NIMCET 2019 Question Paper with solutions

CONCEPT:
Z+ is the set of all positive integers (1,2,3,...)
If quadratic equation ax2+bx+c=0 , has real roots then its discriminant (b2−4ac) should be positive.
CALCULATIONS:
Given equation is [tan2x]−tan‌x−a=0 .
∴D=b2−4ac=(−1)2−4(1)(−a)=1+4a
So D must be an odd perfect square -
⇒√1+4a=2λ+1
⇒1+4a=4λ2+1+4λ
⇒a=λ(λ+1)
So, For different values of λ(1,2,3,4,5,6,7,8,9 but λ can not be 0
as {a∈Z+:a≤100}) we will get different values of 'a'.
So, a=2,6,12,20,30,42,56,72,90 .
Hence there are 9 values of 'a'.