Consider Q=ax^2+bx+c where a, b, c are real and distinct. Also, Q=0 has real roots Column 1 Co

step 1 I have you given a quite equation. ABC are real and distinct. 2 .0 has real roots. A .B .C in AP

Consider Q=ax^2+bx+c where a, b, c are real and distinct....

Consider $\mathrm{Q}=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real and distinct. Also, $\mathrm{Q}=0$ has real roots Column 1 $\quad$ Column II (a) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, then the least value of $\left|\frac{\mathrm{d}}{\mathrm{b}}\right|$ (where $\mathrm{d}$ is the common (p) $-\sec ^{2} \frac{\pi}{4}$ difference), is (b) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then $\mathrm{b}$ cannot be equal to (q) $\cos \frac{\pi}{3}$ (c) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in AP with common difference 'd', and if zero is a root of (r) $\sin \frac{\pi}{3}$ $\mathrm{Q}=0$, then $\left|\frac{\mathrm{d}}{\mathrm{a}}\right|$ equals (d) If $a, b, c$ are in $A P$, and $b, c$, a are in GP, then the common ratio $r$ equals (s) $\sin \frac{\pi}{2}$

Consider $\mathrm{Q}=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real and distinct. Also, $\mathrm{Q}=0$ has real roots Column 1 $\quad$ Column II
(a) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, then the least value of $\left|\frac{\mathrm{d}}{\mathrm{b}}\right|$ (where $\mathrm{d}$ is the common
(p) $-\sec ^{2} \frac{\pi}{4}$
difference), is
(b) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then $\mathrm{b}$ cannot be equal to
(q) $\cos \frac{\pi}{3}$
(c) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in AP with common difference 'd', and if zero is a root of
(r) $\sin \frac{\pi}{3}$
$\mathrm{Q}=0$, then $\left|\frac{\mathrm{d}}{\mathrm{a}}\right|$ equals
(d) If $a, b, c$ are in $A P$, and $b, c$, a are in GP, then the common ratio $r$ equals
(s) $\sin \frac{\pi}{2}$

Key Concepts

Quadratic Equations and Discriminant

A quadratic equation is a second?degree polynomial with the form ax² + bx + c = 0, where the nature of its roots is determined by the discriminant (b² - 4ac). If the discriminant is non-negative, the quadratic equation has real roots. This concept is fundamental in algebra as it connects the coefficients of the quadratic to its solution characteristics.

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which each term differs from the previous one by a constant difference. When the coefficients of a quadratic equation are in AP, they satisfy a linear relation providing additional structure to the equation. Understanding this property can lead to various constraints and relations between the coefficients, which are useful in further analysis of the equation.

Geometric Progression (GP)

A geometric progression is a sequence in which each term is obtained by multiplying the previous term by a constant ratio. For quadratic coefficients in GP, there exists a multiplicative relationship among them typically expressed as b² = ac for a quadratic equation. This property is central to establishing specific relationships and restrictions on the coefficients, influencing the behavior and solvability of the quadratic.

Transcript

00:01 I have you given a quite equation.

00:03 Abc are real and distinct.

00:05 2 .0 has real roots.

00:07 A .b .c in ap, we need to find the least value of b over b.

00:11 Let's work on this.

00:12 So, a, b in ap, so we can write this as, let's say, a equals b negative d, then b equals b and c equals b positive d.

00:22 That's an ap.

00:23 D is a common difference.

00:26 Now we have this real root, quiet equation.

00:28 So b squared negative 4a c squared a than, equal to 0 so b square negative 4 ac we have b negative d and b b b b b b b b squared that's given as b square negative d squared that is greater than equal to 0 so now just work on this we have a negative 3 b square plus 4 d square is greater than equal to 0 from here we have 4 d square is greater than equal to 3 d square or we have d square over b squared than equal to 3 over 4 this will give take a square root so more d over b is squared than equal to we have root 3 over 2 that equals to sign by by 3 3 so we can say option a for response to r now we have we seen gp so we can go with a equals b over r r r the common ratio c equals b r u same condition b squared negative 4 ac squared than equal to 0 we have b squared negative 4 times b over r times b f greater than equal to 0 so we get b square negative 4 b squared than equal to 0 negative 3 b squared greater than equal to 0 or we have b square less than equal to 0 now b square is always positive so i mean b squared less than equal to 0 we can have only uh one value that is b equal to 0 look at the options so no values uh so sine pi over 2 is giving 1, sine pi over 3 is giving root 3 over 2.

02:14 So no value is giving the value as 0.

02:17 So b equal to 0.

02:19 You don't have any value in a, b, c, and b...

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