Consider two function `y=f(x) and y=g(x)` defined as
`f(x)={{:(ax^(2)+b,,0lexle1),(bx+2b,,1ltxle3),((a-1)x+2c-3,,3ltxle4):}`
`and" "g(x)={{:(cx+d,,0lexle2),(ax+3-c,,2ltxlt3),(x^(2)+b+1,,3gexle4):}`
Let f be differentiable at x = 1 and g(x) be continuous at x = 3. If the roots of the quadratic equation `x^(2)+(a+b+c)alphax+49(k+kalpha)=0` are real distinct for all values of `alpha` then possible values of k will be

Options

A. `k in (-1,0)`
B. `k in (oo,0)`
C. `k in (1,5)`
D. `kin (-1,1)`

Answer

A

Explanation

Given two functions : `f(X)={{:(ax^(2)+b,,0lexle1),(bx+2b,, 1ltxle3),((a-1)x+2x-3,,3ltxle4):}`
and `g(x)={{:(cx+d,, 0le xle2),(ax+3-c,,2ltxlt3),(x^(2)+b+1,,xlexle4):}`
Condition for continuity of `f(x):f(1^(-))=f(1)=f(1^(+))and f(3^(-))=f(3)=f(3^(+))`
`rArr" "a+b=3b and 5b=3a+2c-6`
`rArr" "a=2b and c=3-(b)/(2)`
condition for continuity of g(x),
`g(2)=g(2^(-))=g(2^(+))and g(3^(-))=g(3)=g(3^(+))`
`rArr" "2c+d=2a+3-c and 3a+3 -c =10+b`
`rArr" "3c+d-2a=3 and b+c -3a =-7`
Also `f'(x)={{:(2ax,,0ltxlt1),(b,,1ltxlt3),(a-1,,3ltxlt4):}andg'(x)={{:(c,,0ltxlt2),(a,,2ltxlt3),(2x,,3ltxlt4):}`
f is differentiable at x = 1 and g(x) is continues at x = 3.
i.e. a = 2v and 2a = b
Also `3a+3-c=10+b`
`rArr" "a=b=0 and c=-7`
`x^(2)-7alphax+49k (1+alpha)=0` has real and distinct roots for `AA alpha in R.`
`rArr" "49alpha^(2)-4(49k)(1+alpha)gt 0 AA alpha in R`
`rArr" "alpha^(2)-4kalpha-4k gt 0 AA alpha in R`
`rArr" "16k^(2)+16klt0`
`rArr" "k(k+1)lt0`
`" "kin (-1,0)`