JEE Mathematics Progressions Related Inequalities and Series MCQs Set 03

Choose the most appropriate option (a, b, c or d).

Question. If \( a, b, c \) are in AP then \( a + \frac{1}{bc}, b + \frac{1}{ca}, c + \frac{1}{ab} \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (a) AP

Question. The AM of two given positive numbers is 2. If the larger number is increased by 1, the GM of the numbers becomes equal to the AM of the given numbers. Then the HM of the given numbers is
(a) \( \frac{3}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (a) \( \frac{3}{2} \)

Question. Let \( a, b \) are two positive numbers, where \( a > b \) and \( 4 \times \text{GM} = 5 \times \text{HM} \) for the numbers. Then a is
(a) 4b
(b) \( \frac{1}{4} b \)
(c) 2b
(d) b
Answer: (a) 4b

Question. If \( a, a_1, a_2, a_3, \dots a_{2n}, b \) are in AP and \( a, g_1, g_2, g_3, \dots, g_{2n}, b \) are in GP and h is the HM of a and b then \( \frac{a_1 + a_{2n}}{g_1 g_{2n}} + \frac{a_2 + a_{2n-1}}{g_2 g_{2n-1}} + \dots + \frac{a_n + a_{n+1}}{g_n g_{n+1}} \) is equal to
(a) \( \frac{2n}{h} \)
(b) \( 2nh \)
(c) \( nh \)
(d) \( \frac{n}{h} \)
Answer: (a) \( \frac{2n}{h} \)

Question. Let \( a_1= 0 \) and \( a_1, a_2, a_3, \dots, a_n \) be real numbers such that \( |a_i| = |a_{i-1} + 1| \) for all i then the AM of the numbers \( a_1, a_2, a_3, \dots, a_n \) has the value A where
(a) \( A < -\frac{1}{2} \)
(b) \( A < -1 \)
(c) \( A \geq -\frac{1}{2} \)
(d) \( A = -\frac{1}{2} \)
Answer: (c) \( A \geq -\frac{1}{2} \)

Question. Let there be a GP whose first term is a and the common ratio is r. If A and H are the arithmetic mean and the harmonic mean respectively for the first n terms of the GP, \( A \cdot H \) is equal to
(a) \( a^2 r^{n-1} \)
(b) \( a r^n \)
(c) \( a^2 r^n \)
(d) None of the options
Answer: (a) \( a^2 r^{n-1} \)

Question. If the first and the \( (2n - 1) \)th terms of an AP, a GP and an HP are equal and their nth terms are a, b and c respectively then
(a) \( a = b = c \)
(b) \( a \geq b \geq c \)
(c) \( a + c = b \)
(d) \( ac - b^2 = 0 \)
Answer: (d) \( ac - b^2 = 0 \)

Question. \( \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \) is the HM between a and b if n is
(a) 0
(b) \( \frac{1}{2} \)
(c) \( -\frac{1}{2} \)
(d) 1
Answer: (a) 0

Question. If the harmonic mean between P and Q be H then \( H \left( \frac{1}{P} + \frac{1}{Q} \right) \) is equal to
(a) 2
(b) \( \frac{PQ}{P + Q} \)
(c) \( \frac{P + Q}{PQ} \)
(d) \( \frac{1}{2} \)
Answer: (a) 2

Question. Let x be the AM and y, z be two GMs between two positive numbers. Then \( \frac{y^3 + z^3}{xyz} \) is equal to
(a) 1
(b) 2
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (b) 2

Question. If HM : GM = 4 : 5 for two positive numbers then the ratio of the numbers is
(a) 4 : 1
(b) 3 : 2
(c) 3 : 4
(d) 2 : 3
Answer: (a) 4 : 1

Question. In a GP of alternately positive and negative terms, any term is the AM of the next two terms. Then the common ratio is
(a) -1
(b) -3
(c) -2
(d) \( -\frac{1}{2} \)
Answer: (c) -2

Question. If \( a, b, c \) are in AP, and \( p, p' \) are the AM and GM respectively between \( a \) and \( b \), while \( q, q' \) are the AM and GM respectively between \( b \) and \( c \), then
(a) \( p^2 + q^2 = p'^2 + q'^2 \)
(b) \( pq = p'q' \)
(c) \( p^2 - q^2 = p'^2 - q'^2 \)
(d) None of the options
Answer: (c) \( p^2 - q^2 = p'^2 - q'^2 \)

Question. If \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \) then the minimum value of \( \cos^3 \theta + \sec^3 \theta \) is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (b) 2

Question. If \( a > 1, b > 1 \) then the minimum value of \( \log_b a + \log_a b \) is
(a) 0
(b) 1
(c) 2
(d) None of the options
Answer: (c) 2

Question. The minimum value of \( 4^x + 4^{1-x}, x \in R, \) is
(a) 2
(b) 4
(c) 1
(d) None of the options
Answer: (b) 4

Question. If \( x = \log_5 3 + \log_7 5 + \log_9 7 \) then
(a) \( x \geq \frac{3}{2} \)
(b) \( x \geq \frac{1}{\sqrt[3]{2}} \)
(c) \( x \geq \frac{3}{\sqrt[3]{2}} \)
(d) None of the options
Answer: (c) \( x \geq \frac{3}{\sqrt[3]{2}} \)

Question. If \( a_n > 1 \) for all \( n \in N \) then \( \log_{a_2} a_1 + \log_{a_3} a_2 + \dots + \log_{a_n} a_{n-1} + \log_{a_1} a_n \) has the minimum value
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (d) None of the options

