Chapter 10

If \(a_{1}, a_{2}, a_{3}, a_{4}\) are in H.P., then \(\frac{1}{a_{1} a_{4}} \sum_{r=1}^{3} a_{r} a_{r+1}\) is a root of (A) \(x^{2}+2 x+15=0\) (B) \(x^{2}+2 x-15=0\) (C) \(x^{2}-6 x-8=0\) (D) \(x^{2}-9 x+20=0\)

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) \(3^{n}-1\)

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

Let the harmonic mean and the geometric mean of two positive numbers be in the ratio \(4: 5\). The two numbers are in the ratio (A) \(1: 1\) (B) \(2: 1\) (C) \(3: 1\) (D) \(4: 1\)

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

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

For any odd integer \(n \geq 1\), \(n^{3}-(n-1)^{3}+\ldots+(-1)^{n-1} 1^{3}=\) (A) \(\frac{1}{2}(n-1)^{2}(2 n-1)\) (B) \(\frac{1}{4}(n-1)^{2}(2 n-1)\) (C) \(\frac{1}{2}(n+1)^{2}(2 n-1)\) (D) \(\frac{1}{4}(n+1)^{2}(2 n-1)\)

For a positive integer \(n\), let \(a(n)=\) \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{\left(2^{n}\right)-1}\), Then (A) \(a(100) \leq 100\) (B) \(a(100)>100\) (C) \(a(200) \leq 100\) (D) \(a(200)>100\)

Let \(\alpha, \beta, \gamma\) be the roots of the equation \(3 x^{3}-x^{2}-3 x+1=0 .\) If \(\alpha, \beta, \gamma\) are in H.P. then \(|\alpha-\gamma|=\) (A) \(\frac{1}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{4}{3}\) (D) None of these

The coefficient of \(x^{n}\) in the product \((1-x)(1-2 x)\left(1-2^{2} \cdot x\right)\left(1-2^{3} \cdot x\right) \ldots\left(1-2^{n}+x\right)\) is equal to \((\) A \()\left(1-2^{n+1}\right) 2^{\frac{n(n-1)}{2}}\). (B) \(\left(2^{n+1}-1\right) \cdot 2^{\frac{n(n-1)}{2}}\) (C) \(\left(1-2^{n}\right) 2^{\frac{n(n-1)}{2}}\). (D) None of these

If \(0.272727 \ldots, x\) and \(0.727272 \ldots\) are in H.P., then \(x\) must be (A) rational (B) integer (C) irrational (D) None of these

If \(a_{1}=0\) and \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) are real numbers such that \(\left|a_{i}\right|=\left|a_{i-1}+1\right|\) for all \(i\) then the A.M. of the numbers \(a_{1}, a_{2}, \ldots, a_{n}\) has value \(x\) where (A) \(x \leq-\frac{1}{2}\) (B) \(x \geq-\frac{1}{2}\) (C) \(x<-\frac{1}{2}\) (D) None of these

If \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) are in H.P, then \(\frac{a_{1}}{a_{2}+a_{3}+\ldots+a_{n}}, \frac{a_{2}}{a_{1}+a_{3}+\ldots+a_{n}} \cdots\) \(\frac{a_{n}}{a_{1}+a_{2}+\ldots+a_{n-1}}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

The consecutive numbers of a three digit number form a G.P. If we subtract 792 from this number, we get a number consisting of the same digits written in the reverse order and if we increase the second digit of the required number by 2, the resulting number forms an A.P. The number is (A) 139 (B) 193 (C) 931 (D) None of these

The largest term of the sequence \(\frac{1}{503}, \frac{4}{524}, \frac{9}{581}, \frac{16}{692}, \ldots\) is (A) \(\frac{16}{692}\) (B) \(\frac{4}{524}\) (C) \(\frac{49}{1520}\) (D) None of these

The coefficient of \(x^{99}\) and \(x^{98}\) in the polynomial \((x-1)(x-2)(x-3) \ldots(x-100)\) are (A) \(-5050\) and 12482075 (B) \(-4050\) and 12582075 (C) \(-5050\) and 12582075 (D) None of these

The three successive terms of a G.P. will form the sides of a triangle if the common ratio \(r\) satisfies the inequality (A) \(\frac{\sqrt{3}-1}{2}

If the sides of a right angled triangle are in G.P., then the cosine of the greater acute angle is (A) \(\frac{1}{1+\sqrt{5}}\) (B) \(\frac{1}{1-\sqrt{5}}\) (C) \(\frac{1+\sqrt{5}}{2}\) (D) None of these

If, in a G.P. of \(3 n\) terms, \(S_{1}\) denotes the sum of the first \(n\) terms, \(S_{2}\) the sum of the second block of \(n\) terms and \(S_{3}\) the sum of the last \(n\) terms, then \(S_{1}, S_{2}, S_{3}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

In a geometric series, the first term is \(a\) and common ratio is \(r\). If \(\mathrm{S}_{n}\) denotes the sum of \(n\) terms and \(U_{n}\) \(=\sum_{n=1}^{n} \mathrm{~S}_{n}\), then \(r S_{n}+(1-r) u_{n}=\) (A) \(n a\) (B) \((n-1) a\) (C) \((n+1) a\) (D) None of these

