Chapter 9

The product \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \ldots\) to \(\infty\) is equal to: [Jan. 9, 2020 (I)] (a) \(2^{\frac{1}{2}}\) (b) \(2^{\frac{1}{4}}\) (c) 1 (d) 2

Let \(a_{n}\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum_{n=1}^{100} a_{2 n+1}=200\) and \(\sum_{n=1}^{100} a_{2 n}=100\), then \(\sum_{n=1}^{200} a_{n}\) is equal to: [Jan. 9, 2020 (II)] (a) 300 (b) 225 (c) 175 (d) 150

The greatest positive integer \(k\), for which \(49^{k}+1\) is a factor of the sum \(49^{125}+49^{124}+\ldots+49^{2}+49+1\), is: [Jan. 7, 2020 (I)] (a) 32 (b) 63 (c) 60 (d) 65

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be \(a \mathrm{G} .\) P. such that \(a_{1}<0, a_{1}+a_{2}=4\) and \(a_{3}+a_{4}=16 .\) If \(\sum_{i=1}^{9} a_{i}=4 \lambda\), then \(\lambda\) is equal to: [Jan. 7, 2020 (II)] (a) \(-513\) (b) \(-171\) (c) 171 (d) \(\frac{511}{3}\)

The coefficient of \(x^{7}\) in the expression \((1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}\) is: [Jan. 7, 2020 (II)] (a) 210 (b) 330 (c) 120 (d) 420

If \(\alpha, \beta\) and \(\gamma\) are three consecutive terms of a nonconstant G.P. such that the equations \(\alpha x^{2}+2 \beta x+\gamma=0\) and \(x^{2}+x-1=0\) have a common root, then \(\alpha(\beta+\gamma)\) is equal to : [April 12, 2019 (II)] (a) 0 (b) \(\alpha \beta\) (c) \(\alpha \gamma\) (d) \(\beta \gamma\)

Let \(a, b\) and \(c\) be in G.P. with common ratio \(r\), where \(a \neq 0\) and \(0

The product of three consecutive terms of a G.P. is 512 . If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is : [Jan. 12, 2019(I)] (a) 36 (b) 32 (c) 24 (d) 28

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(x^{2} \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0<\theta<45^{\circ}\right)\), and \(\alpha<\beta\). Then \(\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)\) is equal to : [Jan. 11, 2019 (II)] (a) \(\frac{1}{1-\cos \theta}-\frac{1}{1+\sin \theta}\) (b) \(\frac{1}{1+\cos \theta}+\frac{1}{1-\sin \theta}\) (c) \(\frac{1}{1-\cos \theta}+\frac{1}{1+\sin \theta}\) (d) \(\frac{1}{1+\cos \theta}-\frac{1}{1-\sin \theta}\)

Let \(a_{1}, a_{2}, \ldots, a_{10}\) be a G.P. If \(\frac{a_{3}}{a_{1}}=25\), then \(\frac{a_{9}}{a_{5}}\) equals : [Jan. 11, 2019 (I)] (a) \(5^{4}\) (b) \(4\left(5^{2}\right)\) (c) \(5^{3}\) (d) \(2\left(5^{2}\right)\)

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is \(\frac{27}{19}\). Then the common ratio of this series is: (a) \(\frac{1}{3}\) (b) \(\frac{2}{3}\) (c) \(\frac{2}{9}\) (d) \(\frac{4}{9}\)

Let \(\mathrm{S}_{\mathrm{n}}=1+\mathrm{q}+\mathrm{q}^{2}+\ldots .+\mathrm{q}^{\mathrm{n}}\) and \(\mathrm{T}_{\mathrm{n}}=1+\left(\frac{\mathrm{q}+1}{2}\right)+\left(\frac{\mathrm{q}+1}{2}\right)^{2}+\ldots+\left(\frac{\mathrm{q}+1}{2}\right)^{\mathrm{n}}\) where \(\mathrm{q}\) is a real number and \(\mathrm{q} \neq 1\). If \({ }^{101} \mathrm{C}_{1}+{ }^{101} \mathrm{C}_{2} \mathrm{~S}_{1}+\ldots .+{ }^{101} \mathrm{C}_{101} \cdot \mathrm{S}_{100}=\alpha \mathrm{T}_{100}\), then \(\alpha\) is equal to: [Jan. 11, 2019 (II)] (a) \(2^{99}\) (b) 202 (c) 200 (d) \(2^{100}\)

