Chapter 9

The common difference of the A.P. \(b_{1}, b_{2}, \ldots, b_{m}\) is 2 more than the common difference of A.P. \(a_{1}, a_{2}, \ldots, a_{n}\). If \(\mathrm{a}_{40}=-159, \mathrm{a}_{100}=-399\) and \(\mathrm{b}_{100}=\mathrm{a}_{70}\), then \(\mathrm{b}_{1}\) is equal to: [Sep. 06, 2020 (II)] (a) 81 (b) \(-127\) (c) \(-81\) (d) 127

(a) 81 (b) \(-127\) (c) \(-81\) (d) 127 If \(3^{2 \sin 2 \alpha-1}, 14\) and \(3^{4-2 \sin 2 \alpha}\) are the first three terms of an A.P. for some \(\alpha\), then the sixth term of this A.P is: [Sep. \(05,2020(\mathrm{I})]\) (a) 66 (b) 81 (c) 65 (d) 78

If the sum of the first 20 terms of the series \(\log _{\left(7^{1 / 2}\right)} x+\log _{\left(7^{1 / 3}\right)} x+\log _{\left(7^{1 / 4}\right)} x+\ldots\) is 460 , then \(x\) is equal to : [Sep. 05, 2020 (II)] (a) \(7^{2}\) (b) \(7^{1 / 2}\) (c) \(e^{2}\) (d) \(7^{46 / 21}\)

Let \(a_{1}, a_{2}, \ldots ., a_{n}\) be a given A.P. whose common difference is an integer and \(S_{n}=a_{1}+a_{2}+\ldots+a_{n} .\) If \(a_{1}=1, a_{n}=300\) and \(15 \leq n \leq 50\), then the ordered pair \(\left(S_{n-4}, a_{n-4}\right)\) is equal to : (a) \((2490,249)\) (b) \((2480,249)\) (c) \((2480,248)\) (d) \((2490,248)\)

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is: \(\quad\) [Sep. 03, 2020 (I)] (a) \(\frac{1}{6}\) (b) \(\frac{1}{5}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{7}\)

If the sum of first 11 terms of an A.P., \(a_{1}, a_{2}, a_{3}, \ldots\) is 0 \(\left(a_{1} \neq 0\right)\), then the sum of the A.P., \(a_{1}, a_{3}, a_{5}, \ldots, a_{23}\) is \(k a_{1}\), where \(k\) is equal to : [Sep. 02, 2020 (II)] (a) \(-\frac{121}{10}\) (b) \(\frac{121}{10}\) (c) \(\frac{72}{5}\) (d) \(-\frac{72}{5}\)

The number of terms common to the two A.P's \(3,7,11, \ldots\), 407 and \(2,9,16, \ldots, 709\) is \(\quad\) [NAJan. 9, 2020 (II)]

If the \(10^{\text {th }}\) term of an A.P. is \(\frac{1}{20}\) and its \(20^{\text {th }}\) term is \(\frac{1}{10}\), then the sum of its first 200 terms is: \(\quad\) [Jan. 8, \(\mathbf{2 0 2 0}\) (II)] (a) 50 (b) \(50 \frac{1}{4}\) (c) 100 (d) \(100 \frac{1}{2}\)

Let \(f: R \rightarrow R\) be such that for all \(x \in R,\left(2^{1+x}+2^{1-x}\right), f(x)\) and \(\left(3^{x}+3^{-x}\right)\) are in A.P., then the minimum value of \(f(x)\) is: [Jan. 8, 2020 (I)] (a) 2 (b) 3 (c) \(\underline{0}\) (d) 4

Five numbers are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers is \(-\frac{1}{2}\), then the greatest number amongst them is: [Jan. 7, 2020 (I)] (a) 27 (b) 7 (c) \(\frac{21}{2}\) (d) 16

