# Engineering Mathematics

 Question 1
A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is ______. (round off to three decimal places)
 A 0.045 B 0.078 C 0.018 D 0.025
GATE CE 2022 SET-2      Probability and Statistics
Question 1 Explanation:
Event, E = {(1, 3)(3, 1)(2, 2)}
n(E) = 3
n(S) = 36
$p=P(E)=\frac{3}{36}=\frac{1}{12}$
$q=P(\bar{E})=1-\frac{1}{12}=\frac{11}{12}$
$P(x)=3C_2(p^2)(q^1)=3 \times \left (\frac{1}{12} \right )^2\left (\frac{11}{12} \right )=0.02$
 Question 2
Let $y$ be a non-zero vector of size 2022 x 1. Which of the following statement(s) is/are TRUE?
 A $yy^T$ is a symmetric matrix. B $y^Ty$ is an eigenvalue of $yy^T$ C $yy^T$ has a rank of 2022. D $yy^T$ is invertible.
GATE CE 2022 SET-2      Linear Algebra
Question 2 Explanation:
Let vector
\begin{aligned} y&=\begin{bmatrix} 4\\ 4\\ 4 \end{bmatrix}_{3 \times 1}\\ y^T&=\begin{bmatrix} 4& 4& 4 \end{bmatrix}_{1 \times 3}\\ yy^T&=\begin{bmatrix} 4\\ 4\\ 4 \end{bmatrix}\begin{bmatrix} 4& 4& 4 \end{bmatrix}\\ &=\begin{bmatrix} 16 & 16 & 16\\ 16 & 16 & 16\\ 16 & 16 & 16 \end{bmatrix}\\ y^Ty&=[4^2+4^2+4^2]_{1 \times 1}\\ &=_{1 \times 1}\\ \rho (y)&=\rho (y^T)=\rho (yy^T)=\rho (y^Ty)=1 \end{aligned}
From above information
$yy^T$ is asymmetric.
$y^Ty$ is an eigen value of $yy^T$.
$yy^T$ has rank 1
$det(yy^T) =0$ so, $yy^T$ is not invertible.
 Question 3
Consider the polynomial $f(x)=x^3-6x^2+11x-6$ on the domain S given by $1 \leq x \leq 3$. The first and second derivatives are $f'(x)$ and $f''(x)$.

Consider the following statements:
I. The given polynomial is zero at the boundary points x = 1 and x = 3.
II. There exists one local maxima of $f(x)$ within the domain S.
III. The second derivative $f''(x) \gt 0$ throughout the domain S.
IV. There exists one local minima of $f(x)$ within the domain S.

