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Solved Objective Question on Progression and Series Set 1

Posted on - 16-05-2017

JEE Math PS

IIT JEE

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.

 
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