QuizSolver
  • Bank PO
  • CBSE
  • IIT JEE
 
 

Solved Objective Question on Progression and Series Set 2

Posted on - 19-05-2017

JEE Math PS

IIT JEE

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. .

 
Quadratic Equations - Solved Objective Questions Part 2 for Conceptual Clarity
Quadratic Equations - Solved Objective Questions Part 1 for Conceptual Clarity
Solved Objective Question on Probability Set 2
Solved Objective Question on Probability Set 1
Solved Objective Question on Permutations and Combinations Set 3
Solved Objective Question on Permutations and Combinations Set 2
Solved Objective Question on Progression and Series Set 1
Solved Objective Question on Permutations and Combinations Set 1
Quadratic Equations - Solved Subjective Questions Part 4
Quadratic Equations - Solved Subjective Questions Part 2
Quadratic Equations - Solved Subjective Questions Part 3
Quadratic Equations - Solved Subjective Questions Part 1
Solved Subjective Questions on Circle Set 9
Solving Equations Reducible to Quadratic Equations
Theory of Polynomial Equations and Remainder Theorem
Solved Subjective Questions on Circle Set 8
Solving Quadratic Inequalities Using Wavy Curve Methods
Division and Distribution of Objects - Permutation and Combination
Basics of Quadratic Inequality or Inequations
Basic Concepts Of Combinations for IIT and Other Engineering Exams

Comments