Most Important Selected Qs for JEE AdvancedMathematicsQuadratic Equation
Let f(x)=x^3+p x^2+q x+r , where p, q and r are integers, f(0) and f(-1) are odd integers. Which of the following is/are CORRECT ?
Options
- Af(1) is an even integer
- Bf(1) is an odd integer
- Cf(x)=0 has three distinct integer roots
- Df(x)=0 cannot have three integer roots.
Correct answer
D. f(x)=0 cannot have three integer roots.
Step-by-step solution
f(0)=r is odd. Let r=2 n+1, n I . f(-1)=-1+p-q+2 n+1=p-q+2 n is odd exactly one of p, q is odd f(1)=1+p+q+2 n+1=p+q+2 n+2 is odd If possible suppose , , I be zeros of f(x) . x^3+p x^2+q x+r=x^3-( + + ) x^2+( + + ) x- r=- , , are odd integers p =-( + + ) is odd and q= + + is also odd. It is a contradiction. Hence f(x)=0 cannot have three integer roots.