Q1.
Given
p A.P’s, each of which consists of n terms . If their first terms
are 1, 2, 3, -----, p and common differences are
1, 3, 5, ---, 2p –1 repectively , then sum of the terms of all the
progressions is .
(A) 
np(np+1)
(B) n(p+1)
(C) np(n+1)
(D) none of these .
Solution:
The rth A. P.
has first term r and common difference 2r-1. Hence sum of its n
terms = 
.
The required sum = 
= 
= 
= 
. Hence
(A) is the correct answer.
Q2.
If log2, log(2x-1) and
log(2x+3) are in A.P. , then the value of x is .
(A) 5/2
(B) log25
(C) log35
(D) log53
Solution:
2 log(2x–1)
= log2 + log(2x +3)
&⇒ ( 2x –1)2 =
2. (2x +3)
&⇒ (2x)2
– 4.2x – 5 = 0.
&⇒ ( 2x – 5) ( 2x
+1) = 0
&⇒ x = log2 5 , as 2x
+1 > 0
Hence (B) is the
correct answer.
Q3.
If a, b and c are distinct
positive real numbers and a2 +b2 +c2 =1,
then ab + bc +ca is
(A) less than 1
(B) equal to 1
(C) greater than 1
(D) any real number.
Solution:
Since a and b are
unequal , 
( A.M. > G.M.
for unequal numbers)
&⇒ a2 +b2>
2ab
Similarly b2
+c2 > 2bc and c2 +a2 > 2ca.
Hence 2(a2
+b2 +c2 ) > 2(ab + bc +ca)
&⇒ ab +bc +ca < 1
Hence (A) is the
correct answer.
Q4.
If a, b and c are
positive real numbers , then least value of (a+b+c) 
is
(A) 9
(B) 3
(C) 10/3
(D) none of these
Solution:
Using A.M. ≥ G.M. , .
≥ (abc)1/3 and 
≥ 
&⇒ .≥ 1
&⇒ 
≥ 9 .
Equality will
hold when a= b = c
Hence (A) is the
correct answer. .
Q5.
If first and (2n-1)th terms of
an A.P. , G. P. and H.P. , are equal and their nth terms are a, b, c
respectively , then .
(A) a+c = 2b
(B) a+c = b
(C) a ≥ b ≥
c
(D) ac –b2 = 0
Solution:
Let a be the first and b
be the (2n-1)th term of an A.P. , G.P. and H.P. , then a,
a, b will be in A.P. , a, b, b
will be G.P. a, c, b will be in
H.P. .
Hence a, b, c
are respectively A. M. , G.M. and H.M. of a and b. Since A.M. ≥
G.M. ≥H.M. , a ≥ b ≥ c. .
Again a = 
, b2 = ab and c = 
.Hence ac-b2
=0.
Hence (C) and (D) are
correct answers.
Q6.
Let p, q, r ∈
R+ and 27 pqr ≥ ( p + q + r)3 and 3p + 4q
+ 5r = 12 then
p3 + q4 + r5 is equal to
(A) 3
(B) 6
(C) 2
(D) none of these
Solution:
27 pqr ≥
( p + q + r )3
&⇒
( pqr)1/3 ≥ 
&⇒
p = q = r
Also 3 p + 4q + 5r =12
&⇒ p = q = r =1 .
Hence (A) is the
correct answer.
Q7.
If xi >
0, i = 1, 2, . . . ., 50 and x1 +x2 + . . . + x50
= 50, then the minimum value of 
equals
to
(A) 50
(B) (50)2
(C) (50)3
(D) (50)4
Solution:
We have (x1 +x2 +x3+
.. + x50) 
≥ (50)2
[since A.M. ≥ H.M.].
&⇒
≥ 50.
Hence (A) is the
correct answer.
Q8.
The sum of first n terms of the
series 
is
equal to
(A) 2n - n - 1
(B) 1-2-n
(C) n + 2-n-1
(D) 2n -1
Solution:
= n - 
= n +
2-n –1 .
Hence (C) is the
correct answer. .
Q9.
If a, b, c are the pth, qth
rth terms respectively of an H.P. then .
ab(p - q) + bc(q - r) + ca(r - p)
equals to
(A) 1
(B) –1
(C) 0
(D) None of these
Solution:
Let x be the first
term and y be the c. d. of corresponding A.P. , then .
Multiplying (1) ,
(2) and (3) receptively by abc(q – r) , abc(r – p) , abc(p – q) and then
adding we get
bc( q – r) + ca( r –
p) + ab(p – q) = 0
Hence (C) is the
correct answer. .
Q10.
For 0 < q
< p/2 , if
x = 
then
(A) xyz = xz + y
(B) xyz = xy +z
(C) xyz = x+y+z
(D) xyz = yz +x .
Solution:
x = 
=
cosec2q ,
y = sec2
q , z = 
&⇒
z
=
&⇒ xyz = xy + z . .
Also x + y = cosec2
q + sec2 q
= 
= cosec2q sec2q = xy
&⇒ x + y + z = xy + z = xyz.
Hence (B), (C) are
the correct answers. .
