IIT JEE

The sum of the first n terms of the
series 1^{2}+2.2^{2}+3^{2}+2.4^{2}+5^{2}+2.6^{2}+
. . . is _{},
when n is even. When n is odd, the sum is

(A) _{}

(B) _{}

(C) _{}

(D) _{}

If n is odd, n-1
is even. Sum of (n-1 ) terms will be _{}.

The nth term will be
n^{2} . Hence the required sum .

= _{}+n^{2} = _{}

Hence (A) is the correct answer. .

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.

Let the first
term of a G.P. be A and common rario be R . Then pth, qth and rth
terms are AR^{p-1}, AR^{q-1} and AR^{r-1}.
Obviously AR^{p-1}´
AR^{r-1 }= AR^{p+r—2} = (AR^{q-1})^{2}, as p
+ r = 2q. .

Hence terms are in G.P. .

Hence (B) is the correct answer.

If a, b, c are in H.P. , then
the value of _{}is

(A) 0

(B) 1

(C) 2

(D) 3

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.

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.

Let the numbers
be a_{1}; a_{2}; a_{3} ;. . . ; a_{n }. Then a_{1};
a_{2}; a_{3} . . . ; a_{n} =1. Using A.M. ≥ G.M , we get .

_{}

&⇒ a_{1 }. a_{2 }. a_{3}
. . . . . a_{n }≥
n .

Hence (D) is the correct answer.

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 .

Using A. M. ≥ G. M. .

_{}

&⇒ _{}≥ 3.

Hence (A) is the correct answer.

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 .

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.

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) log_{a}e, log_{b}e,
log_{c}e, log_{d}e are in H.P. .

(D) a, b, c, d are in G.P.

Consider _{}= r
(say)

&⇒
_{}

&⇒
_{}

or _{}

&⇒
a, b, c, d from a G. P.

&⇒
ln a , lnb , lnc, lnd form an A.P. .

&⇒
log_{a}e , log_{b}e, log_{c}e and log_{d}e
from H. P.

Hence (C), (D) are the correct answers.

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

_{}and _{}are in
A.P. , d = _{}

T_{n}
= _{}

&⇒ n £ 5

&⇒ 5^{th} term
has least positive value.

Therefore largest term of H.P. = 6 .

Hence (B) is the correct answer.

Coefficient of x^{99} in
the polynomial (x-1) (x-2) . . . ( x- 100) is .

(A) 100!

(B) – 5050

(C) 5050

(D) –100!

For the
coefficient of x^{99}, 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.

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
ax^{2}+2bx+c=0

It is easy to
see that D = (b^{2}
–ac)(aa^{2} +2ba +c) . .

Hence D =0 if a, b, c are in G.P. .

or a is a root of ax^{2} +2bx
+c = 0.

Hence (B) and (D) are the correct answers.