Let \(a, b\) and \(c\) be the \(7^{\text {th }}, 11^{\text {th }}\) and \(13^{\text {th }}\) terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then \(\frac{a}{c}\) is equal to: [Jan. 09,2019 (II)] (a) 2 (b) \(\frac{1}{2}\) (c) \(\frac{7}{13}\) (d) 4

If \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) be three distinct real numbers in GP. and \(\mathrm{a}+\mathrm{b}+\mathrm{c}=x \mathrm{~b}\), then \(x\) cannot be: \(\quad\) [Jan. \(\mathbf{0 9}, \mathbf{2 0 1 9}(\mathrm{I})]\) (a) \(-2\) (b) \(-3\) (c) 4 (d) 2

If \(b\) is the first term of an infinite G. P whose sum is five, then \(b\) lies in the interval. \(\quad\) [Online April 15, 2018] (a) \((-\infty,-10)\) (b) \((10, \infty)\) (c) \((0,10)\) (d) \((-10,0)\)

Let \(A_{n}=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^{2}+\left(\frac{3}{4}\right)^{3}-\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^{n}\) and \(B_{n}=1-A_{n}\). Then, the least odd natural number \(p\), so that \(B_{n}>A_{n}\), for all \(n \geq p\) is [Online April 15, 2018] (a) 5 (b) 7 (c) 11 (d) 9

If \(a, b, c\) are in A.P. and \(a^{2}, b^{2}, c^{2}\) are in G.P. such that \(a

If the \(2^{\text {nd }}, 5^{\text {th }}\) and \(9^{\text {th }}\) terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is : (a) 1 (b) \(\frac{7}{4}\) (c) \(\frac{8}{5}\) (d) \(\frac{4}{3}\)

Let \(z=1+a i\) be a complex number, \(a>0\), such that \(z^{3}\) is areal number. Then the sum \(1+z+z^{2}+\ldots+z^{11}\) is equal to: (Online April 10,2016) (a) \(1365 \sqrt{3} \mathrm{i}\) (b) \(-1365 \sqrt{3} \mathrm{i}\) (c) \(-1250 \sqrt{3} \mathrm{i}\) (d) \(1250 \sqrt{3} \mathrm{i}\)

The sum of the \(3^{\text {rd }}\) and the \(4^{\text {th }}\) terms of a GP. is 60 and the product of its first three terms is 1000 . If the first term of this G.P. is positive, then its \(7^{\text {th }}\) term is : [Online April 11, 2015] (a) 7290 (b) 640 (c) 2430 (d) 320

Three positive numbers form an increasing G P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is: (a) \(2-\sqrt{3}\) (b) \(2+\sqrt{3}\) (c) \(\sqrt{3}, \sqrt{3}\) (d) \(3+\sqrt{2}\)

The least positive integer \(\mathrm{n}\) such that \(1-\frac{2}{3}-\frac{2}{3^{2}}-\ldots .-\frac{2}{3^{n-1}}<\frac{1}{100}\), is: [Online April 12, 2014] (a) \(\underline{4}\) (b) 5 (c) 6 (d) 7

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49 , and the sum of the first and the third term is 35 . Then the first term of this geometric progression is: [Online April 11, 2014] (a) 7 (b) 21 (c) 28 (d) 42

The coefficient of \(x^{50}\) in the binomial expansion of \((1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+\ldots .\) \(+x^{1000}\) is: \(\quad\) [Online April 11, 2014] (a) \(\frac{(1000) !}{(50) !(950) !}\) (b) \(\frac{(1000) !}{(49) !(951) !}\) (c) \(\frac{(1001) !}{(51) !(950) !}\) (d) \(\frac{(1001) !}{(50) !(951) !}\)

Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is : \(\quad\) [Online April 25, 2013] (a) 16 (b) 8 (c) 4 (d) 2

If \(a, b, c, d\) and \(p\) are distinct real numbers such that \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2 p(a b+b c+c d)+\left(b^{2}+c^{2}+d^{2}\right) \leq 0\), then \mathrm{\\{} O n l i n e ~ M a y ~ 1 2 , ~ 2 0 1 2 ] ~ (a) \(a, b, c, d\) are in A.P. (b) \(a b=c d\) (c) \(a c=b d\) (d) \(a, b, c, d\) are in G.P.

