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Recent questions tagged isi2017-mma

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ISI2017-MMA-2
If $(x_1, y_1)$ and $(x_2, y_2)$ are the opposite end points of a diameter of a circle, then the equation of the circle is given by$(x-x_1)(y-y_1)+(x-x_2)(y-y_2)=0$(x-x_1)( ... x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$(x-x_1)(x-x_2)=(y-y_1)(y-y_2)=0$
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ISI2017-MMA-4
Let $S\subseteq \mathbb{R}$. Consider the statement “There exists a continuous function $f:S\rightarrow S$ such that $f(x) \neq x$ for all $x \in S.$ ”This statement is false if $S$ equals$[2,3]$(2,3]$[-3,-2] \cup [2,3]$(-\infty,\infty)$
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ISI2017-MMA-5
If $A$ is a $2 \times 2$ matrix such that $trace \: A = det \: A =3$, then what is the trace of $A^{-1}$?$1$1/3$1/6$1/2$
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ISI2017-MMA-6
In a class of $80$ students, $40$ are girls and $40$ are boys. Also, exactly $50$ students wear glasses. Then the set of all possible numbers of boys without glasses is$\{0, \dots , 30\}$\{10, \dots , 30\}$\{0, \dots , 40\}$none of these
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ISI2017-MMA-8
Let $X_1$, and $X_2$ and $X_3$ be chosen independently from the set $\{0, 1, 2, 3, 4\}$ ... 2}\\$\frac{1}{5^3}\\$\frac{3!}{5^3}\\$\frac{3}{5^3}$
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4 answers
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ISI2017-MMA-10
The inequality $\mid x^2 -5x+4 \mid > (x^2-5x+4)$ holds if and only if$1 < x < 4$x \leq 1$ and $x \geq 4$1 \leq x \leq 4$x$ takes any value except $1$ and $4$
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ISI2017-MMA-11
The digit in the unit's place of the number $2017^{2017}$ is$1$3$7$9$
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ISI2017-MMA-12
Which of the following statements is true?There are three consecutive integers with sum $2015$There are four consecutive integers with sum $2015$There are ... integers with sum $2015$There are three consecutive integers with product $2015$
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ISI2017-MMA-13
An even function $f(x)$ has left derivative $5$ at $x=0$. Thenthe right derivative of $f(x)$ at $x=0$ need not existthe right derivative of $f(x)$ at $x=0$ ... $f(x)$ at $x=0$ exists and equal to $-5$none of the above is necessarily true
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ISI2017-MMA-14
Let $(v_n)$ be a sequence defined by $v_1=1$ and $v_{n+1}=\sqrt{v_n^2 +(\frac{1}{5})^n}$ for $n\geq1$. Then $\lim _{n\rightarrow \infty} v_n$ is$\sqrt{5/3}$\sqrt{5/4}$1$nonexistent
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ISI2017-MMA-15
The diagonal elements of a square matrix $M$ are odd integers while the off-diagonals are even integers. Then$M$ must be singular$M$ must be ... $M$ must have a positive eigenvalue
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ISI2017-MMA-16
Let $(x_n)$ be a sequence of real numbers such that the subsequences $(x_{2n})$ and $(x_{3n})$ converge to limits $K$ and $L$ respectively. Then$(x_n)$ always ... $(x_n)$ may not converge, but $K=L$it is possible to have $K \neq L$
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ISI2017-MMA-17
Suppose that $X$ is chosen uniformly from $\{1, 2, \dots , 100\}$ and given $X=x, \: Y$ is chosen uniformly from $\{1, 2, \dots , x\}$. Then $P(Y =30)=$\frac{1}{ ... $\frac{1}{100} \times (\frac{1}{1}+ \dots +\frac{1}{30})$
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ISI2017-MMA-19
If $\alpha, \beta$ and $\gamma$ are the roots of $x^3-px+q=0$, then the value of the determinant $\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix}$ is$p$p^2$0$p^2+6q$
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ISI2017-MMA-20
The number of ordered pairs $(X, Y)$, where $X$ and $Y$ are $n \times n$ real, matrices such that $XY-YX=I$ is$0$1$n$infinite
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ISI2017-MMA-21
There are four machines and it is known that exactly two of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. The probability that ... 2}\\$\frac{1}{3}\\$\frac{1}{4}\\$\frac{1}{6}$
