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Recent questions tagged cmi2021-datascience

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5
CMI-2021-DataScience-B: 5
Show that every selection of $503$ numbers from $\{1,2,3, \dots,987\}$ has two numbers with g.c.d. $1$. Recall that the g.c.d. of two positive integers $x, y$ is the largest positive integer smaller than $x, y$ which divides both $x$ and $y$.
Show that every selection of $503$ numbers from $\{1,2,3, \dots,987\}$ has two numbers with g.c.d. $1$. Recall that the g.c.d. of two positive integers $x, y$ is the larg...
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6
CMI-2021-DataScience-B: 6
Let $S$ be the sphere given by the equation $x^{2} + y^{2} + z^{2} – 6x + 4y – 4z – 272=0$. What are the centre and radius of $S?$ What is the minimum distance between $S$ and the plane $2x + 2y + z = 214?$
Let $S$ be the sphere given by the equation $x^{2} + y^{2} + z^{2} – 6x + 4y – 4z – 272=0$. What are the centre and radius of $S?$ What is the minimum distance betw...
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8
CMI-2021-DataScience-B: 8
Let $T_{n}$ denote the set of all sequences over $\{ 0, 1, 2 \}$ of length $n$. If we pick an element $x$ in $T_{n}$ uniformly at random what is the expected number of $\text{1's}$ in $x?$
Let $T_{n}$ denote the set of all sequences over $\{ 0, 1, 2 \}$ of length $n$. If we pick an element $x$ in $T_{n}$ uniformly at random what is the expected number of $\...
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9
CMI-2021-DataScience-B: 9
What is the smallest possible value of $\displaystyle{}\sum_{i=1}^{n} \frac{1}{d_{i}+1}$, if the $d_{i} \text{'s}$ are constrained to be non-negative real numbers and satisfy $\displaystyle{}\sum_{i=1}^{n} d_{i}=N.$
What is the smallest possible value of $\displaystyle{}\sum_{i=1}^{n} \frac{1}{d_{i}+1}$, if the $d_{i} \text{'s}$ are constrained to be non-negative real numbers and sat...
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10
CMI-2021-DataScience-B: 10
Let $A$ be the $3 \times 3$ real matrix $\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & I \end{array}\right)$. Suppose $x^{T} Ax \geq 0$ for every $x \in \mathbb{R}^{3}$. Then show that all of $a, e$ and $i$ are non-negative.
Let $A$ be the $3 \times 3$ real matrix $\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & I \end{array}\right)$. Suppose $x^{T} Ax \geq 0$ for every $x \in \mat...
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14
CMI-2021-DataScience-B: 14
Let $Y$ be a continuous random variable that takes values in $[0,1]$. $g(y)$ is a probability density function that can be used to model $Y$, $ g(y)=c \cdot y^{2}(1-y), \quad 0 \leq y \leq 1 $ Determine $c$. Using $g(y)$, find the probability that $Y$ is greater than $0.8$.
Let $Y$ be a continuous random variable that takes values in $[0,1]$.$g(y)$ is a probability density function that can be used to model $Y$, $$ g(y)=c \cdot y^{2}(1-y), \...
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16
CMI-2021-DataScience-B: 16
If a polynomial $f(x)$ has roots $r_{1}, r_{2}, \dots r_{n}$, what are the roots of the polynomial $f(x+1)?$ Let $f$ be a polynomial of degree three. Set $g(x)=f(x-1)$. Assume that $-1$ is not a root of $f$. What are the roots of the polynomial $x^{3} g(1 / x)?$
If a polynomial $f(x)$ has roots $r_{1}, r_{2}, \dots r_{n}$, what are the roots of the polynomial $f(x+1)?$ Let $f$ be a polynomial of degree three. Set $g(x)=f(x-1)$. A...
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17
CMI-2021-DataScience-B: 17
Is the following statement true? If $f(x) \geq 0$ for all $x$, and $\displaystyle{}\int_{-\infty}^{\infty} f(x) d x<\infty$ then $\displaystyle{}\int_{-\infty}^{\infty} x^{2} f(x) d x \geq \epsilon^{2} \int_{\epsilon}^{\infty} f(x) d x$, for all $\epsilon>0$.
Is the following statement true?If $f(x) \geq 0$ for all $x$, and $\displaystyle{}\int_{-\infty}^{\infty} f(x) d x<\infty$ then $\displaystyle{}\int_{-\infty}^{\infty} x^...
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18
CMI-2021-DataScience-B: 18
Consider the function $f(x)=\frac{\ln x}{x}$. Find the critical point(s) of $f$ and say whether $f$ has a maximum or minimum at that point. Prove that for integers $a>b \geq 3$, we must have $a^{b}<b^{a}$.
Consider the function $f(x)=\frac{\ln x}{x}$. Find the critical point(s) of $f$ and say whether $f$ has a maximum or minimum at that point. Prove that for integers $a>b \...
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22
CMI-2021-DataScience-A: 2
The roots of the polynomial $p(x)=x^{4}-2x^{3}-2x^{2}+8x-8$ are: $1, -1, 2, 2+3 i$ $1+i, 1-i, 2, -2$ $1, -1+i, 2, 2+3 i$ $1+i, -1+i, 2, -2$
The roots of the polynomial $p(x)=x^{4}-2x^{3}-2x^{2}+8x-8$ are:$1, -1, 2, 2+3 i$$1+i, 1-i, 2, -2$$1, -1+i, 2, 2+3 i$$1+i, -1+i, 2, -2$
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27
CMI-2021-DataScience-A: 7
Fifteen telephones are received at a service center. Of these, $5$ are mobile, $6$ are cordless, and $4$ are wired. These $15$ phones are randomly numbered from $1$ to $15$ to establish the order in which they are serviced. Which of the following statement(s) is/are ... .
Fifteen telephones are received at a service center. Of these, $5$ are mobile, $6$ are cordless, and $4$ are wired. These $15$ phones are randomly numbered from $1$ to $1...
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28
CMI-2021-DataScience-A: 8
Which of the following statement(s) is/are true? For $k=69597$ and $x_{1}, x_{2}, \dots, x_{k} \in (0, \infty), \dfrac{x_{1}+ x_{2}+ \dots + x_{2^{k}}} {2^{k}} \leq (x_{1} x_{2} \dots x_{2^k})^{2^\frac{1}{k}} .$ ... $(n^{3} - n)$ is divisible by $3$ for all positive integers $n$.
Which of the following statement(s) is/are true?For $k=69597$ and $x_{1}, x_{2}, \dots, x_{k} \in (0, \infty), \dfrac{x_{1}+ x_{2}+ \dots + x_{2^{k}}} {2^{k}} \leq (x_{1}...
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30
CMI-2021-DataScience-A: 10
Let $A, B$ be $n \times n$ matrices. Which of the following properties of $A$ and $B$ is/are preserved under matrix multiplication? Being upper triangular. All diagonal elements being zero. Being diagonal. Being symmetric.
Let $A, B$ be $n \times n$ matrices. Which of the following properties of $A$ and $B$ is/are preserved under matrix multiplication?Being upper triangular.All diagonal ele...
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