# IGNOU MST-011 Solved Assignment 2023 | MSCAST

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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## IGNOU MST-011 Assignment Question Paper 2023

1. Solve the following problems.
(a) A ball is thrown in an upward direction. If the variable $x$$x$xx$x$ represents the velocity of the ball when it strikes the ground. Classify variable $x$$x$xx$x$ as discrete or continuous. Justify your answer with a proper explanation.
(b) In R we have a built-in data set “trees”. A screenshot of the first four rows together with the $\mathrm{R}$$\mathrm{R}$R\mathrm{R}$\mathrm{R}$ code to obtain it is given as follows. To get more detail about this data set you can run? trees command on $\mathrm{R}$$\mathrm{R}$R\mathrm{R}$\mathrm{R}$ console.
$>\mathrm{head}\left($$>\mathrm{head}\left($> head(>\operatorname{head}($>\mathrm{head}\left($ trees, 4)
Girth Height Volume
$\begin{array}{rrrr}1& 8.3& 70& 10.3\\ 2& 8.6& 65& 10.3\\ 3& 8.8& 63& 10.2\\ 4& 10.5& 72& 16.4\end{array}$$\begin{array}{r}1 8.3 70 10.3\\ 2 8.6 65 10.3\\ 3 8.8 63 10.2\\ 4 10.5 72 16.4\end{array}${:[1,8.3,70,10.3],[2,8.6,65,10.3],[3,8.8,63,10.2],[4,10.5,72,16.4]:}\begin{array}{rrrr}1 & 8.3 & 70 & 10.3 \\ 2 & 8.6 & 65 & 10.3 \\ 3 & 8.8 & 63 & 10.2 \\ 4 & 10.5 & 72 & 16.4\end{array}$\begin{array}{rrrr}1& 8.3& 70& 10.3\\ 2& 8.6& 65& 10.3\\ 3& 8.8& 63& 10.2\\ 4& 10.5& 72& 16.4\end{array}$
Note that all the three variables of this data set are numeric. So, assuming each row of this data set is a point in 3-dimension. Find the distances between the points corresponding to the first and the third rows using the Manhattan and Chebyshev distance formula.
(c) Find the equation of a line passing through points $A\left(2,3,5\right)$$A\left(2,3,5\right)$A(2,3,5)A(2,3,5)$A\left(2,3,5\right)$ and $B\left(5,8,9\right)$$B\left(5,8,9\right)$B(5,8,9)B(5,8,9)$B\left(5,8,9\right)$. Also, find the coordinates of a point on this line which is at a distance of 10 units from point $A$$A$AA$A$ opposite to the side of point B.
(d) Give an example of a set which is convex but not affine. Justify your claim with a proper explanation.
1. (a) Test the convergence of the series
$\frac{3}{5.9.7}x+\frac{5}{10.12.11}{x}^{2}+\frac{7}{15.15.15}{x}^{3}+\frac{9}{20.18.19}{x}^{4}+\cdots ,x>0$$\frac{3}{5.9.7}x+\frac{5}{10.12.11}{x}^{2}+\frac{7}{15.15.15}{x}^{3}+\frac{9}{20.18.19}{x}^{4}+\cdots ,x>0$(3)/(5.9.7)x+(5)/(10.12.11)x^(2)+(7)/(15.15.15)x^(3)+(9)/(20.18.19)x^(4)+cdots,x > 0\frac{3}{5.9 .7} x+\frac{5}{10.12 .11} x^2+\frac{7}{15.15 .15} x^3+\frac{9}{20.18 .19} x^4+\cdots, x>0$\frac{3}{5.9.7}x+\frac{5}{10.12.11}{x}^{2}+\frac{7}{15.15.15}{x}^{3}+\frac{9}{20.18.19}{x}^{4}+\cdots ,x>0$
(b) If $f:\left[0,5\right]\to \mathbb{R}$$f:\left[0,5\right]\to \mathbb{R}$f:[0,5]rarrRf:[0,5] \rightarrow \mathbb{R}$f:\left[0,5\right]\to \mathbb{R}$ be a function defined by $f\left(x\right)={x}^{2}+2x+1,x\in \left[0,5\right]$$f\left(x\right)={x}^{2}+2x+1,x\in \left[0,5\right]$f(x)=x^(2)+2x+1,x in[0,5]f(x)=x^2+2 x+1, x \in[0,5]$f\left(x\right)={x}^{2}+2x+1,x\in \left[0,5\right]$. Show that $f$$f$ff$f$ is Riemann integrable using both definitions. Also, verify that the results of both definition match.
1. (a) Evaluate the integral ${\iint }_{D}{e}^{4x+5y}dxdy$${\iint }_{D} {e}^{4x+5y}dxdy$∬_(D)e^(4x+5y)dxdy\iint_D e^{4 x+5 y} d x d y${\iint }_{D}{e}^{4x+5y}dxdy$, where $D=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y\le 1\right\}$$D=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y\le 1\right\}$D={(x,y):x >= 0,y >= 0,x+y <= 1}D=\{(x, y): x \geq 0, y \geq 0, x+y \leq 1\}$D=\left\{\left(x,y\right):x\ge 0,y\ge 0,x+y\le 1\right\}$, by considering $\mathrm{D}$$\mathrm{D}$D\mathrm{D}$\mathrm{D}$ as a region of Type I and then as a region of Type II.
(b) Evaluate the integral ${\int }_{0}^{4}\frac{dx}{\sqrt{4x-{x}^{2}}}$${\int }_{0}^{4} \frac{dx}{\sqrt{4x-{x}^{2}}}$int_(0)^(4)(dx)/(sqrt(4x-x^(2)))\int_0^4 \frac{d x}{\sqrt{4 x-x^2}}${\int }_{0}^{4}\frac{dx}{\sqrt{4x-{x}^{2}}}$ using beta and gamma functions.
$$cos\left(\theta +\phi \right)=cos\:\theta \:cos\:\phi -sin\:\theta \:sin\:\phi$$

