Chapter 10

If \(a, b, c\) are positive numbers in A.P. such that their product is 64 , then the minimum value of \(b\) \((\mathrm{A})=2\) (B) \(=4\) \((\mathrm{C})=1\) (D) Does not exist

If three successive terms of a G.P. with common ratio \(r(r>1)\) form the sides of a \(\triangle A B C\) and \([r]\) denotes greatest integer function, then \([r]+[-r]=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

Number of increasing geometrical progression(s) with first term unity, such that any three consecutive terms, on doubling the middle become an A.P, is (A) 0 (B) (C) 2 (D) infinity

Let \(S_{n}(1 \leq n \leq 9)\) denotes the sum of \(n\) terms of series \(1+22+333+\ldots+999999999\), then for \(2 \leq n \leq 9\) (A) \(S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)\) (B) \(S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)\) (C) \(9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)\) (D) None of these

If \(\log _{\sqrt{5}} x+\log _{5^{n}} x+\log _{5^{4}} x+\ldots\) upto 7 terms \(=35\), then \(x\) is equal to (A) 5 (B) 25 (C) 125 (D) None of these

If \(\sum_{n=1}^{\infty} x^{n-1}=a\) and \(\sum_{n=1}^{\infty} y^{n-1}=b\) where \(|x|,|y|<1\), then \(\sum_{n=1}^{\infty}(x y)^{n-1}=\) (A) \(a b\) (B) \(\frac{a+b-1}{a b}\) (C) \(\frac{1}{1-a b}\) (D) \(\frac{a b}{a+b-1}\)

Let \(p, q, r \in R^{+}\)and \(27 p q r \geq(p+q+r)^{3}\) and \(3 p+4 q\) \(+5 r=12\) then \(p^{3}+q^{4}+r^{5}\) is equal to (A) 3 (B) 6 (C) 2 (D) None of these

The sum of the series \(\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots\) to \(n\) terms is (A) \(\frac{n\left(n^{2}+1\right)}{n^{2}+n+1}\) (B) \(\frac{n(n+1)}{2\left(n^{2}+n+1\right)}\) (C) \(\frac{n\left(n^{2}-1\right)}{2\left(n^{2}+n+1\right)}\) (D) None of these

\(a, b, c\) are three distinct real numbers, which are in G.P. and \(a+b+c=x b\). Then (A) \(x<-1\) or \(x>3\) (B) \(-1

The sum of the first hundred terms of an A.P. is \(x\) and the sum of the hundred terms starting from the third term is \(y\). Then the common difference is (A) \(\frac{y-x}{2}\) (B) \(\frac{y-x}{50}\) (C) \(\frac{y-x}{100}\) (D) \(\frac{y-x}{200}\)

If \(\lambda=\sum_{i=1}^{\infty} \frac{1}{i^{4}}\), then \(\sum_{i=1}^{\infty} \frac{1}{(2 i-1)^{4}}\) is (A) \(\frac{14}{15} \lambda\) (B) \(\frac{\lambda}{2}\) (C) \(\frac{16}{15} \lambda\) (D) \(\frac{15}{16} \lambda\)

The sum of all possible products of the first \(n\) natural numbers taken two at a time is (A) \(\frac{1}{2}\left[\Sigma n^{2}-\Sigma n\right]\) (B) \(\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n\right]\) (C) \(\frac{1}{2}\left[\Sigma n^{2}-\Sigma(n+1)\right]\) (D) \(\frac{1}{2}\left[(\Sigma n)^{2}-\Sigma n^{2}\right]\)

The minimum value of \(8^{\sin x^{\prime} 8}+8^{\cos x^{\prime} 8}\) is (A) \(2^{\frac{1}{3-\sqrt{2} / \sqrt{2}}}\) (B) \(2^{\frac{3+\sqrt{2}}{\sqrt{2}}}\) (C) \(2^{\frac{1}{3+\sqrt{2} / \sqrt{2}}}\) (D) \(2^{\frac{3-\sqrt{2}}{\sqrt{2}}}\)

If \(\log _{2^{12}} a+\log _{2^{n}} a+\log _{2^{n}} a+\log _{2^{n}} a+\ldots\) upto 20 terms is 840 , then \(a\) is equal to(A) 2 (B) 1 (C) 4 (D) \(\sqrt{2}\)

If \(a, b, c\) are distinct positive real numbers and \(a^{2}+b^{2}\) \(+c^{2}=1\), then \(a b+b c+c a\) is (A) less than 1 (B) equal to 1 (C) greater than 1 (D) any real number

The value of \((n-2)^{2}+(n-4)^{2}+(n-6)^{2}+\ldots\) to \(n\) terms is (A) \(\frac{n}{3}\left(n^{2}+2\right)\) (B) \(\frac{n}{2}\left(n^{2}+3\right)\) (C) \(\frac{n}{3}\left(n^{2}-2\right)\) (D) \(\frac{n}{2}\left(n^{2}-3\right)\)

