Let M = [ 0 − α α 0 ] , where a is a non-zero real number an N = ∑ k = 1 4 9 M 2 k . If (I – M2)N = -2I, then the positive integral value of α is…………….

f ( x ) = a x 2 + b x + c g (x) = px + q f ( g ( x ) ) = a ( p x + q ) 2 + b ( p ) ( + q ) + c ⇒ 8 x 2 − 2 x = a ( p x + q ) 2 + b ( p x + q ) + c Compare 8 = ap2 …………… (i) -2 = a (2pq) + bp 0 = aq2 + bq + c 9 ( f ( x ) ) = p ( a x 2 + b x + c ) + q =>4x2 + 6x + 1 = apx2 + bpx + cp + q => Andhra Pradesh = 4 ……………. (ii) 6 = bp 1 = cp + q From (i) & (ii), p = 2, q = -1 => b = 3, c = 1, a = 2 f (x) = 2x2 + 3x + 1 f (2) = 8 + 6 + 1 = 15 g (x) = 2x – 1 g (2) = 3

Let $M = [\begin{matrix} 0 & - α \\ α & 0 \end{matrix}],$ where a is a non-zero real number an $N = \sum_{k = 1}^{49} M^{2 k} .$ If (I – M²)N = -2I, then the positive integral value of α is…………….

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Vishal Baghel | Contributor-Level 10

4 months ago

$f (x) = a x^{2} + b x + c$

g (x) = px + q

$f (g (x)) = a {(p x + q)}^{2} + b (p) (+ q) + c$

$\Rightarrow 8 x^{2} - 2 x = a {(p x + q)}^{2} + b (p x + q) + c$

Compare 8 = ap² …………… (i)

-2 = a (2pq) + bp

0 = aq² + bq + c

$9 (f (x)) = p (a x^{2} + b x + c) + q$

=>4x² + 6x + 1 = apx² + bpx + cp + q

=> Andhra Pradesh = 4 ……………. (ii)

6 = bp

1 = cp + q

From (i) & (ii), p = 2, q = -1

=> b = 3, c = 1, a = 2

f (x) = 2x² + 3x + 1

f (2) = 8 + 6 + 1 = 15

g (x) = 2x – 1

g (2) = 3

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