Q1.
The sum of the first n terms of the
series 12+2.22+32+2.42+52+2.62+
. . . is 
,
when n is even. When n is odd, the sum is
(A) 
(B) 
(C)
(D) 
Solution:
If n is odd, n-1
is even. Sum of (n-1 ) terms will be 
.
The nth term will be
n2 . Hence the required sum .
= 
+n2 = 
Hence (A) is the
correct answer. .
Q2.
If p, q, r are in A.P. , then pth,
qth and rth terms of any G.P. are in.
(A) A.P. .
(B) G. P.
(C) H. P. .
(D) A.G.P.
Solution:
Let the first
term of a G.P. be A and common rario be R . Then pth, qth and rth
terms are ARp-1, ARq-1 and ARr-1.
Obviously ARp-1´
ARr-1 = ARp+r—2 = (ARq-1)2, as p
+ r = 2q. .
Hence terms are
in G.P. .
Hence (B) is the
correct answer.
Q3.
If a, b, c are in H.P. , then
the value of 
is
(A) 0
(B) 1
(C) 2
(D) 3
Solution:
a, b, c are in
H.P. .
&⇒ b = 
&⇒ 
&⇒ 
. . .
. (A)
Again a, b, c are
in H.P. .
&⇒ b =
&⇒ 
&⇒ 
. . .
. (B)
From (A) and (B)
=2.
Hence (C) is the
correct answer.
Q4.
If the product of n positive
numbers is unity , then their sum is
(A) a positive integer
(B) divisible by n
(C) equal to n +1/n
(D) never less than n.
Solution:
Let the numbers
be a1; a2; a3 ;. . . ; an . Then a1;
a2; a3 . . . ; an =1. Using A.M. ≥ G.M , we get .
&⇒ a1 . a2 . a3
. . . . . an ≥
n .
Hence (D) is the
correct answer.
Q5.
If a, b and c are positive
real numbers then 
is
greater than or equal to
(A) 3
(B) 6
(C) 27
(D) none of those .
Solution:
Using A. M. ≥ G. M. .
&⇒ 
≥ 3.
Hence (A) is the
correct answer.
Q6.
If a, b,
c and d are distinct positive numbers in H.P. , then .
(A) a+b > c+d
(B) a+c > b+d
(C) a+d > b+c
(D) none of these .
Solution:
Since b is the H.M.
of a and c, 
> b (A.M. >
H.M.)
Again c is the
H.M. of b and d , 
> c ( A.M. >
H.M.)
Adding, we get + > b+c
&⇒ a + d > b+ c.
Hence (C) is
the correct answer.
Q7.
If 
where
k > 0; a, b, c, d > 0 > 1, then
(A) d, a, c, b are in A.P. .
(B) b, a, d, c, are in H.P.
(C) logae, logbe,
logce, logde are in H.P. .
(D) a, b, c, d are in G.P.
Solution:
Consider 
= r
(say)
&⇒
&⇒
or 
&⇒
a, b, c, d from a G. P.
&⇒
ln a , lnb , lnc, lnd form an A.P. .
&⇒
logae , logbe, logce and logde
from H. P.
Hence (C), (D) are
the correct answers.
Q8.
The first two terms of
an H.P. are 
and 
. The
value of the largest term of the H.P. is
(A) 
(B) 6
(C) 
(D) none of these
Solution:
and 
are in
A.P. , d = 
Tn
= 
&⇒ n £ 5
&⇒ 5th term
has least positive value.
Therefore
largest term of H.P. = 6 .
Hence
(B) is the correct answer.
Q9.
Coefficient of x99 in
the polynomial (x-1) (x-2) . . . ( x- 100) is .
(A) 100!
(B) – 5050
(C) 5050
(D) –100!
Solution:
For the
coefficient of x99, we have to choose constant from one
bracket and x from all other brackets in (x – 1) (x – 2)…(x –100) . .
Hence the required coefficient
= – (1+ 2+ 3+ … +
100) = – 5050. .
Hence (B) is the
correct answer.
Q10.
The determinant D
= 
is
equal to zero, if
(A) a, b, c are in A.P. .
(B) a, b, c are in G. P. .
(C) a, b, c are in H.P. .
(D) a is a root of
ax2+2bx+c=0
Solution:
It is easy to
see that D = (b2
–ac)(aa2 +2ba +c) . .
Hence D =0 if a, b, c are in G.P. .
or a is a root of ax2 +2bx
+c = 0.
Hence (B) and (D)
are the correct answers.
