Chapter 8

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

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

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

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 \((1-x)(1-2 x)\left(1-2^{2} \cdot x\right)\left(1-2^{3} \cdot x\right) \ldots\left(1-2^{n} \cdot x\right)\) 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 \(\frac{a_{1}}{a_{2}+a_{3}+\ldots+a_{n}}, \frac{a_{2}}{a_{1}+a_{3}+\ldots+a_{n}} \ldots\) \(\frac{a_{n}}{a_{1}+a_{2}+\ldots+a_{n-1}}\) 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) \(\frac{49}{1529}\) (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) \(\frac{1+\sqrt{5}}{\underline{\phantom{xx}}}\) (D) None of these

If the \(p\) th, \(q\) th and \(r\) th terms of both an A.P. and a G.P. be respectively \(a, b\) and \(c\), then (A) \(a^{c} \cdot c^{b} \cdot b^{a}=a^{c} \cdot b^{c} \cdot a^{b}\) (B) \(a^{b-1} \cdot b^{c+1} \cdot c^{a-1}=a^{c-1} \cdot b^{a-1} \cdot c^{b+1}\) (C) \(a^{b} \cdot b^{c} \cdot c^{a}=a^{c} \cdot b^{a} \cdot c^{b}\) (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

In a \(\Delta \mathrm{abc}\), if \(\cot A, \cot B, \cot C\) are in A.P. then \(a^{2}, b^{2}\), \(c^{2}\) are in (A) A.P. (B) G.P. (C) H.P. (D) A.G. P.

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 \(a, b, c, d\) are distinct integers in A.P. such that \(d=a^{2}\) \(+b^{2}+c^{2}\), then \(a+b+c+d=\) (A) 2 (B) 1 (C) 0 (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) \(\mathbb{1}\) (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) \(y=10^{5}, 10^{-5 / 7}\)