Mathematics : Binomial Theorem

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series 1² + 2.2² + 3² + 2.4² + 5² + 2.6² + .....
If B – 2A = 100λ, then λis equal to

• 496

• 232

• 248

• 464

If $\left({\log }_5\left(\mathrm{x}\right)\right){\log }_{\mathrm{x}}\left(3\mathrm{x}\right){\log }_{3\mathrm{x}}\left(\mathrm{y}\right) = {\log }_{\mathrm{x}}\left({\mathrm{x}}^3\right)$, then y equals

• 125

• 25

• 5/3

• 243

In the expansion of (x - 1) (x - 2) ... (x - 18), the coefficient of x¹⁷ is

• 684

• - 171

• 171

• - 342

$1 + {}^{\mathrm{n}}\mathrm{C}_1\cos \left(\mathrm{\theta }\right) + {}^{\mathrm{n}}\mathrm{C}_2\cos \left(2\mathrm{\theta }\right) + ... + {}^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}\cos \left(\mathrm{n\theta }\right)$ equals

• ${\left(2\cos \left(\frac{\mathrm{\theta }}{2}\right)\right)}^{\mathrm{n}}\cos \left(\frac{\mathrm{n\theta }}{2}\right)$

• $2{\cos }^2\left(\frac{\mathrm{n\theta }}{2}\right)$

• $2{\cos }^{2\mathrm{n}}\left(\frac{\mathrm{\theta }}{2}\right)$

• ${\left(2{\cos }^2\left(\frac{\mathrm{\theta }}{2}\right)\right)}^{\mathrm{n}}$

The number of irrational terms in the binomial expansion of ${\left(3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} + 7^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}^{100}$

• 90

• 88

• 94

• 95

Let P(x) be a polynomial, which when divided by (x - 3) and (x - 5) leaves remainders 10 and 6, respectively. If the polynomial is divided by (x - 3) (x - 5), then the remainder is

• - 2x + 16

• 16

• 2x - 16

• 60

the coefficient of x⁸ in ${\left({\mathrm{ax}}^2 + \frac{1}{\mathrm{bx}}\right)}^{13}$ is equal to the coefficient of x^{- 8} in ${\left(\mathrm{ax} - \frac{1}{{\mathrm{bx}}^2}\right)}^{13}$ then a and b will satisfy the relation

• ab + 1 = 0

• ab = 1

• a = 1 - b

• a + b = - 1

Let ${\left(1 + \mathrm{x}\right)}^{10} = \sum _{\mathrm{r} = 0}^{10}{\mathrm{c}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{T}} \mathrm{and} \sum _{\mathrm{r} = 0}^7{\left(1 + \mathrm{x}\right)}^7 = {\mathrm{d}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{T}}$. If $\mathrm{P} = \sum _{\mathrm{r} = 0}^5{\mathrm{c}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{T}} \mathrm{and} \mathrm{Q} = \sum _{\mathrm{r} = 0}^3{\mathrm{d}}_{2\mathrm{r} + 1 }, \mathrm{then} \frac{\mathrm{P}}{\mathrm{Q}}$ is equal to

• 4

• 8

• 16

• 32

The coefficient of xⁿ in the expansion of $\frac{{\mathrm{e}}^{7\mathrm{x}} + {\mathrm{e}}^{\mathrm{x}}}{{\mathrm{e}}^{3\mathrm{x}}}$ is

• $\frac{4^{\mathrm{n} - 1} - {\left(- 2\right)}^{\mathrm{n} - 1}}{\mathrm{n}!}$

• $\frac{4^{\mathrm{n} - 1} - 2^{\mathrm{n} - 1}}{\mathrm{n}!}$

• $\frac{4^{\mathrm{n}} - 2^{\mathrm{n}}}{\mathrm{n}!}$

• $\frac{4^{\mathrm{n}} + {\left(- 2\right)}^{\mathrm{n}}}{\mathrm{n}!}$

The number (101)¹⁰⁰ - 1 is divisible by

• 10⁴

• 10⁶

• 10⁸

• 10¹²