# Recent questions tagged isi2017-mma 1
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 $(x-x_1)(y-y_1)+(x-x_2)(y-y_2)=0$ $(x-x_1)(y-y_2)+(x-x_2)(y-y_1)=0$ $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$ $(x-x_1)(x-x_2)=(y-y_1)(y-y_2)=0$
2
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 $[2,3]$ $(2,3]$ $[-3,-2] \cup [2,3]$ $(-\infty,\infty)$
3
If $A$ is a $2 \times 2$ matrix such that $trace \: A = det \: A =3$, then what is the trace of $A^{-1}$? $1$ $1/3$ $1/6$ $1/2$
4
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
1 vote
5
Let $X_1$, and $X_2$ and $X_3$ be chosen independently from the set $\{0, 1, 2, 3, 4\}$, each value being equally likely. What is the probability that the arithmetic mean of $X_1, X_2$ and $X_3$ is the same as their geometric mean? $\frac{1}{5^2}\\$ $\frac{1}{5^3}\\$ $\frac{3!}{5^3}\\$ $\frac{3}{5^3}$
6
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 $4$
7
The digit in the unit's place of the number $2017^{2017}$ is $1$ $3$ $7$ $9$
8
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 five consecutive integers with sum $2015$ There are three consecutive integers with product $2015$
9
An even function $f(x)$ has left derivative $5$ at $x=0$. Then the right derivative of $f(x)$ at $x=0$ need not exist the right derivative of $f(x)$ at $x=0$ exists and is equal to $5$ the right derivative of $f(x)$ at $x=0$ exists and equal to $-5$ none of the above is necessarily true
10
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
11
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 nonsingular there is not enough information to decide the singularity of $M$ $M$ must have a positive eigenvalue
12
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 converges if $K=L$, then $(x_n)$ converges $(x_n)$ may not converge, but $K=L$ it is possible to have $K \neq L$
13
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 $P(Y =30)=$ $\frac{1}{100}$ $\frac{1}{100} \times (\frac{1}{30}+ \dots + \frac{1}{100})$ $\frac{1}{30}$ $\frac{1}{100} \times (\frac{1}{1}+ \dots +\frac{1}{30})$
14
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 $p$ $p^2$ $0$ $p^2+6q$
15
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
16
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 only two tests are required is $\frac{1}{2}\\$ $\frac{1}{3}\\$ $\frac{1}{4}\\$ $\frac{1}{6}$
1 vote
17
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 $m$ be the number of ways in which the same $12$ persons can stand in a queue such that exactly $6$ women stand consecutively. Then the value of $\frac{m}{n}$ is $5$ $7$ $\frac{5}{7}$ $\frac{7}{5}$
18
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 $1$ $3$ $1/2$ $1/3$
19
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 $k$? $3$ $0$ $-7$ $-3$
20
Let $n \geq 3$ be an integer.Then the statement $(n!)^{1/n} \leq \frac{n+1}{2}$ is true 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 $n \geq 5$
1 vote
21
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 $y^3-11y+11=0$ $y^3-11y-11=0$ $y^3+13y+13=0$ $y^3+6y^2+y-3=0$
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, there must be at least $3$ $X’s$. The number of ways in which this can be done is $1200$ $1800$ $2400$ $3000$
23
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 \end{bmatrix}$=$\begin{bmatrix} 0\\ 0\\ 0\\ 0 \end{bmatrix}$The locus of all$(a,b)\in\mathbb{R}^{2 ... system has at least two distinct solution for ($x_{1},x_{2},x_{3},x_{4}$) is a parabola a straight line entire $\mathbb{R}^{2}$ a point
24
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.] $5$ $7$ $9$ $10$
25
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
26
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
27
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 $N$ $H\cap N$ is a normal subgroup of $H$ but not necessarily of $N$ $H\cap N$ is a normal subgroup of $N$ but not necessarily of $H$ $H\cap N$ need not to be a normal subgroup of either $H$ or $N$
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 is drawn at random from the box ( without replacing the first one). Given that the first coin has shown head, the ... that the second coin is fair is $\frac{20}{39}\\$ $\frac{20}{37}\\$ $\frac{1}{2}\\$ $\frac{7}{13}$
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 $\frac{\pi}{3}+\frac{\sqrt{3}}{2}$ $\frac{\pi}{6}+\frac{\sqrt{3}}{4}$ $\frac{\pi}{3}-\frac{\sqrt{3}}{2}$ $\frac{\pi}{6}+\frac{\sqrt{3}}{2}$
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 $1$ $3$ $1/2$ $1/3$