## MST-011 Sample Solution 2023

1. Solve the following problems.
(a) A ball is thrown in an upward direction. If the variable $x$$x$xx$x$ represents the velocity of the ball when it strikes the ground. Classify variable $x$$x$xx$x$ as discrete or continuous. Justify your answer with a proper explanation.

### Introduction

The problem asks us to classify the variable $x$$x$xx$x$, which represents the velocity of the ball when it strikes the ground, as either discrete or continuous. To do this, we need to understand the nature of the variable and how it can take values.

### Work/Calculations

#### Understanding Discrete and Continuous Variables

• Discrete Variable: A variable is considered discrete if it can take only specific, separate values. For example, the number of students in a classroom can only be a whole number; it cannot be a fraction or a decimal.
• Continuous Variable: A variable is considered continuous if it can take any value within a given range. For example, the height of a person can be any value within a certain range, including fractions and decimals.

#### Classifying the Variable $x$$x$xx$x$

The velocity $x$$x$xx$x$ of the ball when it strikes the ground can take any value within a given range, depending on various factors like the initial velocity, angle of projection, air resistance, and gravitational force. It is not restricted to specific, separate values. Therefore, $x$$x$xx$x$ can be any real number within a certain range.
Let’s classify the variable $x$$x$xx$x$:
$\text{Variable}x=\text{Continuous}$“Variable “x=”Continuous”\text{Variable $$x$$} = \text{Continuous}

### Conclusion

The variable $x$$x$xx$x$, which represents the velocity of the ball when it strikes the ground, is a continuous variable. This is because it can take any value within a given range and is not restricted to specific, separate values.
$$cos\left(2\theta \right)=cos^2\theta -sin^2\theta$$

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$$2\:cos\:\theta \:sin\:\phi =sin\:\left(\theta +\phi \right)-sin\:\left(\theta -\phi \right)$$

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