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Maths-

General

Easy

Question

If $open vertical bar table row cell 1 plus x end cell x cell x to the power of 2 end exponent end cell row x cell 1 plus x end cell cell x to the power of 2 end exponent end cell row cell x to the power of 2 end exponent end cell x cell 1 plus x end cell end table close vertical bar$ = ax⁵ + bx⁴ + cx³ + dx² + $lambda$ x + $mu$ be an identity in x, where a, b, c, d, $lambda$ , $mu$ are independent of x. Then the value of $lambda$ is

3
2
4
None of these

hint Hint:

In this question, to find the value of $lambda$ we need to break the determinant.

The correct answer is: 3

Step by step solution:
$open vertical bar table row cell 1 plus x end cell x cell x to the power of 2 end exponent end cell row x cell 1 plus x end cell cell x to the power of 2 end exponent end cell row cell x to the power of 2 end exponent end cell x cell 1 plus x end cell end table close vertical bar$
$equals open parentheses 1 plus x close parentheses left square bracket open parentheses 1 plus x close parentheses squared$ $negative x cubed right square bracket minus x left square bracket x open parentheses 1 plus x close parentheses$ $negative x to the power of 4 right square bracket plus x squared left square bracket x squared$ $negative x squared open parentheses 1 plus x close parentheses right square bracket$
$equals open parentheses 1 plus x close parentheses left square bracket 1 plus 2 x plus x squared$ $negative x cubed right square bracket minus x left square bracket x plus x squared minus$ $x to the power of 4 right square bracket plus x squared left square bracket x squared minus x squared minus x cubed right square bracket$
$equals 1 plus 2 x plus x squared$ $negative x cubed plus x plus 2 x squared$ $plus x cubed minus x to the power of 4 minus x squared$ $negative x cubed plus x to the power of 5 minus x to the power of 5$
$equals negative x to the power of 4 minus x cubed plus$ $2 x squared plus 3 x plus 1$
Comparing this expression with $a x to the power of 5 plus b x to the power of 4 plus c x cubed$ $plus d x squared plus lambda x plus mu$ , we get,
$lambda$ =3
Hence, option(c) is the correct option.

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parallel

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parallel

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parallel

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