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Recent questions tagged isi2020-mma
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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 $(4, 1)$ $(-2, -3)$ $(3, 0)$ $(-2, 1)$
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...
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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
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 isAn ellipseA ...
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Jul 23, 2022
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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 $e^{4} - a$ $2e^{4} - a$ $e^{4} - e – 4a$ $2e^{4} - e – a$
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$$2e^{4} - e – a$
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Jul 23, 2022
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ISI2020-MMA: 4
The area bounded by the curves $y = e^{x}, y = xe^{x}$ and the $y$ - axis is $e – 2$ $e + 2$ $e – 1$ $2e – 3$
The area bounded by the curves $y = e^{x}, y = xe^{x}$ and the $y$ - axis is$e – 2$$e + 2$$e – 1$$2e – 3$
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Jul 23, 2022
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ISI2020-MMA: 5
The set of all solutions of the inequality $\frac{1}{2^{x} - 1} > \frac{1}{1 - 2^{x - 1}}$ is. $\left(1, \infty \right)$ $\left(0, \log_{2} \left ( \frac{4}{3} \right )\right)$ $\left(0, \log_{2} \left ( \frac{4}{3} \right )\right) \cup \left(1, \infty \right)$ $\left(-1, \infty \right)$
The set of all solutions of the inequality $\frac{1}{2^{x} - 1} \frac{1}{1 - 2^{x - 1}}$is.$\left(1, \infty \right)$$\left(0, \log...
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Jul 23, 2022
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ISI2020-MMA: 6
If $\displaystyle{}\lim_{x \rightarrow 0} \frac{ae^{x} - b \cos x}{x} = 5$, then. $a$ and $b$ are uniquely determined. $a$ is uniquely determined, but not $b$. $b$ is uniquely determined, but not $a$. neither $a$ nor $b$ is uniquely determined.
If $\displaystyle{}\lim_{x \rightarrow 0} \frac{ae^{x} - b \cos x}{x} = 5$, then.$a$ and $b$ are uniquely determined.$a$ is uniquely determined, but not $b$.$b$ is unique...
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Jul 23, 2022
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ISI2020-MMA: 7
Consider four events $\text{P, Q, R},$ and $\text{S}$ such that if any of $\text{P}$ and $\text{Q}$ occurs, then either $\text{R}$ occurs or $\text{S}$ doesn't occur. If exactly one of $\text{R}$ and $\text{S}$ always occurs, which of the following ... $\text{R} \Longrightarrow \text{P}^{c}$ $\text{R}^{c} \Longrightarrow \text{Q}^{c}$ $\text{R}^{c} \Longrightarrow \text{Q}$
Consider four events $\text{P, Q, R},$ and $\text{S}$ such that if any of $\text{P}$ and $\text{Q}$ occurs, then either $\text{R}$ occurs or $\text{S}$ doesn’t occur. I...
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Jul 23, 2022
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ISI2020-MMA: 8
The particular solution of $\log_{e}\left ( \frac{dy}{dx} \right ) = 5x + 7y, \;y(0)= 0$ is. $e^{5x}+5e^{-7y}=7$ $7e^{5x}-5e^{-7y}=5$ $5e^{5x}+7e^{7y}=12$ $7e^{5x}+5e^{-7y}=12$
The particular solution of$$\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}=1...
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Jul 23, 2022
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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, $0$ $\frac{1}{2}$ $\frac{2}{3}$ $1$
Define $\text{A}_{j} =\displaystyle{} \sum ^{n}_{i=1} i^{j}, j = 0, 1, 2, 3.$ Then.$$\lim_{n \rightarrow \infty } \frac{\text{A}_{1} \text{A}_{2} }{\text{A}_{0} \text{A}...
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Jul 23, 2022
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ISI2020-MMA: 10
Let $p, q, r \in \mathbb{R}$. If $f(x) = px^{2} + qx + r$ be such that $p + q + r = 3$ and $f (x + y) = f(x) + f(y) + xy$, for all $x, y \in \mathbb{R}$. Then the value of $f(5)$ is. $25$ $30$ $35$ $40$
Let $p, q, r \in \mathbb{R}$. If $f(x) = px^{2} + qx + r$ be such that $p + q + r = 3$ and $f (x + y) = f(x) + f(y) + xy$, for all $x, y \in \mathbb{R}$. Then the value o...
