JEE Mathematics Progressions Related Inequalities and Series MCQs Set 02

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

Question. Let \( \{t_n\} \) be a sequence of integers in GP in which \( t_4 : t_6 = 1 : 4 \) and \( t_2 + t_5 = 216 \). Then \( t_1 \) is
(a) 12
(b) 14
(c) 16
(d) None of the options
Answer: (a) 12

Question. If \( \log \left( \frac{5c}{a} \right), \log \left( \frac{3b}{5c} \right) \) and \( \log \left( \frac{a}{3b} \right) \) are in AP, where a, b, c are in GP, then a, b, c are the lengths of sides of
(a) an isosceles triangle
(b) an equilateral triangle
(c) a scalene triangle
(d) None of the options
Answer: (d) None of the options

Question. Let S be the sum, P be the product and R be the sum of the reciprocals of n terms of a GP. Then \( P^2 R^n : S^n \) is equal to
(a) 1 : 1
(b) \( (\text{common ratio})^n : 1 \)
(c) \( (\text{first term})^2 : (\text{common ratio})^n \)
(d) None of the options
Answer: (a) 1 : 1

Question. If the pth, qth and rth terms of an AP are in GP then the common ratio of the GP is
(a) \( \frac{p+q}{r+q} \)
(b) \( \frac{r-q}{q-p} \)
(c) \( \frac{p-r}{p-q} \)
(d) None of the options
Answer: (b) \( \frac{r-q}{q-p} \)

Question. The number of terms common between the series 1 + 2 + 4 + 8 + ….. to 100 terms and 1 + 4 + 7 + 10 + …….. to 100 terms is
(a) 6
(b) 4
(c) 5
(d) None of the options
Answer: (c) 5

Question. The 10th common term between the series 3 + 7 + 11 + …. and 1 + 6 + 11 + …… is
(a) 191
(b) 193
(c) 211
(d) None of the options
Answer: (a) 191

Question. Three consecutive terms of a progression are 30, 24, 20. The next term of the progression is
(a) 18
(b) \( 17\frac{1}{7} \)
(c) 16
(d) None of the options
Answer: (b) \( 17\frac{1}{7} \)

Question. If three numbers are in GP then the numbers obtained by adding the middle number to each of the three numbers are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (c) HP

Question. If \( a_1, a_2, a_3 \) are in AP, \( a_2, a_3, a_4 \) are in GP and \( a_3, a_4, a_5 \) are in HP then \( a_1, a_3, a_5 \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. If \( a, b, c, d \) are four numbers such that the first three are in AP while the last three are in HP then
(a) \( bc = ad \)
(b) \( ac = bd \)
(c) \( ab = cd \)
(d) None of the options
Answer: (a) \( bc = ad \)

Question. If the first two terms of an HP be 2/5 and 12/23 then the largest positive term of the progression is the
(a) 6th term
(b) 7th term
(c) 5th term
(d) 8th term
Answer: (c) 5th term

Question. If \( x, 2y, 3z \) are in AP, where the distinct numbers \( x, y, z \) are in GP, then the common ratio of the GP is
(a) 3
(b) \( \frac{1}{3} \)
(c) 2
(d) \( \frac{1}{2} \)
Answer: (b) \( \frac{1}{3} \)

Question. If \( x > 1, y > 1, z > 1 \) are three numbers in GP then \( \frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z} \) are in
(a) AP
(b) HP
(c) GP
(d) None of the options
Answer: (b) HP

Question. If \( a, a_1, a_2, a_3, \dots a_{2n-1}, b \) are in AP, \( a, b_1, b_2, b_3, \dots, b_{2n-1}, b \) are in GP and \( a, c_1, c_2, c_3, \dots, c_{2n-1}, b \) are in HP, where a, b are positive, then the equation \( a_n x^2 - b_n x + c_n = 0 \) has its roots
(a) real and unequal
(b) real and equal
(c) imaginary
(d) None of the options
Answer: (c) imaginary

Question. If \( a, x, b \) are in AP, \( a, y, b \) are in GP and \( a, z, b \) are in HP such that \( x = 9z \) and \( a > 0, b > 0 \) then
(a) \( |y| = 3z \)
(b) \( x = 3|y| \)
(c) \( 2y = x + z \)
(d) None of the options
Answer: (b) \( x = 3|y| \)

Question. If three numbers are in HP then the number obtained by subtracting half of the middle number from each of them are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. \( a, b, c, d, e \) are five numbers in which the first three are in AP and the last three are in HP. If the three numbers in the middle are in GP then the numbers in the odd places are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. Let \( a_1, a_2, a_3, \dots, a_{10} \) be in AP and \( h_1, h_2, h_3, \dots, h_{10} \) be in HP. If \( a_1 = h_1 = 2 \) and \( a_{10} = h_{10} = 3 \) then \( a_4 h_7 \) is
(a) 2
(b) 3
(c) 5
(d) 6
Answer: (d) 6