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1 votes
ISI2017-MMA-26
Let $n$ be the number of ways in which $5$ men and $7$ women can stand in a queue such that all the women stand consecutively. Let $m$ ... $ is$5$7$\frac{5}{7}$\frac{7}{5}$
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ISI2017-MMA-29
Suppose the rank of the matrix $\begin{pmatrix} 1 & 1 & 2 & 2 \\ 1 & 1 & 1 & 3 \\ a & b & b & 1 \end{pmatrix}$ is 2 for some real numbers $a$ and $b$. Then the $b$ equals$1$3$1/2$1/3$
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783
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3 answers
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ISI2017-MMA-30
The graph of a cubic polynomial $f(x)$ is shown below. If $k$ is a constant such that $f(x)=k$ has three real solutions, which of the following could be a possible value of $k$?$3$0$-7$-3$
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2 answers
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ISI2017-MMA-7
Let $n \geq 3$ be an integer.Then the statement $(n!)^{1/n} \leq \frac{n+1}{2}$ istrue for every $n \geq 3$true if and only if $n \geq 5$not true for $n \geq 10$true for even integers $n \geq 6$, not true for odd $n \geq 5$
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675
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1 answers
1 votes
ISI2017-MMA-3
If $\alpha,\beta $ and $ \gamma$ are the roots of the equation $x^3+3x^2-8x+1=0$, then an equation whose roots are $\alpha +1 , \beta +1 , \gamma +1 $ is given by$y^3-11y+11=0$y^3-11y-11=0$y^3+13y+13=0$y^3+6y^2+y-3=0$
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3 answers
7 votes
ISI2017-MMA-22
The five vowels-$A, E, I, O, U$-along with $15$ $X's$ are to be arranged in a row such that no $X$ is at an extreme position. Also, between any two vowels, ... $ $X's$. The number of ways in which this can be done is$1200$1800$2400$3000$
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ISI2017-MMA-18
Consider following system of equations:$\begin{bmatrix} 1 &2 &3 &4 \\ 5&6 &7 &8 \\ a&9 &b &10 \\ 6&8 &10 & 13 \end{bmatrix}$\begin{bmatrix} x1\\ x2\\ x3\\ x4 ... $\mathbb{R}^{2}$a point
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ISI2017-MMA-23
What is the smallest degree of a polynomial with real coefficients and having root $2\omega , 2 + 3\omega , 2\omega^{2} , -1 -3\omega$ and $2-\omega - \omega^{2}?$ [Here $\omega\neq$1 is a cube root of unity.]$5$7$9$10$
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5 answers
10 votes
ISI2017-MMA-24
The number of polynomial function $f$ of degree $\geq$ 1 satisfying $f(x^{2})=(f(x))^{2}=f(f(x))$ for all real $x$, is$0$1$2$infinitely many
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ISI2017-MMA-25
For $a,b \in \mathbb{R}$ and $b > a$ , the maximum possible value of the integral $\int_{a}^{b}(7x-x^{2}-10)dx$ is$\frac{7}{2}\\$\frac{9}{2}\\$\frac{11}{2}\\$none of these
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3 answers
3 votes
ISI2017-MMA-28
Let $H$ be a subgroup of group $G$ and let $N$ be a normal subgroup of $G$. Choose the correct statement :$H\cap N$ is a normal subgroup of both $H$ and $N$H\ ... $H\cap N$ need not to be a normal subgroup of either $H$ or $N$
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4 answers
8 votes
ISI2017-MMA-27
A box contains $5$ fair and $5$ biased coins. Each biased coin has a probability of head $\frac{4}{5}$. A coin is drawn at random from the box and tossed. Then the second coin ... frac{20}{39}\\$\frac{20}{37}\\$\frac{1}{2}\\$\frac{7}{13}$
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3 answers
5 votes
ISI2017-MMA-1
The area lying in the first quadrant and bounded by the circle $x^{2}+y^{2}=4$ and the lines $x= 0$ and $x=1$ is given by$\frac{\pi}{3}+\frac{\sqrt{3}}{2}$\frac{\pi}{ ... $\frac{\pi}{6}+\frac{\sqrt{3}}{2}$
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8 answers
10 votes
ISI2017-MMA-29
Suppose the rank of the matrix$\begin{pmatrix}1&1&2&2\\1&1&1&3\\a&b&b&1\end{pmatrix}$is $2$ for some real numbers $a$ and $b$. Then $b$ equals$1$3$1/2$1/3$
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