Recent questions in Engineering Mathematics

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1

ISI2015-DCG-47
The Taylor series expansion of $f(x)= \text{ln}(1+x^2)$ about $x=0$ is $\sum _{n=1}^{\infty} (-1)^n \frac{x^n}{n}$ $\sum _{n=1}^{\infty} (-1)^{n+1} \frac{x^{2n}}{n}$ $\sum _{n=1}^{\infty} (-1)^{n+1} \frac{x^{2n+1}}{n+1}$ $\sum _{n=0}^{\infty} (-1)^{n+1} \frac{x^{n+1}}{n+1}$

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 19 views

+1 vote

1 answer

2

ISI2015-DCG-48
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 25 views

+1 vote

1 answer

3

ISI2015-DCG-49
The domain of the function $\text{ln}(3x^2-4x+5)$ is set of positive real numbers set of real numbers set of negative real numbers set of real numbers larger than $5$

asked Sep 18, 2019 in Set Theory & Algebra by gatecse Boss (17.5k points) | 32 views

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4

ISI2015-DCG-50
The piecewise linear function for the following graph is $f(x) = \begin{cases} = x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$ $f(x) = \begin{cases} = x-2, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x-1, \: x \geq 3 \end{cases}$ ... $f(x) = \begin{cases} = 2-x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 20 views

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1 answer

5

ISI2015-DCG-51
The area bounded by $y=x^2-4$, $y=0$ and $x=4$ is $\frac{64}{3}$ $6$ $\frac{16}{3}$ $\frac{32}{3}$

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 21 views

+2 votes

1 answer

6

ISI2015-DCG-52
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 31 views

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7

ISI2015-DCG-54
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals $-1$ $0$ $1$ Does not exist

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 20 views

+1 vote

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8

ISI2015-DCG-55
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals $-1$ $0$ $1$ Does not exist

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 17 views

+1 vote

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9

ISI2015-DCG-56
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals $-1$ $0$ $1$ Does not exist

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 22 views

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10

ISI2015-DCG-57
Let $y=\lfloor x \rfloor$, where $\lfloor x \rfloor$ is greatest integer less than or equal to $x$. Then $y$ is continuous and many-one $y$ is not differentiable and many-one $y$ is not differentiable $y$ is differentiable and many-one

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 12 views

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11

ISI2015-DCG-58
$\underset{x \to 1}{\lim} \dfrac{x^{16}-1}{\mid x-1 \mid}$ equals $-1$ $0$ $1$ Does not exist

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 14 views

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1 answer

12

ISI2016-DCG-3
The value of $\begin{vmatrix} 1+a& 1& 1& 1\\ 1&1+b &1 &1 \\ 1&1 &1+c &1 \\ 1&1 &1 &1+d \end{vmatrix}$ is $abcd(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$ $abcd(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})$ $1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ None of these

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 23 views

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1 answer

13

ISI2016-DCG-4
If $f(x)=\begin{bmatrix}\cos\:x & -\sin\:x & 0 \\ \sin\:x & \cos\:x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then the value of $\big(f(x)\big)^2$ is $f(x)$ $f(2x)$ $2f(x)$ None of these

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 20 views

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1 answer

14

ISI2016-DCG-11
Let two systems of linear equations be defined as follows: $\begin{array}{lll} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ ... $P$ and $Q$ are inconsistent $P$ and $Q$ are consistent $P$ is consistent but $Q$ is inconsistent None of the above

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 16 views

+2 votes

1 answer

15

ISI2016-DCG-21
The value of the term independent of $x$ in the expansion of $(1-x)^{2}(x+\frac{1}{x})^{7}$ is $-70$ $70$ $35$ None of these

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 30 views

+1 vote

1 answer

16

ISI2016-DCG-22
The value of $\:\:\begin{vmatrix} 1&\log_{x}y &\log_{x}z \\ \log_{y}x &1 &\log_{y}z \\\log_{z}x & \log_{z}y&1 \end{vmatrix}\:\:$ is $0$ $1$ $-1$ None of these

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 28 views

+2 votes

1 answer

17

ISI2016-DCG-24
If the letters of the word $\text{COMPUTER}$ be arranged in random order, the number of arrangements in which the three vowels $O, U$ and $E$ occur together is $8!$ $6!$ $3!6!$ None of these

asked Sep 18, 2019 in Combinatory by gatecse Boss (17.5k points) | 22 views

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1 answer

18

ISI2016-DCG-27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of triangle $x,$ is one-one and into one-one and onto many-one and onto many-one and into

asked Sep 18, 2019 in Set Theory & Algebra by gatecse Boss (17.5k points) | 15 views

