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0
votes
0
answers
1
ISI2014DCG61
If $l=1+a+a^2+ \dots$, $m=1+b+b^2+ \dots$, and $n=1+c+c^2+ \dots$, where $\mid a \mid <1, \: \mid b \mid < 1, \: \mid c \mid <1$ and $a,b,c$ are in arithmetic progression, then $l, m, n$ are in arithmetic progression geometric progression harmonic progression none of these
asked
Sep 23, 2019
in
Numerical Ability

24
views
isi2014dcg
numericalability
arithmeticseries
0
votes
1
answer
2
ISI2014DCG62
If the sum of the first $n$ terms of an arithmetic progression is $cn^2$, then the sum of squares of these $n$ terms is $\frac{n(4n^21)c^2}{6}$ $\frac{n(4n^2+1)c^2}{3}$ $\frac{n(4n^21)c^2}{3}$ $\frac{n(4n^2+1)c^2}{6}$
asked
Sep 23, 2019
in
Numerical Ability

22
views
isi2014dcg
numericalability
arithmeticseries
+2
votes
1
answer
3
ISI2014DCG63
If $^nC_{r1}=36$, $^nC_r=84$ an $^nC_{r+1}=126$ then $r$ is equal to $1$ $2$ $3$ none of these
asked
Sep 23, 2019
in
Combinatory

28
views
isi2014dcg
permutationandcombination
0
votes
1
answer
4
ISI2014DCG64
The value of $\lambda$ such that the system of equation $\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & 4 \\ x & + & y & + & \lambda z & = & 4 \end{array}$ has no solution is $3$ $1$ $0$ $3$
asked
Sep 23, 2019
in
Linear Algebra

59
views
isi2014dcg
linearalgebra
matrices
systemofequations
+1
vote
0
answers
5
ISI2014DCG65
The sum $\dfrac{n}{n^2}+\dfrac{n}{n^2+1^2}+\dfrac{n}{n^2+2^2}+ \cdots + \dfrac{n}{n^2+(n1)^2} + \cdots \cdots$ is $\frac{\pi}{4}$ $\frac{\pi}{8}$ $\frac{\pi}{6}$ $2 \pi$
asked
Sep 23, 2019
in
Numerical Ability

52
views
isi2014dcg
numericalability
summation
nongate
+1
vote
1
answer
6
ISI2014DCG66
Consider all possible words obtained by arranging all the letters of the word $\textbf{AGAIN}$. These words are now arranged in the alphabetical order, as in a dictionary. The fiftieth word in this arrangement is $\text{IAANG}$ $\text{NAAGI}$ $\text{NAAIG}$ $\text{IAAGN}$
asked
Sep 23, 2019
in
Combinatory

29
views
isi2014dcg
permutationandcombination
arrangements
0
votes
1
answer
7
ISI2014DCG67
Let $y=[\:\log_{10}3245.7\:]$ where $[ a ]$ denotes the greatest integer less than or equal to $a$. Then $y=0$ $y=1$ $y=2$ $y=3$
asked
Sep 23, 2019
in
Numerical Ability

16
views
isi2014dcg
numericalability
logarithms
0
votes
1
answer
8
ISI2014DCG68
The number of integer solutions for the equation $x^2+y^2=2011$ is $0$ $1$ $2$ $3$
asked
Sep 23, 2019
in
Numerical Ability

23
views
isi2014dcg
numericalability
integersolutions
+1
vote
1
answer
9
ISI2014DCG69
The number of ways in which the number $1440$ can be expressed as a product of two factors is equal to $18$ $720$ $360$ $36$
asked
Sep 23, 2019
in
Numerical Ability

34
views
isi2014dcg
numericalability
numbersystem
factors
0
votes
0
answers
10
ISI2014DCG70
For the matrices $A = \begin{pmatrix} a & a \\ 0 & a \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $(B^{1}AB)^3$ is equal to $\begin{pmatrix} a^3 & a^3 \\ 0 & a^3 \end{pmatrix}$ ... $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$
asked
Sep 23, 2019
in
Linear Algebra

