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Previous GATE
+2
votes
2
answers
1
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
by
Arjun
Veteran
(
431k
points)

54
views
isi2015mma
calculus
limits
0
votes
1
answer
2
ISI2015MMA19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

32
views
isi2015mma
calculus
limits
nongate
+1
vote
1
answer
3
ISI2015MMA20
The limit $\underset{n \to \infty}{\lim} \left( 1 \frac{1}{n^2} \right) ^n$ equals $e^{1}$ $e^{1/2}$ $e^{2}$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

20
views
isi2015mma
calculus
limits
nongate
0
votes
1
answer
4
ISI2015MMA22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

17
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
5
ISI2015MMA23
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ ... $f(x,A)+f(x,B)\:  f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\:  f(x,B) \mid $
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

11
views
isi2015mma
sets
functions
nongate
+1
vote
1
answer
6
ISI2015MMA25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

33
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
7
ISI2015MMA26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

18
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
8
ISI2015MMA30
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $Kxy$ $K+xy$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

16
views
isi2015mma
calculus
functions
nongate
0
votes
0
answers
9
ISI2015MMA31
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is nonempty none of the above
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

19
views
isi2015mma
sets
nongate
+1
vote
1
answer
10
ISI2015MMA33
If $f(x)$ is a real valued function such that $2f(x)+3f(x)=154x,$ for every $x \in \mathbb{R}$, then $f(2)$ is $15$ $22$ $11$ $0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

17
views
isi2015mma
calculus
functions
nongate
0
votes
1
answer
11
ISI2015MMA34
If $f(x) = \dfrac{\sqrt{3}\sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[1, \sqrt{3}/2]$ the interval $[ \sqrt{3}/2, 1]$ the interval $[1, 1]$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

25
views
isi2015mma
calculus
functions
range
trigonometry
nongate
+1
vote
1
answer
12
ISI2015MMA36
For nonnegative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m1, 1) & \text{ if } m \neq 0, n=0 \\ f(m1, f(m,n1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

17
views
isi2015mma
calculus
functions
nongate
+1
vote
1
answer
13
ISI2015MMA37
Let $a$ be a nonzero real number. Define $f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

53
views
isi2015mma
linearalgebra
determinant
functions
0
votes
0
answers
14
ISI2015MMA38
A real $2 \times 2$ matrix $M$ such that $M^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \varepsilon \end{pmatrix}$ exists for all $\varepsilon > 0$ does not exist for any $\varepsilon > 0$ exists for some $\varepsilon > 0$ none of the above is true
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

25
views
isi2015mma
linearalgebra
matrices
+2
votes
2
answers
15
ISI2015MMA39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

78
views
isi2015mma
linearalgebra
matrices
eigenvalue
0
votes
0
answers
16
ISI2015MMA40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ ... $(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

36
views
isi2015mma
linearalgebra
matrices
rankofmatrix
0
votes
0
answers
17
ISI2015MMA42
Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix $A \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 &  \sin t & \cos t \end{pmatrix}.$ If $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$ ... $\{  \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{  \frac{\pi}{3}, \frac{\pi}{3} \}$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

40
views
isi2015mma
linearalgebra
matrices
eigenvalue
0
votes
1
answer
18
ISI2015MMA43
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta = 1, 2$ $\eta = 1, 2$ $\eta = 3, 3$ $\eta = 1, 2$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

27
views
isi2015mma
linearalgebra
systemofequations
0
votes
1
answer
19
ISI2015MMA44
Let $P_1$, $P_2$ and $P_3$ denote, respectively, the planes defined by $\begin{array} {} a_1x +b_1y+c_1z=\alpha _1 \\ a_2x +b_2y+c_2z=\alpha _2 \\ a_3x +b_3y+c_3z=\alpha _3 \end{array}$ It is given that $P_1$, $P_2$ and $P_3$ ... then the planes do not have any common point of intersection intersect at a unique point intersect along a straight line intersect along a plane
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

30
views
isi2015mma
linearalgebra
systemofequations
0
votes
1
answer
20
ISI2015MMA51
A permutation of $1,2, \dots, n$ is chosen at random. Then the probability that the numbers $1$ and $2$ appear as neighbour equals $\frac{1}{n}$ $\frac{2}{n}$ $\frac{1}{n1}$ $\frac{1}{n2}$
asked
Sep 23, 2019
in
Probability
by
Arjun
Veteran
(
431k
points)

