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5 votes
3 answers
1
ISI2004-MIII: 13
Let $X =\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\ldots+\frac{1}{3001}$. Then $X< 1$ $X>\frac{3}{2}$ $1< X< \frac{3}{2}$ none of the above
Let $X =\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\ldots+\frac{1}{3001}$. Then$X< 1$$X>\frac{3}{2}$$1< X< \frac{3}{2}$none of the above
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9 votes
2 answers
2
ISI2004-MIII: 12
The maximum possible value of $xy^2z^3$ subjected to condition $x,y,z \geq 0$ and $x+y+z=3$ is $1$ $\frac{9}{8}$ $\frac{9}{4}$ $\frac{27}{16}$
The maximum possible value of $xy^2z^3$ subjected to condition $x,y,z \geq 0$ and $x+y+z=3$ is$1$$\frac{9}{8}$$\frac{9}{4}$$\frac{27}{16}$
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14 votes
5 answers
3
ISI 2004 MIII
The number of permutation of $\{1,2,3,4,5\}$ that keep at least one integer fixed is. $81$ $76$ $120$ $60$
The number of permutation of $\{1,2,3,4,5\}$ that keep at least one integer fixed is.$81$$76$$120$$60$
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1 votes
2 answers
4
ISI 2004 MIII
The inequality $\frac{2-gx+x^{2}}{1-x+x^{2}}\leq 3$ is true for all the value of $x$ if and only if $1\leq g\leq 7$ $-1\leq g\leq 1$ $-6\leq g\leq 7$ $-1\leq g\leq 7$
The inequality $\frac{2-gx+x^{2}}{1-x+x^{2}}\leq 3$ is true for all the value of $x$ if and only if$1\leq g\leq 7$$-1\leq g\leq 1$$-6\leq g\leq 7$$-1\leq g\leq 7$
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19 votes
5 answers
6
ISI 2004 MIII
In how many ways can three person, each throwing a single die once, make a score of $11$ $22$ $27$ $24$ $38$
In how many ways can three person, each throwing a single die once, make a score of $11$$22$$27$$24$$38$
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6 votes
2 answers
7
ISI2004-MIII
The equation $\frac{1}{3}+\frac{1}{2}s^{2}+\frac{1}{6}s^{3}=s$ has exactly three solution in $[0.1]$ exactly one solution in $[0,1]$ exactly two solution in $[0,1]$ no solution in $[0,1]$
The equation $\frac{1}{3}+\frac{1}{2}s^{2}+\frac{1}{6}s^{3}=s$hasexactly three solution in $[0.1]$exactly one solution in $[0,1]$exactly two solution in $[0,1]$no solu...
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19 votes
4 answers
8
ISI 2004 MIII
A subset $S$ of set of numbers $\{2,3,4,5,6,7,8,9,10\}$ is said to be good if has exactly $4$ elements and their $gcd=1$, Then number of good subset is $126$ $125$ $123$ $121$
A subset $S$ of set of numbers $\{2,3,4,5,6,7,8,9,10\}$ is said to be good if has exactly $4$ elements and their $gcd=1$, Then number of good subset is$126$$125$$123$$121...
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5 votes
2 answers
10
ISI2004-MIII: 7
The equation $x^{6}-5x^{4}+16x^{2}-72x+9=0$ has exactly two distinct real roots exactly three distinct real roots exactly four distinct real roots six different real roots
The equation $x^{6}-5x^{4}+16x^{2}-72x+9=0$ hasexactly two distinct real rootsexactly three distinct real rootsexactly four distinct real rootssix different real roots
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5 votes
1 answer
11
ISI2004-MIII: 23
If $\textit{f}(x)=x^{2}$ and $g(x)=x \sin x +\cos x$ then $f$ and $g$ agree at no point $f$ and $g$ agree at exactly one point $f$ and $g$ agree at exactly two point $f$ and $g$ agree at more then two point
If $\textit{f}(x)=x^{2}$ and $g(x)=x \sin x +\cos x$ then$f$ and $g$ agree at no point$f$ and $g$ agree at exactly one point$f$ and $g$ agree at exactly two point$f$ and ...
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7 votes
1 answer
12
ISI2004-MIII: 11
If $\alpha 1,\alpha 2,\dots,\alpha n$ are the positive numbers then $\frac{a1}{a2}+\frac{a2}{a3}+\dots+\frac{an-1}{an}+\frac{an}{a1}$ is always $\geq n$ $\leq n$ $\leq n^{\frac{1}{2}}$ None of the above
If $\alpha 1,\alpha 2,\dots,\alpha n$ are the positive numbers then$\frac{a1}{a2}+\frac{a2}{a3}+\dots+\frac{an-1}{an}+\frac{an}{a1}$ is always$\geq n$$\leq n$$\leq n^{\fr...
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6 votes
1 answer
13
ISI2004-MIII: 22
If $\textit{f}(x)=\frac{\sqrt{3}\sin x}{2+\cos x}$ then the range of $\textit{f}(x)$ is the interval $\left[-1,\frac{\sqrt{3}}{2}\right]$ the interval $\left[\frac{-\sqrt{3}}{2},1\right]$ the interval $\left[-1,1\right]$ none of the above
If $\textit{f}(x)=\frac{\sqrt{3}\sin x}{2+\cos x}$ then the range of $\textit{f}(x)$ isthe interval $\left[-1,\frac{\sqrt{3}}{2}\right]$the interval $\left[\frac{-\sqrt{3...
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2 votes
1 answer
15
ISI 2004 MIII
$Q8$ If $\alpha_{1},\alpha_{2},\alpha_{3}, \dots , \alpha_{n}$ be the roots of $x^{n}+1=0$, then $\left ( 1-\alpha_{1} \right )\left ( 1-\alpha_{2} \right ) \dots \left ( 1-\alpha_{n} \right )$ is equal to $1$ $0$ $n$ $2$
$Q8$ If $\alpha_{1},\alpha_{2},\alpha_{3}, \dots , \alpha_{n}$ be the roots of $x^{n}+1=0$, then $\left ( 1-\alpha_{1} \right )\left ( 1-\alpha_{2} \right ) \dots \left (...
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1 votes
1 answer
16
ISI 2004 MIII
If the equation $x^{4}+ax^{3}+bx^{2}+cx+1=0$ (where $a,b,c$ are real number) has no real roots and if at least one of the root is of modulus one, then $b=c$ $a=c$ $a=b$ none of this
If the equation $x^{4}+ax^{3}+bx^{2}+cx+1=0$ (where $a,b,c$ are real number) has no real roots and if at least one of the root is of modulus one, then$b=c$$a=c$$a=b$none ...
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0 votes
1 answer
17
ISI 2004 MIII
$x^{2}+x+1$ is a factor of $\left ( x+1 \right )^{n}-x^{n}-1$ whenever $n$ is odd $n$ is odd and multiple of $3$ $n$ is an even multiple of $3$ $n$ is odd and not a multiple of $3$
$x^{2}+x+1$ is a factor of $\left ( x+1 \right )^{n}-x^{n}-1$ whenever $n$ is odd$n$ is odd and multiple of $3$$n$ is an even multiple of $3$$n$ is odd and not a multiple...
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