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ISI2015MMA69
Consider the function $f(x) = \begin{cases} \int_0^x \{5+ \mid 1y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$ Then $f$ is not continuous at $x=2$ $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at $x=1$ $f$ is continuous everywhere but not differentiable at $x=2$
asked
Sep 23, 2019
in
Calculus
by
Arjun
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431k
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20
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isi2015mma
calculus
continuity
differentiation
definiteintegrals
nongate
0
votes
0
answers
2
ISI2015MMA72
The map $f(x) = a_0 \cos \mid x \mid +a_1 \sin \mid x \mid +a_2 \mid x \mid ^3$ is differentiable at $x=0$ if and only if $a_1=0$ and $a_2=0$ $a_0=0$ and $a_1=0$ $a_1=0$ $a_0, a_1, a_2$ can take any real value
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

12
views
isi2015mma
calculus
differentiation
0
votes
0
answers
3
ISI2015MMA73
$f(x)$ is a differentiable function on the real line such that $\underset{x \to \infty=}{\lim} f(x) =1$ and $\underset{x \to \infty=}{\lim} f’(x) =\alpha$. Then $\alpha$ must be $0$ $\alpha$ need not be $0$, but $\mid \alpha \mid <1$ $\alpha >1$ $\alpha < 1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
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(
431k
points)

14
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isi2015mma
calculus
limits
differentiation
0
votes
0
answers
4
ISI2015MMA74
Let $f$ and $g$ be two differentiable functions such that $f’(x)\leq g’(x)$for all $x<1$ and $f’(x) \geq g’(x)$ for all $x>1$. Then if $f(1) \geq g(1)$, then $f(x) \geq g(x)$ for all $x$ if $f(1) \leq g(1)$, then $f(x) \leq g(x)$ for all $x$ $f(1) \leq g(1)$ $f(1) \geq g(1)$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

10
views
isi2015mma
calculus
differentiation
0
votes
0
answers
5
ISI2015MMA76
Given that $\int_{\infty}^{\infty} e^{x^2} dx = \sqrt{\pi}$, the value of $ \int_{\infty}^{\infty} \int_{\infty}^{\infty} e^{(x^2+xy+y^2)} dxdy$ is $\sqrt{\pi/3}$ $\pi/\sqrt{3}$ $\sqrt{2 \pi/3}$ $2 \pi / \sqrt{3}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

13
views
isi2015mma
calculus
definiteintegrals
nongate
+1
vote
0
answers
6
ISI2015MMA77
Let $R$ be the triangle in the $xy$ – plane bounded by the $x$axis, the line $y=x$, and the line $x=1$. The value of the double integral $ \int \int_R \frac{\sin x}{x}\: dxdy$ is $1\cos 1$ $\cos 1$ $\frac{\pi}{2}$ $\pi$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

16
views
isi2015mma
integration
nongate
0
votes
1
answer
7
ISI2015MMA78
The value of $\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$ is $0$ $\ln 2$ $\ln 3$ $\infty$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

19
views
isi2015mma
calculus
limits
definiteintegrals
nongate
0
votes
0
answers
8
ISI2015MMA80
Let $0 < \alpha < \beta < 1$. Then $ \Sigma_{k=1}^{\infty} \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x} dx$ is equal to $\log_e \frac{\beta}{\alpha}$ $\log_e \frac{1+ \beta}{1 + \alpha}$ $\log_e \frac{1+\alpha }{1+ \beta}$ $\infty$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

21
views
isi2015mma
calculus
definiteintegrals
summation
nongate
0
votes
0
answers
9
ISI2015MMA81
If $f$ is continuous in $[0,1]$ then $\displaystyle \lim_ {n \to \infty} \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$ (where $[y]$ is the largest integer less than or equal to $y$) does not exist exists and is equal to $\frac{1}{2} \int_0^1 f(x) dx$ exists and is equal to $ \int_0^1 f(x) dx$ exists and is equal to $\int_0^{1/2} f(x) dx$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

17
views
isi2015mma
limits
definiteintegrals
nongate
0
votes
0
answers
10
ISI2015MMA92
Consider the group $G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$ ... is of finite order $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

