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1
ISI2015DCG47
The Taylor series expansion of $f(x)= \text{ln}(1+x^2)$ about $x=0$ is $\sum _{n=1}^{\infty} (1)^n \frac{x^n}{n}$ $\sum _{n=1}^{\infty} (1)^{n+1} \frac{x^{2n}}{n}$ $\sum _{n=1}^{\infty} (1)^{n+1} \frac{x^{2n+1}}{n+1}$ $\sum _{n=0}^{\infty} (1)^{n+1} \frac{x^{n+1}}{n+1}$
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

19
views
isi2015dcg
calculus
taylorseries
nongate
+1
vote
1
answer
2
ISI2015DCG48
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}1}{x^{\frac{1}{4}}1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

25
views
isi2015dcg
calculus
limits
+1
vote
1
answer
3
ISI2015DCG49
The domain of the function $\text{ln}(3x^24x+5)$ is set of positive real numbers set of real numbers set of negative real numbers set of real numbers larger than $5$
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

32
views
isi2015dcg
functions
0
votes
0
answers
4
ISI2015DCG50
The piecewise linear function for the following graph is $f(x) = \begin{cases} = x, \: x \leq 2 \\ =4, \: 2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$ $f(x) = \begin{cases} = x2, \: x \leq 2 \\ =4, \: 2<x<3 \\ =x1, \: x \geq 3 \end{cases}$ ... $f(x) = \begin{cases} = 2x, \: x \leq 2 \\ =4, \: 2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

20
views
isi2015dcg
calculus
functions
0
votes
1
answer
5
ISI2015DCG51
The area bounded by $y=x^24$, $y=0$ and $x=4$ is $\frac{64}{3}$ $6$ $\frac{16}{3}$ $\frac{32}{3}$
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

21
views
isi2015dcg
integration
definiteintegrals
+2
votes
1
answer
6
ISI2015DCG52
$\underset{x \to 1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

31
views
isi2015dcg
calculus
limits
0
votes
0
answers
7
ISI2015DCG54
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals $1$ $0$ $1$ Does not exist
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

20
views
isi2015dcg
calculus
limits
+1
vote
0
answers
8
ISI2015DCG55
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals $1$ $0$ $1$ Does not exist
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

17
views
isi2015dcg
calculus
limits
+1
vote
0
answers
9
ISI2015DCG56
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals $1$ $0$ $1$ Does not exist
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

22
views
isi2015dcg
calculus
limits
0
votes
0
answers
10
ISI2015DCG57
Let $y=\lfloor x \rfloor$, where $\lfloor x \rfloor$ is greatest integer less than or equal to $x$. Then $y$ is continuous and manyone $y$ is not differentiable and manyone $y$ is not differentiable $y$ is differentiable and manyone
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

12
views
isi2015dcg
calculus
continuity
differentiation
0
votes
0
answers
11
ISI2015DCG58
$\underset{x \to 1}{\lim} \dfrac{x^{16}1}{\mid x1 \mid}$ equals $1$ $0$ $1$ Does not exist
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

14
views
isi2015dcg
calculus
limits
0
votes
1
answer
12
ISI2016DCG3
The value of $\begin{vmatrix} 1+a& 1& 1& 1\\ 1&1+b &1 &1 \\ 1&1 &1+c &1 \\ 1&1 &1 &1+d \end{vmatrix}$ is $abcd(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$ $abcd(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$ $1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ None of these
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

23
views
isi2016dcg
linearalgebra
determinant
0
votes
1
answer
13
ISI2016DCG4
If $f(x)=\begin{bmatrix}\cos\:x & \sin\:x & 0 \\ \sin\:x & \cos\:x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then the value of $\big(f(x)\big)^2$ is $f(x)$ $f(2x)$ $2f(x)$ None of these
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

20
views
isi2016dcg
linearalgebra
matrices
0
votes
1
answer
14
ISI2016DCG11
Let two systems of linear equations be defined as follows: $\begin{array}{lll} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ ... $P$ and $Q$ are inconsistent $P$ and $Q$ are consistent $P$ is consistent but $Q$ is inconsistent None of the above
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

16
views
isi2016dcg
linearalgebra
systemofequations
+2
votes
1
answer
15
ISI2016DCG21
The value of the term independent of $x$ in the expansion of $(1x)^{2}(x+\frac{1}{x})^{7}$ is $70$ $70$ $35$ None of these
asked
Sep 18, 2019
in
Combinatory
by
gatecse
Boss
(
17.5k
points)

30
views
isi2016dcg
permutationandcombination
binomialtheorem
+1
vote
1
answer
16
ISI2016DCG22
The value of $\:\:\begin{vmatrix} 1&\log_{x}y &\log_{x}z \\ \log_{y}x &1 &\log_{y}z \\\log_{z}x & \log_{z}y&1 \end{vmatrix}\:\:$ is $0$ $1$ $1$ None of these
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

28
views
isi2016dcg
linearalgebra
determinant
+2
votes
1
answer
17
ISI2016DCG24
If the letters of the word $\text{COMPUTER}$ be arranged in random order, the number of arrangements in which the three vowels $O, U$ and $E$ occur together is $8!$ $6!$ $3!6!$ None of these
asked
Sep 18, 2019
in
Combinatory
by
gatecse
Boss
(
17.5k
points)