If \(\frac{1}{1^{4}}+\frac{1}{2^{4}}+\frac{1}{3^{4}}+\ldots .\) up to \(\infty=\frac{\pi^{4}}{90}\), then the value of \(\frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\ldots .\) up to \(\infty\) is (A) \(\frac{\pi^{4}}{45}\) (B) \(\frac{\pi^{4}}{96}\) (C) \(\frac{\pi^{4}}{124}\) (D) None of these

If the \((m+1)\) th, \((n+1)\) th and \((r+1)\) th terms of an A.P. are in G.P. and \(m, n, r\) are in H.P., then the ratio of the first term of the A.P. to its common difference is (A) \(\frac{n}{3}\) (B) \(-\frac{n}{3}\) (C) \(\frac{n}{2}\) (D) \(-\frac{n}{2}\)

Let there be \(n\) numbers in G.P. whose common ratio is \(r\) and \(S_{m}\) denotes the sum of their first \(m\) terms. The sum of their products taken two at a time is \(k S_{n} S_{n-1}\) where \(k=\) (A) \(\frac{r-1}{r}\) (B) \(\frac{r-1}{r+1}\) (C) \(\frac{r}{r+1}\) (D) None of these

If \(H_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots .+\frac{1}{n}\), then the value of \(1+\frac{3}{2}+\frac{5}{3}+\ldots+\frac{2 n-1}{n}\) is (A) \(n-H_{n}\) (B) \(2 n-H_{n}\) (C) \((n-1)-H_{n}\) (D) \(n-2 H_{n}\)

If \(a_{m}\) be the \(m\) th term of an A.P., then \(a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+\ldots .+a_{2 n-1}^{2}-a_{2 n}^{2}=\) (A) \(\frac{n-1}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)\) (B) \(\frac{n}{2 n-1}\left(a_{2 n}^{2}-a_{1}^{2}\right)\) (C) \(\frac{n}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)\) (D) None of these

If \(a_{n+1}=\frac{1}{1-a_{n}}\) for \(n \geq 1\) and \(a_{3}=a_{1}\), then \(\left(a_{2001}\right)^{2001}=\) (A) 1 (B) \(-1\) (C) 0 (D) None of these

If \(a, b, c\) are positive numbers in G.P. and log \(\left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)\) and \(\log \left(\frac{a}{3 b}\right)\) are in A.P. then \(a, b, c\) (A) form the sides of an equilateral triangle (B) form the sides of an isosceles triangle (C) form the sides of a right angled triangle (D) can not form the sides of a triangle

If \(a, b, c\) are in G.P. and \(\log a-\log 2 b, \log 2 b-\log 3 c\) and \(\log 3 c-\log a\) are in A.P., then \(a, b, c\) are the sides of a triangle which is (A) right angled (B) acute angled (C) obtuse angled (D) None of these

In a sequence of \(4 n+1\) terms, the first \(2 n+1\) terms are in A.P. having common difference 2 and the last \(2 n+1\) terms are in G.P. having common ratio \(\frac{1}{2}\), If the middle term of the A.P. is equal to the middle term of the G.P. then the middle term of the sequence is (A) \(\frac{n \cdot 2^{n+1}}{2^{n}+1}\) (B) \(\frac{n \cdot 2^{n+1}}{2^{n}-1}\) (C) \(\frac{n \cdot 2^{n}}{2^{n}-1}\) (D) None of these

If \(S_{1}, S_{2}\) and \(S_{3}\) denote the sums up to \(n>1\) terms of three sequences in A.P. whose first terms are unity and common differences are in H.P. then \(n=\) (A) \(\frac{2 S_{3} S_{1}+S_{1} S_{2}+S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}\) (B) \(\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}+2 S_{2}+S_{3}}\) (C) \(\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}\) (D) None of these

Let \(a\) be a fixed real number such that \(\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}\)If \(p, q, \mathrm{r}\) are in A.P. then \(x, y, z\) are in (A) A.P. (B) G.P. (C) H. P (D) None of these

If \(|a|<1\) and \(|b|<1\), then the sum of the series \(1+(1+a) b+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+\) \(\ldots \infty\) is equal to (A) \(\frac{1}{(1-b)(1-a b)}\) (B) \(\frac{1}{(1-a)(1-a b)}\) (C) \(\frac{1}{(1-a)(1-b)}\) (D) None of these

The sixth term of an A.P. is equal to 2 . The value of the common difference of the A.P. which makes the product \(a_{1} a_{4} a_{5}\) greatest, is (A) \(\frac{8}{5}\) (B) \(\frac{2}{3}\) (C) \(\frac{3}{5}\) (D) \(\frac{3}{4}\)

If the first and the \((2 n-1)\) th terms of an A.P., G.P. and H.P. are equal and their \(n\)th terms are \(a, b, c\) respectively, then (A) \(a=b=c\) (B) \(a \geq b \geq c\) (C) \(a+c=b\) (D) \(a c-b^{2}=0\)

If \(a, b, c\) are in A.P. and \(a^{2}, b^{2}, c^{2}\) arc in H.P. then (A) \(a=b=c\) (B) \(-\frac{a}{2}, b, c\) are in G.P. (C) \(-\frac{c}{2}, b, a\) are in G.P. (D) \(-\frac{a}{2}, b, c\) are in H.P.

