Sequences & Series · NDA previous year questions

1. A.$7x^2+4x+2=0$
2. B.$7x^2-4x+2=0$
3. C.$7x^2+8x+2=0$
4. D.$7x^2-8x+2=0$

Consider the following for the next two (02) items that follow : Given that $m(\theta )={\cot }^2\theta +n^2{\tan }^2\theta +2n$, where n is a fixed positive real number. What is the least value of $m(\theta )$ ?

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

Consider the following for the next two (02) items that follow : Given that $m(\theta )={\cot }^2\theta +n^2{\tan }^2\theta +2n$, where n is a fixed positive real number. Under what condition does m attain the least value ?

1. A.$n={\tan }^2\theta$
2. B.$n={\cot }^2\theta$
3. C.$n={\sin }^2\theta$
4. D.$n={\cos }^2\theta$

If a, b, c are in GP where $a>0$, $b>0$, $c>0$, then which of the following are correct? 1. $a^2$, $b^2$, $c^2$ are in GP 2. $\frac{1}{a}$, $\frac{1}{b}$, $\frac{1}{c}$ are in GP 3. $\sqrt{a}$, $\sqrt{b}$, $\sqrt{c}$ are in GP 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

If $\frac{a+b}{2},b,\frac{b+c}{2}$ are in HP, then which one of the following is correct?

1. A.a, b, c are in AP
2. B.a, b, c are in GP
3. C.a + b, b + c, c + a are in GP
4. D.a + b, b + c, c + a are in AP

What is value of ${\cot }^{2}15^{\circ }+{\tan }^{2}15^{\circ }$ ?

1. A.12
2. B.14
3. C.$8\sqrt{3}$
4. D.4

If the sum of the first 9 terms of an AP is equal to sum of the first 11 terms, then what is the sum of the first 20 terms ?

1. A.20
2. B.10
3. C.2
4. D.0

If the $5^{\text{th}}$ term of an AP is $\frac{1}{10}$ and its $10^{\text{th}}$ term is $\frac{1}{5}$, then what is the sum of first 50 terms?

1. A.25
2. B.$25\cdot 5$
3. C.26
4. D.$26\cdot 5$

What is $(1110011{)}_2÷(10111{)}_2$ equal to?

1. A.$(101{)}_2$
2. B.$(1001{)}_2$
3. C.$(111{)}_2$
4. D.$(1011{)}_2$

If $x^3+y^3=(100010111{)}_2$ and $x+y=(11111{)}_2$, then what is $(x-y{)}^2+xy$ equal to?

1. A.$(1101{)}_2$
2. B.$(1001{)}_2$
3. C.$(1011{)}_2$
4. D.$(1111{)}_2$

Consider the inequations 5x - 4y + 12 < 0, x + y < 2, x < 0 and y > 0. Which one of the following points lies in the common region?

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

If $x^2+x+1=0$, then what is the value of $x^{199}+x^{200}+x^{201}$?

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

If x, y, z are in GP, then which of the following is/are correct? 1. ln(3x), ln(3y), ln(3z) are in AP 2. xyz + ln(x), xyz + ln(y), xyz + ln(z) are in HP 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 ${\log }_{10}2$, ${\log }_{10}(2^x-1)$, ${\log }_{10}(2^x+3)$ are in AP, then what is x equal to?

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

Let S = {2, 3, 4, 5, 6, 7, 9}. How many different 3-digit numbers (with all digits different) from S can be made which are less than 500?

1. A.30
2. B.49
3. C.90
4. D.147

If $p=(1111\dots \text{up to }n\text{ digits})$, then what is the value of $9p^2+p$?

1. A.$10^{n}p$
2. B.$2p\cdot 10^n$
3. C.$10^{n}p-1$
4. D.$10^{n}p+1$

$\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ are in HP, then which of the following is/are correct? 1. a, b, c are in AP 2. $(b+c{)}^2,(c+a{)}^2,(a+b{)}^2$ are in GP. 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 C(n, 4), C(n, 5) and C(n, 6) are in AP, then what is the value of n?

1. A.7
2. B.8
3. C.9
4. D.10

How many 4 - letter words (with or without meaning) containing two vowels can be constructed using only the letters (without repetition) of the word 'LUCKNOW'?

1. A.240
2. B.200
3. C.150
4. D.120

Let $S_k$ denote the sum of first k terms of an AP. What is $\frac{S_{30}}{S_{20}-S_{10}}$ equal to?

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

If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then what is the sum of the first ten terms?

1. A.-500
2. B.-250
3. C.500
4. D.250

Consider the following statements: 1. If each term of a GP is multiplied by same non-zero number, then the resulting sequence is also a GP. 2. If each term of a GP is divided by same non-zero number, then the resulting sequence is also a GP. 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

How many 5-digit prime numbers can be formed using the digits 1, 2, 3, 4, 5 if the repetition of digits is not allowed?

1. A.5
2. B.4
3. C.3
4. D.0

If $f(x+1)=x^2-3x+2$, then what is $f(x)$ equal to?

1. A.$x^2-5x+4$
2. B.$x^2-5x+6$
3. C.$x^2+3x+3$
4. D.$x^2-3x+1$

If $x^2$, $x$, $-8$ are in AP, then which one of the following is correct?

1. A.$x\in \{-2\}$
2. B.$x\in \{4\}$
3. C.$x\in \{-2,4\}$
4. D.$x\in \{-4,2\}$