Sequences & Series · NDA previous year questions

Passage: Let $X(a,p),Y(b,q)$ and $Z(c,r)$ be the points such that $a,b$ and $c$ are in AP. Question: If $p,q$ and $r$ are not in AP and $b=c$, then the line joining the points $X,Y$ and $Z$ is parallel to

1. A.$y$-axis
2. B.$x$-axis
3. C.$y=x$
4. D.$y=-x$

If $p^{\text{th}}$ term of an AP is $k$, then what is the sum of $p^{\text{th}}$ term, $(p+q{)}^{\text{th}}$ term and $(p-q{)}^{\text{th}}$ term of the AP ?

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

The sum of the first $n$ terms of an AP is $3n^2+5n$. If the $m^{\text{th}}$ term of the AP is 68, then what is the value of $m$ ?

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

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

If $p^x=q^y=r^z$ where x, y and z are in GP, then consider the following statements: I. p, q and r are in AP. II. $\ln p,\ln q$ and $\ln r$ are in GP. Which of the statements given above is/are correct?

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

If $(10+{\log }_{10}x),(10+{\log }_{10}y)$ and $(10+{\log }_{10}z)$ are in AP, then consider the following statements: I. The GM of x and z is $y^2$ II. The AM of ${\log }_{10}x$ and ${\log }_{10}z$ is ${\log }_{10}y$ Which of the statements given above is/are correct?

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

How many terms of the series 1 + 3 + 5 + 7 +... amount to a sum equal to 12345678987654321?

1. A.11111111
2. B.110000011
3. C.111101111
4. D.111111111

How many terms are identical in the two APs 19, 21, 23,... up to 110 terms and 19, 22, 25, 28,... up to 75 terms?

1. A.35
2. B.36
3. C.37
4. D.38

If $\alpha =\frac{-1+\sqrt{-3}}{2}$ then what is the value of $(1+{\alpha }^{19}-{\alpha }^{35}{)}^{100}-(1-3{\alpha }^{25}+{\alpha }^{38}{)}^{50}$?

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

What is the remainder when $5^{99}$ is divided by 13?

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

If the roots of the equation are equal, then a, b, c are in

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

The sum of the first k terms of a series S is $3k^2+5k$. Which one of the following is correct?

1. A.The terms of S form an arithmetic progression with common difference 14.
2. B.The terms of S form an arithmetic progression with common difference 6.
3. C.The terms of S form a geometric progression with common ratio 10/7.
4. D.The terms of S form a geometric progression with common ratio 11/4.

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r≠1 is the common ratio, then what is the number of possible real values of r?

1. A.One
2. B.Two
3. C.Three
4. D.More than three

If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

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

If p, 1, q are in AP and p, 2, q are in GP, then which of the following statements is/are correct? I. p, 4, q are in HP. II. (1/p), (1/4), (1/q) are in AP. Select the answer using the code given below.

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

If $x=(1111{)}_2$, $y=(1001{)}_2$ and $z=(110{)}_2$, then what is $x^3-y^3-z^3-3xyz$ equal to?

1. A.$(111001{)}_2$
2. B.$(100111{)}_2$
3. C.$(1{)}_2$
4. D.$(0{)}_2$

1. A.0
2. B.1
3. C.Δ
4. D.−Δ

If p times the pth term of an AP is equal to q times the qth term (p ≠ q), then what is the (p + q)th term equal to?

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

Let $p=\ln (x),q=\ln (x^3)$ and $r=\ln (x^5)$, where $x>1$. Which of the following statements is/are correct? I. p, q and r are in AP. II. p, q and r can never be in GP. Select the answer using the code given below.

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

Let $x>1,y>1,z>1$ be in GP. Then $\frac{1}{1+\ln x}$, $\frac{1}{1+\ln y}$, $\frac{1}{1+\ln z}$ are

1. A.in AP
2. B.in GP
3. C.in HP
4. D.neither in AP nor in GP nor in HP

If the sum of the first n terms of a series is n(2n+1), then what is the nth term?

1. A.4n - 1
2. B.4n
3. C.4n + 1
4. D.4n + 3

In an AP, the ratio of the sum of the first p terms to the sum of the first q terms is $p^2:q^2$. Which one of the following is correct?

1. A.The first term is equal to the common difference
2. B.The first term is equal to twice the common difference
3. C.The common difference is equal to twice the first term
4. D.The first term is equal to square of the common difference

Direction : Consider the following for the items that follow : The product of 5 consecutive terms of an AP is 229635. The first, second and fifth terms are in GP. What is the common difference?

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

What is the sum of all five terms?

1. A.60
2. B.65
3. C.75
4. D.80

= W, where U is an integer and 0 < V < 1. What is V + W equal to?

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