If the G.M. between \(a\) and \(b\) be twice the H.M., then \(\frac{a}{b}\) is equal to (A) \(\frac{2+\sqrt{3}}{2-\sqrt{3}}\) (B) \(\frac{2-\sqrt{3}}{2+\sqrt{3}}\) (C) \(\frac{4+\sqrt{3}}{4-\sqrt{3}}\) (D) \(\frac{4-\sqrt{3}}{4+\sqrt{3}}\)

If \(a, b, c\) are in G.P. and \(x\) is the A.M. between \(a\) and \(b, y\) the A.M. between \(b\) and \(c\), then (A) \(\frac{a}{x}+\frac{c}{y}=1\) (B) \(\frac{a}{x}+\frac{c}{y}=2\) (C) \(\frac{1}{x}+\frac{1}{y}=\frac{2}{b}\) (D) None of these

The solution of the equations \(\log x+\log x^{1 / 2}+\log x^{1 / 4}\) \(+\ldots=y\) and \(\frac{1+3+5+\ldots . .+(2 y-1)}{4+7+10+\ldots+(3 y+1)}\) \(=\frac{20}{7 \log x}\) is (A) \(x=10^{5}, 10^{-5 / 7}\) (B) \(y=10,-\frac{10}{7}\) (C) \(x=10,-\frac{10}{7}\) (D) \(y=10^{5}, 10^{-5 / 7}\)

The sum of of first ten terms of an A.P. is equal to 155 and the sum of first two terms of a G.P. is 9 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P, then (A) first term of the G.P. is \(\frac{2}{3}, 3\) (B) first term of the A.P. is \(\frac{2}{3}, 3\) (C) Common ratio of the G.P. is \(\frac{25}{2}, 2\) (D) Common difference of the A.P is \(\frac{2}{3}, 3\)

Let \(\left(1+x^{2}\right)^{2}(1+x)^{n}=\sum_{k=0}^{n+4} a_{k} x^{k}\). If \(a_{1}, a_{2}, a_{3}\), are in A.P., then \(n\) is equal to (A) 1 (B) 2 (C) 3 (D) 4

If \(a, b, c\) are non-zero real numbers such that 3 \(\left(a^{2}+b^{2}+c^{2}+1\right)=2(a+b+c+a b+b c+c a)\), then, \(a, b, c\) are in (A) A.P. (B) G. P. (C) H.P. (D) all equal

Let \(t_{n}=\underbrace{1.1 \ldots 1}_{n \text { times }}\), then (A) \(t_{912}\) is not prime (B) \(t_{951}\) is not prime (C) \(t_{480}\) is not prime (D) \(t_{91}\) is not prime

Sum to \(n\) terms of the series \(\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots .\) is (A) \(\frac{n}{2 n+1}\) (B) \(\frac{n}{2 n-1}\) (C) \(\frac{n-1}{2 n+1}\) (D) None of these

Sum to infinite terms of the series \(\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots .\) is (A) \(\frac{1}{4}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{2}\) (D) None of these

The sum to infinity of the series \(1+\frac{3}{2}+\frac{5}{2^{2}}+\frac{7}{2^{3}}+\ldots\) is (A) 4 (B) 6 (C) 8 (D) None of these

If the sum to infinity of the series \(3+(3+d) \frac{1}{4}+(3+2 d) \frac{1}{4^{2}}+\ldots\) is \(\frac{44}{9}\), then \(d=\) (A) 1 (B) 2 (C) 4 (D) None of these

\(3^{1 / 3} \cdot 9^{1 / 9} \cdot 27^{1 / 27} \cdot 81^{1 / 81} \ldots\) upto \(\infty=\) (A) \(\sqrt{27}\) (B) \(\sqrt[3]{27}\) (C) \(\sqrt[4]{27}\) (D) None of these

\begin{tabular}{l} Column-I & Column-II \\ \hline I. If \(a, b, c\) are in A.P., \(b, c, d\) are in G.P. and \(c, d, e\) are in H.P., then (A) A.P. \\ \(\qquad a, c, e\) are in \\ II. If \(2(y-a)\) is the H.M. between \(y-x, y-z\) then \(x-a, y-a, z-a\) & (B) G.P. \\\ are in \\ III. If three numbers are in H.P., then the numbers obtained by subtract- (C) H.P. \\ ing half of the middle number from each of them are in \\ IV. If \(a, b, c\) are in G.P., then the equations \(a x^{2}+2 b x+c=0\) and \(d x^{2}+\) (D) A.G.P. \\ \(2 e x+f=0\) have a common root, if \(\frac{d}{a}, \frac{e}{b}\) and \(\frac{f}{c}\) are in \end{tabular}

Assertion: 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 less than 1 . Reason: A.M. >G.M. for unequal numbers