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Question

I The sum of the binomial coefficients of the expansion $open parentheses table row 1 row cell x plus negative end cell row x end table close parentheses$ is $2 to the power of n end exponent$
II The term independent of x in the expansion of $open parentheses table row 1 row cell x plus negative end cell row x end table close parentheses$ is $0$ when is even.
Which of the above statements is correct?

Only I
Only II
Both I and II
neither I nor II

The correct answer is: Only I

Related Questions to study

$S subscript 1 end subscript$ : The fourth term in the expansion of $left parenthesis 2 x plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction right parenthesis to the power of 9 end exponent$ is equal to the second term in the expansion of $left parenthesis 1 plus x to the power of 2 end exponent right parenthesis to the power of 84 end exponent$ then the positive value of $x$ is $fraction numerator 1 over denominator 2 square root of 3 end fraction$
$S subscript 2 end subscript$ :In the expansion of $left parenthesis x to the power of 2 end exponent plus fraction numerator a over denominator x to the power of 3 end exponent end fraction right parenthesis to the power of 10 end exponent$ , the coefficients of $x to the power of 5 end exponent$ and $x to the power of 15 end exponent$ are equal, then the positive value of a is 8

$S subscript 1 end subscript$ : The fourth term in the expansion of $left parenthesis 2 x plus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction right parenthesis to the power of 9 end exponent$ is equal to the second term in the expansion of $left parenthesis 1 plus x to the power of 2 end exponent right parenthesis to the power of 84 end exponent$ then the positive value of $x$ is $fraction numerator 1 over denominator 2 square root of 3 end fraction$
$S subscript 2 end subscript$ :In the expansion of $left parenthesis x to the power of 2 end exponent plus fraction numerator a over denominator x to the power of 3 end exponent end fraction right parenthesis to the power of 10 end exponent$ , the coefficients of $x to the power of 5 end exponent$ and $x to the power of 15 end exponent$ are equal, then the positive value of a is 8

S1: If the coefficients of $x to the power of 6 end exponent$ and $x to the power of 7 end exponent$ in the expansion of $left parenthesis fraction numerator x over denominator 4 end fraction plus 3 right parenthesis to the power of n end exponent$ are equal, then the number ofdivisors ofn is 12.
S2: If the expansion of $left parenthesis x to the power of 2 end exponent plus fraction numerator 2 over denominator x end fraction right parenthesis to the power of n end exponent$ for positive integer n has 13 ^th term independent of x Then the sum of divisors of $n$ is 39.

S1: If the coefficients of $x to the power of 6 end exponent$ and $x to the power of 7 end exponent$ in the expansion of $left parenthesis fraction numerator x over denominator 4 end fraction plus 3 right parenthesis to the power of n end exponent$ are equal, then the number ofdivisors ofn is 12.
S2: If the expansion of $left parenthesis x to the power of 2 end exponent plus fraction numerator 2 over denominator x end fraction right parenthesis to the power of n end exponent$ for positive integer n has 13 ^th term independent of x Then the sum of divisors of $n$ is 39.

I Three consecutive binomial coefficients cannot be in GP.
II Three consecutive binomial coefficients cannot be in A.P.
Which of the above statement is correct?

I Three consecutive binomial coefficients cannot be in GP.
II Three consecutive binomial coefficients cannot be in A.P.
Which of the above statement is correct?

parallel

I The no of distinct terms in the expansion of $left parenthesis x subscript 1 end subscript plus x subscript 2 end subscript plus horizontal ellipsis. plus x subscript n end subscript right parenthesis to the power of 3 end exponent$ is $n plus 2 c subscript 3 end subscript$
II The no of irrational terms in the expansion $left parenthesis 2 to the power of 1 divided by 5 end exponent plus 3 to the power of 1 divided by 10 end exponent right parenthesis to the power of 55 end exponent$ is 55

I The no of distinct terms in the expansion of $left parenthesis x subscript 1 end subscript plus x subscript 2 end subscript plus horizontal ellipsis. plus x subscript n end subscript right parenthesis to the power of 3 end exponent$ is $n plus 2 c subscript 3 end subscript$
II The no of irrational terms in the expansion $left parenthesis 2 to the power of 1 divided by 5 end exponent plus 3 to the power of 1 divided by 10 end exponent right parenthesis to the power of 55 end exponent$ is 55

