The ratio of the
co-efficient of x^{15} to the term independent of x in [x^{2}+2/x]^{15}
is

(A) 12: 32

(B) 1:32

(C) 32 :12

(D) 32:1

General term in the expansion is ^{15}C_{r}(x^{2})^{15}^{-}^{r} i.e. ^{15}C_{r}
x^{30}^{-}^{3r}×2^{r}

Coefficient of x^{15}
is ^{15}C_{5}^{ }2^{5} (r = 5)

Coefficient of
constant term is ^{15}C_{10}^{ }2^{10} (r =
10)

Ratio is 1 : 32. .

If f(x) = x^{n}, then the value of
f(1) + , where f^{r} (x) denotes
the rth order derivative of f(x) with respect to x, is

(A) n

(B) 2^{n}

(C) 2^{n –1}

(D) none of these

We have f (x) = x^{n}. So,.

Now,

The value of

(A) 1

(B) 2

(C) 3

(D) none of these

The numerator is of the form

a^{3} + b^{3} + 3ab (a+b) =
(a+b)^{3}

Where a = 18, and b = 7

\
N^{r} = (18+7)^{3} = (25)^{3}

Denominator can be written as

3^{6} + ^{6}C_{1}.3^{5}.2^{1}
+ ^{6}C_{2}.3^{4}.2^{2} +^{6}C_{3}
3^{3}.2^{3}+^{ 6}C_{4} 3^{2}.2^{4}+
^{6}C_{5} 3.2^{5} + ^{6}C_{6}2^{6}.

= (3+2)^{6} = 5^{6} = (25)^{3}

\
_{}= _{}

The term independent of x in is

(A) 1

(B) 5/12

(C) ^{10}C_{1}

(D) None of these

General term in the expansion is

=

For constant term,

&⇒

which is not an integer. Therefore, there will be no constant term .

If the sum of the
coefficients in the expansion of (1 +2x)^{n} is 6561, the greatest term
in the expansion for x = is

(A) 4^{th}

(B) 5^{th}

(C) 6^{th}

(D) none of these

sum of the coefficient in the expansion of
(1 +2x)^{n} = 6561

&⇒ (1 +2x)^{n} =
6561, when x = 1

&⇒ 3^{n} = 6561

&⇒ 3^{n} = 3^{8}

&⇒ n
= 8

Now,

&⇒ [x = ½]

\

Hence, 5^{th} term
is the greatest term.

Given the integers r>1, n> 2, and
co-efficients of (3r) ^{th} and (r+2)^{nd} term in the
binomial expansion of (1+x)^{2n} are equal, then

(A) n = 2r

(B) n =3r

(C) n = 2r+1

(D) None of these

Coefficients of (3r)^{th} and (r +
2)^{th} terms will be ^{2n}C_{3r}_{-}_{1} and ^{2n}C_{r+1}

These are equal

&⇒ (3r - 1) + (r + 1) = 2n

&⇒
n = 2r

If x^{m}
occurs in the expansion of , then the
co-efficient of x^{m} is

(A)

(B)

(C)

(D) None of these

General term in the expansion is =

For x^{m},
2n - 3r = m

&⇒

So coefficient of x^{m}
is

The co-efficients of
x^{p} and x^{q} (p and q are positive integers) in the
expansion of (1+x)^{p+q} are

(A) equal

(B) equal with opposite signs

(C) reciprocals to each other

(D) None of these

The coefficients of x^{p} and x^{q}
are

Both of which will be equal.

The value of the expression

is

(A) 2

(B) 1

(C) 3

(D) 0

The expression can be divided into two parts.

+

=

= == 0 .

In the usual
notations C_{1}+2C_{2}x +3C_{3}x^{2}+-----+nC_{n}x^{n-1}
is equal to

(A) n(1+x)^{n-1}

(B) n(1+x)^{n}

(C) (n-1)(1+x)^{n-1}

(D) (n-1)(1+x)^{n}

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

Differentiating,

n(1+ x)^{n-1}
= C_{1} + 2C_{2}x + 3C_{3}x^{2} + . . .nC_{n}
x^{n-1}.