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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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Jul 23
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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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Jul 23
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isi2021-mma
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3
ISI2021-MMA: 3
Consider the system of linear equations: $x + y + z = 5, \quad 2x + 2y + 3z = 4$. Then the system is inconsistent the system has a unique solution the system has infinitely many solutions none of the above is true
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Jul 23
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isi2021-mma
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4
ISI2021-MMA: 4
If $g’ (x) = f(x)$ then $\int x^{3} f(x^{2}) dx$ is given by $x^{2} g(x^{2}) – \int xg(x^{2}) dx + C$ $ \frac{1}{2} x^{2} g(x^{2}) – \int xg(x^{2}) dx + C$ $2x^{2} g(x^{2}) – \int xg(x^{2}) dx + C$ $x^{2} g(x^{2}) – \frac{1}{2} \int xg(x^{2}) dx + C$
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Jul 23
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isi2021-mma
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5
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 to $\frac{(n+1)^{n}}{n!}$ $\frac{n^{n}}{n!}$ $\frac{(n+1)^{n}}{nn!}$ $\frac{(n+1)^{n+1}}{n!}$
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Jul 23
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isi2021-mma
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6
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 if and only if $x$ belongs to the interval $( – 1, 1)$ $[ – 1, 1)$ $[0, 1]$ $( – 1, 1]$
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Jul 23
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isi2021-mma
1
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7
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 $S$ can be chosen such that $X_{1} \subseteq X_{2} \dots \subseteq X_{n},$ is $\left( \begin{array} c n \\ 1 \end{array} \right) \left( \begin{array} c n \\ 2 \end{array} \right) \dots \left( \begin{array} c n \\ n \end{array} \right)$ $1$ $n!$ $2^{n}$
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Jul 23
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isi2021-mma
0
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1
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8
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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Jul 23
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isi2021-mma
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9
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 $( – 5, – 2)$ $(1, 4)$ $(0, 2)$ $(0, 4)$
Lakshman Patel RJIT
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Jul 23
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isi2021-mma
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10
ISI2021-MMA: 10
Let $C_{0}$ be the set of all continuous functions $f:[0,1] \rightarrow \mathbb{R}$ and $C_{1}$ be the set of all differentiable functions $g:[0,1] \rightarrow \mathbb{R}$ such that the derivative $g^{\prime}$ is continuous. (Here, differentiability at $0$ means right ... -one and onto $T$ is one-to-one but not onto $T$ is onto but not one-to-one $T$ is neither one-to-one nor onto.
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Jul 23
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isi2021-mma
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1
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11
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 $a$ is $\frac{1}{2 \sqrt{2}}$ $ – \frac{1}{2 \sqrt{2}}$ $\frac{1}{2} – \frac{1}{\sqrt{3}}$ $\frac{1}{2} – \frac{1}{\sqrt{2}}$
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
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isi2021-mma
0
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1
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12
ISI2021-MMA: 12
The number of distinct even divisors of $\prod_{k=1}^{5} k!$ is $24$ $32$ $64$ $72$
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Jul 23
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isi2021-mma
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13
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$ $\ln 2$
Lakshman Patel RJIT
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Jul 23
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isi2021-mma
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14
ISI2021-MMA: 14
Given a real number $\alpha \in (0, 1),$ define a sequence $\{ x_{n}\}_{n \geq 0}$ by the following recurrence relation: $x_{n+1} = \alpha x_{n} + (1 - \alpha) x_{n - 1}, n \geq 1.$ If $\lim_{n \rightarrow \infty} x_{n} = \ell$ then the value of $\ell$ ... $\frac{\alpha x_{0} + x_{1}} {2 - \alpha}$ $\frac{(1 - \alpha) x_{1} + x_{0}} {2 - \alpha}$
Lakshman Patel RJIT
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Jul 23
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40
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isi2021-mma
0
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15
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 equation of the straight line is $23x – 23y + 11 = 0$ $23x – 23y - 11 = 0$ $23x + 23y + 11 = 0$ $23x + 23y – 11 = 0$
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
70
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isi2021-mma
0
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16
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 $x>1$ converges for all $x>0$ converges for $0<x<\frac{1}{2}$ and diverges for $x \geq \frac{1}{2}$ converges for $\frac{1}{2}<x<1$ and diverges for $0<x \leq \frac{1}{2}, x \geq 1$.
