Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If <math><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>8</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>,</mo></mrow></math> and <math><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>4</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo></mrow></math> then the value of f(2) + g(2) is

<!- [if gte mso 9]><xml> </xml><! [endif]-> f ( x ) = a x 2 + b x + c g (x) = px + q f ( g ( x ) ) = a ( p x + q ) 2 + b ( p ) ( + q ) + c <!- [if gte mso 9]><xml> </xml><! [endif]-> ⇒ 8 x 2 − 2 x = a ( p x + q ) 2 + b ( p x + q ) + c <!- [if gte mso 9]><xml> </xml><! [endif]-> Compare 8 = ap2 …………… (i)-2 = a (2pq) + bp0 = aq2 + bq + c 9 ( f ( x ) ) = p ( a x 2 + b x + c ) + q <!- [if gte mso 9]><xml> </xml><! [endif]-> ? 4x2 + 6x + 1 = apx2 + bpx + cp + q? Andhra Pradesh = 4 ……………. (ii)6 = bp1 = cp + qFrom (i) & (ii), p = 2, q = -1? b = 3, c = 1, a = 2f (x) = 2x2 + 3x + 1f (2) = 8 + 6 + 1 = 15g (x) = 2x – 1g (2) = 3

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Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f (g (x)) = 8 x^{2} - 2 x,$ and $g (f (x)) = 4 x^{2} + 6 x + 1,$ then the value of f(2) + g(2) is

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Payal Gupta | Contributor-Level 10

2 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

<!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1814701626"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mrow> </math>  g (x) = px + q <math> <mrow> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>g</mi> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mi>p</mi> <mi>x</mi> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mrow> <mo> (</mo> <mrow> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo> (</mo> <mrow> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1814701627"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mo>⇒</mo> <mn>8</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mi>a</mi> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mi>p</mi> <mi>x</mi> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mrow> <mo> (</mo> <mrow> <mi>p</mi> <mi>x</mi> <mo>+</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1814701628"> </o:OLEObject></xml><! [endif]-> Compare 8 = ap2 …………… (i)-2 = a (2pq) + bp0 = aq2 + bq + c <math> <mrow> <mn>9</mn> <mrow> <mo> (</mo> <mrow> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mrow> <mo> (</mo> <mrow> <mi>a</mi> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>q</mi> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1814701629"> </o:OLEObject></xml><! [endif]-> ? 4x2 + 6x + 1 = apx2 + bpx + cp + q? Andhra Pradesh = 4 ……………. (ii)6 = bp1 = cp + qFrom (i) & (ii), p = 2, q = -1? b = 3, c = 1, a = 2f (x) = 2x2 + 3x + 1f (2) = 8 + 6 + 1 = 15g (x) = 2x – 1g (2) = 3

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Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f (g (x)) = 8 x^{2} - 2 x,$ and $g (f (x)) = 4 x^{2} + 6 x + 1,$ then the value of f(2) + g(2) is

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Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x))=8x2−2x, and g(f(x))=4x2+6x+1, then the value of f(2) + g(2) is

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Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $f (g (x)) = 8 x^{2} - 2 x,$ and $g (f (x)) = 4 x^{2} + 6 x + 1,$ then the value of f(2) + g(2) is