Binomial Theorem

If the expansion in powers of x of the function  is a₀ + a₁x + a₂x² + a₃x³ + … , then an is

For natural numbers m, n if (1 − y)m (1 + y)n = 1 + a1y + a²y² + … , and a₁ = a₂ = 10, then (m, n) is

• (20, 45)

• (35, 20)

• (45, 35)

• (45, 35)

The value of ,where [x] denotes the greatest integer not exceeding x is

• af(a) − {f(1) + f(2) + … + f([a])}

• [a] f(a) − {f(1) + f(2) + … + f([a])}

• [a] f([a]) − {f(1) + f(2) + … + f(a)}

• [a] f([a]) − {f(1) + f(2) + … + f(a)}

If a₁, a₂, … , an are in H.P., then the expression a₁a₂ + a₂a₃ + … + a_n−1a_n is equal to

• n(a₁ − a_n)

• (n − 1) (a₁ − a_n)

• na₁a_n

• na₁a_n

If x^m.yⁿ = (x+y)^m+n, then dy/dx is

• y/x

• x+y/xy

• xy

• xy

The coefficient of the middle term in the binomial expansion in powers of x of (1 +αx)⁴  and of (1−αx )⁶  is the same if α equals

• -5/3

• 3/5

• -3/10

• -3/10

• 
• 

• n-1

• n-1

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