Let \(\mathrm{S}_{\mathrm{n}}\) denote the sum of the first \(n\) terms of an A.P. If \(\mathrm{S}_{4}=16\) and \(S_{6}=-48\), then \(S_{10}\) is equal to : [April 12, 2019 (I)\\} (a) \(-260\) (b) \(-410\) (c) \(-320\) (d) \(-380\)

If \(a_{1}, a_{2}, a_{3}, \ldots \ldots\) are in A.P. such that \(a_{1}+a_{7}+a_{16}=40\), then the sum of the first 15 terms of this A.P. is : [April 12, 2019 (II)] (a) 200 (b) 280 (c) 120 (d) 150

If \(a_{1}, a_{2}, a_{3}, \ldots . . a_{n}\) are in A.P. and \(a_{1}+a_{4}+a_{7}+\ldots+a_{16}=114\) then \(a_{1}+a_{6}+a_{11}+a_{16}\) is equal to : \(\quad\) April 10,2019 (I)] (a) 98 (b) 76 (c) 38 (d) 64

Let the sum of the first \(\mathrm{n}\) terms of a non-constant A.P., \(a_{1}, a_{2}, a_{3}, \ldots \ldots \ldots \ldots . .\) be \(50 n+\frac{n(n-7)}{2} A\), where \(A\) is a constant. If \(\mathrm{d}\) is the common difference of this A.P., then the ordered pair \(\left(\mathrm{d}, \mathrm{a}_{50}\right)\) is equal to: \(\quad\) [April \(\mathbf{0 9}, \mathbf{2 0 1 9}\) (I)] (a) \((50,50+46 \mathrm{~A})\) (b) \((50,50+45 \mathrm{~A})\) (c) \((\mathrm{A}, 50+45 \mathrm{~A})\) (d) \((\mathrm{A}, 50+46 \mathrm{~A})\)

Let \(\sum_{\mathrm{k}=1}^{10} \mathrm{f}(\mathrm{a}+\mathrm{k})=16\left(2^{10}-1\right)\), where the function \(\mathrm{f}\) satisfies \(f(x+y)=f(x) f(y)\) for all natural numbers \(x, y\) and \(f(a)=2\). Then the natural number ' \(\mathrm{a}\) ' is: [April 09, 2019 (I)] (a) 2 (b) 16 (c) 4 (d) 3

If the sum and product of the first three terms in an A.P. are 33 and 1155 , respectively, then a value of its \(11^{\text {th }}\) term is: \(\quad\) [April09,2019 (II)] (a) \(-35\) (b) 25 (c) \(-36\) (d) \(-25\)

The sum of all natural numbers ' \(n\) ' such that \(1001\) is: [April 08, 2019 (I)] (a) 3203 (b) 3303 (c) 3221 (d) 3121

If \({ }^{n} C_{4},{ }^{n} C_{5}\) and \({ }^{n} C_{6}\) are in A.P., then \(n\) can be : [Jan.12, 2019 (II)] (a) 9 (b) 14 (c) 11 (d) 12

If 19 th term of a non-zero A.P. is zero, then its (49th term): ( 29 th term ) is: [Jan. 11, 2019 (II)] (a) \(4: 1\) (b) \(1: 3\) (c) \(3: 1\) (d) \(2: 1\)

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is: [Jan. 10, 2019 (I)] (a) 1256 (b) 1465 (c) 1365 (d) 1356

Let \(a_{1}, a_{2}, \ldots ., a_{30}\) be an A.P., \(S=\sum_{i=1}^{30} a_{i}\) and \(\mathrm{T}=\sum_{i=1}^{15} \mathrm{a}_{(2 i-1)^{*}}\) If \(\mathrm{a}_{5}=27\) and \(\mathrm{S}-2 \mathrm{~T}=75\), then \(\mathrm{a}_{10}\) is equal to: [Jan. 09, 2019 (I)] (a) 52 (b) 57 (c) 47 (d) 42

