Question

Fill in the blanks.
(i) If $25^{15}$ is divided by $13$, then the remainder is P.
(ii) The coefficient of $x$ in the expansion of $(1 - 3x + 7x^2)(1 - x)^{16}$ is Q.
(iii) If $n$ is even and the middle term in the expansion of $\left(x^{2}+\frac{1}{x}\right)^{n}$ is $924x^6$, then n is equal to R.
(iv) The number of terms in the expansion of $[(2x + y^3)^4]^7$ is S.

	P	Q	R	S
(a)$\,\,\,\,$	14$\,\,\,\,$	-15$\,\,\,\,$	12$\,\,\,\,$	29
(b)	14	-15	10	18
(c)	12	-19	10	29
(d)	12	-19	12	29

• A. a
• B. b
• C. c
• D. d

Answer 1

Answer: (D)

Let $25^{15} = (26 - 1)^{15}$
$=\,{}^{15}C_026^{15}- \,{}^{15}C_126^{14}+ ......-\,{}^{15}C_{15}$
$= \,{}^{15}C_026^{15}- \,{}^{15}C_126^{14} + ......-1-13 + 13$
$= \,{}^{15}C_026^{15}- \,{}^{15}C_126^{14} + ..... -13 + 12$
$= 13k + 12$, where $k \in N$
Hence, when $25^{15}$ is divided by $13$, then remainder will be $12$.
(ii) Given expansion is $(1 - 3x + 7x^2) (1 - x)^{16}$
$= (1 - 3x + 7x^2)(^{16}C_0 - \,{}^{16}C_1x^1 + \,{}^{16}C_2x^2 +....+ \,{}^{16}C_{16}x1^{6})$
$= (1 - 3x + 7x^2)(1 - 16x + 120x^2 + ..... + x^{16})$
$\therefore $ Coefficient of $x = -3 -16 = - 19$
(iii) Since, $n$ is even, therefore $\left(\frac{n}{2}+1\right)^{th}$ term is the middle term
$\therefore T_{\frac{n}{2}+1} = \,{}^{n}C_{n/2}\left(x^{2}\right)^{n/2} \left(\frac{1}{x}\right)^{n/2} = 924x^{6}\,\,$ (given)
On comparing, we get $x^{n/2} = x^{6}$
$\Rightarrow n=12$
(iv) We have, $[(2x + y^3)^4]^7 = (2x + y^3)^{28}$
$\therefore $ This expansion has $29$ terms.