Recent questions in Others / GO Electronics

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161

TIFR ECE 2011 | Question: 15
Consider a channel where $x_{n} \in\{0,1\}$ is the input and $y_{n}=x_{n} * z_{n}$ is the output, where $*$ is $\text{EX-OR}$ operation, and $P\left(z_{n}=x_{n-1}\right)=P\left(z_{n}=y_{n-1}\right)=\frac{1}{2}$ ... $\frac{1}{2}.$ $1$. $<1.$ $\geq 0.$ Both $(c)$ and $(d)$.

Consider a channel where $x_{n} \in\{0,1\}$ is the input and $y_{n}=x_{n} * z_{n}$ is the output, where $*$ is $\text{EX-OR}$ operation, and $P\left(z_{n}=x_{n-1}\right)=...

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162

TIFR ECE 2011 | Question: 16
Consider a triangular shaped pulse $x$ of base $2 T$ and unit height centered at $0$ , i.e. $x(t)=0$ for $|t|>T, x(t)=1-|t|$ for $t \in[-T, T]$. Then if $x$ is convolved with itself, the output is Square shape. Triangular shape. Bell shape. Inverted $\text{U}$ shape. None of the above.

Consider a triangular shaped pulse $x$ of base $2 T$ and unit height centered at $0$ , i.e. $x(t)=0$ for $|t|>T, x(t)=1-|t|$ for $t \in[-T, T]$. Then if $x$ is convolved ...

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163

TIFR ECE 2011 | Question: 17
Let $x[n]$ and $y[n]$ be the input and output of a linear time invariant $\text{(LTI)}$ system. Then which of following system is $\text{LTI}$. $z[n]=y[n]+c$ for a constant $c$. $z[n]=x[n] y[n]$. $z[n]=y[n]+x[n]+c$ for a constant $c$. $z[n]=y[n]+x[n]$. None of the above.

Let $x[n]$ and $y[n]$ be the input and output of a linear time invariant $\text{(LTI)}$ system. Then which of following system is $\text{LTI}$.$z[n]=y[n]+c$ for a constan...

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164

TIFR ECE 2011 | Question: 18
Which of the following statements is TRUE. The cascade of a non-causal linear time invariant $\text{(LTI)}$ system with a causal $\text{LTI}$ system can be causal. If $h[n] \leq 2$ for all $n$, then the $\text{LTI}$ system with $h[n]$ as its impulse response is stable and ... $u[n]=1, n \geq 0, u[n]=0, n<0$, the $\text{LTI}$ system is stable. Both $(c)$ and $(d)$.

Which of the following statements is TRUE.The cascade of a non-causal linear time invariant $\text{(LTI)}$ system with a causal $\text{LTI}$ system can be causal.If $h[n]...

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165

TIFR ECE 2011 | Question: 19
Let $R_{X}(\tau)$ be the autocorrelation function of a zero mean stationary random process $X(t)$. Which of following statements is FALSE. If $R_{X}(\tau)=0, \forall \tau, X(n)$ and $X(m), n \neq m$ are independent. $R_{X}(\tau)=R_{X}(-\tau)$. $R_{X}(0)=E\left[X^{2}\right]$, where $E$ denotes the expectation. $R_{X}(0) \geq R_{X}(\tau), \forall \tau.$ None of the above.

Let $R_{X}(\tau)$ be the autocorrelation function of a zero mean stationary random process $X(t)$. Which of following statements is FALSE.If $R_{X}(\tau)=0, \forall \tau,...

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166

TIFR ECE 2011 | Question: 20
Let $x(t)$ be a signal whose Fourier transform $X(f)$ is zero for $|f|>W$. Using a sampler with sampling frequency $4 W$, which of the following filters can be used to exactly reconstruct $x(t)$ ... $\text{[-3W} \;5 \mathrm{W}]$. All the above. None of the above.