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{49}\) be in A.P. such that \(\sum_{\mathrm{k}=0}^{12} \mathrm{a}_{4 \mathrm{k}+1}=416\) and \(\mathrm{a}_{9}+\mathrm{a}_{43}=66 .\) If \(\mathrm{a}_{1}^{2}+\mathrm{a}_{2}^{2}+\ldots+\mathrm{a}_{17}^{2}=140 \mathrm{~m}\), then \(\mathrm{m}\) is equal to: (a) 68 (b) 34 (c) 33 (d) 66

For any three positive real numbers \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\), \(9\left(25 a^{2}+b^{2}\right)+25\left(c^{2}-3 a c\right)=15 b(3 a+c)\). Then (a) \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) are in G.P. (b) \(\mathrm{b}, \mathrm{c}\) and a are in G.P. (c) \(\mathrm{b}, \mathrm{c}\) and \(\mathrm{a}\) are in A.P. (d) \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) are in A.P.

If three positive numbers \(a, b\) and \(c\) are in A.P. such that \(\mathrm{abc}=8\), then the minimum possible value of \(b\) is : [OnlineApril9, 2017] (a) 2 (b) \(4^{\frac{1}{3}}\) (c) \(4^{\frac{2}{3}}\) (d) 4

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\), be in A.P. If \(a_{3}+a_{7}+a_{11}+a_{15}=72\), then the sum of its first 17 terms is equal to: [Online April 10, 2016] (a) 306 (b) 204 (c) 153 (d) 612

Let \(\alpha\) and \(\beta\) be the roots of equation \(\mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}=0, \mathrm{p} \neq 0\). If \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are in A.P and \(\frac{1}{\alpha}+\frac{1}{\beta}=4\), then the value of \(|\alpha-\beta|\) is: (a) \(\frac{\sqrt{34}}{9}\) (b) \(\frac{2 \sqrt{13}}{9}\) (c) \(\frac{\sqrt{61}}{9}\) (d) \(\frac{2 \sqrt{17}}{9}\)

The sum of the first 20 terms common between the series 3 \(+7+11+15+\ldots \ldots . . .\) and \(1+6+11+16+\ldots \ldots\), is |Online April 11, 2014] (a) 4000 (b) 4020 (c) 4200 (d) 4220

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220 . If the second term in it is 12 , then its \(4^{\text {th }}\) term is: |Online April 9, 2014] (a) 8 (b) 16 (c) 20 (d) 24

If \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots .\) are in A.P. such that \(a_{4}-a_{7}+a_{10}=m\), then the sum of first 13 terms of this A.P., is: [Online April 23, 2013] (a) \(10 \mathrm{~m}\) (b) \(12 \mathrm{~m}\) (c) \(13 \mathrm{~m}\) (d) \(15 \mathrm{~m}\)

Given sum of the first \(n\) terms of an A.P. is \(2 n+3 n^{2}\). Another A.P. is formed with the same first term and double of the common difference, the sum of \(n\) terms of the new A.P. is : [Online April 22, 2013] (a) \(n+4 n^{2}\) (b) \(6 n^{2}-n\) (c) \(n^{2}+4 n\) (d) \(3 n+2 n^{2}\)

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an A.P, such that \(\frac{a_{1}+a_{2}+\ldots+a_{p}}{a_{1}+a_{2}+a_{3}+\ldots+a_{q}}=\frac{p^{3}}{q^{3}} ; p \neq q\). Then \(\frac{a_{6}}{a_{21}}\) is equal to: [Online April 9, 2013] (a) \(\frac{41}{11}\) (b) \(\frac{31}{121}\) (c) \(\frac{11}{41}\) (d) \(\frac{121}{1861}\)

If 100 times the \(100^{\text {th }}\) term of an AP with non zero common difference equals the 50 times its \(50^{\text {th }}\) term, then the \(150^{\text {th }}\) term of this AP is : (a) \(-150\) (b) 150 times its \(50^{\text {th }}\) term (c) 150 (d) Zero

