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Recent questions tagged isi2020-mma
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1
ISI2020-MMA: 1
Let $\text{A} = (1, -1), \text{B} = (-2, 0), \text{C} = (1, 2)$ and $\text{D}$ be the vertices of a parallelogram in the $\text{X – Y}$ plane listed clockwise. Then the point $\text{D}$ is $(4, 1)$ $(-2, -3)$ $(3, 0)$ $(-2, 1)$
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2
ISI2020-MMA: 2
Let $z = (1 – t^{2}) + i \sqrt{1 - t^{2}}$ be a complex number where $t$ is a real number such that $|t| < 1$. Then the locus of $z$ in the complex plane is An ellipse A hyperbola A parabola A pair of straight lines
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3
ISI2020-MMA: 3
Let $\int ^{2}_{1} e^{x^{2}} dx = a$. Then the value of $\int ^{e^{4}}_{e} \sqrt{\log_{e} x} dx$ is $e^{4} - a$ $2e^{4} - a$ $e^{4} - e – 4a$ $2e^{4} - e – a$
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4
ISI2020-MMA: 4
The area bounded by the curves $y = e^{x}, y = xe^{x}$ and the $y$ - axis is $e – 2$ $e + 2$ $e – 1$ $2e – 3$
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5
ISI2020-MMA: 5
The set of all solutions of the inequality $\frac{1}{2^{x} - 1} > \frac{1}{1 - 2^{x - 1}}$ is. $\left(1, \infty \right)$ $\left(0, \log_{2} \left ( \frac{4}{3} \right )\right)$ $\left(0, \log_{2} \left ( \frac{4}{3} \right )\right) \cup \left(1, \infty \right)$ $\left(-1, \infty \right)$
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isi2020-mma
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6
ISI2020-MMA: 6
If $\displaystyle{}\lim_{x \rightarrow 0} \frac{ae^{x} - b \cos x}{x} = 5$, then. $a$ and $b$ are uniquely determined. $a$ is uniquely determined, but not $b$. $b$ is uniquely determined, but not $a$. neither $a$ nor $b$ is uniquely determined.
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Jul 23
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7
ISI2020-MMA: 7
Consider four events $\text{P, Q, R},$ and $\text{S}$ such that if any of $\text{P}$ and $\text{Q}$ occurs, then either $\text{R}$ occurs or $\text{S}$ doesn't occur. If exactly one of $\text{R}$ and $\text{S}$ always occurs, which of the following ... $\text{R} \Longrightarrow \text{P}^{c}$ $\text{R}^{c} \Longrightarrow \text{Q}^{c}$ $\text{R}^{c} \Longrightarrow \text{Q}$
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8
ISI2020-MMA: 8
The particular solution of $\log_{e}\left ( \frac{dy}{dx} \right ) = 5x + 7y, \;y(0)= 0$ is. $e^{5x}+5e^{-7y}=7$ $7e^{5x}-5e^{-7y}=5$ $5e^{5x}+7e^{7y}=12$ $7e^{5x}+5e^{-7y}=12$
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Jul 23
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9
ISI2020-MMA: 9
Define $\text{A}_{j} =\displaystyle{} \sum ^{n}_{i=1} i^{j}, j = 0, 1, 2, 3.$ Then. $\lim_{n \rightarrow \infty } \frac{\text{A}_{1} \text{A}_{2} }{\text{A}_{0} \text{A}_{3}}$ is, $0$ $\frac{1}{2}$ $\frac{2}{3}$ $1$
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Jul 23
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10
ISI2020-MMA: 10
Let $p, q, r \in \mathbb{R}$. If $f(x) = px^{2} + qx + r$ be such that $p + q + r = 3$ and $f (x + y) = f(x) + f(y) + xy$, for all $x, y \in \mathbb{R}$. Then the value of $f(5)$ is. $25$ $30$ $35$ $40$
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Jul 23
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11
ISI2020-MMA: 11
If ${ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}$ denote the binomial coefficients in the expansion of $(1+x)^{n}, p>0$ is a real number and $q=1-p$, then $ \sum_{r=0}^{n} r^{2}{ }^{n} C_{r} p^{n-r} q^{r} $ is $n p^{2} q^{2}$ $n^{2} p^{2} q^{2}$ $n p q+n^{2} p^{2}$ $n p q+n^{2} q^{2}$
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Jul 23
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12
ISI2020-MMA: 12