\(a_{1}, a_{2}, a_{3}, \ldots\) are in A.P. with common difference not a multiple of 3 . Then, maximum number of consecutive terms so that all the terms are prime numbers is (A) 2 (B) 3 (C) 5 (D) infinite

The coefficient of \(x^{49}\) in the product \((x-1)(x-3) \ldots\) \((x-99)\) is (A) \(-99^{2}\) (B) 1 (C) \(-2500\) (D) None of these

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) \([2, \infty)\) (B) \([3, \infty)\) (C) \((3, \infty)\) (D) \((-\infty, 3)\)

In a G.P. of alternating positive and negative terms, any term is the A.M. of the next two terms. Then the common ratio is (A) \(-1\) (B) \(-3\) (C) \(-2\) (D) \(\frac{-1}{2}\)

If the sum of \(n\) terms of an A.P. is cn \((n-1)\), where \(c \neq 0\), then sum of the squares of these terms is (A) \(c^{2} n^{2}(n+1)^{2}\) (B) \(\frac{2}{3} c^{2} n(n-1)(2 n-1)\) (C) \(\frac{2 c^{2}}{3} n(n+1)(2 n+1)\) (D) None of these

If \(b_{1}, b_{2}\) and \(b_{3}\left(b_{1}>0\right)\) are three successive terms of a G.P. with common ratio \(r\), the value of \(r\) for which the inequality \(b_{3}>4 b_{2}-3 b_{1}\) holds, is given by (A) \(r>3\) (B) \(r<1\) (C) \(r=2.5\) (D) \(r=1.7\)

If \(p, q, r\) are positive and are in A.P., the roots of quadratic equation \(p x^{2}+q x+r=0\) are all real for (A) \(\left|\frac{r}{p}-7\right| \geq 4 \sqrt{3}\) (B) \(\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}\) (C) all \(p\) and \(r\) (D) no \(p\) and \(r\)

The sum to \(n\) terms of the series \(\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots\) is (A) \(n-\frac{\left(3^{n}-2^{n}\right)}{2^{n}}\) (B) \(n-\frac{2\left(3^{n}-2^{n}\right)}{3^{n}}\) (C) \(2^{n}-1\) (D) \(3^{n}-1\)

Sum to \(n\) terms of the series \(\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots\) is (A) \(\frac{2}{5 !}-\frac{1}{(n+1) !}\) (B) \(\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)\) (C) \(\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)\) (D) None of these

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 \(a, b, c, d\) are in (A) A.P. (B) G.P. (C) H.P. (D) \(a b=c d\)

If \(a+b+c=3\) and \(a>0, b>0, c>0\), then the greatest value of \(a^{2} b^{3} c^{2}\) is (A) \(\frac{3^{10} \cdot 2^{4}}{7^{7}}\) (B) \(\frac{3^{9} \cdot 2^{4}}{7^{7}}\) (C) \(\frac{3^{8} \cdot 2^{4}}{7^{7}}\) (D) None of these

If \(\left|\begin{array}{ccc}a & b & a \alpha-b \\ b & c & b \alpha-c \\ 2 & 1 & 0\end{array}\right|=0\) and \(\alpha \neq \frac{1}{2}\), then (A) \(a, b, c\) are in A.P. (B) \(a, b, c\) are in G.P. (C) \(a, b, c\) are in H.P. (D) None of these

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

If \(a_{1}, a_{2}, \ldots, a_{n}\) are in A.P. with common difference \(d \neq 0\), then sum of the series \(\sin d\left[\sec a_{1} \sec a_{2}+\sec \right.\) \(\left.a_{2} \sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right]\) is (A) \(\tan a_{n}-\tan a_{1}\) (B) \(\cot a_{n}-\cot a_{1}\) (C) \(\sec a_{n}-\sec a_{1}\) (D) \(\operatorname{cosec} a_{n}-\operatorname{cosec} a_{1}\)

The first and last term of an A.P. are \(a\) and \(l\) respectively. If \(S\) is the sum of all the terms of the A.P. and the common difference is \(\frac{l^{2}-a^{2}}{k-(l+a)}\), then \(k\) is equal to (A) \(S\) (B) \(2 S\) (C) \(3 S\) (D) None of these

If \(a, b, c, d\) are in G.P., then \(\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=\) (A) \((a b+a c+b c)^{2}\) (B) \((a c+c d+a d)^{2}\) (C) \((a b+b c+c d)^{2}\) (D) None of these

A man saves ? 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ? 40 more than the saving of immediately previous months. His total saving from the start of service will be ? 11040 after (A) 21 months (B) 18 months (C) 19 months (D) 20 months

Statement-1: The sum of the series \(1+(1+2+4)+\) \((4+6+9)+(9+12+16)+\ldots+(361+380+400)\) is \(8000 .\) \(\begin{array}{l}\text { Statement-2: } \\ \text { number } n .\end{array}_{k=1}^{n}\left(k^{3}-(k-1)^{3}\right)=n^{3}\), for any natural (A) Statement- 1 is false, Statement-2 is true. (B) Statement- 1 is true, statement- 2 is true; statement- 2 is a correct explanation for Statement- 1 (C) Statement- 1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1. (D) Statement- 1 is true, statement- 2 is false.

