Sequences & Series · NDA previous year questions

If a, b, c are in AP or GP or HP, then $\frac{a-b}{b-c}$ is equal to

1. A.$\frac{b}{a}$ or 1 or $\frac{b}{c}$
2. B.$\frac{c}{a}$ or $\frac{c}{b}$ or 1
3. C.1 or $\frac{a}{b}$ or $\frac{a}{c}$
4. D.1 or $\frac{a}{b}$ or $\frac{c}{a}$

If $\sec (\theta -\alpha )$, $\sec \theta$ and $\sec (\theta +\alpha )$ are in AP, where $\cos \alpha \ne 1$, then what is the value of ${\sin }^2\theta +\cos \alpha$?

1. A.$0$
2. B.$1$
3. C.$-1$
4. D.$\frac{1}{2}$

If $A+B+C=180^{\circ }$, then what is $\sin 2A-\sin 2B-\sin 2C$ equal to?

1. A.$-4\sin A\sin B\sin C$
2. B.$-4\cos A\sin B\cos C$
3. C.$-4\cos A\cos B\sin C$
4. D.$-4\sin A\cos B\cos C$

If x, 3/2, z are in AP; x, 3, z are in GP; then which one of the following will be in HP ?

1. A.x, 6, z
2. B.x, 4, z
3. C.x, 2, z
4. D.x, 1, z

If $x$, $x-y$ and $x+y$ are the angles of a triangle (not an equilateral triangle) such that $\tan (x-y)$, $\tan x$ and $\tan (x+y)$ are in GP, then what is $x$ equal to?

1. A.$\pi /4$
2. B.$\pi /3$
3. C.$\pi /6$
4. D.$\pi /2$

A person is to count 4500 notes. Let $a_n$ denote the number of notes he counts in the $n\text{th}$ minute. If $a_1=a_2=a_3=\cdots =a_{10}=150$, and $a_{10},a_{11},a_{12},\dots$ are in AP with the common difference $-2$, then the time taken by him to count all the notes is

1. A.24 minutes
2. B.34 minutes
3. C.125 minutes
4. D.135 minutes

If $y=x+x^2+x^3+\dots$ up to infinite terms, where $x<1$, then which one of the following is correct?

1. A.$x=\frac{y}{1+y}$
2. B.$x=\frac{y}{1-y}$
3. C.$x=\frac{1+y}{y}$
4. D.$x=\frac{1-y}{y}$

The value of $\frac{1}{{\log }_{3}e}+\frac{1}{{\log }_3{e}^2}+\frac{1}{{\log }_3{e}^4}+\dots$ up to infinite terms is

1. A.${\log }_{e}9$
2. B.0
3. C.1
4. D.${\log }_{e}3$

If $1.3+2.3^2+3.3^3+\cdots +n.3^n=\frac{(2n-1)3^a+b}{4}$ then $a$ and $b$ are respectively

1. A.$n,2$
2. B.$n,3$
3. C.$n+1,2$
4. D.$n+1,3$

If $S_n=nP+\frac{n(n-1)Q}{2}$, where $S_n$ denotes the sum of the first $n$ terms of an AP, then the common difference is

1. A.$P+Q$
2. B.$2P+3Q$
3. C.$2Q$
4. D.$Q$

What is the sum of the series $0.3+0.33+0.333+\dots n$ terms?

1. A.$\frac{1}{3}\left[n-\frac{1}{9}\left(1-\frac{1}{10^n}\right)\right]$
2. B.$\frac{1}{3}\left[n-\frac{2}{9}\left(1-\frac{1}{10^n}\right)\right]$
3. C.$\frac{1}{3}\left[n-\frac{1}{3}\left(1-\frac{1}{10^n}\right)\right]$
4. D.$\frac{1}{3}\left[n-\frac{1}{9}\left(1+\frac{1}{10^n}\right)\right]$

If the sum of $m$ terms of an AP is $n$ and the sum of $n$ terms is $m$, then the sum of $(m+n)$ terms is

1. A.$mn$
2. B.$m+n$
3. C.$2(m+n)$
4. D.$-(m+n)$

The fifth term of an AP of $n$ terms, whose sum is $n^2-2n$, is

1. A.5
2. B.7
3. C.8
4. D.15

The sum of all the two-digit odd numbers is

1. A.2475
2. B.2530
3. C.4905
4. D.5049

The sum of the first $n$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\dots$ is equal to

1. A.$2^n-n-1$
2. B.$1-2^{-n}$
3. C.$2^{-n}+n-1$
4. D.$2^n-1$

The sum of the roots of the equation $x^2+bx+c=0$ (where $b$ and $c$ are non-zero) is equal to the sum of the reciprocals of their squares. Then $\frac{1}{c},b,\frac{c}{b}$ are in

1. A.AP
2. B.GP
3. C.HP
4. D.None of the above

The sum of the roots of the equation $a{x}^2+x+c=0$ (where $a$ and $c$ are non-zero) is equal to the sum of the reciprocals of their squares. Then $a,c{a}^2,c^2$ are in

1. A.AP
2. B.GP
3. C.HP
4. D.None of the above

Let $x,y,z$ be positive real numbers such that $x,y,z$ are in GP and ${\tan }^{-1}x$, ${\tan }^{-1}y$ and ${\tan }^{-1}z$ are in AP. Then which one of the following is correct?

1. A.$x=y=z$
2. B.$xz=1$
3. C.$x\ne y$ and $y=z$
4. D.$x=y$ and $y\ne z$

How many geometric progressions is/are possible containing 27, 8 and 12 as three of its/their terms ?

1. A.One
2. B.Two
3. C.Four
4. D.Infinitely many

Consider the following for the next three (03) items that follow : Let a, x, y, z, b be in AP, where $x+y+z=15$. Let a, p, q, r, b be in HP, where $p^{-1}+q^{-1}+r^{-1}=5/3$. What is the value of ab ?

1. A.10
2. B.9
3. C.8
4. D.6

Consider the following for the next three (03) items that follow : Let a, x, y, z, b be in AP, where $x+y+z=15$. Let a, p, q, r, b be in HP, where $p^{-1}+q^{-1}+r^{-1}=5/3$. What is the value of xyz ?

1. A.120
2. B.105
3. C.90
4. D.Cannot be determined

Consider the following for the next three (03) items that follow : Let a, x, y, z, b be in AP, where $x+y+z=15$. Let a, p, q, r, b be in HP, where $p^{-1}+q^{-1}+r^{-1}=5/3$. What is the value of pqr ?

1. A.35/243
2. B.81/35
3. C.243/35
4. D.Cannot be determined

What is the common difference of the AP so that the product of the first, fourth and fifth terms is greatest ?

1. A.8/5
2. B.9/5
3. C.2
4. D.11/5

What is the first term of the AP so that the product of the first, fourth and fifth terms is greatest ?

1. A.-4
2. B.-6
3. C.-8
4. D.-10

The interior angles of a polygon of n sides are in AP. The smallest angle is $120^{\circ }$ and the common difference is $5^{\circ }$. How many possible values can n have ?

1. A.One
2. B.Two
3. C.Three
4. D.Infinitely many