Engineering Mathematics

Q.

What are the $4$ concepts of calculus?

Q.

How to find the Maclaurin series of $f\left(x\right)=\cos \left(x\right)$

Q.

The domain of the function $f\left(x\right)={\sin }^{-1}\left[\frac{\left|x\right|+5}{x^2+1}\right]$ is $(-\infty ,-a]\cup [a,\infty )$. Then a is equal to:

1. $\frac{\sqrt{17-1}}{2}$

2. $\frac{\sqrt{17}}{2}$

3. $\frac{1+\sqrt{17}}{2}$

4. $\frac{\sqrt{17}}{2}+1$

Q. A single dice is thrown twice. What is the probability that the sum is neither 8 nor 9?

1. 1/9
2. 5/36
3. 1/4
4. 3/4

Q. The real root of $xlo{g}_{10}x=1.2$ (upto 4 decimal places) by Regula-Falsi method is

1. 2.7210
2. 2.7400
3. 2.7406
4. 2.7409

Q. Three cards were drawn from a pack of 52 cards. The probability that they are a king, a queen, and a jack is

1. $\frac{16}{5525}$
2. $\frac{64}{2197}$
3. $\frac{3}{13}$
4. $\frac{8}{16575}$

Q.

From a pack of playing cards three cards are drawn simultaneously. The probability that these are one king, one queen and one jack is

1. $\frac{64}{5525}$

2. $\frac{16}{5525}$

3. $\frac{128}{5525}$

4. $\frac{64}{625}$

Q.

Explain the procedure to find $\frac{d{}^{2}y}{d{x}^2}$ [second order derivative] of any function y=f(x).

Q.

A fair coin is tossed a fixed number of items. If the probability of getting$7$ heads is equal to that of getting $9$ heads, then the probability of getting $3$ heads is

1. $\frac{35}{2^{12}}$

2. $\frac{35}{2^{14}}$

3. $\frac{7}{2^{12}}$

4. None of these

Q. The particular integral of the differential equation $\frac{d^{2}y}{d{x}^2}-\frac{2dy}{dx}=e^{2x}sinx$

1. $\frac{-e^{2x}}{10}(3\cos x-2\sin x)$
2. $\frac{e^{2x}}{10}(3\cos x-2\sin x)$
3. $\frac{-e^{2x}}{5}(2\cos x+\sin x)$
4. $\frac{e^{2x}}{5}(2\cos x-\sin x)$

Q. For $\frac{dy}{dx}=x+y^2$, given that $y=1$ at $x=0$; value of $y$ at $x=0.2$, using Runge Kutta fourth order method (h = 0.2) will be _____ (upto 4 decimal places)

1. 1.2735

Q.

How do you find the exact value for $\cos (240°)$?

Q. The particular integral of $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=x^2+2x+4$ is

1. $\frac{x^3}{3}+x^2+4x+C$
2. $\frac{x^3}{3}+x^2+C$
3. $\frac{x^3}{3}+4x+C$
4. $x^2+4x+C$

Q. A coin is tossed 4 times. What is the probability of getting heads exactly 3 times?

1. 1/4
2. 3/8
3. 1/2
4. 3/4

Q. $IfthematrixA=\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]then{A}^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+Iwillbe$______

1. $\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]$
2. $\left[\begin{array}{ccc}5& 4& 4\\ 0& 1& 0\\ 4& 4& 5\end{array}\right]$
3. $\left[\begin{array}{ccc}7& 5& 5\\ 0& 2& 0\\ 5& 5& 7\end{array}\right]$
4. $\left[\begin{array}{ccc}8& 5& 5\\ 0& 3& 0\\ 5& 5& 8\end{array}\right]$

Q. A continuous random variable X has a probability density function $f\left(x\right)=e^{-x},0<x<\infty .$ Then P{X > 1} is

1. 0.368
2. 0.5
3. 0.632
4. 1.0

Q. The value of $\left[{lim}_{x\to \infty }\frac{sinx}{x}\right]$ is :

1. $\infty$
2. 2
3. 1
4. 0

Q. The three characteristic roots of the following matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 0& 2\end{array}\right]$ are

1. $1,2,3$
2. $1,2,2$
3. $1,0,0$
4. $0,2,3$

Q. Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

1. $8/\left(2e^3\right)$
2. $9/\left(2e^3\right)$
3. $17/\left(2e^3\right)$
4. $26/\left(2e^3\right)$

Q. If X is a normal variate with mean 30 and standard deviation 5, what is Probability $(26\le X\le 34),$. Given A (z = 0.8) = 0.2881, where A represents area.

1. 0.2881
2. 0.5762
3. 0.8181
4. 0.1616

Q. The value of ${\int }_c\frac{z^2}{z^4-1}dz$, using Cauchy's integral formula, around the circle $|z+1|=1$ where $z=x+iy$ is

1. $2\pi i$
2. $-\pi i/2$
3. $-3\pi i/2$
4. ${\pi }^{2}i$

Q. The probability that a student knows the correct answer to a multiple choice question is 2/3. If the student does not know the answer, then the student guesses the answer. The probability of the gussed answer being correct is 1/4. Given that the student has answreed the question correctly, the conditional probability that the student knows the correct answer is

1. $\frac{2}{3}$
2. $\frac{3}{4}$
3. $\frac{5}{6}$
4. $\frac{8}{9}$

Q. If A and B are non-zero square matices, then AB=$0$ implies

1. A and B are othogonal
2. A and B are singular
3. B is singular
4. A is singular

Q.

A random variable X has probability density function f(x) as give below:
$f\left(x\right)=(\begin{array}{c}a+\mathrm{bx}\mathrm{for}0<x<1\\ 0\mathrm{otherwise}\end{array}$

If the expected value E[X] = 2/3, then Pr[X < 0.5] is .

1. 0.25

Q.

A die is thrown three times.

Getting a $3$ or a $6$ is considered a success.

Then the probability of at least two successes is

1. $\frac{2}{9}$

2. $\frac{7}{27}$

3. $\frac{1}{27}$

4. None of these

Q. Two dice are thrown together then the probability that the sum of the number on the two faces is divisible by 4 or 6 is_________

1. $\frac{5}{18}$
2. $\frac{13}{18}$
3. $\frac{7}{18}$
4. $\frac{17}{18}$

Q. The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is

1. 0.26

Q. The inverse Laplace transform of $F\left(s\right)=\frac{s+3}{s^2+2s+1}$ for $t\ge 0$ is

1. $e^{-t}+3t{e}^{-t}$
2. $e^{-t}+4t{e}^{-t}$
3. $e^{-t}+2t{e}^{-t}$
4. $3t{e}^{-t}$

Q. The partial differential equation that can be formed from $z=ax+by+ab$ has the form $(\text{with}p=\frac{\partial z}{\partial x}\text{and}q=\frac{\partial z}{\partial y})$

1. $z=px+qy$
2. $z=px+pq$
3. $z=px+qy+pq$
4. $z=qy+pq$

Q. The Laplace Transform of the following function is
f(t)=$\{\begin{array}{c}\sin tfor0\le t\le \pi \\ 0fort>\pi \end{array}$

1. $\frac{1}{1+s^2}foralls>0$
2. $\frac{1}{1+s^2}foralls<\pi$
3. $\frac{1+e^{-\pi s}}{1+s^2}foralls>0$
4. $\frac{e^{-\pi s}}{1+s^2}foralls>0$