Question. The product of n positive numbers is 1. Their sum is
(a) a positive integer
(b) divisible by n
(c) equal to \( n + \frac{1}{n} \)
(d) greater than or equal to n
Answer: (d) greater than or equal to n

Question. If \( x, y, z \) are three real numbers of the same sign then the value of \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \) lies in the interval
(a) [2, +∞)
(b) [3, +∞)
(c) (3, +∞)
(d) (-∞, 3)
Answer: (b) [3, +∞)

Question. The least value of \( 2\log_{100} a - \log_a 0.0001, a > 1 \) is
(a) 2
(b) 3
(c) 4
(d) None of the options
Answer: (c) 4

Question. If \( 0 < x < \pi/2 \) then the minimum value of \( (\sin x + \cos x + \csc 2x)^3 \) is
(a) 27
(b) 13.5
(c) 6.75
(d) None of the options
Answer: (b) 13.5

Question. If \( x, y, z \) are positive then the minimum value of \( x^{\log y - \log z} + y^{\log z - \log x} + z^{\log x - \log y} \) is
(a) 3
(b) 1
(c) 9
(d) 16
Answer: (a) 3

Question. \( a, b, c \) are three positive numbers and \( abc^2 \) has the greatest value \( \frac{1}{64} \). Then
(a) \( a = b = \frac{1}{2}, c = \frac{1}{4} \)
(b) \( a = b = \frac{1}{4}, c = \frac{1}{2} \)
(c) \( a = b = c = \frac{1}{3} \)
(d) None of the options
Answer: (b) \( a = b = \frac{1}{4}, c = \frac{1}{2} \)

Question. If \( a > 0, b > 0, c > 0 \) and the minimum value of \( a(b^2 + c^2) + b(c^2 + a^2) + c(a^2 + b^2) \) is \( \lambda abc \) then \( \lambda \) is
(a) 2
(b) 1
(c) 6
(d) 3
Answer: (c) 6

Question. The value of \( \sum_{n=1}^{10} \int_0^n x \, dx \) is
(a) an even integer
(b) an odd integer
(c) a rational number
(d) an irrational number
Answer: (c) a rational number

Question. The sum of 0.2 + 0.004 + 0.00006 + 0.0000008 + ….. to \( \infty \) is
(a) \( \frac{200}{891} \)
(b) \( \frac{2000}{9801} \)
(c) \( \frac{1000}{9801} \)
(d) None of the options
Answer: (b) \( \frac{2000}{9801} \)

Question. If \( (2n + r)r, n \in N, r \in N \) is expressed as the sum of k consecutive odd natural numbers then k is equal to
(a) r
(b) n
(c) r + 1
(d) n + 1
Answer: (a) r

Choose the correct options. One or more options may be correct.

Question. If a, b, c, d are four positive numbers then
(a) \( \left(\frac{a}{b} + \frac{b}{c}\right) \left(\frac{c}{d} + \frac{d}{e}\right) \geq 4 \sqrt{\frac{a}{e}} \)
(b) \( \left(\frac{a}{b} + \frac{c}{d}\right) \left(\frac{b}{c} + \frac{d}{e}\right) \geq 4 \sqrt{\frac{a}{e}} \)
(c) \( \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{e} + \frac{e}{a} \geq 5 \)
(d) \( \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{e}{d} + \frac{a}{e} \geq \frac{1}{5} \)
Answer: (a) \( \left(\frac{a}{b} + \frac{b}{c}\right) \left(\frac{c}{d} + \frac{d}{e}\right) \geq 4 \sqrt{\frac{a}{e}} \)

Question. Let \( f(x) = \frac{1 - x^{n+1}}{1 - x} \) and \( g(x) = 1 - \frac{2}{x} + \frac{3}{x^2} - \dots + (-1)^n \frac{n + 1}{x^n} \). Then the constant term in \( f'(x) \times g(x) \) is equal to
(a) \( \frac{n(n^2 - 1)}{6} \) when n is even
(b) \( \frac{n(n + 1)}{2} \) when n is odd
(c) \( -\frac{n}{2} (n + 1) \) when n is even
(d) \( -\frac{n(n - 1)}{2} \) when n is odd
Answer: (b) \( \frac{n(n + 1)}{2} \) when n is odd

Question. Let \( a_n \) = product of the first n natural numbers. Then for all n \( \in \) N
(a) \( a^n \geq a_n \)
(b) \( \left(\frac{n + 1}{2}\right)^n \geq n! \)
(c) \( n^n \geq a_{n+1} \)
(d) None of the options
Answer: (a) \( a^n \geq a_n \)

Question. Let the sets A = {2, 4, 6, 8,….} and B = {3, 6, 9, 12,….} and n(A) = 200, n(B) = 250. Then
(a) n(A \( \cap \) B) = 67
(b) n(A \( \cup \) B) = 450
(c) n(A \( \cap \) B) = 66
(d) n(A \( \cup \) B) = 384
Answer: (c) n(A \( \cap \) B) = 66

Question. Let \( a, x, b \) be in AP; \( a, y, b \) be in GP and \( a, z, b \) be in HP. If x = y + 2 and a = 5z then
(a) \( y^2 = xz \)
(b) x > y > z
(c) a = 9, b = 1
(d) \( a = \frac{1}{4}, b = \frac{9}{4} \)
Answer: (a) \( y^2 = xz \)