Sequences & Series · NDA previous year questions

If $p^2$, $q^2$ and $r^2$ (where $p,q,r>0$) are in GP, then which of the following is/are correct? 1. $p,q$ and $r$ are in GP. 2. $\ln p$, $\ln q$ and $\ln r$ are in AP. Select the correct answer using the code given below:

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

If 3rd, 8th and 13th terms of a GP are $p,q$ and $r$ respectively, then which one of the following is correct?

1. A.$q^2=pr$
2. B.$r^2=pq$
3. C.$pqr=1$
4. D.$2q=p+r$

Let $S_n$ be the sum of the first $n$ terms of an AP. If $S_{2n}=3n+14n^2$, then what is the common difference?

1. A.5
2. B.6
3. C.7
4. D.9

How many two-digit numbers are divisible by 4?

1. A.21
2. B.22
3. C.24
4. D.25

If the sum of first $n$ terms of a series is $(n+12)$, then what is its third term?

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

Let $a,b,c$ be in AP and $k\ne 0$ be a real number. Which of the following are correct? 1. $ka,kb,kc$ are in AP 2. $k-a,k-b,k-c$ are in AP 3. $\frac{a}{k},\frac{b}{k},\frac{c}{k}$ are in AP Select the correct answer using the code given below :

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

Let $m$ and $n$ ($m<n$) be the roots of the equation $x^2-16x+39=0$. If four terms $p,q,r$ and $s$ are inserted between $m$ and $n$ to form an AP, then what is the value of $p+q+r+s$?

1. A.29
2. B.30
3. C.32
4. D.35

A geometric progression (GP) consists of 200 terms. If the sum of odd terms of the GP is $m$, and the sum of even terms of the GP is $n$, then what is its common ratio?

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

What is the value of $1-2+3-4+5-\dots +101$?

1. A.51
2. B.55
3. C.110
4. D.111

If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal, then which one of the following is correct?

1. A.$a$, $b$ and $c$ are in AP
2. B.$a$, $b$ and $c$ are in GP
3. C.$a$, $b$ and $c$ are in HP
4. D.$a$, $b$ and $c$ do not follow any regular pattern

If $|x^2-3x+2|>x^2-3x+2$, then which one of the following is correct?

1. A.$x\le 1$ or $x\ge 2$
2. B.$1\le x\le 2$
3. C.$1<x<2$
4. D.$x$ is any real value except 3 and 4

What is $C(47,4)+C(51,3)+C(50,3)+C(49,3)+C(48,3)+C(47,3)$ equal to?

1. A.$C(47,4)$
2. B.$C(52,5)$
3. C.$C(52,4)$
4. D.$C(47,5)$

If $n!$ has 17 zeros, then what is the value of $n$?

1. A.95
2. B.85
3. C.80
4. D.No such value of $n$ exists

What is the number of diagonals of an octagon?

1. A.48
2. B.40
3. C.28
4. D.20

If $P(n,r)=2520$ and $C(n,r)=21$, then what is the value of $C(n+1,r+1)$?

1. A.7
2. B.14
3. C.28
4. D.56

If the angles of a triangle $ABC$ are in AP and $b:c=\sqrt{3}:\sqrt{2}$, then what is the measure of angle $A$?

1. A.$30^{\circ }$
2. B.$45^{\circ }$
3. C.$60^{\circ }$
4. D.$75^{\circ }$

1. A.$|2m\sin (\alpha -\beta )|$
2. B.$|2m\cos (\alpha -\beta )|$
3. C.$|m\sin (2\alpha -2\beta )|$
4. D.$|m\cos (2\alpha -2\beta )|$

An equilateral triangle has one vertex at $(-1,-1)$ and another vertex at $(-\sqrt{3},\sqrt{3})$. The third vertex may lie on

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

The numbers 1, 5 and 25 can be three terms (not necessarily consecutive) of

1. A.only one AP
2. B.more than one but finite numbers of APs
3. C.infinite number of APs
4. D.finite number of GPs

The sum of $(p+q{)}^{th}$ and $(p-q{)}^{th}$ terms of an AP is equal to

1. A.$(2p{)}^{th}$ term
2. B.$(2q{)}^{th}$ term
3. C.Twice the $p^{th}$ term
4. D.Twice the $q^{th}$ term

What is the fourth term of an AP of n terms whose sum is $n(n+1)$?

1. A.6
2. B.8
3. C.12
4. D.20

If an infinite GP has the first term x and the sum 5, then which one of the following is correct?

1. A.$x<-10$
2. B.$-10<x<0$
3. C.$0<x<10$
4. D.$x>10$

The sum of the series $3-1+\frac{1}{3}-\frac{1}{9}+\dots$ is equal to

1. A.$\frac{20}{9}$
2. B.$\frac{9}{20}$
3. C.$\frac{9}{4}$
4. D.$\frac{4}{9}$

Let $T_r$ be the $r^{th}$ term of an AP for $r=1,2,3,\dots$. If for some distinct positive integers m and n we have $T_m=1/n$ and $T_n=1/m$, then what is $T_{mn}$ equal to?

1. A.$(mn{)}^{-1}$
2. B.$m^{-1}+n^{-1}$
3. C.1
4. D.0

If the second term of a GP is 2 and the sum of its infinite terms is 8, then the GP is

1. A.$8,2,\frac{1}{2},\frac{1}{8},\dots$
2. B.$10,2,\frac{2}{5},\frac{2}{25},\dots$
3. C.$4,2,1,\frac{1}{2},\frac{1}{2^2},\dots$
4. D.$6,3,\frac{3}{2},\frac{3}{4},\dots$