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Standard XI

Mathematics

Sum of n Terms

If p,q,r∈ℝ+ a...

Question

If $p, q, r \in R^{+}$ and $p, q, r$ are in A.P with non zero common difference, then the roots of $p x^{2} + 2 q x + r = 0$ are

rational and equal

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imaginary

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real and equal

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real and distinct

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Solution

The correct option is D real and distinct
$p, q, r$ are in A.P.
$\Rightarrow 2 q = p + r$
$p x^{2} + 2 q x + r = 0 D = 4 q^{2} - 4 p r = (p + r)^{2} - 4 p r = (p - r)^{2} > 0 (∵ Common difference is non zero)$

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Similar questions

Q. If

p, q, r \in R^{+}

and

p, q, r

are in A.P with non zero common difference, then the roots of

p x^{2} + 2 q x + r = 0

are

Q. If

p, q, r

are in

G . P .

and the equations,

p x^{2} + 2 q x + r = 0

and

d x^{2} + 8 e x + f = 0

have a common root, then show that

\frac{d}{p}, \frac{e}{q}, \frac{f}{r}

are in

A . P .

If p, q and r are in GP and the equations $p x^{2} + 2 q x + r = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, show that $\frac{d}{p}, \frac{e}{q}$ and $\frac{f}{r}$ are in AP.

Q. If

p \neq 0, q \neq 0 a n d ∣ ∣ ∣ ∣ \begin{matrix} p & q & p α + q q & r & q α + r p α + q & q α + r & 0 \end{matrix} ∣ ∣ ∣ ∣ = 0

, then using properties of determinants prove that either p,q,r are in G.P. or

α

is a root of the equation

p x^{2} + 2 q x + r = 0.

Q. If

a x^{2} + 2 b x + c = 0

and

p x^{2} + 2 q x + r = 0

have one and only one root in common and

a, b, c, p, q, r

are rational, then

b^{2} - a c

and

q^{2} - p r

are

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If p,q,r∈R+ and p,q,r are in A.P with non zero common difference, then the roots of px2+2qx+r=0 are

The correct option is D real and distinctp,q,r are in A.P. ⇒2q=p+r px2+2qx+r=0D=4q2−4pr =(p+r)2−4pr =(p−r)2>0 (∵Common difference is non zero)

If $p, q, r \in R^{+}$ and $p, q, r$ are in A.P with non zero common difference, then the roots of $p x^{2} + 2 q x + r = 0$ are

The correct option is D real and distinct
$p, q, r$ are in A.P.
$\Rightarrow 2 q = p + r$
$p x^{2} + 2 q x + r = 0 D = 4 q^{2} - 4 p r = (p + r)^{2} - 4 p r = (p - r)^{2} > 0 (∵ Common difference is non zero)$