If 100 times the \(100^{\text {th }}\) term of an \(A P\) with non-zero common difference equals the 50 times its \(50^{\text {th }}\) term, then the \(150^{\text {th }}\) term of this \(A P\) is (A) \(-150\) (B) 150 times its \(50^{\text {th }}\) term (C) 150 (D) zero

If the sum of first \(n\) terms of two A.P's are in the ratio \(3 n+8: 7 n+15\), then the ratio of their 12 th terms is (A) \(8: 7\) (B) \(7: 16\) (C) \(74: 169\) (D) \(13: 47\)

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

The sum of the products of the \(2 n\) numbers \(\pm 1, \pm 2, \pm 3\). \(\ldots . \pm n\) taking two at a time is (A) \(\frac{n(n+1)}{2}\) (B) \(-\frac{n(n+1)}{2}\) (C) \(\frac{n(n+1)(2 n+1)}{6}\) (D) \(-\frac{n(n+1)(2 n+1)}{6}\)

If \(a\) is the first term, \(d\) the common difference and \(S_{k}\) the sum to \(k\) terms of an A.P., then for \(\frac{S_{k x}}{S_{x}}\) to be inde- pendent of \(x\) (A) \(a=2 d\) (B) \(a=d\) (C) \(2 a=d\) (D) None of these

Given that \(\alpha, \gamma\) are roots of the equation \(A x^{2}-4 x+1=0\) and \(\beta, \delta\) are roots of the equation \(B x^{2}-6 x+1=0\). If \(\alpha, \beta, \gamma\) and \(\delta\) are in H.P., then (A) \(A=5\) (B) \(A=-3\) (C) \(B=8\) (D) \(B=-8\)

The sum of \(n\) terms of \(m\) A.P.s are \(S_{1}, S_{2}, S_{3}, \ldots, S_{m}\), If the first term and common difference are \(1,2,3, \ldots, m\) respectively, then \(S_{1}+S_{2}+S_{3}+\ldots+S_{m}=\) (A) \(\frac{1}{4} m n(m+1)(n+1)\) (B) \(\frac{1}{2} m n(m+1)(n+1)\) (C) \(m n(m+1)(n+1)\) (D) None of these

If three positive numbers \(a, b, c\) are in H.P., then \(a^{n}+c^{n}\) \((\mathrm{A})>2 b^{n}\) \((\mathrm{B})=2 b^{n}\) \((\mathrm{C})<2 b^{n}\) \((\mathrm{D})>b^{n}\)

The sum of first \(n\) terms of the series \(1 \cdot 1 !+2 \cdot 2 !+3 \cdot 3 !+4 \cdot 4 !+\ldots\) is (A) \((n+1) !-1\) (B) \(n !-1\) (C) \((n-1) !-1\) (D) None of these

If \(a, b, c\) are digits, then the rational number represented by \(0 \cdot c a b a b a b \ldots\) is (A) \(\frac{99 c+a b}{990}\) (B) \(\frac{99 c+10 a+b}{99}\) (C) \(\frac{99 c+10 a+b}{990}\) (D) None of these

The sum of first \(n\) terms of the series \(1^{2}+2.2^{2}+3^{2}+2.4^{2}+5^{2}+5.6^{2}+\ldots\) is \(\frac{n(n+1)^{2}}{2}\) when \(n\) is even. When \(n\) is odd, the sum is (A) \(\frac{n^{2}(n+1)}{2}\) (B) \(\frac{n(n+1)^{2}}{2}\) (C) \(\left[\frac{n(n+1)}{2}\right]^{2}\) (D) \(\frac{n(n+1)}{2}\)

The sum of the series \(1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+5 \cdot 2^{4}+\ldots+100 \cdot 2^{99}\) is (A) \(99 \cdot 2^{100}+1\) (B) \(100 \cdot 2^{100}\) (C) \(99 \cdot 2^{100}\) (D) \(99 \cdot 2^{100}+1\)

Four different integers form an increasing A.P. If one of these numbers is equal to the sum of the squares of the other three numbers, then the numbers are (A) \(-2,-1,0,1\) (B) \(0,1,2,3\) (C) \(-1,0,1,2\) (D) None of these

If three successive terms of a G.P. with common ratio \(r(r>1)\) form the sides of a \(\Delta A B C\) and \([r]\) denotes greatest integer function, then \([r]+[-r]=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

Let \(S_{n}(1 \leq n \leq 9)\) denotes the sum of \(n\) terms of series \(1+22+333+\ldots+999999999\), then for \(2 \leq n \leq 9\) (A) \(S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)\) (B) \(S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)\) (C) \(9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)\) (D) None of these

\(a, b, c\) are three distinct real numbers, which are in G.P. and \(a+b+c=x b\). Then, (A) \(x<-1\) or \(x>3\) (B) \(-1