22
views
isi2016dcg
permutationandcombination
arrangements
0
votes
1
answer
18
ISI2016DCG27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of triangle $x,$ is oneone and into oneone and onto manyone and onto manyone and into
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

15
views
isi2016dcg
sets
functions
+1
vote
1
answer
19
ISI2016DCG31
Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$ stands for determinant of matrix $A.$ Then $\mid\:(A)\mid=1$ $\mid\:(A)\mid=0\:\text{or}\:1$ $\mid\:(A)\mid=1,0\:\text{or}\:1$ $\mid\:(A)\mid=1\:\text{or}\:1$
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

22
views
isi2016dcg
linearalgebra
matrices
determinant
0
votes
0
answers
20
ISI2016DCG32
The set of vectors constituting an orthogonal basis in $\mathbb{R}^{3}$ is $\begin{Bmatrix} \begin{pmatrix} 1\\ 1 \\0 \end{pmatrix}&,\begin{pmatrix} 1\\ 1 \\0 \end{pmatrix}&,\begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} \end{Bmatrix}$ ... None of these
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

12
views
isi2016dcg
linearalgebra
matrices
orthogonalmatrix
eigenvectors
0
votes
0
answers
21
ISI2016DCG33
Suppose $A$ and $B$ are orthogonal $n\times n$ matrices. Which of the following is also an orthogonal matrix? Assume that $O$ is the null matrix of order $n\times n$ and $I$ is the identity matrix of order $n.$ $ABBA$ $\begin{pmatrix} A & O \\ O & B \end{pmatrix}$ $\begin{pmatrix} A & I \\ I & B \end{pmatrix}$ $A^{2}B^{2}$
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

13
views
isi2016dcg
linearalgebra
matrices
orthogonalmatrix
0
votes
0
answers
22
ISI2016DCG34
Let $A_{ij}$ denote the minors of an $n\times n$ matrix $A.$ What is the relationship between $\mid A_{ij}\mid$ and $\mid A_{ji}\mid$? They are always equal. $\mid A_{ij}\mid=\mid A_{ji}\mid$ if $i\neq j.$ They are equal if $A$ is a symmetric matrix. If $\mid A_{ij}\mid=0$ then $\mid A_{ji}\mid=0.$
asked
Sep 18, 2019
in
Linear Algebra
by
gatecse
Boss
(
17.5k
points)

12
views
isi2016dcg
linearalgebra
matrices
minors
0
votes
0
answers
23
ISI2016DCG35
Let $A,B$ and $C$ be three non empty sets. Consider the two relations given below: $A(BC)=(AB)\cup C$ $A(B\cup C)=(AB)C$ Both (1) and (2) are correct. (1) is correct but (2) is not. (2) is correct but (1) is not. Both (1) and (2) are incorrect.
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

12
views
isi2016dcg
sets
0
votes
1
answer
24
ISI2016DCG36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is $n^{n}$ $n\log_{2}n$ $n^{2}$ $n!$
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

16
views
isi2016dcg
sets
functions
0
votes
0
answers
25
ISI2016DCG37
Suppose $f_{\alpha}\::\:[0,1]\rightarrow[0,1],\:1<\alpha<\infty$ is given by $f_{\alpha}(x)=\dfrac{(\alpha+1)x}{\alpha x+1}.$ Then $f_{\alpha}$ is A bijective (oneone and onto) function. A surjective (onto) function. An injective (oneone) function. We can not conclude about the type.
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

11
views
isi2016dcg
sets
functions
+2
votes
1
answer
26
ISI2016DCG45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:xx\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

42
views
isi2016dcg
limits
0
votes
0
answers
27
ISI2016DCG46
Let $I=\int(\sin\:x\cos\:x)(\sin\:x+\cos\:x)^{3}dx$ and $K$ be a constant of integration. Then the value of $I$ is $(\sin\:x+\cos\:x)^{4}+K$ $(\sin\:x+\cos\:x)^{2}+K$ $\frac{1}{4}(\sin\:x+\cos\:x)^{4}+K$ None of these
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

18
views
isi2016dcg
calculus
integration
nongate
0
votes
0
answers
28
ISI2016DCG47
The Taylor series expansion of $f(x)=\ln(1+x^{2})$ about $x=0$ is $\sum_{n=1}^{\infty}(1)^{n}\frac{x^{n}}{n}$ $\sum_{n=1}^{\infty}(1)^{n+1}\frac{x^{2n}}{n}$ $\sum_{n=1}^{\infty}(1)^{n+1}\frac{x^{2n+1}}{n+1}$ $\sum_{n=1}^{\infty}(1)^{n+1}\frac{x^{n+1}}{n+1}$
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

13
views
isi2016dcg
calculus
taylorseries
nongate
0
votes
0
answers
29
ISI2016DCG48
The piecewise linear function for the following graph is $f(x)=\begin{cases} = x,x\leq2 \\ =4,2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = x2,x\leq2 \\ =4,2<x<3 \\ = x1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2x,x\leq2 \\ =x,2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2x,x\leq2 \\ =4,2<x<3 \\ = x+1,x\geq 3\end{cases}$
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

13
views
isi2016dcg
calculus
functions
curves
nongate
0
votes
1
answer
30
ISI2016DCG49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}1}{x^{\frac{1}{4}}1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

19
views
isi2016dcg
calculus
limits
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