21
views
isi2015mma
grouptheory
subgroups
normal
nongate
0
votes
1
answer
11
ISI2015MMA93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

27
views
isi2015mma
grouptheory
0
votes
1
answer
12
ISI2015MMA94
Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is $4$ $1$ $2$ $3$
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
Veteran
(
431k
points)

20
views
isi2015mma
grouptheory
nongate
+1
vote
1
answer
13
Mean Value Theorem
f(x) is a differentiable function that satisfies 5 ≤ f′(x) ≤ 14 for all x. Let a and b be the maximum and minimum values, respectively, that f(11)−f(3) can possibly have, then what is the value of a+b?
asked
Sep 22, 2019
in
Calculus
by
Nirmal Gaur
Active
(
2.3k
points)

59
views
0
votes
2
answers
14
ISI2015DCG3
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)

58
views
isi2015dcg
linearalgebra
determinant
0
votes
1
answer
15
ISI2015DCG5
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)

39
views
isi2015dcg
linearalgebra
matrices
0
votes
1
answer
16
ISI2015DCG11
Let two systems of linear equations be defined as follows: $\begin{array} {} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ ... $P$ 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)

26
views
isi2015dcg
linearalgebra
systemofequations
0
votes
2
answers
17
ISI2015DCG17
The set $\{(x,y): \mid x \mid + \mid y \mid \leq 1\}$ is represented by the shaded region in
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

19
views
isi2015dcg
sets
0
votes
1
answer
18
ISI2015DCG21
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)

22
views
isi2015dcg
permutationandcombination
binomialtheorem
0
votes
1
answer
19
ISI2015DCG22
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)

23
views
isi2015dcg
linearalgebra
determinant
+2
votes
1
answer
20
ISI2015DCG24
If the letters of the word $\textbf{COMPUTER}$ 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)

26
views
isi2015dcg
permutationandcombination
arrangements
+1
vote
1
answer
21
ISI2015DCG27
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)

28
views
isi2015dcg
functions
0
votes
1
answer
22
ISI2015DCG31
Let $A$ be an $n \times n$ matrix such that $\mid A^{2} \mid\: =1$. Here $\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)

35
views
isi2015dcg
linearalgebra
matrices
determinant
0
votes
1
answer
23
ISI2015DCG32
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)

57
views
isi2015dcg
linearalgebra
matrices
eigenvectors
0
votes
1
answer
24
ISI2015DCG33
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)

23
views
isi2015dcg
linearalgebra
matrices
orthogonalmatrix
0
votes
1
answer
25
ISI2015DCG34
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 $\mid A_{ij} \mid = – \mid A _{ji} \mid \text{ if } i \neq j$ 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)

24
views
isi2015dcg
linearalgebra
matrices
determinant
0
votes
1
answer
26
ISI2015DCG35
Let $A$, $B$ and $C$ be three non empty sets. Consider the two relations given below: $\begin{array}{lll} A(BC)=(AB) \cup C & & (1) \\ A – (B \cup C) = (A B)C & & (2) \end{array}$ 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)

13
views
isi2015dcg
sets
+2
votes
1
answer
27
ISI2015DCG36
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)

23
views
isi2015dcg
functions
0
votes
0
answers
28
ISI2015DCG37
Suppose $f_{\alpha} : [0,1] \to [0,1],\:\: 1 < \alpha < \infty$ is given by $f_{\alpha} (x) = \frac{(\alpha +1)x}{\alpha x+1}$ Then $f_{\alpha}$ is A bijective (oneone and onto) function A surjective (onto ) function An injective (oneone) function We cannot conclude about the type
asked
Sep 18, 2019
in
Set Theory & Algebra
by
gatecse
Boss
(
17.5k
points)

20
views
isi2015dcg
sets
functions
+1
vote
1
answer
29
ISI2015DCG45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan ^2 x – x \tan x }{\sin x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
asked
Sep 18, 2019
in
Calculus
by
gatecse
Boss
(
17.5k
points)

25
views
isi2015dcg
calculus
limits
0
votes
1
answer
30
ISI2015DCG46
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)

22
views
isi2015dcg
calculus
integration
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