If $left parenthesis 3 plus x to the power of 2008 end exponent plus x to the power of 2009 end exponent right parenthesis to the power of 2010 end exponent$ $equals a subscript 0 end subscript plus a subscript 1 end subscript x plus a subscript 2 end subscript x to the power of 2 end exponent plus$ $plus a subscript n end subscript x to the power of n end exponent$ then $a subscript 0 end subscript minus fraction numerator a subscript 1 end subscript over denominator 2 end fraction minus fraction numerator a subscript 2 end subscript over denominator 2 end fraction plus a subscript 3 end subscript minus fraction numerator a subscript 4 end subscript over denominator 2 end fraction minus fraction numerator a subscript 5 end subscript over denominator 2 end fraction plus a subscript 6 end subscript$

If $left parenthesis 3 plus x to the power of 2008 end exponent plus x to the power of 2009 end exponent right parenthesis to the power of 2010 end exponent$ $equals a subscript 0 end subscript plus a subscript 1 end subscript x plus a subscript 2 end subscript x to the power of 2 end exponent plus$ $plus a subscript n end subscript x to the power of n end exponent$ then $a subscript 0 end subscript minus fraction numerator a subscript 1 end subscript over denominator 2 end fraction minus fraction numerator a subscript 2 end subscript over denominator 2 end fraction plus a subscript 3 end subscript minus fraction numerator a subscript 4 end subscript over denominator 2 end fraction minus fraction numerator a subscript 5 end subscript over denominator 2 end fraction plus a subscript 6 end subscript$

If $left curly bracket x right curly bracket$ denotes fractional part of $x$ then $left curly bracket fraction numerator 2 to the power of 2003 end exponent over denominator 17 end fraction right curly bracket equals$

If $left curly bracket x right curly bracket$ denotes fractional part of $x$ then $left curly bracket fraction numerator 2 to the power of 2003 end exponent over denominator 17 end fraction right curly bracket equals$

parallel

$404 C subscript 4 end subscript minus C subscript 4 end subscript. C subscript 1 end subscript plus 2024 minus 1014 equals$

$404 C subscript 4 end subscript minus C subscript 4 end subscript. C subscript 1 end subscript plus 2024 minus 1014 equals$

The number of rational terms in the expansion of $left parenthesis 1 plus square root of 2 plus root index 3 of 3 end root right parenthesis to the power of 6 end exponent$ is

The number of rational terms in the expansion of $left parenthesis 1 plus square root of 2 plus root index 3 of 3 end root right parenthesis to the power of 6 end exponent$ is

If $2 to the power of 2006 end exponent minus 2006$ divided by 7, the remainder is

If $2 to the power of 2006 end exponent minus 2006$ divided by 7, the remainder is

parallel

The remainder when $2 3 to the power of 23 end exponent$ is divided by 53 is

The remainder when $2 3 to the power of 23 end exponent$ is divided by 53 is

If $C subscript r end subscript equals 32 C subscript r end subscript$ , then $not stretchy sum subscript r equals 0 end subscript superscript 5 end superscript C subscript 6 r end subscript equals$

If $C subscript r end subscript equals 32 C subscript r end subscript$ , then $not stretchy sum subscript r equals 0 end subscript superscript 5 end superscript C subscript 6 r end subscript equals$

$bullet$ If $C subscript r end subscript equals to the power of 30 end exponent C subscript r end subscript$ , then $C subscript 0 end subscript plus C subscript 4 end subscript plus C subscript 8 end subscript plus$ $plus C subscript 28 end subscript equals$

$bullet$ If $C subscript r end subscript equals to the power of 30 end exponent C subscript r end subscript$ , then $C subscript 0 end subscript plus C subscript 4 end subscript plus C subscript 8 end subscript plus$ $plus C subscript 28 end subscript equals$

parallel

A particle goes from point A to B. Its displacement is X and pathlength is y. So $x over y$

A particle goes from point A to B. Its displacement is X and pathlength is y. So $x over y$

A car goes from one end to the other end of a semicircular path of diameter ' d '. Find the ratio between path length and displacement.

A car goes from one end to the other end of a semicircular path of diameter ' d '. Find the ratio between path length and displacement.

A particle moves from A to P and then it moves from P to B as shown in the figure. Find path length and displacement.

A particle moves from A to P and then it moves from P to B as shown in the figure. Find path length and displacement.

parallel

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