If the A.M. between \(p^{\text {th }}\) and \(q^{\text {th }}\) terms of an A.P. is equal to the A.M. between \(r^{\text {th }}\) and \(s^{\text {th }}\) terms of the same A.P., then \(p+q\) is equal to Online May 26, 2012] (a) \(r+s-1\) (b) \(r+s-2\) (c) \(r+s+1\) (d) \(r+s\)

Suppose \(\theta\) and \(\phi(\neq 0)\) are such that \(\sec (\theta+\phi), \sec \theta\) and \(\sec (\theta-\phi)\) are in A.P. If \(\cos \theta=k \cos \left(\frac{\phi}{2}\right)\) for some \(k\), then \(k\) is equal to [Online May 19, 2012] (a) \(\pm \sqrt{2}\) (b) \(\pm 1\) (c) \(\pm \frac{1}{\sqrt{2}}\) (d) \(\pm 2\)

Let \(a_{n}\) be the \(n^{\text {th }}\) term of an A.P. If \(\sum_{r=1}^{100} a_{2 r}=\alpha\) and \(\sum_{r=1}^{100} a_{2 r-1}=\beta\), then the common difference of the A.P. is [2011] (a) \(\alpha-\beta\) (b) \(\frac{\alpha-\beta}{100}\) (c) \(\beta-\alpha\) (d) \(\frac{\alpha-\beta}{200}\)

A man saves \(\ 200\) in each of the first three months of his service. In each of the subsequent months his saving increases by \(\mathbf{7} 40\) more than the saving of immediately previous month. His total saving from the start of service will be \(¥ 11040\) after (a) 19 months (b) 20 months (c) 21 months (d) 18 months

A person is to count 4500 currency notes. Let \(a_{n}\) denote the number of notes he counts in the \(n^{\text {th }}\) minute. If \(a_{1}=a_{2}\) \(=\ldots=a_{10}=150\) and \(a_{10}, a_{11}, \ldots\) are in an AP with common difference \(-2\), then the time taken by him to count all notes is \([\mathbf{2 0 1 0}]\) (a) 34 minutes (b) 125 minutes (c) 135 minutes (d) 24 minutes

Let \(a_{1}, a_{2}, a_{3} \ldots \ldots \ldots . .\) be terms on A.P. If \(\frac{a_{1}+a_{2}+\ldots \ldots \ldots a_{p}}{a_{1}+a_{2}+\ldots \ldots \ldots .+a_{q}}=\frac{p^{2}}{q^{2}}, p \neq q\), then \(\frac{a_{6}}{a_{21}}\) equals \([2006]\) (a) \(\frac{41}{11}\) (b) \(\frac{7}{2}\) (c) \(\frac{2}{7}\) (d) \(\frac{11}{41}\)

Let \(T_{\mathrm{r}}\) be the rth term of an A.P. whose first term is a and common difference is \(d\). If for some positive integers \(m, n, m \neq n, T_{m}=\frac{1}{n}\) and \(T_{n}=\frac{1}{m}\), then \(a-d\) equals [2004] (a) \(\frac{1}{m}+\frac{1}{n}\) (b) 1 (c) \(\frac{1}{m n}\) (d) 0

Let \(T_{\mathrm{r}}\) be the rth term of an A.P. whose first term is a and common difference is \(d\). If for some positive integers \(m, n, m \neq n, T_{m}=\frac{1}{n}\) and \(T_{n}=\frac{1}{m}\), then \(a-d\) equals [2004] (a) \(\frac{1}{m}+\frac{1}{n}\) (b) 1 (c) \(\frac{1}{m n}\) (d) 0

If \(1, \log _{9}\left(3^{1-x}+2\right), \log _{3}\left(4.3^{x}-1\right)\) are in A.P. then \(x\) equals [2002] (a) \(\log _{3} 4\) (b) \(1-\log _{3} 4\) (c) \(1-\log _{4} 3\) (d) \(\log _{4} 3\)

