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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 $a_n=4, \: b_n=4^n$ $a_n=4, \: b_n=4n^2$ $a_n=4^{(n1)!}, \: b_n=4^n$ $a_n=4^{(n1)!}, \: b_n=4n^2$
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Sep 23
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isi2015mma
0
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1
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2
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
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Sep 23
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33
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isi2015mma
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1
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3
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
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Sep 23
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26
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isi2015mma
+2
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3
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4
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
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Sep 23
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Arjun
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422k
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29
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isi2015mma
0
votes
1
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5
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
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Arjun
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422k
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15
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isi2015mma
+1
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2
answers
6
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
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Arjun
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422k
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13
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isi2015mma
+1
vote
2
answers
7
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
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by
Arjun
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(
422k
points)

27
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isi2015mma
0
votes
1
answer
8
ISI2015MMA9
Let $(1+x)^n = C_0+C_1x+C_2x^2+ \cdots +C_nx^n, \: n$ being a positive integer. The value of $\bigg( 1+\frac{C_0}{C_1} \bigg) \bigg( 1+\frac{C_1}{C_2} \bigg) \cdots \bigg( 1+\frac{C_{n1}}{C_n} \bigg)$ is $\bigg( \frac{n+1}{n+2} \bigg) ^n$ $ \frac{n^n}{n!} $ $\bigg( \frac{n}{n+1} \bigg) ^n$ $\frac{(n+1)^n}{n!}$
asked
Sep 23
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by
Arjun
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422k
points)

8
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isi2015mma
+2
votes
2
answers
9
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
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by
Arjun
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(
422k
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28
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isi2015mma
+1
vote
1
answer
10
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
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by
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422k
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9
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isi2015mma
0
votes
1
answer
11
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 $524/65$ $524/65$ $1/65$ $1/65$
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Sep 23
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422k
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13
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isi2015mma
0
votes
1
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12
ISI2015MMA13
The number of real roots of the equation $2 \cos \bigg( \frac{x^2+x}{6} \bigg) = 2^x +2^{x} \text{ is }$ $0$ $1$ $2$ infinitely many
asked
Sep 23
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by
Arjun
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7
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isi2015mma
0
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1
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13
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
in
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by
Arjun
Veteran
(
422k
points)

7
views
isi2015mma
0
votes
1
answer
14
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
in
Others
by
Arjun
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(
422k
points)

9
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isi2015mma
+1
vote
1
answer
15
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
in
Others
by
Arjun
Veteran
(
422k
points)

26
views
isi2015mma
0
votes
1
answer
16
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
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by
Arjun
Veteran
(
422k
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8
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isi2015mma
+1
vote
1
answer
17
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
in
Others
by
Arjun
Veteran
(
422k
points)

17
views
isi2015mma
0
votes
1
answer
18
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
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by
Arjun
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(
422k
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21
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isi2015mma
+1
vote
1
answer
19
ISI2015MMA20
The limit $\underset{n \to \infty}{\lim} \bigg( 1 \frac{1}{n^2} \bigg) ^n$ equals $e^{1}$ $e^{1/2}$ $e^{2}$ $1$
asked
Sep 23
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by
Arjun
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(
422k
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6
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isi2015mma
+1
vote
1
answer
20
ISI2015MMA21
Let $\omega$ denote a complex fifth root of unity. Define $b_k =\sum_{j=0}^{4} j \omega^{kj},$ for $0 \leq k \leq 4$. Then $ \sum_{k=0}^{4} b_k \omega ^k$ is equal to $5$ $5 \omega$ $5(1+\omega)$ $0$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
422k
points)

11
views
isi2015mma
0
votes
1
answer
21
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
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by
Arjun
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(
422k
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8
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isi2015mma
0
votes
0
answers
22
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
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by
Arjun
Veteran
(
422k
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3
views
isi2015mma
0
votes
1
answer
23
ISI2015MMA24
The series $\sum_{k=2}^{\infty} \frac{1}{k(k1)}$ converges to $1$ $1$ $0$ does not converge
asked
Sep 23
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by
Arjun
Veteran
(
422k
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9
views
isi2015mma
+1
vote
1
answer
24
ISI2015MMA25
The limit $\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
422k
points)

21
views
isi2015mma
0
votes
0
answers
25
ISI2015MMA26
$\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
422k
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10
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isi2015mma
0
votes
0
answers
26
ISI2015MMA27
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is $0$ $1/32$ $15/32$ $10/32$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
422k
points)

10
views
isi2015mma
0
votes
1
answer
27
ISI2015MMA28
In a triangle $ABC$, $AD$ is the median. If length of $AB$ is $7$, length of $AC$ is $15$ and length of $BC$ is $10$ then length of $AD$ equals $\sqrt{125}$ $69/5$ $\sqrt{112}$ $\sqrt{864}/5$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
422k
points)

10
views
isi2015mma
0
votes
0
answers
28
ISI2015MMA29
The set $\{x \: : \begin{vmatrix} x+\frac{1}{x} \end{vmatrix} \gt6 \}$ equals the set $(0,32\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $( \infty, 32\sqrt{2}) \cup (3+2 \sqrt{2}, \infty)$ $( \infty, 32\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $( \infty, 32\sqrt{2}) \cup (3+2 \sqrt{2},32\sqrt{2}) \cup (3+2 \sqrt{2}, \infty )$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
422k
points)

7
views
isi2015mma
0
votes
0
answers
29
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
in
Others
by
Arjun
Veteran
(
422k
points)

4
views
isi2015mma
0
votes
0
answers
30
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
in
Others
by
Arjun
Veteran
(
422k
points)

4
views
isi2015mma
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