Let alpha, beta be the roots of the equation | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) For the equation $x^2-px+r=0$,the roots are $\alpha$ and $\beta$. Thus,$\alpha+\beta=p$ and $\alpha\beta=r$.For the equation $x^2-qx+r=0$,the root

Vedclass · Accepted Answer

$\frac{2}{9}(2p-q)(2q-p)$

Vedclass · Answer

(D) For the equation $x^2-px+r=0$,the roots are $\alpha$ and $\beta$. Thus,$\alpha+\beta=p$ and $\alpha\beta=r$.For the equation $x^2-qx+r=0$,the roots are $\frac{\alpha}{2}$ and $2\beta$. Thus,$\frac{\alpha}{2}+2\beta=q$ and $\left(\frac{\alpha}{2}\right)(2\beta)=r$.From the product of roots,we have $\alpha\beta=r$,which is consistent.We have the system of linear equations:$1) \alpha+\beta=p$$2) \frac{\alpha}{2}+2\beta=q \Rightarrow \alpha+4\beta=2q$Subtracting $(1)$ from $(2)$,we get $3\beta=2q-p \Rightarrow \beta=\frac{2q-p}{3}$.Substituting $\beta$ into $(1)$,$\alpha=p-\frac{2q-p}{3} = \frac{3p-2q+p}{3} = \frac{4p-2q}{3} = \frac{2(2p-q)}{3}$.Since $r=\alpha\beta$,we have $r = \left(\frac{2(2p-q)}{3}\right) \left(\frac{2q-p}{3}\right) = \frac{2}{9}(2p-q)(2q-p)$.

Let $\alpha, \beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is

Similar Questions

$A$ horizontal pipe of non-uniform cross-section allows water to flow through it with a velocity $1 \ m/s$ when pressure is $50 \ kPa$ at a point. If the velocity of flow has to be $2 \ m/s$ at some other point,the pressure at that point should be (in $kPa$)

If one end of the diameter is $(1, 1)$ and the other end lies on the line $x + y = 3$,then the locus of the centre of the circle is

The angle between the lines represented by $y^2 \sin^2 \theta - xy \sin^2 \theta + x^2(\cos^2 \theta - 1) = 0$ is

$A$ steel ring of radius $r$ and cross-section area $A$ is fitted onto a wooden disc of radius $R$ $(R > r)$. If Young's modulus is $E$,then the force with which the steel ring is expanded is:

Plants follow ...$A$... pathways in response to environment or phases of life to form different kinds of structures. This ability is called ...$B$.... Complete the given statement with the correct combination of options.

Vedclass Test Series

Exam Paper Generator

Online Exam Module