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Recent questions tagged isi2021-mma

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2 answers
1
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}$
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 $f(0)$ ...
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2 answers
2
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 $1 / 9$ $4 / 7$ $1 / 18$ $7 / 36$
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 d...
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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
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 ...
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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$ $2x^{2} g(x^{2}) – \int xg(x^{2}) dx + C$ $x^{2} g(x^{2}) – \frac{1}{2} \int xg(x^{2}) dx + C$
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...
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1 votes
1 answer
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 $\frac{(n+1)^{n}}{n!}$ $\frac{n^{n}}{n!}$ $\frac{(n+1)^{n}}{nn!}$ $\frac{(n+1)^{n+1}}{n!}$
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+...
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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 $( – 1, 1)$ $[ – 1, 1)$ $[0, 1]$ $( – 1, 1]$
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 i...
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1 votes
1 answer
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 $1$ $2$ $3$ $4$
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$$4$
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1 votes
1 answer
9
ISI2021-MMA: 9
The set of all $a$ satisfying the inequality $\frac{1}{\sqrt{a}} \int_{1}^{a} \left( \frac{3}{2} \sqrt{x} + 1 – \frac{1}{\sqrt{x}} \right) dx < 4$ Is equal to the interval $( – 5, – 2)$ $(1, 4)$ $(0, 2)$ $(0, 4)$
The set of all $a$ satisfying the inequality$$\frac{1}{\sqrt{a}} \int_{1}^{a} \left( \frac{3}{2} \sqrt{x} + 1 – \frac{1}{\sqrt{x}} \right) dx < 4$$Is equal to the inte...
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11
ISI2021-MMA: 11
Suppose $a, b, c$ are in $\text{A.P.}$ and $a^{2}, b^{2}, c^{2}$ are in $\text{G.P.}$ If $a < b < c$ and $a + b + c = \frac{3}{2},$ then the value of $a$ is $\frac{1}{2 \sqrt{2}}$ $ – \frac{1}{2 \sqrt{2}}$ $\frac{1}{2} – \frac{1}{\sqrt{3}}$ $\frac{1}{2} – \frac{1}{\sqrt{2}}$
Suppose $a, b, c$ are in $\text{A.P.}$ and $a^{2}, b^{2}, c^{2}$ are in $\text{G.P.}$ If $a < b < c$ and $a + b + c = \frac{3}{2},$ then the value of $a$ is$\frac{1}{2 \s...
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12
ISI2021-MMA: 12
The number of distinct even divisors of $\prod_{k=1}^{5} k!$ is $24$ $32$ $64$ $72$
The number of distinct even divisors of $$\prod_{k=1}^{5} k!$$ is$24$$32$$64$$72$
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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$ $\ln 2$
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}{...
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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 $23x – 23y + 11 = 0$ $23x – 23y - 11 = 0$ $23x + 23y + 11 = 0$ $23x + 23y – 11 = 0$
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 a...
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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
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 positivem...
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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
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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20
ISI2021-MMA: 20
The number of real roots of the polynomial $x^{3}-2 x+7$ is $0$ $1$ $2$ $3$
The number of real roots of the polynomial $x^{3}-2 x+7$ is$0$$1$$2$$3$
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2 votes
1 answer
24
ISI2021-MMA: 24
A fair die is rolled five times. What is the probability that the largest number rolled is $5$? $5 / 6$ $1 / 6$ $1-(1 / 6)^{6}$ $(5 / 6)^{5}-(2 / 3)^{5}$
A fair die is rolled five times. What is the probability that the largest number rolled is $5$?$5 / 6$$1 / 6$$1-(1 / 6)^{6}$$(5 / 6)^{5}-(2 / 3)^{5}$
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1 answer
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 $n!$ $n! / 2$ $2^{n} n!$ $2 n !$
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 opposit...
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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
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...
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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 $q$ $-q$ $\frac{p}{q}$ $\frac{p^{2}}{q}$
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}$$\frac{p^{2}...
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30
ISI2021-MMA: 30
Let $ A=\left[\begin{array}{lll} a & 1 & 1 \\ b & a & 1 \\ 1 & 1 & 1 \end{array}\right] . $ The number of elements in the set $ \left\{(a, b) \in \mathbb{Z}^{2}: 0 \leq a, b \leq 2021, \operatorname{rank}(A)=2\right\} $ is $2021$ $2020$ $2021^{2}-1$ $2020 \times 2021$
Let $$ 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 \le...
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