JEE Mathematics Progressions Related Inequalities and Series MCQs Set 01

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

Question. If \( a_1, a_2, a_3, \dots \) are in AP then \( a_p, a_q, a_r \) are in AP if \( p, q, r \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (a) AP

Question. Let \( t_r \) denote the rth term of an AP. If \( t_m = \frac{1}{n} \) and \( t_n = \frac{1}{m} \) then \( t_{mn} \) equals
(a) \( \frac{1}{mn} \)
(b) \( \frac{1}{m} + \frac{1}{n} \)
(c) 1
(d) 0
Answer: (c) 1

Question. If \( p, q, r, s \in N \) and they are four consecutive terms of an AP then the pth, qth, rth, sth terms of a GP are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. If in a progression \( a_1, a_2, a_3, \dots, \) etc., \( (a_r - a_{r+1}) \) bears a constant ratio with \( a_r \cdot a_{r+1} \) then the terms of the progression are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. If \( \frac{a_2 a_3}{a_1 a_4} = \frac{a_2 + a_3}{a_1 + a_4} = 3 \left( \frac{a_2 - a_3}{a_1 - a_4} \right) \) then \( a_1, a_2, a_3, a_4 \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. Let \( x, y, z \) be three positive prime numbers. The progression in which \( \sqrt{x}, \sqrt{y}, \sqrt{z} \) can be three terms (not necessarily consecutive) is
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (d) None of the options

Question. Let \( f(x) = 2x + 1 \). Then the number of real values of \( x \) for which the three unequal number \( f(x), f(2x), f(4x) \) are in GP is
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (c) 0

Question. If \( a_r > 0, r \in N \) and \( a_1, a_2, a_3, \dots, a_{2n} \) are in AP then \( \frac{a_1 + a_{2n}}{\sqrt{a_1} + \sqrt{a_2}} + \frac{a_2 + a_{2n-1}}{\sqrt{a_2} + \sqrt{a_3}} + \frac{a_3 + a_{2n-2}}{\sqrt{a_3} + \sqrt{a_4}} + \dots + \frac{a_n + a_{n+1}}{\sqrt{a_n} + \sqrt{a_{n+1}}} \) is equal to
(a) \( n - 1 \)
(b) \( \frac{n(a_1 + a_{2n})}{\sqrt{a_1} + \sqrt{a_{n+1}}} \)
(c) \( \frac{n - 1}{\sqrt{a_1} + \sqrt{a_{n+1}}} \)
(d) None of the options
Answer: (b) \( \frac{n(a_1 + a_{2n})}{\sqrt{a_1} + \sqrt{a_{n+1}}} \)

Question. If \( a_1, a_2, a_3, \dots, a_{2n+1} \) are in AP then \( \frac{a_{2n+1} - a_1}{a_{2n+1} + a_1} + \frac{a_{2n} - a_2}{a_{2n} + a_2} + \dots + \frac{a_{n+2} - a_n}{a_{n+2} + a_n} \) is equal to
(a) \( \frac{n(n+1)}{2} \cdot \frac{a_2 - a_1}{a_{n+1}} \)
(b) \( \frac{n(n+1)}{2} \)
(c) \( (n+1)(a_2 - a_1) \)
(d) None of the options
Answer: (a) \( \frac{n(n+1)}{2} \cdot \frac{a_2 - a_1}{a_{n+1}} \)

Question. Let \( a_1, a_2, a_3, \dots \) be in AP and \( a_p, a_q, a_r \) be in GP. Then \( a_q : a_p \) is equal to
(a) \( \frac{r - p}{q - p} \)
(b) \( \frac{q - p}{r - q} \)
(c) \( \frac{r - q}{q - p} \)
(d) None of the options
Answer: (c) \( \frac{r - q}{q - p} \)

Question. If \( a, b, c \) are in GP then \( a + b, 2b, b + c \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. If \( a, b, c, d \) are nonzero real numbers such that \( (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) \leq (ab + bc + cd)^2 \). Then \( a, b, c, d \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. If \( 4a^2 + 9b^2 + 16c^2 = 2(3ab + 6bc + 4ca) \), where \( a, b, c \) are nonzero numbers, then \( a, b, c \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. If \( a, b, c \) are in AP then \( \frac{a}{bc}, \frac{1}{c}, \frac{2}{b} \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (d) None of the options

Question. If in an AP, \( t_1 = \log_{10} a, t_{n+1} = \log_{10} b \) and \( t_{2n+1} = \log_{10} c \) then \( a, b, c \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. If \( n!, 3 \times n! \) and \( (n + 1)! \) are in GP then \( n!, 5 \times n! \) and \( (n + 1)! \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (a) AP

