Algebra

Let $\displaystyle -\frac { \pi }{ 6 } <\theta <-\frac { \pi }{ 12 } $, Suppose $\displaystyle { \alpha }{ 1 }$ and $\displaystyle { \beta }{ 1 }$ are the roots of the equation $\displaystyle { x }^{ 2 }-2x\sec { \theta } +1=0$ and $\displaystyle { \alpha }{ 2 }$ and $\displaystyle { \beta }{ 2 }$ are the roots of the equation $\displaystyle { x }^{ 2 }+2x\tan { \theta } -1=0$. If $\displaystyle { \alpha }{ 1 }>{ \beta }{ 1 }$ and $\displaystyle { \alpha }{ 2 }>{ \beta }{ 2 }$, then $\displaystyle { \alpha }{ 1 }+{ \beta }{ 2 }$ equals to

If $A,G$ and $H$ are respectively arithmetic, geometric and harmonic means between $a$ and $b$ both being unequal and positive, then $A=\dfrac{a+b}{2}\Rightarrow a+b=2A, G=\sqrt{ab} \Rightarrow ab={G}^{2}$ and $H=\dfrac{2ab}{a+b}\Rightarrow{G}^{2}=AH$ From the above discussion we can say that $a,b$ are the roots of the equation ${x}^{2}-2Ax+{G}^{2}=0$ Now,quadratic equation, ${x}^{2}-Px+Q=0$ and quadratic equation $a\left(b-c\right){x}^{2}+b\left(c-a\right)x+c\left(a-b\right)=0$ have a root common and satisfy the relation $b=\dfrac{2ac}{a+c},$ where $a,b,c$ are real numbers. On the basis of the above information, answer the following questions: The ratio of A.M, G.M and H.M of the roots of the given quadratic equation is:

Let $A\left(\alpha, \dfrac{1}{\alpha}\right), B\left(\beta, \dfrac{1}{\beta}\right), C\left(\gamma, \dfrac{1}{\gamma}\right)$ be the vertices of a $\triangle ABC$, where $\alpha, \beta$ are the roots of the equation $x^{2}-6P_{1}x+2=0$; $\beta, \gamma$ are the roots of the equation $x^{2}-6P_{2}x+3=0$ and $\gamma, \alpha$ are the roots of the equation $x^{2}-6P_{3}x+6=0, P_{1}, P_{2}$ and $P_{3}$ being positive then the coordinates of the centroid of $\triangle ABC$ is :

Lets consider quadratic equation $ax^2 + bx + c = 0$ where $a, b, c \in R$ and $a\ne 0$. If above equation has roots $\alpha, \beta$, then $\displaystyle{\alpha + \beta = -\frac{b}{a} , \alpha\beta = \frac{c}{a}}$ and the equation can be written as $ax^2+ bx + c = a(x - \alpha)(x - \beta)$. Also, if $a_1, a_2, a_3, a_4$, ..... are in A.P., then $a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = ... \ne 0$ and if $b_1, b_2, b_3, b_4$, ... are in G.P., then $\displaystyle{\frac{b_2}{b_1} = \frac{b_3}{b_2} = \frac{b_4}{b_3}} =$ ... $\ne1$ Now if $c_1$, $c_2$, $c_3$, $c_4$, ... are in HP, then $\displaystyle{\frac{1}{c_2} - \frac{1}{c_1} = \frac{1}{c_3} - \frac{1}{c_2} = \frac{1}{c_4} - \frac{1}{c_3}} = $... $\ne 0$. If the roots of equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ are equal, then $a, b, c$ are in

(i) lf $\alpha,\beta$ are the roots of $\mathrm{a}\mathrm{x}^{2}+\mathrm{b}\mathrm{x}+\mathrm{c}=0$. lf $\mathrm{K}$ is a real number $[\mathrm{K}\neq 0,1 ]$ then the equation whose roots are $ K\alpha,\ K\beta$ is $\mathrm{a}\mathrm{x}^{2}+$ Kbx $+\mathrm{K}^{2}\mathrm{c}=0$. (ii) 8,2 are roots of $\mathrm{x}^{2}+\mathrm{a}\mathrm{x}+\beta=0$ and 3, 3 are the roots of $\mathrm{x}^{2}+\alpha \mathrm{x}+\mathrm{b}=0$ then the roots of $\mathrm{x}^{2}+\mathrm{a}\mathrm{x}+\mathrm{b}=0$ are 9, 1. Which of the above statement is correct

Let the equation $x^2$ + 2(a - 1)x + a + 5 = 0, where 'a' is a parameter, match the real value of 'a' so that the given equation has :- Column - I Column - II (A) Imaginary roots (P) $\displaystyle{\left(-\infty, -\frac{8}{7} \right)}$ (B) One root less than 3 other root is greater than 3 (Q) (-1, 4) (C) One root less than 1 & other root is greater than 3 (R) $\displaystyle{\left(-\infty, -\frac{4}{3} \right)}$