Sequences & Series · NDA previous year questions

What is the value of (U + V)W?

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

$a^2(b^2-c^2)x^2+b^2(c^2-a^2)x+c^2(a^2-b^2)=0$ are equal $(a^2\ne b^2\ne c^2)$. Which one of the following statements is correct?

1. A.$a^2$, $b^2$, $c^2$ are in AP.
2. B.$a^2$, $b^2$, $c^2$ are in GP.
3. C.$a^2$, $b^2$, $c^2$ are in HP.
4. D.$a^2$, $b^2$, $c^2$ are neither in AP nor in GP nor in HP.

If $a,b,c$ are in AP; $b,c,d$ are in GP; $c,d,e$ are in HP, then which of the following is/are correct? 1. $a,c,$ and $e$ are in GP 2. $\frac{1}{a},\frac{1}{c},\frac{1}{e}$ 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

What is the number of solutions of ${\log }_4(x-1)={\log }_2(x-3)$?

1. A.Zero
2. B.One
3. C.Two
4. D.Three

For $x\ge y>1$, let ${\log }_x\left(\frac{x}{y}\right)+{\log }_y\left(\frac{y}{x}\right)=k$, then the value of $k$ can never be equal to :

1. A.$-1$
2. B.$-\frac{1}{2}$
3. C.0
4. D.1

If $a,b,c$ are in HP, then what is $\frac{1}{b-a}+\frac{1}{b-c}$ equal to? 1. $\frac{2}{b}$ 2. $\frac{1}{a}+\frac{1}{c}$ 3. $\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$ Select the correct answer using the code given below:

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

If (a + b), 2b, (b + c) are in HP, then which one of the following is correct?

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

Let $t_1,t_2,t_3\dots$ be in GP. What is $(t_1{t}_3\dots t_{21}{)}^{\frac{1}{11}}$ equal to ?

1. A.$t_{10}$
2. B.$t_{10}^2$
3. C.$t_{11}$
4. D.$t_{11}^2$

Which one of the following is a square root of $-\sqrt{-1}$?

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

equal to ?

1. A.0
2. B.25
3. C.125
4. D.250

Let $a_1,a_2,a_3\dots$ be in AP such that $a_1+a_5+a_{10}+a_{15}+a_{20}+a_{25}+a_{30}+a_{34}=300$. What is ${\sum }_{n=1}^{34}{a}_n$ equal to ?

1. A.900
2. B.1025
3. C.1200
4. D.1275

Consider the following for the next items that follow: Let $p=\cos \left(\frac{\pi }{5}\right)\cos \left(\frac{2\pi }{5}\right)$ and $q=\cos \left(\frac{4\pi }{5}\right)\cos \left(\frac{8\pi }{5}\right)$. What is the value of p + q?

1. A.$-\frac{1}{2}$
2. B.$-\frac{1}{4}$
3. C.0
4. D.$\frac{1}{2}$

Consider the following for the next items that follow: Let $p=\cos \left(\frac{\pi }{5}\right)\cos \left(\frac{2\pi }{5}\right)$ and $q=\cos \left(\frac{4\pi }{5}\right)\cos \left(\frac{8\pi }{5}\right)$. What is the value of pq?

1. A.$-\frac{1}{16}$
2. B.$-\frac{1}{4}$
3. C.$\frac{1}{4}$
4. D.$\frac{1}{16}$

Consider the following for the next items that follow: Let $p=\frac{1}{3}-\frac{\tan 3x}{\tan x}$ and $q=1-3{\tan }^{2}x$, $0<x<\pi$, $x\ne \frac{\pi }{2}$. What is pq equal to ?

1. A.1
2. B.2
3. C.$\frac{8}{3}$
4. D.$-\frac{8}{3}$

Consider the following for the next items that follow: Let $p=\frac{1}{3}-\frac{\tan 3x}{\tan x}$ and $q=1-3{\tan }^{2}x$, $0<x<\pi$, $x\ne \frac{\pi }{2}$. For how many values of x does $\frac{1}{p}$ become zero?

1. A.No value
2. B.Only one value
3. C.Only two values
4. D.Only three values

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

If ${\log }_{x}a$, $a^x$ and ${\log }_{b}x$ are in GP, then what is x equal to ?

1. A.${\log }_a({\log }_{b}a)$
2. B.${\log }_b({\log }_{a}b)$
3. C.$\frac{{\log }_a({\log }_{b}a)}{2}$
4. D.$\frac{{\log }_b({\log }_{a}b)}{2}$

If $2^{\frac{1}{c}},2^{\frac{b}{ac}},2^{\frac{1}{a}}$ are in GP, 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, c are in HP
4. D.ab, bc, ca are in AP

The first and the second terms of an AP are $\frac{5}{2}$ and $\frac{23}{12}$ respectively. If ${\mathrm{n}}^{\mathrm{th}}$ term is the largest negative term, what is the value of n ?

1. A.5
2. B.6
3. C.7
4. D.n cannot be determined

In an AP, the first term is x and the sum of the first n terms is zero. What is the sum of next m terms ?

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

Consider the following statements : 1. (25)! + 1 is divisible by 26 2. (6)! + 1 is divisible by 7 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

p, q, r and s are in AP such that p + s = 8 and qr = 15. What is the difference between largest and smallest numbers ?

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

Consider the following statements for a fixed natural number n: 1. C(n, r) is greatest if n = 2r 2. C(n, r) is greatest if n = 2r - 1 and n = 2r + 1 Which of the statements given above is/are correct ?

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

m parallel lines cut n parallel lines giving rise to 60 parallelograms. What is the value of (m + n) ?

1. A.6
2. B.7
3. C.8
4. D.9

Consider the following for the next two (02) items that follow : Let $\Delta (a,b,c,\alpha )=\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|$ If $\Delta (a,b,c,\alpha )=0$ for every $\alpha >0$, 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, 2b, c are in AP
4. D.a, 2b, c are in GP