Question. If in an AP, \( S_n = p \cdot n^2 \) and \( S_m = p \cdot m^2 \), where \( S_r \) denotes the sum of r terms of the AP, then \( S_p \) is equal to
(a) \( \frac{1}{2} p^3 \)
(b) \( mnp \)
(c) \( p^3 \)
(d) \( (m + n)p^2 \)
Answer: (c) \( p^3 \)

Question. If \( S_r \) denotes the sum of the \( \frac{S_{3r} - S_{r-1}}{S_{2r} - S_{2r-1}} \) is equal to
(a) \( 2r - 1 \)
(b) \( 2r + 1 \)
(c) \( 4r + 1 \)
(d) \( 2r + 3 \)
Answer: (b) \( 2r + 1 \)

Question. \( S_r \) denotes the sum of the first r terms of a GP. Then \( S_n, S_{2n} - S_n, S_{3n} - S_{2n} \) are in
(a) AP
(b) GP
(c) HP
(d) None of the options
Answer: (b) GP

Question. If \( (1 - p)(1 + 3x + 9x^2 + 27x^3 + 81x^4 + 243x^5) = 1 - p^6, p \neq 1 \) then the value of \( \frac{p}{x} \) is
(a) \( \frac{1}{3} \)
(b) 3
(c) \( \frac{1}{2} \)
(d) 2
Answer: (b) 3

Question. If the sum of series \( 1 + \frac{2}{x} + \frac{4}{x^2} + \frac{8}{x^3} + \dots \text{to } \infty \) is a finite number then
(a) \( x < 2 \)
(b) \( x > \frac{1}{2} \)
(c) \( x > -2 \)
(d) \( x < -2 \) or \( x > 2 \)
Answer: (d) \( x < -2 \) or \( x > 2 \)

Question. Let \( S_n \) denote the sum of the first n terms of an AP. If \( S_{2n} = 3S_n \) then \( S_{3n} : S_n \) is equal to
(a) 4
(b) 6
(c) 8
(d) 10
Answer: (b) 6

Question. In a GP of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the GP is
(a) \( -\frac{4}{5} \)
(b) \( \frac{1}{5} \)
(c) 4
(d) None of the options
Answer: (c) 4

Question. In an AP, \( S_p = q, S_q = p \) and \( S_r \) denote the sum of the first r terms. Then \( S_{p+q} \) is equal to
(a) 0
(b) \( -(p + q) \)
(c) \( p + q \)
(d) \( pq \)
Answer: (b) \( -(p + q) \)

Question. The coefficient of \( x^{15} \) in the product \( (1 - x)(1 - 2x)(1 - 2^2 \cdot x)(1 - 2^3 \cdot x)\dots(1 - 2^{15} \cdot x) \) is equal to
(a) \( 2^{105} - 2^{121} \)
(b) \( 2^{121} - 2^{105} \)
(c) \( 2^{120} - 2^{104} \)
(d) None of the options
Answer: (a) \( 2^{105} - 2^{121} \)

Question. The coefficient of \( x^{49} \) in the product \( (x - 1)(x - 3)\dots(x - 99) \) is
(a) \( -99^2 \)
(b) 1
(c) -2 500
(d) None of the options
Answer: (c) -2 500

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

Question. The value of \( \sum_{r=1}^{n} \frac{1}{\sqrt{a + rx} + \sqrt{a + (r - 1)x}} \) is
(a) \( \frac{n}{\sqrt{a} + \sqrt{a + nx}} \)
(b) \( \frac{\sqrt{a + nx} - \sqrt{a}}{x} \)
(c) \( \frac{n(\sqrt{a + nx} - a)}{x} \)
(d) None of the options
Answer: (a) \( \frac{n}{\sqrt{a} + \sqrt{a + nx}} \)

Question. Let \( \sum_{n=1}^{n} r^4 = f(n) \). Then \( \sum_{r=1}^{n} (2r - 1)^4 \) is equal to
(a) f(2n) – 16f(n) for all n \( \in \) N
(b) \( f(n) - 16f\left(\frac{n - 1}{2}\right) \) when n is odd
(c) \( f(n) - 16f\left(\frac{n}{2}\right) \) when n is even
(d) None of the options
Answer: (a) f(2n) – 16f(n) for all n \( \in \) N

Question. If \( 2 \cdot {}^n P_1, {}^n P_2, {}^n P_3 \) are three consecutive terms of an AP then they are
(a) in GP
(b) in HP
(c) equal
(d) None of the options
Answer: (a) in GP

Question. In a GP the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of the GP up to infinite terms is
(a) 8
(b) -8
(c) \( \frac{8}{3} \)
(d) \( -\frac{8}{3} \)
Answer: (c) \( \frac{8}{3} \)

Question. If \( \sum_{k=1}^{n} \left( \sum_{m=1}^{k} m^2 \right) = an^4 + bn^3 + cn^2 + dn + e \) then
(a) \( a = \frac{1}{12} \)
(b) \( b = \frac{1}{6} \)
(c) \( d = \frac{1}{6} \)
(d) e = 0
Answer: (a) \( a = \frac{1}{12} \)