Question. In an AP, the pth term is \( q \) and the \( (p + q) \)th term is 0. Then the qth term is
(a) \( -p \)
(b) \( p \)
(c) \( p + q \)
(d) \( p - q \)
Answer: (b) p

Question. In a sequence of \( (4n + 1) \) terms the first \( (2n + 1) \) terms are in AP whose common difference is 2, and the last \( (2n + 1) \) terms are in GP whose common ratio is 0.5. If the middle terms of the AP and GP are equal then the middle term of the sequence is
(a) \( \frac{n \cdot 2^{n+1}}{2^n - 1} \)
(b) \( \frac{n \cdot 2^{n+1}}{2^{2n} - 1} \)
(c) \( n \cdot 2^n \)
(d) None of the options
Answer: (a) \( \frac{n \cdot 2^{n+1}}{2^n - 1} \)

Question. If \( x^2 + 9y^2 + 25z^2 = xyz \left( \frac{15}{x} + \frac{5}{y} + \frac{3}{z} \right) \) then \( x, y, z \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. If \( a, b, c, d \) and \( p \) are distinct real numbers such that \( (a^2+ b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) \leq 0 \) then \( a, b, c, d \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. The largest term common to the sequences 1, 11, 21, 31, …. to 100 terms and 31, 36, 41, 46, …. to 100 terms is
(a) 381
(b) 471
(c) 281
(d) None of the options
Answer: (d) None of the options

Question. The interior angles of a convex polygon are in AP, the common difference being \( 5^\circ \). If the smallest angle is \( 2\pi/3 \) then the number of sides is
(a) 9
(b) 16
(c) 7
(d) None of the options
Answer: (a) 9

Question. The minimum number of terms of 1 + 3 + 5 + 7 + ……. that add up to a number exceeding 1357 is
(a) 15
(b) 37
(c) 35
(d) 17
Answer: (b) 37

Question. In the value of 100! the number of zeros at the end is
(a) 11
(b) 22
(c) 23
(d) 24
Answer: (d) 24

Question. The sum of all the proper divisors of 9900 is
(a) 33851
(b) 23952
(c) 23951
(d) None of the options
Answer: (c) 23951

Question. The sum of all odd proper divisors of 360 is
(a) 77
(b) 78
(c) 81
(d) None of the options
Answer: (a) 77

Question. In the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ……., where n consecutive terms have the value n, the 150th term is
(a) 17
(b) 16
(c) 18
(d) None of the options
Answer: (a) 17

Question. In the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ….., where n consecutive terms have the value n, the 1025th term is
(a) \( 2^9 \)
(b) \( 2^{10} \)
(c) \( 2^{11} \)
(d) \( 2^8 \)
Answer: (b) \( 2^{10} \)

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

Question. If AM of the number \( 5^{1+x} \) and \( 5^{1-x} \) is 13 then the set of possible real values of x is
(a) \( \{5, \frac{1}{5}\} \)
(b) {1, 1}
(c) \( \{x | x^2 - 1 = 0, x \in R\} \)
(d) None of the options
Answer: (c) \( \{x | x^2 - 1 = 0, x \in R\} \)

Question. If the AM of two positive numbers be three times their geometric mean then the ratio of the numbers is
(a) \( 3 \pm 2\sqrt{2} \)
(b) \( \sqrt{2} \pm 1 \)
(c) \( 17 + 12\sqrt{2} \)
(d) \( (3 - 2\sqrt{2})^{-2} \)
Answer: (c) \( 17 + 12\sqrt{2} \)

Question. If \( a, b, c \) are in HP then \( \frac{1}{b - a} + \frac{1}{b - c} \) is equal to
(a) \( \frac{2}{b} \)
(b) \( \frac{2}{a + c} \)
(c) \( \frac{1}{a} + \frac{1}{c} \)
(d) None of the options
Answer: (c) \( \frac{1}{a} + \frac{1}{c} \)

Question. \( S_r \) denotes the sum of the first r terms of an AP. Then \( S_{3n} : (S_{2n} - S_n) \) is
(a) n
(b) 3n
(c) 3
(d) independent of n
Answer: (c) 3

Question. If \( a^x = b^y = c^z \) and \( x, y, z \) are in GP then \( \log_c b \) is equal to
(a) \( \log_b a \)
(b) \( \log_a b \)
(c) \( \frac{z}{y} \)
(d) None of the options
Answer: (a) \( \log_b a \)