Let left( a + bx + cx ^2right)^{10}=Σ limits_{ i =0}^{20} p _{ i } x ^{ i }, a , b , c ∈ N . If p

English

Gujarati

Hindi

7.Binomial Theorem

hard

Let $\left( a + bx + cx ^2\right)^{10}=\sum \limits_{ i =0}^{20} p _{ i } x ^{ i }, a , b , c \in N$. If $p _1=20$ and $p _2=210$, then $2( a + b + c )$ is equal to

A

$8$

B

$12$

C

$15$

D

$6$

(JEE MAIN-2023)

Solution

$\left(a+b x+c x^2\right)^{10}=\sum_{i=0}^{20} p_i x^i$

Coefficient of $x^1=20$

$20=\frac{10 !}{9 ! 1 !} \times a^9 \times b^1$

$a^9 . b =2$

$a=1, b=2$

Coefficient of $x ^2=210$

$210=\frac{10 !}{9 ! 1 !} \times a^9 \times c^1+\frac{10 !}{8 ! 2 !} \times a^8 b^2$

$210=10 . c+45 \times 4$

$10 c=30$

$c=3$

$2(a+b=c)=12$

Standard 11

Mathematics

Share

Similar Questions

If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$, then $|\alpha|$ equals

hard

(JEE MAIN-2024)

If $x + y = 1$, then $\sum\limits_{r = 0}^n {{r^2}{\,^n}{C_r}{x^r}{y^{n – r}}} $ equals

hard

The coefficient of $x^{49}$ in the expansion of $(x – 1)$$\left( {x\, – \,\frac{1}{2}\,} \right)$$\left( {x\, – \,\frac{1}{{{2^2}}}\,} \right)$ …..$\left( {x\, – \,\frac{1}{{{2^{49}}}}\,} \right)$ is equal to

normal

Coefficient of $x^{64}$ in the expansion of $(x – 1)^2(x – 2)^3(x – 3)^4(x – 4)^5 …. (x – 10)^{11}$

normal

If $(1 -x + x^2)^n = a_0 + a_1x + a_2x^2 + ……. + a_{2n}x^{2n}$, then $a_0 + a_2 + a_4 +……..+ a_{2n}$ is equal to

normal