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Recent questions tagged isi2021-mma

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ISI2021-MMA: 1
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $f(x) = \frac{2 – \sqrt{x+4}}{\sin 2x}$ for all $x \neq 0.$ Then the value of $f(0)$ is$ – \frac{1}{8}$\frac{1}{8}$0$ – \frac{1}{4}$
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ISI2021-MMA: 2
A person throws a pair of fair dice. If the sum of the numbers on the dice is a perfect square, then the probability that the number $3$ appeared on at least one of the dice is$1 / 9$4 / 7$1 / 18$7 / 36$
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ISI2021-MMA: 3
Consider the system of linear equations: $x + y + z = 5, \quad 2x + 2y + 3z = 4$. Thenthe system is inconsistentthe system has a unique solutionthe system has infinitely many solutionsnone of the above is true
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ISI2021-MMA: 4
If $g' (x) = f(x)$ then $\int x^{3} f(x^{2}) dx$ is given by$x^{2} g(x^{2}) - \int xg(x^{2}) dx + C$ ... ) - \int xg(x^{2}) dx + C$x^{2} g(x^{2}) - \frac{1}{2} \int xg(x^{2}) dx + C$
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ISI2021-MMA: 5
If $(^{n}C_{0} + ^{n}C_{1}) (^{n}C_{1} + ^{n}C_{2}) \cdots (^{n}C_{n-1} + ^{n}C_{n}) = k \; ^{n}C_{0} \; ^{n}C_{1} \cdots \; ^{n}C_{n-1},$ then $k$ is equal ... n+1)^{n}}{n!}$\frac{n^{n}}{n!}$\frac{(n+1)^{n}}{nn!}$\frac{(n+1)^{n+1}}{n!}$
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ISI2021-MMA: 6
Let $\{ f_{n}\}$ be a sequence of functions defined as follows:$f_{n}(x) = x^{n} \cos (2 \pi nx), \; x \in [ - 1, 1].$Then $\lim_{x \rightarrow 0} f_{n} (x)$ exists ... $ belongs to the interval$( - 1, 1)$[ - 1, 1)$[0, 1]$( - 1, 1]$
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ISI2021-MMA: 7
Let $S$ be a set of $n$ elements. The number of ways in which $n$ distinct non-empty subsets $X_{1}, \dots, X_{n}$ of $S$ ... ) \dots \left( \begin{array} c n \\ n \end{array} \right)$1$n!$2^{n}$
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ISI2021-MMA: 8
Let $A$ be a $4 \times 4$ matrix such that both $A$ and Adj$(A)$ are non-null. If $\det A = 0,$ then the rank of $A$ is$1$2$3$4$
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ISI2021-MMA: 9
The set of all $a$ satisfying the inequality$\frac{1}{\sqrt{a}} \int_{1}^{a} \left( \frac{3}{2} \sqrt{x} + 1 – \frac{1}{\sqrt{x}} \right) dx < 4$Is equal to the interval$( – 5, – 2)$(1, 4)$(0, 2)$(0, 4)$
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ISI2021-MMA: 10
Let $C_{0}$ be the set of all continuous functions $f:[0,1] \rightarrow \mathbb{R}$ and $C_{1}$ ... not onto$T$ is onto but not one-to-one$T$ is neither one-to-one nor onto.
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ISI2021-MMA: 11
Suppose $a, b, c$ are in $\text{A.P.}$ and $a^{2}, b^{2}, c^{2}$ are in $\text{G.P.}$ If $a < b < c$ and $a + b + c = \frac{3}{2},$ then the value of $a$ ... $\frac{1}{2} - \frac{1}{\sqrt{2}}$
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ISI2021-MMA: 12
The number of distinct even divisors of $\prod_{k=1}^{5} k!$ is$24$32$64$72$
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ISI2021-MMA: 13
Let $D$ be the triangular region in the $xy$-plane with vertices at $(0,0), (0,1)$ and $(1, 1).$ Then the value of $ \iint_{D} \frac{2}{1 + x^{2}} dx dy$ is$\frac{\pi}{2}$\frac{\pi}{2} – \ln 2$ 2 \ln 2$\ln 2$
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ISI2021-MMA: 14
Given a real number $\alpha \in (0, 1),$ define a sequence $\{ x_{n}\}_{n \geq 0}$ ... x_{1}} {2 - \alpha}$\frac{(1 - \alpha) x_{1} + x_{0}} {2 - \alpha}$
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ISI2021-MMA: 15
A straight line passes through the intersection of the lines given by $3x - 4y + 1 = 0$ and $5x + y = 1$ and makes equal intercepts of the same sign on the coordinate axes. The ... $23x + 23y - 11 = 0$
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ISI2021-MMA: 16
The series $ \sum_{n} \frac{3 \cdot 6 \cdot 9 \cdots 3 n}{7 \cdot 10 \cdot 13 \cdots(3 n+4)} x^{n}, \quad x>0 $converges for $0<x \leq 1$ and diverges for $x>1$ ... for $\frac{1}{2}<x<1$ and diverges for $0<x \leq \frac{1}{2}, x \geq 1$.
