Infinite GP

Q.

Let n $\in$N, $n>25$. Let A, G, H denote the arithmetic mean. Geometric mean and harmonic mean of 25 and n. The least value of n for which A, G, H $\in${25, 26, ...... n} is

1. 49

2. 81

3. 169

4. 225

Q. If $(a+bx)e^{\frac{y}{x}}$=x then$x^3\frac{d^{2}y}{d{x}^2}$ equals

1. 
2. 
3. 
4. 

Q.

In ( $\sqrt[3]{32}+\frac{1}{\sqrt[3]{33}}{)}^n$ if the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}$, then n =

1. 7

2. 8

3. 9

4. None of these

Q. If $\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\cdots \text{upto}\infty =\frac{{\pi }^4}{90},$ then the value of $\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\cdots \text{upto}\infty$ is

1. $\frac{{\pi }^4}{45}$
2. $\frac{{\pi }^4}{96}$
3. $\frac{{\pi }^4}{124}$
4. $\frac{{\pi }^4}{196}$

Q.

The middle term in the expansion of ${(\frac{b\sqrt{a}}{5}-\frac{5}{a\sqrt{b}})}^{12}$ is:

1. ¹²C₆(b/a)³

2. -¹²C₆(b/a)³

3. ¹²C₇(b⁵/a)

4. -¹²C₇(b⁵/a)

Q. The sum of infinite terms of a G.P. is x and on squaring the each term of it, the sum will be y, then the common ratio of this series is

1. $\frac{x^2-y^2}{x^2+y^2}$
2. $\frac{x^2+y^2}{x^2-y^2}$
3. $\frac{x^2-y}{x^2+y}$
4. $\frac{x^2+y}{x^2-y}$

Q.

The product of infinite terms in $x^{\frac{1}{2}}.x^{\frac{1}{4}}.x^{\frac{1}{8}}...........\infty$ is

1. 0

2. x

3. infinity

4. 1

Q. This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.

इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियों का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Match the Columns and choose the appropriate answer

कॉलमों का मिलान करें और सही उत्तर चुनें

	Column-I (कॉलम)-I		Column-II (कॉलम)-II
(A)	S = 1 + 2 + 4 + 8 + 16 + … up to n terms S = 1 + 2 + 4 + 8 + 16 + … n पदों तक	(P)	$\frac{2n^2-5n}{2}$
(B)	$S=3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+...uptonterms$ $S=3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+...$ n पदों तक	(Q)	$\frac{n^2}{4}+\frac{9n}{4}$
(C)	$S=\frac{5}{2}+3+\frac{7}{2}+4+...uptonterms$ $S=\frac{5}{2}+3+\frac{7}{2}+4+...$ n पदों तक	(R)	2n – 1
(D)	$S=-\frac{3}{2}+\frac{1}{2}+\frac{5}{2}+...uptonterms$ $S=-\frac{3}{2}+\frac{1}{2}+\frac{5}{2}+...$n पदों तक	(S)	6 – 6·2–n

1. A-S, B-R, C-P, D-Q
2. A-R, B-S, C-Q, D-P
3. A-Q, B-S, C-R, D-P
4. A-Q, B-P, C-P, D-S

Q. Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is
[AIEEE 2002]

1. 5

2. 3/5

3. 8/5
4. 1/5

Q.

The product of infinite terms in $x^{\frac{1}{2}}.x^{\frac{1}{4}}.x^{\frac{1}{8}}...........\infty$ is

1. 0
   

2. infinity
   

3. 1
   

4. x
   
   

Q.

The $4^{th}$ of a G.P. is square of its second term, and the first term is - 3. Determine its $7^{th}$ term.

Q. $\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\frac{1}{a^{i+j+k}}$ is equal to $(\text{where}|a|>1)$

1. $(a-1{)}^{-3}$
2. $\frac{3}{a-1}$
3. $\frac{3}{a^3-1}$
4. None of these

Q. The mean and variance of n observations $x_1,x_2,x_3,..............x_n$ are $5$ and $0$ respectively. If $\sum _{i=1}^n{x}_i^2=400$, then n is

1. $10$
2. $20$
3. $25$
4. $36$

Q.

