Let Px be a quadratic polynomial with real coefficients satisfying x2 -2x - 2 - Px - 2x2-4x -3 for all x - Suppose P7-64- then the value of P13 is

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 P(x) is a quadratic polynomial, we get P(x)= ...

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Byju's Answer

Standard XIII

Mathematics

Factor Theorem

Let Px be a q...

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

242

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252

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253

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201

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Solution

The correct option is C $253$
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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Similar questions

Now for each pair of polynomials p(x) and q(x) given below, find p(x) + q(x) and p(x) − q(x). Also compute p(1) + q(1) and p(1) − q(1) for each.

(i)

(ii)

(iii)

(iv)

(v)

Q. 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)

Q. 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)

Q. Let

P (x)

be a polynomial with real coefficients such that

P ({sin}^{2} x) = P ({cos}^{2} x)

, for all

x ϵ [0, π / 2]

. Consider the following statements:
I.

P (x)

is an even function.
II.

P (x)

can be expressed as a polynomial in

(2 x - 1)^{2}

.
III.

P (x)

is a polynomial of even degree.
Then

Q. Let

Q (x)

be a quadratic polynomial with real coefficients satisfying the inequality

x^{2} - 4 x + 6 \leq Q (x) \leq 3 x^{2} - 12 x + 14, \forall x \in R .

Q (12) = 162

, then the value of

Q (7)

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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 C 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