Let $x(t)$ be a signal whose Fourier transform $X(f)$ is zero for $|f|>W$. Using a sampler with sampling frequency $4 W$, which of the following filters can be used to ex...

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167

TIFR ECE 2010 | Question: 1
A linear system could be a composition of Two non-linear systems a non-causal non-linear system and a linear system a time varying non-linear system and a time varying linear system All of the above None of the above

A linear system could be a composition ofTwo non-linear systemsa non-causal non-linear system and a linear systema time varying non-linear system and a time varying linea...

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168

TIFR ECE 2010 | Question: 4
Let $Y_{n}=s_{n}+W_{n}$ where $\left\{s_{n}\right\}$ is the desired signal bandlimited to $[-W, W]$ and $\left\{W_{n}\right\}$ is a noise component, which is sparse (that is, only few samples are non-zero), bursty (that is, runs of non-zero samples are ... of $\left\{Y_{n+k}\right\}_{k=-K}^{K}$ for suitably chosen $K$ Both $a)$ and $b)$ are better than the other options

Let $Y_{n}=s_{n}+W_{n}$ where $\left\{s_{n}\right\}$ is the desired signal bandlimited to $[-W, W]$ and $\left\{W_{n}\right\}$ is a noise component, which is sparse (that...

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169

TIFR ECE 2010 | Question: 5
Let $Y(t)=\sum_{n=-\infty}^{\infty} x_{n} h(t-n T)$. We sample $Y(t)$ at time instants $n T / 2$ and let $Y_{n}=Y(n T / 2)$. Which of the following is true? $\left\{Y_{n}\right\}$ can be interpreted as the output of a discrete time, ... of a discrete time, linear, time-invariant system with input $\left\{X_{n}\right\}$. Both $a)$ and $b)$ above Both $b)$ and $c)$ above

Let $Y(t)=\sum_{n=-\infty}^{\infty} x_{n} h(t-n T)$. We sample $Y(t)$ at time instants $n T / 2$ and let $Y_{n}=Y(n T / 2)$. Which of the following is true?$\left\{Y_{n}\...

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170

TIFR ECE 2010 | Question: 6
If we convolve $\sin (t) / t$ with itself, then we get $C \sin (t) / t$ for some constant $C$ $C \cos (t) / t$ for some constant $C$ $C \cos (t) / t^{2}$ for some constant $C$ $C_{1} \sin (t) / t^{2}+C_{2} \cos (t) / t^{2}$ for some constants $C_{1}, C_{2}$ None of the above

If we convolve $\sin (t) / t$ with itself, then we get$C \sin (t) / t$ for some constant $C$$C \cos (t) / t$ for some constant $C$$C \cos (t) / t^{2}$ for some constant $...

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171

TIFR ECE 2010 | Question: 7
A voltage source with internal resistance $\text{R}$ is connected to an inductor $\text{L}$ and a capacitor $\text{C}$ connected in parallel. The output is the common voltage across the inductor and the capacitor. What is the nature of the transfer ... depending upon the values of $\text{L}$ and $\text{C}$. The circuit is not stable and no transfer function exists.

A voltage source with internal resistance $\text{R}$ is connected to an inductor $\text{L}$ and a capacitor $\text{C}$ connected in parallel. The output is the common vol...

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172

TIFR ECE 2010 | Question: 8
Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$. The channel operation is such that to produce the output $y_{n}$ it drops one ... we do not make any error If $R<1 / 2$, then there exists a scheme with zero error All of the above None of the above

Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$....

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173

TIFR ECE 2010 | Question: 9
The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the sequence $y_{n}=x_{2 n+1}$ is $Y(z)=z(X(z)-X(-z)) / 2$ ... $Y(z)=z(X(\sqrt{z})-X(-\sqrt{z})) / 2$ $Y(z)=(X(\sqrt{z})-X(-\sqrt{z})) / 2$

The $z$-transform of a sequence $\left\{x_{n}\right\}_{n=-\infty}^{\infty}$ is defined to be $X(z)=\sum_{n=-\infty}^{\infty} x_{n} z^{-n}$. The $z$-transform of the seque...

