(D) Let $f(x) = x^2 + ax + b$. Since $f(x)$ is an integer for every integer $x$,we have: $f(0) = 0^2 + a(0) + b = b$. Since $f(0)$ must be an integer,

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(D) Let $f(x) = x^2 + ax + b$. Since $f(x)$ is an integer for every integer $x$,we have: $f(0) = 0^2 + a(0) + b = b$. Since $f(0)$ must be an integer,$b$ must be an integer. Now,$f(1) = 1^2 + a(1) + b = 1 + a + b$. Since $f(1)$ is an integer and $b$ is an integer,$1 + a + b$ is an integer,which implies $a$ must be an integer. Therefore,both $a$ and $b$ must be integers. Since the options provided do not state that both are integers,the correct choice is $D$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Solution of quadratic equations and Nature of roots

Difficult

If $x^2 + ax + b$ is an integer for every integer $x$,then which of the following is true?

A
$a$ is always an integer,but $b$ need not be an integer.
B
$b$ is always an integer,but $a$ need not be an integer.
C
$a + b$ is always an integer.
D
None of these.

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4-2.Quadratic Equations and Inequations

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Solution of quadratic equations and Nature of roots

When the roots of $x^3+\alpha x^2+\beta x+6=0$ are increased by $1$,if one of the resultant values is the least root of $x^4-6 x^3+11 x^2-6 x=0$,then

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If $a, b, c$ are distinct and the roots of $(b-c)x^2 + (c-a)x + (a-b) = 0$ are equal,then $a, b$ and $c$ are in

EasyAP EAMCET 2015

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Let $A, G$ and $H$ be the arithmetic mean,geometric mean and harmonic mean,respectively,of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A(G-H) x^2 + G(H-A) x + H(A-G) = 0$,then:

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If $f(x)$ is a quadratic expression such that $f(1) + f(2) = 0$,and $-1$ is a root of $f(x) = 0$,then the other root of $f(x) = 0$ is

DifficultJEE MAIN 2018

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When are the roots of the quadratic equation $ax^2 + bx + c = 0$ imaginary?

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If $x^2 + ax + b$ is an integer for every integer $x$,then which of the following is true?

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Let $A, G$ and $H$ be the arithmetic mean,geometric mean and harmonic mean,respectively,of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A(G-H) x^2 + G(H-A) x + H(A-G) = 0$,then:

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