The sum ^{n}C_{0}
+^{n}C_{1} +^{n}C_{2}+----+^{n}C_{n}
is eqaul to

(A)

(B) n^{n}

(C) n!

(D) 2^{n}

(1
+ x)^{n} = ^{n}C_{0} + ^{n}C_{1} x + ^{n}C_{2}
x^{2} + ^{n}C_{3} x^{3} + L ^{n}C_{n} x^{n}

Put x = 1,

2^{n} = ^{n}C_{0} + ^{n}C_{1}
+ ^{n}C_{2} + L
+ ^{n}C_{n}

The digits at unit’s place in the number is

(A) 0

(B) 1

(C) 2

(D) 3

We have

=

= a multiple of 10 + 1

Thus, the unit’s place digits is 1.

The coefficient of x^{5}
in the expansion of (1 + x)^{21} +(1 + x)^{22} + …..+ (1 + x)^{30}
is.

(A) ^{51}C_{5 }

(B) ^{9}C_{5}

(C) ^{31}C_{6}
-^{21}C_{6}

(D) ^{30}C_{5}
+ ^{20}C_{5}

(1 + x)^{21} +(1 + x)^{22} +
…..+(1 + x)^{30}.

=

\ coefficient of x^{5}
in the given expression

= coefficient of x^{5}
in

= coefficient of x^{6}
in

If the co-efficients of x^{7} and x^{8}
in are equal, then n is

(A) 56

(B) 55

(C) 45

(D) 15

&⇒
n = 55

If z = , then

(A) Re(z) =0

(B) Im(z) =0

(C) Re(z) >0, I_{m}(z)
>0

(D) Re(z) >0, I_{m}(z)
<0

z = 2

= Purely real number.

Hence Im(z) = 0

If (1-x +x^{2})^{n}
= a_{0}+a_{1}x+a_{2}x^{2} +------+a_{2n}x^{2n}
then a_{0} +a_{2} +a_{4}+---+a_{2n} equals

(A)

(B)

(C)

(D)

Put
x = 1

&⇒ 1 = a_{0}
+ a_{1} + a_{2} + L
+ a_{2n}

Put x = -1

&⇒ 3^{n} = a_{0}
- a_{1} + a_{2}
- a_{3} + L + a_{2n}

Adding, 3^{n} + 1 = 2 (a_{0} + a_{2}
+ a_{4} + L+ a_{2n})

The positive integer which is just greater
than (1+0.0001)^{1000} is .

(A) 3

(B) 4

(C) 5

(D) 2

Expression on expansion gives

1 + 1000 ´
10^{-}^{4} + < 1 +

=

So integer just greater than the given expression must be 2.

If n is an even natural number and coefficient of x^{r}
in the expansion of is 2^{n}, (|x|
< 1), then

(A) r £ n/2

(B) r ≥

(C) r £

(D) r ≥ n

= (C_{0} + C_{1}x
+C_{2}x^{2} + . . . + C_{n}x^{n}) (1+ x+ x^{2}
+ ….)

The coefficient of x^{r} = C_{0} + C_{1}+C_{2}
+C_{3} + . . . + C_{r} = 2^{n} for r = n.

Moreover coefficient of x^{r} is C_{0} +
C_{1}+C_{2} +C_{3} +. . .+ C_{r} if r > n.
So r ≥ n.

The number of terms
in the expansion of (a+b+c)^{n}, where n∈N, is

(A)

(B) n+1

(C) n+2

(D) (n+1)n

(a
+ (b+c))^{n} = a^{n} +^{n}C_{1} a^{n}^{-}^{1}(b + c)^{1}+^{n}C_{2}a^{n}^{-}^{2} (b + c)^{2}
+ L+ ^{n}C_{n}^{
}(b + c)^{n}

Further expanding each term of R.H.S.,.

First term on expansion gives one term

Second term on expansion gives two terms

Third term on expansion gives three terms and so on.

\ Total no. of terms = 1 + 2 + 3 + L + (n + 1) =

If the co-efficient
of the second, third and fourth terms in the expansion of (1+x)^{n} are
in A.P. then n is equal to .

(A) 2

(B) 7

(C) 9

(D) None of these

^{n}C_{1},
^{n}C_{2,} ^{n}C_{3}^{ }are in A.P.

&⇒ ^{n}C_{1} + ^{n}C_{3}
= 2^{n}C_{2}.

&⇒
n^{2} - 9n + 14 = 0

&⇒ n = 7,2

But n = 2 is rejected as ^{n}C_{3} is
not possible.