Manipal Entrance Test 2020 Math Paper

Three lines of triangle are given by
(x2−y2)(2x+3y−6)=0
⇒(x−y)(x+y)(2x+3y−6)=0
∴ The three lines of triangle are
x−y=0,x+y=0
and ‌‌2x+3y−6=0From given lines of triangle, the required ∆OAB is formed.
∵(−2,λ) lies inside the triangle.
∴2(−2)+3(λ)−6<0
and ‌‌−2+λ>0
⇒‌‌−4+3λ−6<0 and λ>2
⇒‌‌3λ<10 and λ>2
⇒‌‌λ<‌

10

3

and λ>2
∴‌‌λ∈(2,‌

10

3

)‌‌...(i)
Now, (µ,1) lies outside the triangle.
To find value of µ , we find the interval [M,N] for values of x.
x+1≥0 and x−1≤0
⇒‌‌x≥−1 and x≤1
∴‌‌x∈[−1,1]
∵(µ,1) lies outside the triangle.
∴‌‌µ∈(−∞,−1)∪(1,∞)
or ‌‌µ∈R−[−1,1]