63
views
isi2015mma
probability
randomvariable
permutationandcombination
+1
vote
1
answer
21
ISI2015MMA52
Two coins are tossed independently where $P$(head occurs when coin $i$ is tossed) $=p_i, \: i=1,2$. Given that at least one head has occurred, the probability that coins produced different outcomes is $\frac{2p_1p_2}{p_1+p_22p_1p_2}$ $\frac{p_1+p_22p_1p_2}{p_1+p_2p_1p_2}$ $\frac{2}{3}$ none of the above
asked
Sep 23, 2019
in
Probability
by
Arjun
Veteran
(
431k
points)

42
views
isi2015mma
probability
independentevents
+1
vote
1
answer
22
ISI2015MMA53
The number of cars $(X)$ arriving at a service station per day follows a Poisson distribution with mean $4$. The service station can provide service to a maximum of $4$ cars per day. Then the expected number of cars that do not get service per day equals $4$ $0$ $\Sigma_{i=0}^{\infty} i P(X=i+4)$ $\Sigma_{i=4}^{\infty} i P(X=i4)$
asked
Sep 23, 2019
in
Probability
by
Arjun
Veteran
(
431k
points)

45
views
isi2015mma
poissondistribution
expectation
0
votes
0
answers
23
ISI2015MMA55
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if $\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \infty}{\lim} a_{2n+2}$ exists $\underset{n \to \infty}{\lim} a_{2n}$ ... $\underset{n \to \infty}{\lim} a_{3n}$ exist none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

14
views
isi2015mma
calculus
limits
0
votes
0
answers
24
ISI2015MMA57
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ does not exist $\underset{n \to \infty}{\lim} a_n=e$ $\underset{n \to \infty}{\lim} a_n=0$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

14
views
isi2015mma
calculus
limits
0
votes
0
answers
25
ISI2015MMA58
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$, for $n \geq 1$. Then $\underset{n \to \infty}{\lim} \sqrt{n}(A_{n+1}A_n)$ is equal to $0$ $1$ $1$ none of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

13
views
isi2015mma
calculus
limits
nongate
0
votes
0
answers
26
ISI2015MMA59
In the Taylor expansion of the function $f(x)=e^{x/2}$ about $x=3$, the coefficient of $(x3)^5$ is $e^{3/2} \frac{1}{5!}$ $e^{3/2} \frac{1}{2^5 5!}$ $e^{3/2} \frac{1}{2^5 5!}$ none of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

11
views
isi2015mma
calculus
taylorseries
nongate
0
votes
0
answers
27
ISI2015MMA60
Let $\sigma$ be the permutation: $\begin{array} {}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 5 & 6 & 2 & 4 & 9 & 8 & 7 & 1, \end{array}$ $I$ be the identity permutation and $m$ be the order of $\sigma$ i.e. $m=\text{min}\{\text{positive integers }n: \sigma ^n=I \}$. Then $m$ is $8$ $12$ $360$ $2520$
asked
Sep 23, 2019
in
Combinatory
by
Arjun
Veteran
(
431k
points)

19
views
isi2015mma
permutationandcombination
+1
vote
1
answer
28
ISI2015MMA61
Let $ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix} \text{ and } B=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}.$ Then there exists a matrix $C$ ... no matrix $C$ such that $A=BC$ there exists a matrix $C$ such that $A=BC$, but $A \neq CB$ there is no matrix $C$ such that $A=CB$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

39
views
isi2015mma
linearalgebra
matrices
0
votes
1
answer
29
ISI2015MMA62
If the matrix $A = \begin{bmatrix} a & 1 \\ 2 & 3 \end{bmatrix}$ has $1$ as an eigenvalue, then $\textit{trace}(A)$ is $4$ $5$ $6$ $7$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

48
views
isi2015mma
linearalgebra
matrices
eigenvalue
0
votes
1
answer
30
ISI2015MMA63
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\  \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ ... $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
Veteran
(
431k
points)

24
views
isi2015mma
linearalgebra
matrices
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