For the function
f(x) = $\frac{x^{100}}{100}+\frac{x^{99}}{99}+...+\frac{x^2}{2}+x+1$ prove that f'(1) = 100 f'(0).

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

S = 1 + 2x + 4x² + 8x³ + … ∞ is having some finite value, then possible values of x is/are

S = 1 + 2x + 4x² + 8x³ + … ∞ के कुछ परिमित मान हैं, तब x के संभावित मान है/हैं

1. 1
2. $\frac{1}{2}$
3. $\frac{-1}{4}$
4. $\frac{1}{4}$

Q.

If sum of infinite terms of a G.P. is 3 and sum of squares of its terms is 3, then its first term and common ratio are
[RPET 1999]

1. 3/2, 1/2

2. 1, 1/2

3. 3/2, 2

4. None of these

Q. The value of x in $(0,\pi )$ which satisfy the equation
$8^{-+|cosx|+co{s}^{2}x+|co{s}^{3}x|+\cdots to\infty }=4^3$ is

1. $\{\frac{\pi }{2},\frac{3\pi }{4}\}$
2. $\{\frac{\pi }{4},\frac{3\pi }{4}\}$
3. $\{\frac{\pi }{3},\frac{2\pi }{3}\}$
4. $\{\frac{\pi }{6},\frac{5\pi }{6}\}$

Q. Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

A man started investing in a company. He started with Rs. 10 in 1^st month. In every succeeding month, he invested 2 times as of the proceeding month.

एक व्यक्ति ने एक कम्पनी में निवेश करना प्रारम्भ किया। प्रथम माह में उसने 10 रूपये से शुरूआत की, तथा प्रत्येक आने वाले माह में गत माह की तुलना में 2 गुना निवेश किया।

Q. The amount he invested in 6^th month is

प्रश्न - उसके द्वारा 6^th महीने में निवेश की गई राशि है

1. 320
2. 340
3. 440
4. 640

Q. If $11+103+1005+\cdots n\text{terms}=\frac{{10}^{n+1}+x{n}^2+y}{9}$, then the value of $x-y$ is

1. $-19$
2. $19$
3. $1$
4. $-1$

Q. The least value of n ( a natural number), for which the sum S of the series $1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots$ differs from $S_n$ by a quantity $<{10}^{-6}$ is

1. 21
2. 20
3. 19
4. None

Q.

FInd the value of $\sum {}_{k=0}^7$ $(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$

__

Q. The value of x in $(0,\pi )$ which satisfy the equation
$8^{-+|cosx|+co{s}^{2}x+|co{s}^{3}x|+\cdots to\infty }=4^3$ is

1. $\{\frac{\pi }{2},\frac{3\pi }{4}\}$
2. $\{\frac{\pi }{4},\frac{3\pi }{4}\}$
3. $\{\frac{\pi }{3},\frac{2\pi }{3}\}$
4. $\{\frac{\pi }{6},\frac{5\pi }{6}\}$

Q. The solution set of the system of equations $x+y=\frac{2\pi }{3},cosx+cosy=\frac{3}{2}$, where x and y are real, is ___

Q.

Fill in the blanks.

$\frac{13}{15}=\frac{\square }{30}=\frac{39}{\square }$

Q. The sum of the series $(\sqrt{2}+1)+1+(\sqrt{2}-1)+\cdots$ up to infinite term is

1. $\frac{1+3\sqrt{2}}{2}$
2. $\frac{3+\sqrt{2}}{2}$
3. $\frac{5+2\sqrt{2}}{2}$
4. $\frac{4+3\sqrt{2}}{2}$

Q. Let $a+a{r}_1+a{r_1}^2+...$ and $a+a{r}_2+a{r_2}^2+...$ be two infinite series of positive numbers with the same first term, where $r_1,r_2<1$. The sum of the first series is $r_1$ and the sum of the second series is $r_2$ then the value of $r_1+r_2$ is