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1221
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.$$\lim_{n \rightarrow \infty } \frac{\text{A}_{1} \text{A}_{2} }{\text{A}_{0} \text{A}...
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1222
ISI2020-MMA: 10
Let $p, q, r \in \mathbb{R}$. If $f(x) = px^{2} + qx + r$ be such that $p + q + r = 3$ and $f (x + y) = f(x) + f(y) + xy$, for all $x, y \in \mathbb{R}$. Then the value of $f(5)$ is. $25$ $30$ $35$ $40$
Let $p, q, r \in \mathbb{R}$. If $f(x) = px^{2} + qx + r$ be such that $p + q + r = 3$ and $f (x + y) = f(x) + f(y) + xy$, for all $x, y \in \mathbb{R}$. Then the value o...
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1223
ISI2020-MMA: 11
If ${ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}$ denote the binomial coefficients in the expansion of $(1+x)^{n}, p>0$ is a real number and $q=1-p$, then $ \sum_{r=0}^{n} r^{2}{ }^{n} C_{r} p^{n-r} q^{r} $ is $n p^{2} q^{2}$ $n^{2} p^{2} q^{2}$ $n p q+n^{2} p^{2}$ $n p q+n^{2} q^{2}$
If ${ }^{n} C_{0},{ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}$ denote the binomial coefficients in the expansion of $(1+x)^{n}, p>0$ is a real number and $q=1-p$, t...
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1224
ISI2020-MMA: 12
If $| z + 3 – 2i| = 8$ and the maximum and the minimum values of $|2z + 9 – 8i|$ are $\alpha$ and $\beta$, respectively, then the value of $\alpha + \beta$ is. $10$ $21$ $32$ $27$
If $| z + 3 – 2i| = 8$ and the maximum and the minimum values of $|2z + 9 – 8i|$ are $\alpha$ and $\beta$, respectively, then the value of $\alpha + \beta$ is.$10$$21...
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1227
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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1228
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$$8$
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1229
ISI2020-MMA: 17
The number of words that can be constructed using $10$ letters of the English alphabet such that all five vowels appear exactly once in the word is $^{21} \text{C}_{5} \;10!$ $^{21} \text{C}_{5} \;(5!)^{2}$ $^{10} \text{P}_{5} \; ^{21} \text{P}_{5} $ $^{10} \text{P}_{5} \;(21)^{5}$
The number of words that can be constructed using $10$ letters of the English alphabet such that all five vowels appear exactly once in the word is$^{21} \text{C}_{5} \;1...
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1231
ISI2020-MMA: 19
Let $x, y, z$ be the three natural numbers. Then the number of triplets $(x, y, z)$ such that $xyz = 100$ is $36$ $25$ $72$ $18$
Let $x, y, z$ be the three natural numbers. Then the number of triplets $(x, y, z)$ such that $xyz = 100$ is$36$$25$$72$$18$
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1232
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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1234
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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1235
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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1238
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$$11$
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1239
ISI2020-MMA: 27
Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function and $f(1) = 4$. Then the value of $\lim_{x\rightarrow 1}\int_{4}^{f(x)}\frac{2t}{x - 1}dt$ is. $8f’(1)$ $2f’(1)$ $4f’(1)$ $f’(1)$
Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function and $f(1) = 4$. Then the value of$$\lim_{x\rightarrow 1}\int_{4}^{f(x)}\frac{2t}{x - ...
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