38
views
isi2014dcg
linearalgebra
matrices
inverse
+1
vote
2
answers
11
ISI2014DCG71
Five letters $A, B, C, D$ and $E$ are arranged so that $A$ and $C$ are always adjacent to each other and $B$ and $E$ are never adjacent to each other. The total number of such arrangements is $24$ $16$ $12$ $32$
asked
Sep 23, 2019
in
Combinatory

38
views
isi2014dcg
permutationandcombination
arrangements
circularpermutation
+1
vote
1
answer
12
ISI2014DCG72
The sum $\sum_{k=1}^n (1)^k \:\: {}^nC_k \sum_{j=0}^k (1)^j \: \: {}^kC_j$ is equal to $1$ $0$ $1$ $2^n$
asked
Sep 23, 2019
in
Combinatory

38
views
isi2014dcg
permutationandcombination
summation
+1
vote
1
answer
13
ISI2015MMA1
Let $\{f_n(x)\}$ be a sequence of polynomials defined inductively as $ f_1(x)=(x2)^2$ $f_{n+1}(x) = (f_n(x)2)^2, \: \: \: n \geq 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x)$. Then $a_n=4, \: b_n=4^n$ $a_n=4, \: b_n=4n^2$ $a_n=4^{(n1)!}, \: b_n=4^n$ $a_n=4^{(n1)!}, \: b_n=4n^2$
asked
Sep 23, 2019
in
Combinatory

38
views
isi2015mma
recurrencerelations
nongate
0
votes
1
answer
14
ISI2015MMA2
If $a,b$ are positive real variables whose sum is a constant $\lambda$, then the minimum value of $\sqrt{(1+1/a)(1+1/b)}$ is $\lambda \: – 1/\lambda$ $\lambda + 2/\lambda$ $\lambda+1/\lambda$ None of the above
asked
Sep 23, 2019
in
Numerical Ability

52
views
isi2015mma
numericalability
numbersystem
minimumvalue
nongate
+1
vote
1
answer
15
ISI2015MMA3
Let $x$ be a positive real number. Then $x^2+\pi ^2 + x^{2 \pi} > x \pi+ (\pi + x) x^{\pi}$ $x^{\pi}+\pi^x > x^{2 \pi} + \pi ^{2x}$ $\pi x +(\pi+x)x^{\pi} > x^2+\pi ^2 + x^{2 \pi}$ none of the above
asked
Sep 23, 2019
in
Numerical Ability

38
views
isi2015mma
numbersystem
nongate
+2
votes
3
answers
16
ISI2015MMA4
Suppose in a competition $11$ matches are to be played, each having one of $3$ distinct outcomes as possibilities. The number of ways one can predict the outcomes of all $11$ matches such that exactly $6$ of the predictions turn out to be correct is $\begin{pmatrix}11 \\ 6 \end{pmatrix} \times 2^5$ $\begin{pmatrix}11 \\ 6 \end{pmatrix} $ $3^6$ none of the above
asked
Sep 23, 2019
in
Combinatory

46
views
isi2015mma
permutationandcombination
+1
vote
2
answers
17
ISI2015MMA5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is $2^n$ $2^{n+1}$ $2^{n1}$ $2^{2n}$
asked
Sep 23, 2019
in
Set Theory & Algebra

49
views
isi2015mma
sets
subsets
0
votes
1
answer
18
ISI2015MMA6
A club with $x$ members is organized into four committees such that each member is in exactly two committees, any two committees have exactly one member in common. Then $x$ has exactly two values both between $4$ and $8$ exactly one value and this lies between $4$ and $8$ exactly two values both between $8$ and $16$ exactly one value and this lies between $8$ and $16$
asked
Sep 23, 2019
in
Combinatory

26
views
isi2015mma
permutationandcombination
+1
vote
2
answers
19
ISI2015MMA7
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ and $y$ have the same remainder when divided by $3\}$. Then the number of elements in $\mathcal{R}$ is $40$ $36$ $34$ $33$
asked
Sep 23, 2019
in
Set Theory & Algebra

