Sequences & Series · NDA previous year questions

What is the largest interior angle of the polygon ?

1. A.$160^{\circ }$ only
2. B.$195^{\circ }$ only
3. C.Either $160^{\circ }$ or $195^{\circ }$
4. D.Neither $160^{\circ }$ nor $195^{\circ }$

If $x^{\ln \left(\frac{y}{z}\right)}\cdot y^{\ln (xz{)}^2}\cdot z^{\ln \left(\frac{x}{y}\right)}=y^{4\ln y}$ for any $x>1$, $y>1$ and $z>1$, then which one of the following is correct ?

1. A.$\ln y$ is the GM of $\ln x$, $\ln x$, $\ln x$ and $\ln z$
2. B.$\ln y$ is the AM of $\ln x$, $\ln x$, $\ln x$ and $\ln z$
3. C.$\ln y$ is the HM of $\ln x$, $\ln x$, $\ln x$ and $\ln z$
4. D.$\ln y$ is the AM of $\ln x$, $\ln x$, $\ln z$ and $\ln z$

Consider the following for the next two (02) items that follow : $2x^2+3x-\alpha =0$ has roots $-2$ and $\beta$ while the equation $x^2-3mx+2m^2=0$ has both roots positive, where $\alpha >0$ and $\beta >0$. If $\beta$, 2, 2m are in GP, then what is the value of $\beta \sqrt{m}$?

1. A.1
2. B.2
3. C.4
4. D.6

If $c>0$ and $4a+c<2b$, then $a{x}^2-bx+c=0$ has a root in which one of the following intervals ?

1. A.(0, 2)
2. B.(2, 3)
3. C.(3, 4)
4. D.(-2, 0)

What is the number of odd integers between 1000 and 9999 with no digit repeated ?

1. A.2100
2. B.2120
3. C.2240
4. D.3331

For the next two (2) items that follow : Given that ${\log }_{x}y,{\log }_{z}x,{\log }_{y}z$ are in GP, $xyz=64$ and $x^3,y^3,z^3$ are in AP. Which one of the following is correct? $x,y$ and $z$ are

1. A.in AP only
2. B.in GP only
3. C.in both AP and GP
4. D.neither in AP nor in GP

For the next two (2) items that follow : Given that ${\log }_{x}y,{\log }_{z}x,{\log }_{y}z$ are in GP, $xyz=64$ and $x^3,y^3,z^3$ are in AP. Which one of the following is correct? $xy,yz$ and $zx$ are

1. A.in AP only
2. B.in GP only
3. C.in both AP and GP
4. D.neither in AP nor in GP

For the next two (2) items that follow : Given that $$a_n={\int }_0^{\pi }\frac{{\sin }^2\{(n+1)x\}}{\sin 2x}dx$$ Consider the following statements : 1. The sequence $\{a_{2n}\}$ is in AP with common difference zero. 2. The sequence $\{a_{2n-1}\}$ is in AP with common difference zero. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For the next two (2) items that follow : Given that $$a_n={\int }_0^{\pi }\frac{{\sin }^2\{(n+1)x\}}{\sin 2x}dx$$ What is $a_{n-1}-a_{n-4}$ equal to?

1. A.$-1$
2. B.0
3. C.1
4. D.2

If $p,q,r$ are in one geometric progression and $a,b,c$ are in another geometric progression, then $ap,bq,cr$ are in

1. A.Arithmetic progression
2. B.Geometric progression
3. C.Harmonic progression
4. D.None of the above

What is the sum of $n$ terms of the series $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\dots$?

1. A.$\frac{n(n-1)}{\sqrt{2}}$
2. B.$\sqrt{2}n(n+1)$
3. C.$\frac{n(n+1)}{\sqrt{2}}$
4. D.$\frac{n(n-1)}{2}$

What is the sum of the series $0.5+0.55+0.555+\cdots$ to $n$ terms?

1. A.$\frac{5}{9}\left[n-\frac{2}{9}\left(1-\frac{1}{10^n}\right)\right]$
2. B.$\frac{1}{9}\left[5-\frac{2}{9}\left(1-\frac{1}{10^n}\right)\right]$
3. C.$\frac{1}{9}\left[n-\frac{5}{9}\left(1-\frac{1}{10^n}\right)\right]$
4. D.$\frac{5}{9}\left[n-\frac{1}{9}\left(1-\frac{1}{10^n}\right)\right]$

If $${\log }_{10}2,{\log }_{10}(2^x-1)\text{ and }{\log }_{10}(2^x+3)$$ are three consecutive terms of an AP, then the value of $x$ is

1. A.1
2. B.${\log }_{5}2$
3. C.${\log }_{2}5$
4. D.${\log }_{10}5$

For the next two (02) items that follow : Let $S_n$ denote the sum of first n terms of an AP and $3S_n=S_{2n}$. What is $S_{3n}:S_n$ equal to?

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

For the next two (02) items that follow : Let $S_n$ denote the sum of first n terms of an AP and $3S_n=S_{2n}$. What is $S_{3n}:S_{2n}$ equal to?

1. A.$2:1$
2. B.$3:1$
3. C.$4:1$
4. D.$5:1$

For the next three (03) items that follow: Let $f(x)=a{x}^2+bx+c$ such that $f(1)=f(-1)$ and a, b, c are in Arithmetic Progression. $f^{\prime }(a),f^{\prime }(b),f^{\prime }(c)$ are in

1. A.A.P.
2. B.G.P.
3. C.H.P.
4. D.Arithmetico-geometric progression

For the next three (03) items that follow: Let $f(x)=a{x}^2+bx+c$ such that $f(1)=f(-1)$ and a, b, c are in Arithmetic Progression. $f^{\prime \prime }(a),f^{\prime \prime }(b),f^{\prime \prime }(c)$ are

1. A.in A.P. only
2. B.in G.P. only
3. C.in both A.P. and G.P.
4. D.neither in A.P. nor in G.P.

The sum of an infinite GP is $x$ and the common ratio $r$ is such that $|r|<1$. If the first term of the GP is 2, then which one of the following is correct?

1. A.$-1<x<1$
2. B.$-\infty <x<1$
3. C.$1<x<\infty$
4. D.None of the above

The sum of the series formed by the sequence $3,\sqrt{3},1\dots \dots$ upto infinity is:

1. A.$\frac{3\sqrt{3}(\sqrt{3}+1)}{2}$
2. B.$\frac{3\sqrt{3}(\sqrt{3}-1)}{2}$
3. C.$\frac{3(\sqrt{3}+1)}{2}$
4. D.$\frac{3(\sqrt{3}-1)}{2}$

If the positive integers $a,b,c,d$ are in AP, then the numbers $abc,abd,acd,bcd$ are in

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

The sum of the first five terms and the sum of the first ten terms of an AP are same. Which one of the following is the correct statement?

1. A.The first term must be negative
2. B.The common difference must be negative
3. C.Either the first term or the common difference is negative but not both
4. D.Both the first term and the common difference are negative

If the numbers $n-3,4n-2,5n+1$ are in AP, what is the value of n ?

1. A.1
2. B.2
3. C.3
4. D.4

What is the sum of first eight terms of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots \dots$?

1. A.$\frac{89}{128}$
2. B.$\frac{57}{384}$
3. C.$\frac{85}{128}$
4. D.None of the above

FOR THE NEXT TWO (02) QUESTIONS THAT FOLLOW: The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively. What is its first term?

1. A.2
2. B.3
3. C.4
4. D.5

What is the common difference?

1. A.1
2. B.2
3. C.3
4. D.4