The difference between the fourth term and the first term of a Geometrical Progresssion is 52 . If the sum of its first three terms is 26 , then the sum of the first six terms of the progression is [Online May 7, 2012] (a) 63 (b) 189 (c) 728 (d) 364

The first two terms of a geometric progression add up to 12\. the sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is (a) \(-4\) (b) \(-12\) (c) 12 (d) 4

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals [2007] (a) \(\sqrt{5}\) (b) \(\frac{1}{2}(\sqrt{5}-1)\) (c) \(\frac{1}{2}(1-\sqrt{5})\) (d) \(\frac{1}{2} \sqrt{5}\)

The value of \(\sum_{k=1}^{10}\left(\sin \frac{2 k \pi}{11}+i \cos \frac{2 k \pi}{11}\right)\) is [2006] (a) \(i\) (b) 1 (c) \(-1\) (d) \(-i\)

If the expansion in powers of \(x\) of the function \(\frac{1}{(1-a x)(1-b x)}\) is \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3} \ldots .\) then \(a_{n}\) is [2006] (a) \(\frac{b^{n}-a^{n}}{b-a}\) (b) \(\frac{a^{n}-b^{n}}{b-a}\) (c) \(\frac{a^{n+1}-b^{n+1}}{b-a}\) (d) \(\frac{b^{n+1}-a^{n+1}}{b-a}\)

Let two numbers have arithmetic mean 9 and geometric mean 4 . Then these numbers are the roots of the quadratic equation (a) \(x^{2}-18 x-16=0\) (b) \(x^{2}-18 x+16=0\) (c) \(x^{2}+18 x-16=0\) (d) \(x^{2}+18 x+16=0\)

Sum of infinite number of terms of GP is 20 and sum of their square is 100 . The common ratio of GP is (a) 5 (b) \(3 / 5\) (c) \(8 / 5\) (d) \(1 / 5\)

Fifth term of a GP is 2 , then the product of its 9 terms is [2002] (a) 256 (b) 512 (c) 1024 (d) none of these

The sum of all values of \(\theta \in\left(0, \frac{\pi}{2}\right)\) satisfying \(\sin ^{2} 2 \theta+\cos ^{4} 2 \theta=\frac{3}{4}\) is: \(\quad\) [Jan. 10,2019 (I)] (a) \(\pi\) (b) \(\frac{5 \pi}{4}\) (c) \(\frac{\pi}{2}\) (d) \(\frac{3 \pi}{8}\)

If \(m\) arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that \(4^{\text {th }}\) A. M. is equal to \(2^{\text {nd }} \mathrm{G} \mathrm{M}\), then \(m\) is equal to

If the arithmetic mean of two numbers a and \(\mathrm{b}, \mathrm{a}>\mathrm{b}>0\), is five times their geometric mean, then \(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{a}-\mathrm{b}}\) is equal to : [Online April 8, 2017| (a) \(\frac{\sqrt{6}}{2}\) (b) \(\frac{3 \sqrt{2}}{4}\) (c) \(\frac{7 \sqrt{3}}{12}\) (d) \(\frac{5 \sqrt{6}}{12}\)

Let \(x, y, z\) be positive real numbers such that \(x+y+z=12\) and \(x^{3} y^{4} z^{5}=(0.1)(600)^{3}\). Then \(x^{3}+y^{3}+z^{3}\) is equal to: [Online April 9, 2016] (a) 342 (b) 216 (c) 258 (d) 270

