Let P x be a quadratic polynomial with real coefficients satisfying x 2-2 x -2 - P x - 2 x 2-4 x -3 for all x - R- Suppose P 7-64- then the value of P 13 isA- 201B- 242- 252D- 253

The correct option is D 253 nbsp; nbsp;Observe that x^2 2x +2 = (x 1)^2 +1 and 2x^2 4x +3= 2(x 1)^2 +1 Then (x 1)^2 +1 ≤ nbsp; P(x) ≤ 2(x 1)^2 +1 Since ...

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Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

Let P x be a ...

Question

Let $P (x)$ be a quadratic polynomial with real coefficients satisfying $x^{2} - 2 x + 2 \leq P (x) \leq 2 x^{2} - 4 x + 3$ for all $x \in R .$ Suppose $P (7) = 64,$ then the value of $P (13)$ is

201

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242

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252

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253

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Solution

The correct option is D $253$
Observe that $x^{2} - 2 x + 2 = (x - 1)^{2} + 1$
and $2 x^{2} - 4 x + 3 = 2 (x - 1)^{2} + 1$
Then $(x - 1)^{2} + 1 \leq P (x) \leq 2 (x - 1)^{2} + 1$
Since $P (x)$ is a quadratic polynomial, we get
$P (x) = a (x - 1)^{2} + 1$ for some $a \in [1, 2]$

$P (7) = 64 \Rightarrow a = \frac{7}{4}$
$∴ P (x) = \frac{7}{4} (x - 1)^{2} + 1$
Hence, $P (13) = 253$

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Let P(x) be a quadratic polynomial with real coefficients satisfying x2−2x+2≤P(x)≤2x2−4x+3 for all x∈R. Suppose P(7)=64, then the value of P(13) is

The correct option is D 253 Observe that x2−2x+2=(x−1)2+1 and 2x2−4x+3=2(x−1)2+1 Then (x−1)2+1≤P(x)≤2(x−1)2+1 Since P(x) is a quadratic polynomial, we get P(x)=a(x−1)2+1 for some a∈[1,2] P(7)=64⇒a=74 ∴P(x)=74(x−1)2+1 Hence, P(13)=253

Let $P (x)$ be a quadratic polynomial with real coefficients satisfying $x^{2} - 2 x + 2 \leq P (x) \leq 2 x^{2} - 4 x + 3$ for all $x \in R .$ Suppose $P (7) = 64,$ then the value of $P (13)$ is