The correct option is:
 A Only statements I, II and III are correct. B Only statements I, II and IV are correct. C Only statements I and IV are correct. D Only statements II and IV are correct.
GATE CE 2022 SET-2      Calculus
Question 3 Explanation:
\begin{aligned} f(x)&=x^3-6x^2+11x-6 \\ f(1)&=1-6+11-6=0 \\ f(3)&=3^3-6 \times 3^2+11 \times 3-6=0 \\ &\text{Statement (I) is correct.} \\ f'(x)&=3x^2-12x+11 \\ f'(x)&=0\Rightarrow x=2\pm \frac{1}{\sqrt{3}} \end{aligned} $f(x)$ has local maxima at $x=2-\frac{1}{\sqrt{3}}$
Statement (II) is also true.
Now,
\begin{aligned} f''(x)&=6x-12\\ f''(x) &\gt 0\\ 6x-12 &\gt 0\\ x & \gt 0 \end{aligned}
Statement (III) is incorrect statement (IV) is also correct.
$\because x=2+\frac{1}{\sqrt{3}}$ is point of minima.
 Question 4
$P$ and $Q$ are two square matrices of the same order. Which of the following statement(s) is/are correct?
 A If P and Q are invertible, then $[PQ]^{-1}=Q^{-1}P^{-1}$ B If P and Q are invertible, then $[QP]^{-1}=P^{-1}Q^{-1}$ C If P and Q are invertible, then $[PQ]^{-1}=Q^{-1}P^{-1}$ D If P and Q are not invertible, then $[PQ]^{-1}=P^{-1}Q^{-1}$
GATE CE 2022 SET-2      Linear Algebra
Question 4 Explanation:
If P and Q are invertible then $(PQ)^{-1} = Q^{-1}P^{-1}$ is correct. Let,
\begin{aligned} PQ&=C \\ P^{-1}PQ&=P^{-1}C \\ Q&=P^{-1} C\\ Q^{-1}Q&=Q^{-1}P^{-1}C \\ I&=Q^{-1}P^{-1}C \\ IC^{-1}&=Q^{-1}P^{-1}CC^{-1} \\ C^{-1}&= Q^{-1}P^{-1}\\ (PQ)^{-1}&= Q^{-1}P^{-1} \end{aligned}
Hence, proved. Similarly, we can prove if P, Q are invertible then $(QP)^{-1}= P^{-1}Q^{-1}$
 Question 5
Match the following attributes of a city with the appropriate scale of measurements.
$\begin{array}{|l|l|}\hline \text{Attribute}&\text{Scale of measurement} \\ \hline \text{(P) Average temperature } (^{\circ}C)\text{ of a city} & \text{((I) Interval} \\ \hline \text{(Q) Name of a city} & \text{(II) Ordinal}\\ \hline \text{(R) Population density of a city}& \text{(III) Nominal}\\ \hline \text{(S) Ranking of a city based on ease of business}& \text{(IV) Ratio}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P)-(I), (Q)-(III), (R)-(IV), (S)-(II) B (P)-(II), (Q)-(I), (R)-(IV), (S)-(III) C (P)-(II), (Q)-(III), (R)-(IV), (S)-(I) D (P)-(I), (Q)-(II), (R)-(III), (S)-(IV)
GATE CE 2022 SET-2      Probability and Statistics
Question 5 Explanation:
Meaning of
Nominal -> a name or term
Ordinal -> in an ordered sequence
Ratio -> quantitative relation between two things
Interval -> indicates average of a range
 Question 6
$\int \left ( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+... \right )dx$ is equil to
 A $\frac{1}{1+x}+constant$ B $\frac{1}{1+x^2}+constant$ C $-\frac{1}{1-x}+constant$ D $-\frac{1}{1-x^2}+constant$
GATE CE 2022 SET-2      Calculus
Question 6 Explanation:
MTA- Marks to All
$I=\int \left ( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...\infty \right )dx$
$I=\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{12}-\frac{x^5}{20}+...$
Option (A)
$\frac{1}{1+x}=(1+x)^{-1}=1-x+x^2-x^3...\infty$
So, its incorrect.
Option (B)
$\frac{1}{1+x^2}=(1+x^2)^{-1}=1-x^2+x^4-x^6...\infty$
So, its incorrect.
Similarly option (C) and (D) both are incorrect.
No-correct choice given.
 Question 7
The function $f(x, y)$ satisfies the Laplace equation
$\triangledown ^2f(x,y)=0$
on a circular domain of radius $r = 1$ with its center at point P with coordinates $x = 0, y = 0$. The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of $f(0, 0)$ is:
 A 0 B 2 C 3 D 1
GATE CE 2022 SET-2      Partial Differential Equation
Question 7 Explanation:
According to given condition given function f(x,y) is nothing but constant function i.e. f(x,y)=3 because this is the only function whose value is 3 at any point on the boundary of unit circle and it is also satisfying Laplace equation, so
f(0,0)=3
 Question 8
A set of observations of independent variable ($x$) and the corresponding dependent variable ($y$) is given below.
$\begin{array}{|c|c|c|c|c|} \hline x&5&2&4&3 \\ \hline y&16&10&13&12\\ \hline \end{array}$
Based on the data, the coefficient $a$ of the linear regression model
$y = a + bx$
is estimated as 6.1.
The coefficient $b$ is ______________ . (round off to one decimal place)
 A 6.1 B 1.9 C 2.2 D 3.6
GATE CE 2022 SET-1      Probability and Statistics
Question 8 Explanation:
We know that, normal equation for fitting of straight lines are
\begin{aligned} \Sigma y&=na+b\Sigma x\\ \Sigma xy&=a\Sigma x+b\Sigma x^2\\ n&=4\\ 51&=4a+b(14)\\ 188&=a(14)+b(54) \end{aligned}
After solving, a=6.1 and b=1.9
 Question 9
Consider the differential equation
$\frac{dy}{dx}=4(x+2)-y$
For the initial condition $y = 3$ at $x = 1$, the value of $y$ at $x = 1.4$ obtained using Euler's method with a step-size of 0.2 is _________. (round off to one decimal place)
 A 5.4 B 6.4 C 2.8 D 4.2
GATE CE 2022 SET-1      Ordinary Differential Equation
Question 9 Explanation:
\begin{aligned} x_0&=1\rightarrow y_0=3\\ x_1&=1.2\rightarrow y_1=?\\ x_2&=1.4\rightarrow y_2=?\\ \end{aligned}
Using Euler's forward methods
\begin{aligned} y_n&=y_n+hf(x_n,y_n)\\ y_1&=y_0+hf(x_0,y_0)\\ &=3+0.2 \times (4(1+2)-3)\\ &=4.8\\ y_2&=y_1+hf(x_1,y_1)\\ &=4.8+0.2 \times (4(1.2+2)-4.8)\\ &=6.4\\ \end{aligned}
 Question 10
Let $max \{a, b\}$ denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function $f(x) = max\{3 -x, x - 1\}$ ?
 A It is continuous on its domain. B It has a local minimum at x = 2. C It has a local maximum at x = 2. D It is differentiable on its domain.
GATE CE 2022 SET-1      Calculus
Question 10 Explanation:
$f(x)=max{3-x,x-1}$ both intersecting at
\begin{aligned} 3-x&=x-1 \\ 2x&= 4\\ x&=2 \\ y &= max\{3-x,x-1\} \end{aligned} There are 10 questions to complete.

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