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174

TIFR ECE 2010 | Question: 12
Consider a system with input $x(t)$ and the output $y(t)$ is given by \[ y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 . \] The system is Non-linear Non-causal Time varying All of the above None of the above

Consider a system with input $x(t)$ and the output $y(t)$ is given by\[y(t)=x(t)-\sin (t) x(t-1)-0.5 x(t+2)+1 .\]The system isNon-linearNon-causalTime varyingAll of the a...

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175

TIFR ECE 2010 | Question: 13
Output of a linear system with input $x(t)$ is given by \[ y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) . \] The system is time invariant if $h(t, \tau)=h(t-\tau)$ $h(t, \tau)=h(\tau)$ $h(t, \tau)=h(t)$ $h(t, \tau)=$ constant $h(t, \tau)$ is a continuous function of $t$

Output of a linear system with input $x(t)$ is given by\[y(t)=\int_{-\infty}^{\infty} h(t, \tau) x(\tau) .\]The system is time invariant if$h(t, \tau)=h(t-\tau)$$h(t, \ta...

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176

TIFR ECE 2010 | Question: 14
Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x<0$. For $n \geq 0$ ... $-1,1,-1,1, \ldots$. $0,1,-1,1,-1, \ldots$ $0,1,1,1,-1,1,-1,1, \ldots$ None of the above

Define $\operatorname{sign}(x)=0$ for $x=0, \operatorname{sign}(x)=1$ for $x>0$ and $\operatorname{sign}(x)=-1$ for $x<0$. For $n \geq 0$, let\[Y_{n}=\operatorname{sign}\...

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177

TIFR ECE 2022 | Question: 3
Consider two linear time invariant $\text{(LTI)}$ systems $T_{1}$ and $T_{2}$ with impulse responses $h_{1}(n)$ and $h_{2}(n)$, respectively. Let there be two cascades $C_{1}$ and $C_{2}$, where in $C_{1}, T_{2}$ follows after ... statement $1$ is correct Only statement $3$ is correct Both statements $1, 2$ are correct Both statements $2, 3$ are correct None of the above

Consider two linear time invariant $\text{(LTI)}$ systems $T_{1}$ and $T_{2}$ with impulse responses $h_{1}(n)$ and $h_{2}(n)$, respectively. Let there be two cascades $C...

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178

TIFR ECE 2021 | Question: 1
Consider a system with input $x(t)$ and output $y(t)$ such that \[y(t)=t \;x(t) .\] Consider the following statements: The system is linear. The system is time-invariant. The system is causal. Then which of the following is $\text{TRUE?}$ Only ... Only statement $3$ is correct. Only statements $1$ and $3$ are correct. All three statements $1, 2,$ and $3$ are correct.

Consider a system with input $x(t)$ and output $y(t)$ such that\[y(t)=t \;x(t) .\]Consider the following statements:The system is linear.The system is time-invariant.The ...

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179

TIFR ECE 2021 | Question: 5
Recall that \[\operatorname{sinc}(t)=\frac{\sin (\pi t)}{\pi t}\] and convolution of functions $x(t)$ and $y(t)$ is defined as \[x(t) \star y(t)=\int_{-\infty}^{\infty} x(t-\tau) y(\tau) d \tau .\] What is the necessary and sufficient condition on positive real ... \quad \text { for all real } t \text {. }\] $f<a$ $f>a$ $f<a^{-1}$ $f>a^{-1}$ None of the above

Recall that\[\operatorname{sinc}(t)=\frac{\sin (\pi t)}{\pi t}\]and convolution of functions $x(t)$ and $y(t)$ is defined as\[x(t) \star y(t)=\int_{-\infty}^{\infty} x(t-...