If $| z + 3 – 2i| = 8$ and the maximum and the minimum values of $|2z + 9 – 8i|$ are $\alpha$ and $\beta$, respectively, then the value of $\alpha + \beta$ is. $10$ $21$ $32$ $27$
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13
ISI2020-MMA: 13
Consider the cubic equation $x^{3} = 2x + 5$. Which of the following statements about the above equation is true? All its roots are real and positive It has two positive real roots and one negative real root It has two negative real roots and one positive real root It has one real root and a pair of complex roots
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14
ISI2020-MMA: 14
Consider two real-valued sequences $\left \{ x_{n} \right \}$ and $\left \{ y_{n} \right \}$ satisfying the condition $x^{3}_{n} - y^{3}_{n} \rightarrow 0$ as $n \rightarrow \infty $. Then, as $n \rightarrow \infty $, $x_{n} - y_{n} \rightarrow 0$ ... $x_{n} - y_{n} \rightarrow 0$ only if $\left \{ |x^{2}_{n} +x_{n} y_{n} + y^{2}_{n}| \right \}$ converges
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Jul 23
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isi2020-mma
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15
ISI2020-MMA: 15
Let $\frac{d}{dx} \text{P}(x)=\frac{e^{\sin x}}{x}, x > 0$. If $\int ^{2}_{1}\frac{3}{x} e^{\sin x^{3}} dx= \text{P}(k) - \text{P}(1)$, then which of the following is a possible value of $k$? $2$ $4$ $8$ $16$
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Jul 23
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16
ISI2020-MMA: 16
The distance of the point $(1, -2, 3)$ from the plane $x – y + z = 11$ measured along a line parallel to $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$ is. $5$ $6$ $7$ $8$
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Jul 23
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isi2020-mma
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17
ISI2020-MMA: 17
The number of words that can be constructed using $10$ letters of the English alphabet such that all five vowels appear exactly once in the word is $^{21} \text{C}_{5} \;10!$ $^{21} \text{C}_{5} \;(5!)^{2}$ $^{10} \text{P}_{5} \; ^{21} \text{P}_{5} $ $^{10} \text{P}_{5} \;(21)^{5}$
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Jul 23
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isi2020-mma
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18
ISI2020-MMA: 18
Let $f : [0, \infty ) \rightarrow \mathbb{R}$ be a differentiable function with $f(0) = 1$ and $f(x) f' (x) > 0$, for all $x$. Let $\text{A} (n)$ be the area of region bounded by $x$ - axis, $y$ - axis, graph of $f$ and the ... $\text{A} : \mathbb{N}\rightarrow \mathbb{R}$ is increasing None of the above statements is true
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Jul 23
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isi2020-mma
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19
ISI2020-MMA: 19
Let $x, y, z$ be the three natural numbers. Then the number of triplets $(x, y, z)$ such that $xyz = 100$ is $36$ $25$ $72$ $18$
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Jul 23
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20
ISI2020-MMA: 20
How many distinct straight lines can one form that are given by an equation $ax + by = 0$, where $a$ and $b$ are numbers from the set $\left \{ 0, 1, 2, 3, 4, 5, 6, 7 \right \}?$ $63$ $57$ $37$ $49$
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Jul 23
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21
ISI2020-MMA: 21
Consider three non-zero matrices $\text{A}, \text{B}$ and $\text{C}$ such that $\text{ABB}’ = \text{CBB}’$ where $\text{B}’$ is the transpose of $\text{B}$. Which of the following statements is necessarily true? $r(\text{A}) =r(\text{C})$ non-zero eigenvalues of $\text{A}$ and $\text{C}$ are identical. $\text{AB = CB}$ None of the above.
