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1
ISI2014DCG33
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties. $f(0)=0$, $f(1)=1$, and $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$. Then the number of such functions is $0$ $1$ $2$ $\infty$
asked
Sep 23, 2019
in
Calculus
by
Arjun
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431k
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27
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isi2014dcg
calculus
functions
limits
+1
vote
1
answer
2
ISI2014DCG34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$
asked
Sep 23, 2019
in
Combinatory
by
Arjun
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(
431k
points)

49
views
isi2014dcg
permutationandcombination
binomialtheorem
summation
+1
vote
1
answer
3
ISI2014DCG35
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ ... $\begin{pmatrix} m \\ pq \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$
asked
Sep 23, 2019
in
Set Theory & Algebra
by
Arjun
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(
431k
points)

33
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isi2014dcg
sets
disjointsets
+1
vote
1
answer
4
ISI2014DCG37
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^}{2}$ and $f(x) \to – \infty$ as $x \to \dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{1} x$ $\sin x$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
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27
views
isi2014dcg
calculus
functions
limits
continuity
+1
vote
1
answer
5
ISI2014DCG38
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then which one of the following is true? $A’=A$ $A’=A$ $AA’=I$ None of these
asked
Sep 23, 2019
in
Linear Algebra
by
Arjun
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(
431k
points)

59
views
isi2014dcg
linearalgebra
matrices
+1
vote
1
answer
6
ISI2014DCG39
The function $f(x) = x^{1/x}, \: x \neq 0$ has a minimum at $x=e$; a maximum at $x=e$; neither a maximum nor a minimum at $x=e$; None of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

54
views
isi2014dcg
maximaminima
calculus
+1
vote
1
answer
7
ISI2014DCG41
The number of permutations of the letters $a, b, c$ and $d$ such that $b$ does not follow $a,c$ does not follow $b$, and $c$ does not follow $d$, is $11$ $12$ $13$ $14$
asked
Sep 23, 2019
in
Combinatory
by
Arjun
Veteran
(
431k
points)

47
views
isi2014dcg
permutationandcombination
0
votes
0
answers
8
ISI2014DCG42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

22
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
9
ISI2014DCG43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid 1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

15
views
isi2014dcg
calculus
functions
limits
0
votes
1
answer
10
ISI2014DCG44
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$ has a maximum at $x= \pi /3$ has a maximum at $x= \pi$ has a minimum at $x= \pi /3$ has neither a maximum nor a minimum at $x=\pi/3$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

16
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
11
ISI2014DCG45
Which of the following is true? $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$ $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

23
views
isi2014dcg
calculus
functions
logarithms
0
votes
1
answer
12
ISI2014DCG46
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is $2$ $1$ $0$ $\sqrt{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
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26
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
13
ISI2014DCG47
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is $\frac{\pi}{6} + \sqrt{3}$ $\frac{\pi}{6}  \sqrt{3}$ $0$ $\frac{1}{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

19
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
0
answers
14
ISI2014DCG48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2ax+12a^2$ is always greater than zero is $ \frac{2}{3} < a \leq \frac{2}{3}$ $ \frac{2}{3} \leq a < \frac{2}{3}$ $ \frac{2}{3} < a < \frac{2}{3}$ None of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
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13
views
isi2014dcg
calculus
functions
quadraticequations
0
votes
0
answers
15
ISI2014DCG50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1 \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

24
views
isi2014dcg
calculus
limits
0
votes
1
answer
16
ISI2014DCG51
The function $f(x)$ defined as $f(x)=x^36x^2+24x$, where $x$ is real, is strictly increasing strictly decreasing increasing in $( \infty, 0)$ and decreasing in $(0, \infty)$ decreasing in $( \infty, 0)$ and increasing in $(0, \infty)$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

28
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
17
ISI2014DCG53
The value of the integral $\displaystyle{}\int_{1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $ – \frac{1}{2}$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
431k
points)

31
views
isi2014dcg
calculus
integration
definiteintegrals
+2
votes
1
answer
18
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
by
Arjun
Veteran
(
431k
points)

28
views
isi2014dcg
permutationandcombination
0
votes
1
answer
19
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
by
Arjun
Veteran
(
431k
points)

59
views
isi2014dcg
linearalgebra
matrices
systemofequations
+1
vote
1
answer
20
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
by
Arjun
Veteran
(
431k
points)

28
views
isi2014dcg
permutationandcombination
arrangements
0
votes
0
answers
21
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
by
Arjun
Veteran
(
431k
points)

37
views
isi2014dcg
linearalgebra
matrices
inverse
+1
vote
2
answers
22
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
by
Arjun
Veteran
(
431k
points)

38
views
isi2014dcg
permutationandcombination
arrangements
circularpermutation
+1
vote
1
answer
23
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
by
Arjun
Veteran
(
431k
points)

38
views
isi2014dcg
permutationandcombination
summation
+1
vote
1
answer
24
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
by
Arjun
Veteran
(
431k
points)

37
views
isi2015mma
recurrencerelations
nongate
+2
votes
3
answers
25
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
by
Arjun
Veteran
(
431k
points)

46
views
isi2015mma
permutationandcombination
+1
vote
2
answers
26
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
by
Arjun
Veteran
(
431k
points)

49
views
isi2015mma
sets
subsets
0
votes
1
answer
27
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
by
Arjun
Veteran
(
431k
points)

26
views
isi2015mma
permutationandcombination
+1
vote
2
answers
28
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
by
Arjun
Veteran
(
431k
points)

33
views
isi2015mma
sets
cartesianproduct
+1
vote
2
answers
29
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
by
Arjun
Veteran
(
431k
points)

46
views
isi2015mma
permutationandcombination
sets
0
votes
1
answer
30
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
by
Arjun
Veteran
(
431k
points)

16
views
isi2015mma
permutationandcombination
binomialtheorem
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