JEE Advanced · Mathematics · 25. AOD

For a polynomial $g (x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g (x)$ . Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{{(x^{2} - 1)}^{2} (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}) : a_{0}, a_{1}, a_{2}, a_{3} \in R\}$ . For a polynomial $f$ , let $f'$ and $f^{″}$ denote its first and second order derivatives respectively. Then the minimum possible value of $(m_{f^{'}} + m_{f^{″}})$ , where $f \in S$ , is ____

A 10
B 51
C 24
D 5

Verified Solution

Answer & Solution

Correct Answer

(D) 5

Step-by-step Solution

Detailed explanation

f (x)

is a polynomial in

x

such that

f (x) = \{{(x^{2} - 1)}^{2} (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}) : a_{0}, a_{1}, a_{2}, a_{3} \in R\}

So,

f^{'} (x) = (x^{2} - 1) [q (x)]

where

q (x)

is polynomial roots of

f^{'} (x) = - 1, 1

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let

ℓ_{1}

and

ℓ_{2}

be the lines

{\vec{r}}_{1} = λ (\hat{i} + \hat{j} + \hat{k})

and

{\vec{r}}_{2} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})

, respectively, Let

X

be the set of all the planes

H

that contain the line

ℓ_{1}

. For a plane

H

, let

d (H)

denote the smallest possible distance between the points of

ℓ_{2}

and

H

. Let

H_{0}

be a plane in

X

for which

d (H_{0})

is the maximum value of

d (H)

H

varies over all planes in

X

. Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	The value of $d (H_{0})$ is	$(1)$	$\sqrt{3}$
$(Q)$	The distance of the point $(0, 1, 2)$ from $H_{0}$ is	$(2)$	$\frac{1}{\sqrt{3}}$
$(R)$	The distance of origin from $H_{0}$ is	$(3)$	$0$
$(S)$	The distance of origin from the point of intersection of planes $y = z, x = 1$ and $H_{0}$ is	$(4)$	$\sqrt{2}$
		$(5)$	$\frac{1}{\sqrt{2}}$

The correct option is

JEE Advanced 2023 Medium

Let $f, g : [- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of $f$ and $g$ at the points $- 1, 0$ and $2$ be as given in the following table:

$x = - 1$ $x = 0$ $x = 2$

$f (x)$ 3 6 0

$g (x)$ 0 1 -1

In each of the intervals $(- 1, 0)$ and $(0, 2)$ the function $(f - 3 g) "$ never vanishes. Then the correct statement(s) is(are)

JEE Advanced 2015 Medium

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