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Standard XII

Mathematics

Binomial Coefficients

Let a and b b...

Question

Let $a$ and $b$ be the coefficient of $x^{3}$ in $(1 + x + 2 x^{2} + 3 x^{3})^{4}$ and $(1 + x + 2 x^{2} + 3 x^{3} + 4 x^{4})^{4}$ respectively. Then the value of $\frac{6 a}{b}$ is

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Solution

$b = coefficient of x^{3} in {((1 + x + 2 x^{2} + 3 x^{3}) + 4 x^{4})}^{4}$
$= coefficient of x^{3} in [^{4} C_{0} (1 + x + 2 x^{2} + 3 x^{3})^{4} (4 x^{4})^{0} +^{4} C_{1} (1 + x + 2 x^{2} + 3 x^{3})^{3} (4 x^{4})^{1} + . . .]$

Neglect the higher power term, we get
$= coefficient of x^{3} in (1 + x + 2 x^{2} + 3 x^{3})^{4}$
$= a$
Hence, $\frac{6 a}{b} = 6$

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Similar questions

Q. Let

a

and

b

be the coefficient of

x^{3}

(1 + x + 2 x^{2} + 3 x^{3})^{4}

and

(1 + x + 2 x^{2} + 3 x^{3} + 4 x^{4})^{4}

respectively. Then the value of

\frac{6 a}{b}

The coefficient of xⁿin (1+x+2x²+3x³+.....+nxⁿ)²is ?

Q. The quotient

q (x)

is when the polynomial

(x^{4} + x^{3} - 2 x^{2} + x - 1)

is divided by

(x - 1)

$(x + 1)$ is a factor of the polynomial
(a) $x^{3} - 2 x^{2} + x + 2$ (b) $x^{3} + 2 x^{2} + x - 2$
(c) $x^{3} + 2 x^{2} - x - 2$ (d) $x^{3} + 2 x^{2} - x + 2$

Q. If dividend =

x^{4} + x^{3} - 2 x^{2} + x + 1

, divisor

= x - 1

and remainder

= 2

, then the quotient will be .

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Binomial Coefficients

Standard XII Mathematics

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Let a and b be the coefficient of x3 in (1+x+2x2+3x3)4 and (1+x+2x2+3x3+4x4)4 respectively. Then the value of 6ab is

b=coefficient of x3 in ((1+x+2x2+3x3)+4x4)4 =coefficient of x3 in [4C0(1+x+2x2+3x3)4(4x4)0 + 4C1(1+x+2x2+3x3)3(4x4)1+...] Neglect the higher power term, we get =coefficient of x3 in (1+x+2x2+3x3)4 =a Hence, 6ab=6

Let $a$ and $b$ be the coefficient of $x^{3}$ in $(1 + x + 2 x^{2} + 3 x^{3})^{4}$ and $(1 + x + 2 x^{2} + 3 x^{3} + 4 x^{4})^{4}$ respectively. Then the value of $\frac{6 a}{b}$ is