Quadratic Equations · Mathematics · AP EAPCET
MCQ (Single Correct Answer)
If $x^2-4 a x+5+a>0$ for all $x \in R$ whenever $a \in(\alpha, \beta)$, then $4 \beta+\alpha=$
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-12 x^2+k x-18=0$ and one of them is thrice the sum of the other two roots, then $\alpha^2+\beta^2+\gamma^2-k=$
The polynomial equation of degree 5 whose roots are the roots of the equation $x^5-3 x^4-x^3+11 x^2-12 x+4=0$ each increased by 2 , is
If the area of a square is 575 square units, then the approximate value of its side is
If $\alpha$ is the common root of the quadratic equations $x^2-5 x+4 a=0, x^2-2 a x-8=0$, where $a \in R$, then the value $\alpha^4-\alpha^3+68$ is
If $\alpha, \beta$ are the roots of $x^2-5 \gamma x-6 \delta=0$ and $\gamma, \delta$ are the roots of $x^2-5 \alpha x-6 \beta=0$, then $\alpha+\beta+\gamma+\delta=$
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)=$
If the difference of the roots of the equation $x^2-7 x+10=0$ is same as the difference of the roots of the equation $x^2-17 x+k=0$, then a divisor of $k$ is $x^2-7 x+10=0$
The product of all the real roots of the equation $|x|^2-5|x|+6=0$
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-4 x^2+3 x-2=0$, then $\alpha^3+\beta^3+\gamma^3=$
After the roots of the equation $6 x^3+7 x^2-4 x-2=0$ are diminished by $h$, if the transformed equation does not contain $x$ term, then the product of all the possible value of $h$ is
The number of distinct quadratic equations $a x^2+b x+c=0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c(a \neq b \neq c)$ from the set $\{0,1,2,4\}$ is
The number of solutions of the equation $\sqrt{3 x^2+x+5}=x-3$ is
The set of all real values of $x$ for which $\frac{x^2-1}{(x-4)(x-3)} \geq 1$ is
Two roots of the equation, $a x^4+b x^3+c x^2+d x+e=0$ are positive and equal. If the product of the other two real roots is 1 , then
If the roots of the equation $x^2+2 a x+b=0$ are real, distinct and differ atmost by 2 m , then $b$ lies in the interval
The cubic equation whose roots are the squares of the roots of the equation $x^3-2 x^2+3 x-4=0$ is
If $\alpha, \beta$ are the roots of the equation $x^2+b x+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$, then $3 c+2=$
If $\alpha, \beta, \gamma$ are the roots of the equation,
$$ \begin{aligned} & x^3+a x^2+b x+c=0, \text { then }(\alpha+\beta-2 \gamma) \\ & (\beta+\gamma-2 \alpha)(\gamma+\alpha-2 \beta)= \end{aligned} $$
If the sum of two roots of the equation $x^4+2 x^3-7 x^2-8 x+12=0$ is zero, then the sum of the squares of the other two roots is
$f(x)$ is a quadratic polynomial satisfying the condition $f(x)+f\left(\frac{1}{x}\right)=f(x) f\left(\frac{1}{x}\right)$. If $f(-1)=0$, then the range of $f$ is
If $\alpha \neq 0$ and zero are the roots of the equation $x^2-5 k x+\left(6 k^2-2 k\right)=0$, then $\alpha=$
The set of all real values of $x$ satisfying the inequation $\frac{8 x^2-14 x-9}{3 x^2-7 x-6}>2$ is
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
Let ' $a$ ' be a non-zero real number. If the equation whose roots are the squares of the roots of the cubic equation $x^3-a x^2+a x-1=0$ is identical with this cubic equation, then ' $a$ ' =
If $(2 k-1) x^2-2(3 k-2) x+4 k>0$ for every $x \in R$, then the sum of all possible integral values of $k$ is
If $\alpha$ is a repeated root of multiplicity 2 of the equation $18 x^3-33 x^2+20 x-4=0$, then
The equation $6 x^4-5 x^3+13 x^2-5 x+6=0$ will have
The roots $\alpha, \beta$ of the equation $x^2-6(k-1) x+4(k-2)=0$ are equal in magnitude but opposite in sign, if $\alpha>\beta$, then the product of the roots of the equation $2 x^2-\alpha x+6 \beta(\alpha+1)=0$