Let \(\mathrm{G}\) be the geometric mean of two positive numbers a and \(\mathrm{b}\), and \(\mathrm{M}\) be the arithmetic mean of \(\frac{1}{\mathrm{a}}\) and \(\frac{1}{\mathrm{~b}}\). If \(\frac{1}{\mathrm{M}}: \mathrm{G}\) is \(4: 5\), then \(a: b\) can be: \(\quad\) Online April 12, 2014] (a) \(1: 4\) (b) \(1: 2\) (c) \(2: 3\) (d) \(3: 4\)

If the sum of the roots of the quadratic equation \(a x^{2}+b x+c=0\) is equal to the sum of the squares of their reciprocals, then \(\frac{a}{c}, \frac{b}{a}\) and \(\frac{c}{b}\) are in \(\quad\) [2003] (a) Arithmetic -Geometric Progression (b) Arithmetic Progression (c) Geometric Progression (d) Harmonic Progression.

If \(1+\left(1-2^{2} \cdot 1\right)+\left(1-4^{2} \cdot 3\right)+\left(1-6^{2} \cdot 5\right)+\ldots \ldots+\left(1-20^{2},\right.\), 19) \(=\alpha-220 \beta\), then an ordered pair \((\alpha, \beta)\) is equal to: [Sep. 04, 2020 (I)] (a) \((10,97)\) (b) \((11,103)\) (c) \((10,103)\) (d) \((11,97)\)

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a function which satisfies \(f(x+y)=f(x)+f(y), \forall x, y \in \mathbf{R}\). If \(f(a)=2\) and \(g(n)=\sum_{k=1}^{(n-1)} f(k), n \in \mathbf{N}\), then the value of \(n\), for which \(g(n)=20\), is: \(\quad\) [Sep. 02, 2020 (II)] (a) 5 (b) 20 (c) 4 (d) 9

The sum, \(\sum_{n=1}^{7} \frac{n(n+1)(2 n+1)}{4}\) is equal to

The sum \(\sum_{k=1}^{20}(1+2+3+\ldots+k)\) is

For \(x \in \mathrm{R}\), let \([x]\) denote the greatest integer \(\leq x\), then the sum of the series \(\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\cdots+\left[-\frac{1}{3}-\frac{99}{100}\right]\) is [April 12, 2019 (I)] (a) \(-153\) (b) \(-133\) (c) \(-131\) (d) \(-135\)

The sum \(\left.\frac{3 \times 1^{3}}{1^{2}}+\frac{5 \times\left(1^{3}+2^{3}\right)}{1^{2}+2^{2}}+\frac{7 \times\left(1^{3}+2^{3}+3^{3}\right)}{1^{2}+2^{2}+3^{2}}\right)+\ldots \ldots\) upto \(10^{\text {th }}\) term, is : \(\quad\) [April 10, 2019 (I)] (a) 680 (b) 600 (c) 660 (d) 620

The sum \(1+\frac{1^{3}+2^{3}}{1+2}+\frac{1^{3}+2^{3}+3^{3}}{1+2+3}+\ldots \ldots+\) \(\frac{1^{3}+2^{3}+3^{3}+\ldots+15^{3}}{1+2+3+\ldots+15}-\frac{1}{2}(1+2+3+\ldots+15\) is equal to : [April 10, 2019 (II)] (a) 620 (b) 1240 (c) 1860 (d) 660

The sum of the series \(1+2 \times 3+3 \times 5+4 \times 7+\ldots \ldots\) upto \(11^{\text {th }}\) term is: [April 09, 2019 (II)] (a) 915 (b) 946 (c) 945 (d) 916

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is: \(\quad\) [April 09, 2019 (II)] (a) 157 2) 262 (c) 225 (d) 190

The sum \(\sum_{k=1}^{20} k \frac{1}{2^{k}}\) is equal to : \(\quad\) [April08, 2019 (II)] (a) \(2-\frac{3}{2^{17}}\) (b) \(1-\frac{11}{2^{20}}\) (c) \(2-\frac{11}{2^{19}}\) (d) \(2-\frac{21}{2^{20}}\)