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Jul 23, 2022
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ISI2020-MMA: 11
If ${ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}$ denote the binomial coefficients in the expansion of $(1+x)^{n}, p>0$ is a real number and $q=1-p$, then $ \sum_{r=0}^{n} r^{2}{ }^{n} C_{r} p^{n-r} q^{r} $ is $n p^{2} q^{2}$ $n^{2} p^{2} q^{2}$ $n p q+n^{2} p^{2}$ $n p q+n^{2} q^{2}$
If ${ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}$ denote the binomial coefficients in the expansion of $(1+x)^{n}, p>0$ is a real number and $q=1-p$, t...
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Jul 23, 2022
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ISI2020-MMA: 12
If $| z + 3 – 2i| = 8$ and the maximum and the minimum values of $|2z + 9 – 8i|$ are $\alpha$ and $\beta$, respectively, then the value of $\alpha + \beta$ is. $10$ $21$ $32$ $27$
If $| z + 3 – 2i| = 8$ and the maximum and the minimum values of $|2z + 9 – 8i|$ are $\alpha$ and $\beta$, respectively, then the value of $\alpha + \beta$ is.$10$$21...
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Jul 23, 2022
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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
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 positiveIt has two positive re...
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Jul 23, 2022
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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
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 \righta...
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Jul 23, 2022
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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 $k$? $2$ $4$ $8$ $16$
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 p...
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Jul 23, 2022
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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. $5$ $6$ $7$ $8$
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$$8$
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Jul 23, 2022
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ISI2020-MMA: 17
The number of words that can be constructed using $10$ letters of the English alphabet such that all five vowels appear exactly once in the word is $^{21} \text{C}_{5} \;10!$ $^{21} \text{C}_{5} \;(5!)^{2}$ $^{10} \text{P}_{5} \; ^{21} \text{P}_{5} $ $^{10} \text{P}_{5} \;(21)^{5}$
The number of words that can be constructed using $10$ letters of the English alphabet such that all five vowels appear exactly once in the word is$^{21} \text{C}_{5} \;1...
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Jul 23, 2022
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ISI2020-MMA: 18
Let $f : [0, \infty ) \rightarrow \mathbb{R}$ be a differentiable function with $f(0) = 1$ and $f(x) f' (x) > 0$, for all $x$. Let $\text{A} (n)$ be the area of region bounded by $x$ - axis, $y$ - axis, graph of $f$ and the ... $\text{A} : \mathbb{N}\rightarrow \mathbb{R}$ is increasing None of the above statements is true
Let $f : [0, \infty ) \rightarrow \mathbb{R}$ be a differentiable function with $f(0) = 1$ and $f(x) f’ (x) 0$, for all $x$. Let $\text{A} (n)$ be the area of region b...
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Jul 23, 2022
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ISI2020-MMA: 19
Let $x, y, z$ be the three natural numbers. Then the number of triplets $(x, y, z)$ such that $xyz = 100$ is $36$ $25$ $72$ $18$
Let $x, y, z$ be the three natural numbers. Then the number of triplets $(x, y, z)$ such that $xyz = 100$ is$36$$25$$72$$18$
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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 $\left \{ 0, 1, 2, 3, 4, 5, 6, 7 \right \}?$ $63$ $57$ $37$ $49$
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 \ri...
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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.
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 o...
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Jul 23, 2022
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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 $0$ $10$ $16$ $\infty$
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 numbe...
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Jul 23, 2022
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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 $p$
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 $p$Lin...
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Jul 23, 2022
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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. $\left \{ 1 \right \}$ $\left \{ 1, 2\right \}$ $\left \{ 1, 3 \right \}$ $\left \{ 1, 2, 3 \right \}$
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 \...
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Jul 23, 2022
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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
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 n...
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Jul 23, 2022
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ISI2020-MMA: 26
For a cyclic group $\text{G}$ of order $12$, the number of subgroups of $\text{G}$ is $2$ $6$ $8$ $11$
For a cyclic group $\text{G}$ of order $12$, the number of subgroups of $\text{G}$ is$2$$6$$8$$11$
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ISI2020-MMA: 27
Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function and $f(1) = 4$. Then the value of $\lim_{x\rightarrow 1}\int_{4}^{f(x)}\frac{2t}{x - 1}dt$ is. $8f’(1)$ $2f’(1)$ $4f’(1)$ $f’(1)$
Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function and $f(1) = 4$. Then the value of$$\lim_{x\rightarrow 1}\int_{4}^{f(x)}\frac{2t}{x - ...
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Jul 23, 2022
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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 $\left [ \frac{-1}{2}, \frac{1}{2} \right ]$
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 n...
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Jul 23, 2022
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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.
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(\tex...
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Jul 23, 2022
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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
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} \...
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