+1 vote

1 answer

19

ISI2016-DCG-31
Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$ stands for determinant of matrix $A.$ Then $\mid\:(A)\mid=1$ $\mid\:(A)\mid=0\:\text{or}\:1$ $\mid\:(A)\mid=-1,0\:\text{or}\:1$ $\mid\:(A)\mid=-1\:\text{or}\:1$

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 22 views

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20

ISI2016-DCG-32
The set of vectors constituting an orthogonal basis in $\mathbb{R}^{3}$ is $\begin{Bmatrix} \begin{pmatrix} 1\\ -1 \\0 \end{pmatrix}&,\begin{pmatrix} 1\\ 1 \\0 \end{pmatrix}&,\begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} \end{Bmatrix}$ ... None of these

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 12 views

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21

ISI2016-DCG-33
Suppose $A$ and $B$ are orthogonal $n\times n$ matrices. Which of the following is also an orthogonal matrix? Assume that $O$ is the null matrix of order $n\times n$ and $I$ is the identity matrix of order $n.$ $AB-BA$ $\begin{pmatrix} A & O \\ O & B \end{pmatrix}$ $\begin{pmatrix} A & I \\ I & B \end{pmatrix}$ $A^{2}-B^{2}$

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 13 views

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22

ISI2016-DCG-34
Let $A_{ij}$ denote the minors of an $n\times n$ matrix $A.$ What is the relationship between $\mid A_{ij}\mid$ and $\mid A_{ji}\mid$? They are always equal. $\mid A_{ij}\mid=-\mid A_{ji}\mid$ if $i\neq j.$ They are equal if $A$ is a symmetric matrix. If $\mid A_{ij}\mid=0$ then $\mid A_{ji}\mid=0.$

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.5k points) | 12 views

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23

ISI2016-DCG-35
Let $A,B$ and $C$ be three non empty sets. Consider the two relations given below: $A-(B-C)=(A-B)\cup C$ $A-(B\cup C)=(A-B)-C$ Both (1) and (2) are correct. (1) is correct but (2) is not. (2) is correct but (1) is not. Both (1) and (2) are incorrect.

asked Sep 18, 2019 in Set Theory & Algebra by gatecse Boss (17.5k points) | 12 views

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1 answer

24

ISI2016-DCG-36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is $n^{n}$ $n\log_{2}n$ $n^{2}$ $n!$

asked Sep 18, 2019 in Set Theory & Algebra by gatecse Boss (17.5k points) | 16 views

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25

ISI2016-DCG-37
Suppose $f_{\alpha}\::\:[0,1]\rightarrow[0,1],\:-1<\alpha<\infty$ is given by $f_{\alpha}(x)=\dfrac{(\alpha+1)x}{\alpha x+1}.$ Then $f_{\alpha}$ is A bijective (one-one and onto) function. A surjective (onto) function. An injective (one-one) function. We can not conclude about the type.

asked Sep 18, 2019 in Set Theory & Algebra by gatecse Boss (17.5k points) | 11 views

+2 votes

1 answer

26

ISI2016-DCG-45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 42 views

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0 answers

27

ISI2016-DCG-46
Let $I=\int(\sin\:x-\cos\:x)(\sin\:x+\cos\:x)^{3}dx$ and $K$ be a constant of integration. Then the value of $I$ is $(\sin\:x+\cos\:x)^{4}+K$ $(\sin\:x+\cos\:x)^{2}+K$ $-\frac{1}{4}(\sin\:x+\cos\:x)^{4}+K$ None of these

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 18 views

0 votes

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28

ISI2016-DCG-47
The Taylor series expansion of $f(x)=\ln(1+x^{2})$ about $x=0$ is $\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{n}}{n}$ $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n}}{n}$ $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n+1}}{n+1}$ $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{n+1}}{n+1}$

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 13 views

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29

ISI2016-DCG-48
The piecewise linear function for the following graph is $f(x)=\begin{cases} = x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = x-2,x\leq-2 \\ =4,-2<x<3 \\ = x-1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2x,x\leq-2 \\ =x,-2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2-x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 13 views

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1 answer

30

ISI2016-DCG-49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these

asked Sep 18, 2019 in Calculus by gatecse Boss (17.5k points) | 19 views

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