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Binomial Theorem

Short Answer Type

171. The coefficients of the (r - 1)^th, r^th and (r + 1)^th terms in the expansion of (x + 1)ⁿ are in the ratio 1 : 3 : 5. Find n and r.

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Long Answer Type

172.

Find the absolute term in the expansion of: $open parentheses straight x minus 1 over straight x squared close parentheses to the power of 3 straight n end exponent$

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173. The co-efficients of a^r-1, a^r and a^r+1 in the expansion of (1 + a)ⁿ are in arithmetic progression, Show that n² - n(4r + 1) + 4r² - 2 =0.

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Multiple Choice Questions

174.

The sum of coefficients of integral powers of x in the binomial expansion of $left parenthesis 1 minus 2 square root of straight x right parenthesis to the power of 50 space$ is

$1 half left parenthesis 3 to the power of 50 space plus space 1 right parenthesis$
$1 half left parenthesis 30 to the power of 50 right parenthesis$
$1 half left parenthesis 30 to the power of 50 minus 1 right parenthesis$
$1 half left parenthesis 30 to the power of 50 minus 1 right parenthesis$

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175.

If (10)⁹ +2 (11)²(10)⁷ + .....+10 (11)⁹ = K(10)⁹, then k is equal to

121/10
441/100
100
100

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176.

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

17/3⁵
13/3⁵
11/3⁵
11/3⁵

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177.
The term independent of x in $open parentheses fraction numerator straight x plus 1 over denominator straight x to the power of 2 divided by 3 end exponent minus straight x to the power of 1 divided by 3 end exponent space plus 1 end fraction minus fraction numerator straight x minus 1 over denominator straight x minus straight x to the power of 1 divided by 2 end exponent end fraction close parentheses to the power of 10$ is

4

120

18

18

straight T subscript straight r plus 1 end subscript space equals space to the power of straight n straight C subscript straight r straight x to the power of straight n minus straight r end exponent straight a to the power of straight r therefore open square brackets fraction numerator straight x plus 1 over denominator straight x to the power of 2 divided by 3 end exponent minus straight x to the power of 1 divided by 3 end exponent plus 1 end fraction minus fraction numerator left parenthesis straight x minus 1 right parenthesis over denominator straight x minus straight x to the power of 1 divided by 2 end exponent end fraction close square brackets to the power of 10 equals space open square brackets fraction numerator left parenthesis straight x to the power of 1 divided by 3 end exponent right parenthesis cubed plus 1 cubed over denominator straight x to the power of 2 divided by 3 end exponent minus straight x to the power of 1 divided by 3 end exponent plus 1 end fraction minus fraction numerator open curly brackets left parenthesis square root of straight x right parenthesis squared minus 1 close curly brackets over denominator square root of straight x left parenthesis square root of straight x minus 1 end root right parenthesis end fraction close square brackets space The space general space term space is space straight T subscript straight r plus 1 end subscript space equals space to the power of 10 straight C subscript straight r left parenthesis straight x to the power of 1 divided by 3 end exponent right parenthesis to the power of 10 minus straight r end exponent left parenthesis negative straight x to the power of 1 divided by 2 end exponent right parenthesis to the power of straight r space equals space to the power of 10 straight C subscript straight r left parenthesis negative 1 right parenthesis to the power of straight r space straight x to the power of fraction numerator 10 minus straight r over denominator 3 end fraction minus straight r over 2 end exponent space

For independent of x, put

fraction numerator 10 minus straight r over denominator 3 end fraction minus straight r over 2 space equals space 0 20 minus 2 straight r minus 3 straight r space equals space 0 20 space equals space 5 straight r rightwards double arrow space straight r space equals 4 straight T subscript 5 space equals space to the power of 10 straight C subscript 4 space equals space fraction numerator 10 factorial over denominator 6 factorial 4 factorial end fraction space equals space fraction numerator 10 space straight x 9 space straight x space 8 space straight x space 7 space straight x space 6 straight x space 5 straight x space 4 space straight x 3 space straight x space 2 space straight x 1 over denominator 6 space straight x 5 space straight x 4 space straight x 3 space straight x 2 straight x 1 space straight x space left square bracket 4 space straight x 3 space straight x 2 straight x 1 right square bracket space end fraction space equals space 210

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178.

If n is a positive integer, then $open parentheses square root of 3 plus 1 close parentheses to the power of 2 straight n end exponent minus space left parenthesis square root of 3 minus 1 right parenthesis to the power of 2 straight n end exponent space$ is