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
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isi2021-mma
0
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0
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17
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 positive must be negative must be $0$ is none of the above
Lakshman Patel RJIT
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Jul 23
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44
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isi2021-mma
0
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18
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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Jul 23
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40
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isi2021-mma
0
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0
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19
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 $y^{2}=b$. Then, there exists an element $g \in G$ such that $g^{2}=a b$ ... $g^{3}=a b$. the smallest exponent $k>1$ such that $g^{k}=a b$ for some $g \in G$ is $4 .$ none of the above is true.
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Jul 23
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isi2021-mma
0
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0
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20
ISI2021-MMA: 20
The number of real roots of the polynomial $x^{3}-2 x+7$ is $0$ $1$ $2$ $3$
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
54
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isi2021-mma
0
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0
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21
ISI2021-MMA: 21
Suppose that a $3 \times 3$ matrix $A$ has an eigen value $-1$. If the matrix $A+I$ is equal to $ \left[\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] $ then the eigen vectors of $A$ corresponding to the eigenvalue ... $\left[\begin{array}{c}t \\ s \\ 2 t\end{array}\right], s, t \in \mathbb{R}$
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
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isi2021-mma
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0
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22
ISI2021-MMA: 22
Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $f(x, y)=x^{2}(y-1)$. For $\vec{u}=\left(\frac{1}{2}, \frac{1}{2}\right)$ and $\vec{v}=(3,4)$, the value of the limit $ \lim _{t \rightarrow 0} \frac{f(\vec{u}+t \vec{v})-f(\vec{u})}{t} $ is $\frac{3}{4}$ $\frac{6}{13}$ $-\frac{1}{2}$ none of the above
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
48
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isi2021-mma
0
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0
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23
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 $\phi^{\prime}(0)=5$. Then $\phi(x) \rightarrow \infty$ as $|x| \rightarrow \infty$ $\phi(x) \rightarrow-\infty$ as $|x| \rightarrow \infty$ $\phi(x) \rightarrow-\infty$ as $x \rightarrow-\infty$ $\phi(x) \rightarrow \infty$ as $x \rightarrow-\infty$
Lakshman Patel RJIT
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Jul 23
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Lakshman Patel RJIT
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isi2021-mma
2
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1
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24
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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Jul 23
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isi2021-mma
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25
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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Jul 23
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isi2021-mma
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26
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 $c>0$, let $A=\{f(z) \mid \operatorname{Re} z=c\}$ and $B=\{f(z) \mid \operatorname{Im} z=c\}$ ... $A$ is a circle and $B$ is a straight line segment $A$ is a straight line segment and $B$ is a circle both $A$ and $B$ are circles
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Jul 23
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isi2021-mma
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27
ISI2021-MMA: 27
Consider two real valued functions $f$ and $g$ given by $f(x)=\frac{x}{x-1} \; \text{for } x>1, \quad \text{and }\quad g(x)=7-x^{3} \; \text{for } x \in \mathbb{R}.$ Which of the following statements about inverse functions is true? Neither $f^{-1}$ nor $g^{-1}$ exists $f^{-1}$ exists, but not $g^{-1}$ $f^{-1}$ does not exist, but $g^{-1}$ does Both $f^{-1}$ and $g^{-1}$ exist.
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Jul 23
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isi2021-mma
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28
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 touches both the axes only the $x$-axis none of the two axes only the $y$-axis
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Jul 23
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isi2021-mma
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29
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 $q$ $-q$ $\frac{p}{q}$ $\frac{p^{2}}{q}$
Lakshman Patel RJIT
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Jul 23
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isi2021-mma
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30
ISI2021-MMA: 30
Let $ A=\left[\begin{array}{lll} a & 1 & 1 \\ b & a & 1 \\ 1 & 1 & 1 \end{array}\right] . $ The number of elements in the set $ \left\{(a, b) \in \mathbb{Z}^{2}: 0 \leq a, b \leq 2021, \operatorname{rank}(A)=2\right\} $ is $2021$ $2020$ $2021^{2}-1$ $2020 \times 2021$
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Jul 23
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Please upload 4th Mock Test, due date was 4th Dec.
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