If $a x^2+b x+c<0 \forall x \in R$ and the expressions $c x^2+a x+b$ and $a x^2+b x+c$ have their extreme values at the same point $x$, then for the expression $c x^2+a x+b$
If $x^2-5 x+6$ is a factor of $f(x)=x^4-17 x^3+k x^2-247 x+210$, then the other quadratic factor of $f(x)$ is
Given $f(x)=x^2-5 x+4$. Out of first 20 natural numbers, if a number $x$ is chosen at random, then the probability that the chosen $x$ satisfies the inequality $f(x)>10$ is
If the harmonic mean of the roots of the equation $\sqrt{2} x^2-b x+(8-2 \sqrt{5})=0$ is
All the values of $k$ such that the quadratic expression $2 k x^2-(4 k+1) x+2$ is negative for exactly three integrals values of $x$, lie in the interval
If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-13 x^2+k x+189=0$ such that $\beta-\gamma=2$, then $\beta+\gamma: k+\alpha=$
The cubic equation whose roots are the square of the roots of the equation is
$$ 12 x^3-20 x^2+x+3=0 $$
The set of all real values ' $a$ ' for which $-1<\frac{2 x^2+a x+2}{x^2+x+1}<3$ holds for all real values of $x$ is
The quotient, when $3 x^5-4 x^4+5 x^3-3 x^2+6 x-8$ is divided by $x^2+x-3$ is
If $$S=\left\{m \in R: x^2-2(1+3 m) x+7(3+2 m)=0\right.$$ has distinct roots}, then the number of elements in $$S$$ is
The sum of the real roots of the equation $$x^4-2 x^3+x-380=0$$ is
If one root of the cubic equation $$x^3+36=7 x^2$$ is double of another, then the number of negative roots are
If $$f(f(0))=0$$, where $$f(x)=x^2+a x+b, b \neq 0$$, then $$a+b=$$
The sum of the real roots of the equation $$|x-2|^2+|x-2|-2=0$$ is
If the difference between the roots of $$x^2+a x+b=0$$ and that of the roots of $$x^2+b x+a=0$$ is same and $$a \neq b$$, then
For what values of $$a \in Z$$, the quadratic expression $$(x+a)(x+1991)+1$$ can be factorised as $$(x+b)(x+c)$$, where $$b, c \in Z$$ ?
If $$\frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6}$$, then $$A^2+B^2=$$
If $$f(x)=a x^2+b x+c$$ for some $$a, b, c \in R$$ with $$a+b+c=3$$ and $$f(x+y)=f(x)+f(y)+x y, \forall x, y \in R$$. Then, $$\sum_\limits{n=1}^{10} f(n)=$$
The number of positive real roots of the equation $$3^{x+1}+3^{-x+1}=10$$ is
The number of real roots of the equation $$\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$$ is
For $$a\ne b$$, if the equation $$x^2+ax+b=0$$ and $$x^2+bx+a=0$$ have a common root, then the value of $$a+b$$ is equal to
If the product of the roots of $$9x^3+112x^2-120x+a=0$$ is 12, then the value of $$a$$ is
$$2+\sqrt{5}, 1$$ are roots of the cubic equation given by
If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation $$x^2+x+1=0$$, then the equation whose roots are $$\alpha^{2021}, \beta^{2021}$$ is given by
If $$2, 1$$ and $$1$$ are roots of the equation $$x^3-4 x^2+5 x-2=0$$, then the roots of $$\left(x+\frac{1}{3}\right)^3-4\left(x+\frac{1}{3}\right)^2+5\left(x+\frac{1}{3}\right)-2=0$$
If $$f(x)=2x^3+mx^2-13x+n$$ and 2, 3 are the roots of the equation $$f(x)=0$$, then the values of m and n are
If $$\alpha$$ and $$\beta$$ are the roots of $$11 x^2+12 x-13=0$$, then $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}$$ is equal to (approximately close to)
The value of $$a$$ for which the equations $$x^3+a x+1=0$$ and $$x^4+a x^2+1=0$$ have a common root is
If $$a$$ is a positive integer such that roots of the equation $$7 x^2-13 x+a=0$$ are rational numbers, then the smallest possible value of $$a$$ is
The sum of the roots of the equation $$e^{4 t}-10 e^{3 t}+29 e^{2 t}-22 e^t+4=0$$ is