Question

Solve 1,2,3 :

Q. LINKED COMPREHENSION TYPE

Read the following write up carefully and answer the following questions :

$\alpha$ and $\beta$ are the roots of the equation ax² + bx + c = 0 and $\alpha$⁴,$\beta$⁴ are the roots of the equation lx² + mx + n = 0 ($\alpha$ , $\beta$ are real and distinct ). Let f (x) = a² lx² – 4aclx + 2c²l + a²m = 0, then

1. Roots of f (x) = 0 are

(A) real and same in sign (B) real and opposite in sign

(C) equal (D) data is insufficient


2. One root of the f(x) = o is

(A) $\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}$ (B) ​$\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$

(C) b/a (D) a/b


3. If ${\mathrm{\alpha }}^3+{\mathrm{\beta }}^3=0(\mathrm{b}\ne 0)$ , then

(A) a, b, c, are in G. P. (B) 2a , b, c are in G. P.

(C) 3a, b, c are in G.P. (D) 4a , b, c are in G. P.

Answer 1

Dear student


$$\begin{gathered} \mathrm{NOTE}:\mathrm{If}{\mathrm{px}}^2+\mathrm{qx}+\mathrm{r}\mathrm{is}\mathrm{a}\mathrm{quadratic}\mathrm{equations}. \\ \mathrm{Then}\mathrm{Sum}\mathrm{of}\mathrm{roots}=-\frac{\mathrm{Coeff}.\mathrm{of}\mathrm{x}}{\mathrm{Coeff}.\mathrm{of}{\mathrm{x}}^2}=-\frac{\mathrm{q}}{\mathrm{p}} \\ \mathrm{and}\mathrm{product}\mathrm{of}\mathrm{roots}=\frac{\mathrm{Constant}\mathrm{term}}{\mathrm{Coeff}.\mathrm{of}{\mathrm{x}}^2}=\frac{\mathrm{r}}{\mathrm{p}} \\ \mathrm{Que}1 \\ \mathrm{Since}\mathrm{\alpha }\mathrm{and}\mathrm{\beta }\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}{\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0. \\ \mathrm{Then}\mathrm{\alpha }+\mathrm{\beta }=-\frac{\mathrm{b}}{\mathrm{a}} \\ \mathrm{and}\mathrm{\alpha \beta }=\frac{\mathrm{c}}{\mathrm{a}}....\left(1\right) \\ \mathrm{Also},{\mathrm{\alpha }}^4\mathrm{and}{\mathrm{\beta }}^4\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}{\mathrm{lx}}^2+\mathrm{mx}+\mathrm{n}=0 \\ \mathrm{Then}{\mathrm{\alpha }}^4+{\mathrm{\beta }}^4=-\frac{\mathrm{m}}{\mathrm{l}}...\left(2\right) \\ \mathrm{and}{\mathrm{\alpha }}^4{\mathrm{\beta }}^4=\frac{\mathrm{n}}{\mathrm{l}} \\ \mathrm{Now},\mathrm{Consider}, \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{a}}^2{\mathrm{lx}}^2-4\mathrm{aclx}+2{\mathrm{c}}^2\mathrm{l}+{\mathrm{a}}^2\mathrm{m}=0 \\ \mathrm{Then}\mathrm{x}=\frac{4\mathrm{acl}\pm \sqrt{{\left(4\mathrm{acl}\right)}^2-4\left({\mathrm{a}}^2\mathrm{l}\right)\left(2{\mathrm{c}}^2\mathrm{l}+{\mathrm{a}}^2\mathrm{m}\right)}}{2{\mathrm{a}}^2\mathrm{l}} \\ \Rightarrow \mathrm{x}=\frac{4\mathrm{acl}\pm \sqrt{16{\mathrm{a}}^2{\mathrm{c}}^2{\mathrm{l}}^2-8{\mathrm{a}}^2{\mathrm{c}}^2{\mathrm{l}}^2-4{\mathrm{a}}^4\mathrm{lm}}}{2{\mathrm{a}}^2\mathrm{l}} \\ \Rightarrow \mathrm{x}=\frac{4\mathrm{acl}\pm \sqrt{8{\mathrm{a}}^2{\mathrm{c}}^2{\mathrm{l}}^2-4{\mathrm{a}}^4\mathrm{lm}}}{2{\mathrm{a}}^2\mathrm{l}} \\ \Rightarrow \mathrm{x}=\frac{4\mathrm{acl}\pm 2\mathrm{al}\sqrt{2{\mathrm{c}}^2-{\displaystyle \frac{{\mathrm{a}}^2\mathrm{m}}{\mathrm{l}}}}}{2{\mathrm{a}}^2\mathrm{l}} \\ \Rightarrow \mathrm{x}=\frac{2\mathrm{c}}{\mathrm{a}}\pm \frac{1}{a}\sqrt{\frac{2{\mathrm{c}}^2{\mathrm{a}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{a}}^2\mathrm{m}}{\mathrm{l}}} \\ \Rightarrow \mathrm{x}=\frac{2\mathrm{c}}{\mathrm{a}}\pm \frac{\mathrm{a}}{\mathrm{a}}\sqrt{\frac{2{\mathrm{c}}^2}{{\mathrm{a}}^2}-\frac{\mathrm{m}}{\mathrm{l}}} \\ \Rightarrow \mathrm{x}=2\mathrm{\alpha \beta }\pm \sqrt{2{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+{\mathrm{\alpha }}^4+{\mathrm{\beta }}^4}\left[\mathrm{using}1\mathrm{and}2\right] \\ \Rightarrow \mathrm{x}=2\mathrm{\alpha \beta }\pm \sqrt{{\left({\mathrm{\alpha }}^2+{\mathrm{\beta }}^2\right)}^2} \\ \Rightarrow \mathrm{x}=2\mathrm{\alpha \beta }\pm \left({\mathrm{\alpha }}^2+{\mathrm{\beta }}^2\right) \end{gathered}$$
$$\begin{gathered} \Rightarrow \mathrm{x}=2\mathrm{\alpha \beta }\pm \left[{\left(\mathrm{\alpha }+\mathrm{\beta }\right)}^2-2\mathrm{\alpha \beta }\right]\left[\because {\left(\mathrm{x}+\mathrm{y}\right)}^2={\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{xy}\right] \\ \Rightarrow \mathrm{x}=2\mathrm{\alpha \beta }+\left[{\left(\mathrm{\alpha }+\mathrm{\beta }\right)}^2-2\mathrm{\alpha \beta }\right]\mathrm{or}\mathrm{x}=2\mathrm{\alpha \beta }-\left[{\left(\mathrm{\alpha }+\mathrm{\beta }\right)}^2-2\mathrm{\alpha \beta }\right] \\ \mathrm{So},\mathrm{f}\left(\mathrm{x}\right)\mathrm{has}\mathrm{real}\mathrm{and}\mathrm{opposite}\mathrm{in}\mathrm{sign}\mathrm{roots}\mathrm{because}\mathrm{\alpha }\mathrm{and}\mathrm{\beta }\mathrm{are}\mathrm{real}\mathrm{and}\mathrm{distinct}. \end{gathered}$$
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