33
views
isi2015mma
sets
cartesianproduct
+1
vote
2
answers
20
ISI2015MMA8
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals $n^2$ $n^3$ $2^n$ $3^n$
asked
Sep 23, 2019
in
Combinatory

46
views
isi2015mma
permutationandcombination
sets
0
votes
1
answer
21
ISI2015MMA9
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \ldots +C_nx^n, \: n$ being a positive integer. The value of $\left( 1+\frac{C_0}{C_1} \right) \left( 1+\frac{C_1}{C_2} \right) \cdots \left( 1+\frac{C_{n1}}{C_n} \right)$ is $\left( \frac{n+1}{n+2} \right) ^n$ $ \frac{n^n}{n!} $ $\left( \frac{n}{n+1} \right) ^n$ $\frac{(n+1)^n}{n!}$
asked
Sep 23, 2019
in
Combinatory

16
views
isi2015mma
permutationandcombination
binomialtheorem
+2
votes
2
answers
22
ISI2015MMA10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^31}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above
asked
Sep 23, 2019
in
Calculus

54
views
isi2015mma
calculus
limits
+1
vote
1
answer
23
ISI2015MMA11
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals $854$ $153$ $160$ none of the above
asked
Sep 23, 2019
in
Numerical Ability

18
views
isi2015mma
numericalability
numbersystem
remaindertheorem
0
votes
1
answer
24
ISI2015MMA12
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(32i)$ are two two roots of this polynomial then the value of $a$ is $524/65$ $524/65$ $1/65$ $1/65$
asked
Sep 23, 2019
in
Numerical Ability

23
views
isi2015mma
numericalability
numbersystem
polynomial
roots
nongate
0
votes
1
answer
25
ISI2015MMA13
The number of real roots of the equation $2 \cos \left( \frac{x^2+x}{6} \right) = 2^x +2^{x} \text{ is }$ $0$ $1$ $2$ infinitely many
asked
Sep 23, 2019
in
Numerical Ability

20
views
isi2015mma
numericalability
quadraticequations
trigonometry
0
votes
1
answer
26
ISI2015MMA14
Consider the following system of equivalences of integers, $x \equiv 2 \text{ mod } 15$ $x \equiv 4 \text{ mod } 21$ The number of solutions in $x$, where $1 \leq x \leq 315$, to the above system of equivalences is $0$ $1$ $2$ $3$
asked
Sep 23, 2019
in
Numerical Ability

17
views
isi2015mma
numericalability
numbersystem
congruentmodulo
nongate
0
votes
1
answer
27
ISI2015MMA15
The number of real solutions of the equations $(9/10)^x = 3+xx^2$ is $2$ $0$ $1$ none of the above
asked
Sep 23, 2019
in
Numerical Ability

21
views
isi2015mma
numericalability
numbersystem
quadraticequations
nongate
+1
vote
2
answers
28
ISI2015MMA16
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\: (\geq 2)$ and $n\: (\geq 1)$ respectively, satisfy $f(x^2+1)=f(x)g(x),$ for every $x \in \mathbb{R}$, then $f$ has exactly one real root $x_0$ such that $f’(x_0) \neq 0$ $f$ has exactly one real root $x_0$ such that $f’(x_0) = 0$ $f$ has $m$ distinct real roots $f$ has no real root
asked
Sep 23, 2019
in
Numerical Ability

56
views
isi2015mma
numericalability
quadraticequations
functions
nongate
+1
vote
1
answer
29
ISI2015MMA17
Let $X=\frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$. Then, $X \lt1$ $X\gt3/2$ $1\lt X\lt 3/2$ none of the above holds
asked
Sep 23, 2019
in
Numerical Ability

24
views
isi2015mma
numericalability
summation
+1
vote
1
answer
30
ISI2015MMA18
The set of complex numbers $z$ satisfying the equation $(3+7i)z+(102i)\overline{z}+100=0$ represents, in the complex plane, a straight line a pair of intersecting straight lines a point a pair of distinct parallel straight lines
asked
Sep 23, 2019
in
Numerical Ability

29
views
isi2015mma
numericalability
geometry
straightlines
complexnumber
nongate
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