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180

TIFR ECE 2021 | Question: 6
Consider a fair coin (i.e., both heads and tails have equal probability of appearing). Suppose we toss the coin repeatedly until both sides have been seen. What is the expected number of times we would have seen heads? $1$ $5 / 4$ $3 / 2$ $2$ None of the above

Consider a fair coin (i.e., both heads and tails have equal probability of appearing). Suppose we toss the coin repeatedly until both sides have been seen. What is the ex...

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181

TIFR ECE 2021 | Question: 13
Consider a unit Euclidean ball in $4$ dimensions, and let $V_{n}$ be its volume and $S_{n}$ its surface area. Then $S_{n} / V_{n}$ is equal to: $1$ $4$ $5$ $2$ $3$

Consider a unit Euclidean ball in $4$ dimensions, and let $V_{n}$ be its volume and $S_{n}$ its surface area. Then $S_{n} / V_{n}$ is equal to:$1$$4$$5$$2$$3$

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182

TIFR ECE 2020 | Question: 1
Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that \[y[n]=\left\{\begin{array}{ll} 0, & n=-1,-2,-3, \ldots, \\ \alpha y[n-1] ... Non-linear, time-invariant, BIBO stable Linear, time-variant, BIBO unstable Non-linear, time-variant, BIBO stable Cannot be determined from the information given

Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that\[y[n]=\left\{\begin{array}{ll}0, & n=-1,-2...

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183

TIFR ECE 2020 | Question: 2
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=$ $\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$. If $f(t) * g(t)=h(t)$, what is $f(t-1) * g(t+1)?$ $h(2 t)$ $h(t)$ $h(t-1)$ $h(t+1)$ None of the above

Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=$ $\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$. If $f(t) * g(t)=h(t)$, what ...

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184

TIFR ECE 2020 | Question: 5
Let $f(t)$ be a periodic signal of period $1$, i.e. $f(t+1)=f(t) \forall t$. Define the averaging operator depending on a fixed parameter $h>0$ as below: \[g(x)=\frac{1}{2 h} \int_{x-h}^{x+h} f(t) d t .\] Which of the following is ... $\frac{1}{2}$ $g(x)$ is periodic with period $1$ The value of $h$ determines whether or not $g(x)$ is periodic None of the above

Let $f(t)$ be a periodic signal of period $1$, i.e. $f(t+1)=f(t) \forall t$. Define the averaging operator depending on a fixed parameter $h>0$ as below:\[g(x)=\frac{1}{2...

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185

TIFR ECE 2020 | Question: 6
For all values of $r>0$, the area of the set of all points outside the unit square whose Euclidean distance to the unit square is less than $r$ is: $=\pi r^{2}+4 r$ $<4 \pi r^{2}$ $>4 \pi r^{3}+4 r$ $=\frac{4 \pi r^{3}}{3}+6 r+2 \pi r^{2}$ None of the above

For all values of $r>0$, the area of the set of all points outside the unit square whose Euclidean distance to the unit square is less than $r$ is:$=\pi r^{2}+4 r$$<4 \pi...

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186

TIFR ECE 2019 | Question: 1
Consider a discrete-time system which in response to input sequence $x[n]$ ( $n$ integer) outputs the sequence $y[n]$ such that \[y[n]=\left\{\begin{array}{ll} 0, & n=-1,-2,-3, \ldots, \\ \alpha y[2 n-1]+\beta ... -linear, but time-invariant only if $\alpha=0$ (parameters $\beta$ and $\gamma$ can take arbitrary values) Cannot be determined from the information given

Consider a discrete-time system which in response to input sequence $x[n]$ ( $n$ integer) outputs the sequence $y[n]$ such that\[y[n]=\left\{\begin{array}{ll}0, & n=-1,-2...

