beta equals lambda over 4
    Now as per the condition, we have:
    1 less than alpha less than 2 and 2 less than beta less than 3, multiplying both, we get:
    2 less than alpha beta less than 6
2 less than lambda over 4 less than 6
S i m p l i f y i n g space i t comma space w e space g e t colon
8 less than lambda less than 24....................... left parenthesis i i right parenthesis
C o m b i n i n g space i space a n d space i i comma space w e space g e t colon
8 less than lambda less than 16

    Here we used the concept of quadratic equations and solved the problem. We also understood the concept of discriminant and used it in the solution to find the intervals. Therefore, the number of integral solutions of λ is in between 8 less than lambda less than 16

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