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isi2020-mma
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22
ISI2020-MMA: 22
Let $m$ and $n$ be nonzero integers. Define $\text{A}_{m, n}= \left \{ x \in \mathbb{R}:n^{2} x^{3}+ 2020x^{2}+mx = 0\right \}$. Then the number of pairs $(m, n)$ for which $\text{A}_{m, n}$ has exactly two points is $0$ $10$ $16$ $\infty$
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Jul 23
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23
ISI2020-MMA: 23
Consider two independent events with the same probability $p (0 < p < 1)$. Then the probability of occurrence of at least one of the two events is. The same for all $p$ Linearly increasing in $p$ Strictly convex in $p$ Strictly concave in $p$
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isi2020-mma
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24
ISI2020-MMA: 24
Let $\text{S}$ be the set of all $3 \times 3$ real matrices $\text{A} = ((a_{ij}))$ such that the matrix $ ((a^{3}_{ij}))$ has rank one. Define a set $\text{R} = \left \{ \text{rank(A)} : \text{A} \in \text{S}\right \}$. Then $\text{R}$ is equal to. $\left \{ 1 \right \}$ $\left \{ 1, 2\right \}$ $\left \{ 1, 3 \right \}$ $\left \{ 1, 2, 3 \right \}$
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Jul 23
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25
ISI2020-MMA: 25
The function $f : \mathbb{R}\rightarrow \mathbb{R}$ is defined by $f(x)= \left\{\begin{matrix} e^{-\frac{1}{x}}, & x > 0\\ 0,& x \leq 0\;. \end{matrix}\right.$ Then $f$ is not continuous $f$ is continuous, but not differentiable everywhere $f$ is differentiable but $f’$ is not continuous $f$ is differentiable and $f’$ is continuous
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Jul 23
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26
ISI2020-MMA: 26
For a cyclic group $\text{G}$ of order $12$, the number of subgroups of $\text{G}$ is $2$ $6$ $8$ $11$
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isi2020-mma
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27
ISI2020-MMA: 27
Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function and $f(1) = 4$. Then the value of $\lim_{x\rightarrow 1}\int_{4}^{f(x)}\frac{2t}{x - 1}dt$ is. $8f’(1)$ $2f’(1)$ $4f’(1)$ $f’(1)$
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28
ISI2020-MMA: 28
The series $\frac{2x}{1+x^{2}}+\frac{4x^{3}}{1+x^{4}}+\frac{8x^{7}}{1+x^{8}}+\dots$ is uniformly convergent for all $x$ is convergent for all $x$, but the convergence is not uniform is convergent only for $|x| \leq \frac{1}{2}$, but the convergence is not uniform is uniformly convergent on $\left [ \frac{-1}{2}, \frac{1}{2} \right ]$
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29
ISI2020-MMA: 29
Let $\text{S}$ and $\text{T}$ be two non-empty sets and $f : \text{S} \rightarrow \text{T}$ be a function such that $f (\text{A} \cap \text{B}) = f (\text{A}) \cap f(\text{B})$ for all subsets $\text{A}$ and $\text{B}$ of $\text{S}$. Then there ... $\text{S}$ such that $f(\text{A}) \cap f(\text{B}) \neq \phi $ none of the above statements is necessarily true.
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30
ISI2020-MMA: 30
Let $x_{1}, x_{2}, , x_{n} \in \mathbb{R}$ be distinct reals. Define the set $\text{A} = \left \{ \left ( f_{1}(t), f_{2} (t), \dots, f_{n}(t)\right ):t \in\mathbb{R} \right \},$ ... $\text{A}$ contains exactly $n$ distinct elements exactly $(n + 1)$ distinct elements exactly $2^{n}$ distinct elements infinitely many distinct elements
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Please upload 4th Mock Test, due date was 4th Dec.
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