Sequences & Series · NDA previous year questions

What does the series $1+3^{-\frac{1}{2}}+3+\frac{1}{3\sqrt{3}}+\dots$ represent?

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

If p, q, r are in AP as well as GP, then which one of the following is correct?

1. A.$p=q\ne r$
2. B.$p\ne q\ne r$
3. C.$p\ne q=r$
4. D.$p=q=r$

If the sequence $\{s_n\}$ is a geometric progression and $s_2{s}_{11}=s_p{s}_8$, then what is the value of p?

1. A.1
2. B.3
3. C.5
4. D.can not be determined

If 1/4, 1/x, 1/10 are in HP, then what is the value of x?

1. A.5
2. B.6
3. C.7
4. D.8

If $f(xy)=f(x)f(y)$, then $f(t)$ may be of the form: where k, c are constants

1. A.$t+k$
2. B.$ct+k$
3. C.$t^k+c$
4. D.$t^k$

If $A+iB=\frac{4+2i}{1-2i}$ where $i=\sqrt{-1}$, then what is the value of A?

1. A.$-8$
2. B.0
3. C.4
4. D.8

What is the sum of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots$ equal to?

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

What is the $10^{th}$ common term between the series $2+6+10+\dots \dots$ and $1+6+11+\dots \dots$?

1. A.180
2. B.186
3. C.196
4. D.206

If the $10^{th}$ term of a GP is 9 and $4^{th}$ term is 4, then what is its $7^{th}$ term?

1. A.6
2. B.14
3. C.27/14
4. D.56/15

If ${\log }_{10}2$, ${\log }_{10}(2^x-1)$, ${\log }_{10}(2^x+3)$ are three consecutive terms of an AP, then which one of the following is correct?

1. A.$x=0$
2. B.$x=1$
3. C.$x={\log }_{2}5$
4. D.$x={\log }_{5}2$

If $n!$, $3\times (n!)$ and $(n+1)!$ are in GP, then the value of $n$ will be

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

If a, b, c, d, e, f are in AP, then $(e-c)$ is equal to which one of the following ?

1. A.$2(c-a)$
2. B.$2(d-c)$
3. C.$2(f-d)$
4. D.$(d-c)$

If the arithmetic and geometric means of two numbers are 10, 8 respectively, then one number exceeds the other number by

1. A.8
2. B.10
3. C.12
4. D.16

Which one of the following statements is correct?

1. A.${\sin }^{2}30^{\circ },{\sin }^{2}45^{\circ },{\sin }^{2}60^{\circ }$ are in GP
2. B.${\cos }^{2}30^{\circ },{\cos }^{2}45^{\circ },{\cos }^{2}60^{\circ }$ are in GP
3. C.${\cot }^{2}30^{\circ },{\cot }^{2}45^{\circ },{\cot }^{2}60^{\circ }$ are in AP
4. D.${\tan }^{2}30^{\circ },{\tan }^{2}45^{\circ },{\tan }^{2}60^{\circ }$ are in GP

In a GP of positive terms, any term is equal to one-third of the sum of next two terms. What is the common ratio of the GP?

1. A.$\frac{\sqrt{13}+1}{2}$
2. B.$\frac{\sqrt{13}-1}{2}$
3. C.$\frac{\sqrt{13}+1}{3}$
4. D.$\sqrt{13}$

If $(1+\tan \theta )(1+\tan \phi )=2$, then what is $(\theta +\phi )$ equal to?

1. A.$30^{\circ }$
2. B.$45^{\circ }$
3. C.$60^{\circ }$
4. D.$90^{\circ }$

Which term of the series $\frac{1}{4},-\frac{1}{2},1,\dots$ is $-128$?

1. A.9th
2. B.10th
3. C.11th
4. D.12th

What is the number of ordered pairs of non-zero positive integers $(x,y)$ such that $x+y\le 4$?

1. A.4
2. B.5
3. C.6
4. D.8

If $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$ then $a,b,c$ are in

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

If $A,B,C$ are in AP and $b:c=\sqrt{3}:\sqrt{2}$, then what is the value of $\sin C$?

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

The sum of an infinite geometric progression is 6. If the sum of the first two terms is $9/2$, then what is the first term?

1. A.1
2. B.$5/2$
3. C.3 or $3/2$
4. D.9 or 3

If p times the $p^{\text{th}}$ term of an AP is q times the $q^{\text{th}}$ term, then what is the $(p+q{)}^{\text{th}}$ term equal to?

1. A.$p+q$
2. B.$pq$
3. C.1
4. D.0

The $59^{\text{th}}$ term of an AP is 449 and the $449^{\text{th}}$ term is 59. Which term is equal to 0 (zero)?

1. A.$501^{\text{st}}$ term
2. B.$502^{\text{nd}}$ term
3. C.$508^{\text{th}}$ term
4. D.$509^{\text{th}}$ term

If angles A, B, C are in AP, then what is $\sin A+2\sin B+\sin C$ equal to?

1. A.$4\sin B{\cos }^2\left(\frac{A-C}{2}\right)$
2. B.$4\sin B{\cos }^2\left(\frac{A-C}{4}\right)$
3. C.$4\sin (2B){\cos }^2\left(\frac{A-C}{2}\right)$
4. D.$4\sin (2B){\cos }^2\left(\frac{A-C}{4}\right)$

Statement I: If $-1\le x<0$, then $\cos ({\sin }^{-1}x)=-\sqrt{1-x^2}$. Statement II: If $-1\le x<0$, then $\sin ({\cos }^{-1}x)=\sqrt{1-x^2}$. Which one of the following is correct in respect of the above statements?

1. A.Both statements I and II are independently correct and statement II is the correct explanation of statement I
2. B.Both statements I and II are independently correct but statement II is not the correct explanation of statement I
3. C.Statement I is correct but statement II is false
4. D.Statement I is false but statement II is correct