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187

TIFR ECE 2019 | Question: 8
Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set of all points in $\mathbb{R}^{3}$ whose Euclidean distance to $K$ is less than or equal to $t$ ... $V \leq \frac{4}{3} \pi\left(\frac{\sqrt{3}}{2}+t\right)^{3}$ $V \geq(1+2 t)^{3}$

Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set o...

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188

TIFR ECE 2019 | Question: 13
For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is, \[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} x_{i}^{2} \leq t^{2}\right\} .\] Let $N_{t}$ be the number of ... $\lim _{t \rightarrow \infty} \frac{N_{t}}{V_{t}}=1$ $N_{t}$ is a monotonically decreasing function of $t$

For $t>0$, let $S_{t}$ denote the ball of radius $t$ centered at the origin in $\mathbb{R}^{n}$. That is,\[S_{t}=\left\{\mathbf{x} \in \mathbb{R}^{n} \mid \sum_{i=1}^{n} ...

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189

TIFR ECE 2018 | Question: 1
Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that \[y[n]=\left\{\begin{array}{ll} 0, & n=-1,-2,-3, \ldots, \\ \frac{1}{2 ... describes the system? Linear, time-invariant Linear, time-variant Non-linear, time-invariant Non-linear, time-variant Cannot be determined from the information given

Consider a discrete-time system which in response to input sequence $x[n] \;( n$ integer) outputs the sequence $y[n]$ such that\[y[n]=\left\{\begin{array}{ll}0, & n=-1,-2...

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190

TIFR ECE 2018 | Question: 6
Consider the system shown below. If $K>0$, which of the following describes the system? Stable, causal Stable, non-causal Unstable, non-causal Unstable, causal Cannot be determined from the information given

Consider the system shown below.If $K>0$, which of the following describes the system?Stable, causalStable, non-causalUnstable, non-causalUnstable, causalCannot be determ...

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191

TIFR ECE 2018 | Question: 13
Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let \[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\] (i.e., $d_{i j}$ is the number of coordinates where $X_{i}$ ... to $X_{5}$, and argue about the result noting that there are five binary vectors.] $d=10$ $d=9$ $d=8$ $d<8$ Information is not sufficient

Consider five distinct binary vectors $X_{1}, \ldots, X_{5}$ each of length $10$. Let\[d_{i j}=\sum_{k=1}^{10}\left(X_{i k} \text { XOR } X_{j k}\right),\](i.e., $d_{i j}...

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192

TIFR ECE 2018 | Question: 14
Define the $\ell_{p}$ ball in two dimensions as the set of points $(x, y)$ such that $|x|^{p}+|y|^{p} \leq 1$. Which of the following is $\text{FALSE:}$ The $\ell_{2}$ ball is contained in the $\ell_{3}$ ball The $\ell_{2}$ ball is contained in ... $\ell_{2}$ ball is contained in the $\ell_{5}$ ball The $\ell_{1}$ ball is contained in the $\ell_{3}$ ball

Define the $\ell_{p}$ ball in two dimensions as the set of points $(x, y)$ such that $|x|^{p}+|y|^{p} \leq 1$. Which of the following is $\text{FALSE:}$The $\ell_{2}$ bal...

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193

TIFR ECE 2017 | Question: 1
Consider a system which in response to input $x(t)$ outputs \[ y(t)=2 x(t-2)+x(2 t-1)+1 . \] Which of the following describes the system? linear, time-invariant, causal linear, time-invariant, non-causal non-linear, time-invariant, causal non-linear, time-invariant, non-causal non-linear, time-variant

Consider a system which in response to input $x(t)$ outputs\[y(t)=2 x(t-2)+x(2 t-1)+1 .\]Which of the following describes the system?linear, time-invariant, causallinear,...