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ISI2021-MMA: 17
Suppose $A$ and $B$ are two square matrices such that the largest eigenvalue of $(AB – BA)$ is positive. Then the smallest eigen value of $(AB – BA)$must be positivemust be negativemust be $0$is none of the above
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ISI2021-MMA: 18
The number of saddle points of the function $f(x, y) = 2x^{4} – x^{2} + 3y^{2}$ is $1$0$2$none of the above
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ISI2021-MMA: 19
Suppose $G$ is a cyclic group and $a, b \in G$. There does not exist any $x \in G$ such that $x^{2}=a$. Also, there does not exist an $y \in G$ ... such that $g^{k}=a b$ for some $g \in G$ is $4 .$none of the above is true.
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ISI2021-MMA: 20
The number of real roots of the polynomial $x^{3}-2 x+7$ is$0$1$2$3$
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ISI2021-MMA: 21
Suppose that a $3 \times 3$ matrix $A$ has an eigen value $-1$. If the matrix $A+I$ ... R}$\left[\begin{array}{c}t \\ s \\ 2 t\end{array}\right], s, t \in \mathbb{R}$
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ISI2021-MMA: 22
Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $f(x, y)=x^{2}(y-1)$. For $\vec{u}=\left(\frac{1}{2}, \frac{1}{2}\right)$ ... $\frac{3}{4}$\frac{6}{13}$-\frac{1}{2}$none of the above
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ISI2021-MMA: 23
Suppose $\phi$ is a solution of the differential equation $y^{\prime \prime}-y^{\prime}-2 y=0$ such that $\phi(0)=1$ and $\phi^{\prime}(0)=5$ ... -\infty$ as $x \rightarrow-\infty$\phi(x) \rightarrow \infty$ as $x \rightarrow-\infty$
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ISI2021-MMA: 24
A fair die is rolled five times. What is the probability that the largest number rolled is $5$?$5 / 6$1 / 6$1-(1 / 6)^{6}$(5 / 6)^{5}-(2 / 3)^{5}$
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ISI2021-MMA: 25
Two rows of $n$ chairs, facing each other, are laid out. The number of different ways that $n$ couples can sit on these chairs such that each person sits directly opposite to his/her partner is$n!$n! / 2$2^{n} n!$2 n !$
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ISI2021-MMA: 26
Consider the function $f: \mathbb{C} \rightarrow \mathbb{C}$ defined on the complex plane $\mathbb{C}$ by $f(z)=e^{z}$. For a real number $c>0$ ... $A$ is a straight line segment and $B$ is a circleboth $A$ and $B$ are circles
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ISI2021-MMA: 27
Consider two real valued functions $f$ and $g$ ... $g^{-1}$ doesBoth $f^{-1}$ and $g^{-1}$ exist.
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ISI2021-MMA: 28
A circle is drawn with centre at $(-1,1)$ touching $x^{2}+y^{2}-4 x+6 y-3=0$ externally. Then the circle touchesboth the axesonly the $x$-axisnone of the two axesonly the $y$-axis
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ISI2021-MMA: 29
Let $f(x-y)=\frac{f(x)}{f(y)}$ for all $x, y \in \mathbb{R}$ and $f^{\prime}(0)=p, f^{\prime}(5)=q$. Then the value of $f^{\prime}(-5)$ is$q$-q$\frac{p}{q}$\frac{p^{2}}{q}$
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ISI2021-MMA: 30
Let $ A=\left[\begin{array}{lll} a & 1 & 1 \\ b & a & 1 \\ 1 & 1 & 1 \end{array}\right] . $ ... , b \leq 2021, \operatorname{rank}(A)=2\right\} $ is$2021$2020$2021^{2}-1$2020 \times 2021$
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