If \(f(x+y)=f(x) f(y)\) and \(\sum_{x=1}^{\infty} f(x)=2, x, y \in \mathrm{N}\), where \(\mathrm{N}\) is the set of all natural numbers, then the value of \(\frac{f(4)}{f(2)}\) is : [Sep. 06, 2020 (I)] (a) \(\frac{2}{3}\) (b) \(\frac{1}{9}\) (c) \(\frac{1}{3}\) (d) \(\frac{4}{9}\)

Let \(a, b, c, d\) and \(p\) be any non zero distinct real numbers such that \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2(a b+b c+c d) p+\left(b^{2}+c^{2}+\right.\) \(\left.d^{2}\right)=0 .\) Then : (a) \(a, c, p\) are in A.P. (b) \(a, c, p\) are in GP. (c) \(a, b, c, d\) are in G.P. (d) \(a, b, c, d\) are in A.P.

Suppose that a function \(f: \mathrm{R} \rightarrow\) R satisfies \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathrm{R}\) and \(f(\mathrm{a})=3\). If \(\sum_{i=1}^{\mathrm{n}} f(i)=363\), then \(\mathrm{n}\) is equal to \(\quad\) [NA Sep. 06, 2020 (II)]

If \(2^{10}+2^{9} \cdot 3^{1}+2^{8} \cdot 3^{2}+\ldots+2 \times 3^{9}+3^{10}=\mathrm{S}-2^{11}\) then \(\mathrm{S}\) is equal to: [Sep. \(\mathbf{0 5 , 2 0 2 0}\) (I)] (a) \(3^{11}-2^{12}\) (b) \(3^{11}\) (c) \(\frac{3^{11}}{2}+2^{10}\) (d) \(2 \cdot 3^{11}\)

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243 , then the sum of the first 50 terms of this G.P. is: [Sep. \(05, \mathbf{2 0 2 0}\) (II)] (a) \(\frac{1}{26}\left(3^{49}-1\right)\) (b) \(\frac{1}{26}\left(3^{50}-1\right)\) (c) \(\frac{2}{13}\left(3^{50}-1\right)\) (d) \(\frac{1}{13}\left(3^{50}-1\right)\)

Let \(\alpha\) and \(\beta\) be the roots of \(x^{2}-3 x+p=0\) and \(\gamma\) and \(\delta\) be the roots of \(x^{2}-6 x+q=0\). If \(\alpha, \beta, \gamma, \delta\) form a geometric progression. Then ratio \((2 q+p):(2 q-p)\) is: \([\) Sep. \(04,2020(\mathrm{I})]\) (a) \(3: 1\) (b) \(9: 7\) (c) \(5: 3\) (d) \(33: 31\)

The value of \((0.16) \log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots\right.\) to \(\left.\infty\right)\) is equal to [NA Sep. \(\mathbf{0 3}, \mathbf{2 0 2 0}\) (I)]

The sum of the first three terms of a G.P. is \(S\) and their product is \(27 .\) Then all such \(S\) lie in : \(\quad\) [Sep. 02, \(\mathbf{2 0 2 0}\) (I)] (a) \((-\infty,-9] \cup[3, \infty)\) (b) \([-3, \infty)\) (c) \((-\infty,-3] \cup[9, \infty)\) (d) \((-\infty, 9]\)

If \(|x|<1,|y|<1\) and \(x \neq y\), then the sum to infinity of the following series [Sep. 02, 2020 (I)] \((x+y)+\left(x^{2}+x y+y^{2}\right)+\left(x^{3}+x^{2} y+x y^{2}+y^{3}\right)+\ldots .\) is : (a) \(\frac{x+y-x y}{(1+x)(1+y)}\) (b) \(\frac{x+y+x y}{(1+x)(1+y)}\) (c) \(\frac{x+y-x y}{(1-x)(1-y)}\) (d) \(\frac{x+y+x y}{(1-x)(1-y)}\)