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194

TIFR ECE 2017 | Question: 2
Suppose a $1 \mu \mathrm{H}$ inductor and a $1 \Omega$ resistor are connected in series to a $1 \mathrm{ V}$ battery. What happens to the current in the circuit? The current starts at $0 \mathrm{ A}$, and gradually rises to $1 \mathrm{ A}$ The current ... $1 \mathrm{ A}$ The current oscillates over time between $1 \mathrm{ A}$ and $-1 \mathrm{ A}$

Suppose a $1 \mu \mathrm{H}$ inductor and a $1 \Omega$ resistor are connected in series to a $1 \mathrm{ V}$ battery. What happens to the current in the circuit?The curre...

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195

TIFR ECE 2017 | Question: 3
What is the maximum average power that can be dissipated by a load connected to the output terminals of the following circuit with an alternating current source? $23 \mathrm{~W}$ $11.5 \mathrm{~W}$ $8.1317 \mathrm{~W}$ $2.875 \mathrm{~W}$ None of the above

What is the maximum average power that can be dissipated by a load connected to the output terminals of the following circuit with an alternating current source?$23 \math...

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196

TIFR ECE 2017 | Question: 4
A Schmitt trigger circuit is a comparator circuit with a hysteresis. Consider the Schmitt trigger circuit in the figure implemented using an opamp. What are the trigger levels for this circuit? $\pm \frac{R_{1}}{R_{2}} V_{s}$ $\pm \frac{R_{2}}{R_{1}} V_{s}$ $\pm \frac{R_{1}}{R_{1}+R_{2}} V_{s}$ $\pm \frac{R_{2}}{R_{1}+R_{2}} V_{s}$ None of the above

A Schmitt trigger circuit is a comparator circuit with a hysteresis. Consider the Schmitt trigger circuit in the figure implemented using an opamp. What are the trigger l...

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197

TIFR ECE 2017 | Question: 8
Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$ \[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lfloor\frac{c}{n}\right\rfloor\] where $\lfloor t\rfloor$ denotes the ... $0$ The first sequence converges to $1 / c$, while the second sequence converges to $c$

Consider the two positive integer sequences, defined for a fixed positive integer $c \geq 2$\[f(n)=\frac{1}{n}\left\lfloor\frac{n}{c}\right\rfloor, \quad g(n)=n\left\lflo...

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198

TIFR ECE 2017 | Question: 15
Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let \[a_{n}=\sum_{j=1}^{n} 1 / j\] and \[b_{n}=\sum_{j=1}^{n} 1 / j^{2} .\] Which of the following statements is true ... any number $M$ so that $f\left(b_{n}\right)$ and $f\left(a_{n}\right)$ are greater than $M$ for all $n$ None of the above

Suppose that $f(x)$ is a real valued continuous function such that $f(x) \rightarrow \infty$ as $x \rightarrow \infty$. Further, let\[a_{n}=\sum_{j=1}^{n} 1 / j\]and\[b_{...

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199

TIFR ECE 2016 | Question: 4
Consider a system which in response to input $x(t)$ outputs \[y(t)=x\left(t^{2}\right) .\] Which of the following describes the system? linear, time-invariant, causal linear, time-invariant, non-causal linear, time-variant non-linear, time-invariant non-linear, time-variant

Consider a system which in response to input $x(t)$ outputs\[y(t)=x\left(t^{2}\right) .\]Which of the following describes the system?linear, time-invariant, causallinear,...

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200

TIFR ECE 2016 | Question: 5
Consider the opamp circuit in the figure. Approximately what is $V_{0}$? $-\left(\frac{V_{1}}{2}+V_{2}\right)$ $-\left(\frac{V_{1}}{4}+\frac{V_{2}}{2}\right)$ $-\left(V_{1}+2 V_{2}\right)$ $-\left(4 V_{1}+2 V_{2}\right)$ None of the above

Consider the opamp circuit in the figure. Approximately what is $V_{0}$?$-\left(\frac{V_{1}}{2}+V_{2}\right)$$-\left(\frac{V_{1}}{4}+\frac{V_{2}}{2}\right)$$-\left(V_{1}+...

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admin asked Nov 29, 2022