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# Binomial Theorem

#### Multiple Choice Questions

1.

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

• n(a1 − an)

• (n − 1) (a1 − an)

• na1an

• na1an

D.

na1an

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2.

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)}

B.

[a] f(a) − {f(1) + f(2) + … + f([a])}

Let a = k + h, where [a] = k and 0 ≤ h < 1

{f(2) − f(1)} + 2{f(3) − f(2)} + 3{f(4) − f(3)}+…….+ (k−1) – {f(k) − f(k − 1)} + k{f(k + h) − f(k)}

= − f(1) − f(2) − f(3)……. − f(k) + k f(k + h)
= [a] f(a) − {f(1) + f(2) + f(3) + …. + f([a])}

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3.

If xm.yn = (x+y)m+n, then dy/dx is

• y/x

• x+y/xy

• xy

• xy

A.

y/x

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4.

The larger of 9950 + 10050 and 10150 is

• 9950 + 10050

• both are equal

• 10150

• None of the above

C.

10150

We have,

10150 = (100 + 1)50 = 10050 + 50 . 10049 + ...   ...(i)

and 9950 = (100 - 1)50 = 10050 - 50 . 10049 $\frac{}{}$ + ...     ...(ii)

On subtracting Eq. (ii) from Eq. (i), we get

10150 - 9950 = 10050 + ... > 10050

Hence, 10150 - 9950 > 10050

5.

In the binomial expansion of (a - b)n, n ≥ 5, the sum of 5th and 6th terms is zero, then
a/b equals

• 5/n −4

• 6 /n −5

• n -5 /6

• n -5 /6

D.

n -5 /6

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6.

The sum of the series 20C020C1 + 20C220C3 + …… - ….. + 20C10 is-

• 20C10

• 0

• 0

B.

(1 + x)20 = 20C0 + 20C1x + … + 20C10x10 + …+ 20C20x20
put x = − 1,
0 = 20C0 − 20C1 + … − 20C9 + 20C1020C11 + … + 20C20
0 = 2 (20C020C1 + … − 20C9) + 20C10
20C020C1 + … + 20C10 =

1417 Views

7.
• n-1

• n-1

A.

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8.

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)

D.

(45, 35)

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9.

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

D.

(1-ax)-1(1-bx)-1 = (1+ax+a2x2+.....)(1+bx+b2x2+....)
therefore coefficient of xn = bn +abn-1 +a2bn-2 +.....+an-1b +an =

182 Views

10.

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

C.

-3/10

The coefficient of x in the middle term of expansion of (1+ αx)4= 4C22
The coefficient of x in middle term of the expansion of 3(1− αx) = 6C3 (−α)3
According to question

219 Views