Question.1
The ratio of the
co-efficient of x15 to the term independent of x in [x2+2/x]15
is
(A) 12: 32
(B) 1:32
(C) 32 :12
(D) 32:1
Solution
General term in the expansion is 15Cr(x2)15-r
i.e. 15Cr
x30-3r×2r
Coefficient of x15
is 15C5 25 (r = 5)
Coefficient of
constant term is 15C10 210 (r =
10)
Ratio is 1 : 32.
.
Question.2
If f(x) = xn, then the value of
f(1) +
, where fr (x) denotes
the rth order derivative of f(x) with respect to x, is
(A) n
(B) 2n
(C) 2n –1
(D) none of these
Solution
We have f (x) = xn. So,.
![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image004.gif)
Now,
![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image005.gif)
![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image006.gif)
Question.3
The value of ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image007.gif)
(A) 1
(B) 2
(C) 3
(D) none of these
Solution
The numerator is of the form
a3 + b3 + 3ab (a+b) =
(a+b)3
Where a = 18, and b = 7
\
Nr = (18+7)3 = (25)3
Denominator can be written as
36 + 6C1.35.21
+ 6C2.34.22 +6C3
33.23+ 6C4 32.24+
6C5 3.25 + 6C626.
= (3+2)6 = 56 = (25)3
\
=
Question.4
The term independent of x in
is
(A) 1
(B) 5/12
(C) 10C1
(D) None of these
Solution
General term in the expansion is ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image011.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image012.gif)
For constant term, ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image013.gif)
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image014.gif)
which is not an integer.
Therefore, there will be no constant term .
Question.5
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) 4th
(B) 5th
(C) 6th
(D) none of these
Solution
sum of the coefficient in the expansion of
(1 +2x)n = 6561
&⇒ (1 +2x)n =
6561, when x = 1
&⇒ 3n = 6561
&⇒ 3n = 38
&⇒ n
= 8
Now, ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image016.gif)
&⇒
[
x = ½]
\ ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image019.gif)
Hence, 5th term
is the greatest term.
Question.6
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
Solution
Coefficients of (3r)th and (r +
2)th terms will be 2nC3r-1 and 2nCr+1
These are equal
&⇒ (3r - 1) + (r + 1) = 2n
&⇒
n = 2r
Question.7
If xm
occurs in the expansion of
, then the
co-efficient of xm is
(A)
(B) ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image022.gif)
(C)
(D) None of these
Solution
General term in the
expansion is
= ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image025.gif)
For xm,
2n - 3r = m
&⇒ ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image026.gif)
So coefficient of xm
is ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image027.gif)
Question.8
The co-efficients of
xp and xq (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
Solution
The coefficients of xp and xq
are ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image028.gif)
Both of which will
be equal.
Question.9
The value of the
expression
is
(A) 2
(B) 1
(C) 3
(D) 0
Solution
The expression can
be divided into two parts.
+ ![](http://www.quizsolver.com/radix/dth/notif/OBJ_SOL_BIN_1_files/image031.gif)
=
=
=
=
0 .
Question.10
In the usual
notations C1+2C2x +3C3x2+-----+nCnxn-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
Solution
(1 +x)n = C0 +C1x
+ C2x2 +C3x3 + . . . . + Cn
xn.
Differentiating,
n(1+ x)n-1
= C1 + 2C2x + 3C3x2 + . . .nCn
xn-1.