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42
GATE2017 CE-2-26
The tangent to the curve represented by $y=x \text{ ln }x$ is required to have $45^{\circ}$ inclination with the $x$-axis. The coordinates of the tangent point would be $(1,0)$ $(0, 1)$ $(1,1)$ $(\sqrt{2}, (\sqrt{2})$
The tangent to the curve represented by $y=x \text{ ln }x$ is required to have $45^{\circ}$ inclination with the $x$-axis. The coordinates of the tangent point would be$(...
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43
GATE2017 CE-2-27
Consider the following definite integral: $I= \displaystyle{} \int_0^1 \dfrac{(\sin ^{-1}x)^2}{\sqrt{1-x^2}} dx$. The value of the integral is $\dfrac{\pi ^3}{24} \\$ $\dfrac{\pi ^3}{12} \\$ $\dfrac{\pi ^3}{48} \\$ $\dfrac{\pi ^3}{64}$
Consider the following definite integral: $I= \displaystyle{} \int_0^1 \dfrac{(\sin ^{-1}x)^2}{\sqrt{1-x^2}} dx$. The value of the integral is$\dfrac{\pi ^3}{24} \\$$\dfr...
2 votes
1 answer
46
GATE2019 CE-1: 1
Which one of the following is correct? $\displaystyle{} \lim_{x\rightarrow 0} \left( \dfrac{\sin4x}{\sin2x}\right)=2\;\text{and}\: \lim_{x\rightarrow 0} \left( \dfrac{\tan x}{x}\right)=1$ ...
Which one of the following is correct?$\displaystyle{} \lim_{x\rightarrow 0} \left( \dfrac{\sin4x}{\sin2x}\right)=2\;\text{and}\: \lim_{x\rightarrow 0} \left( \dfrac{\tan...
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53
GATE2019 CE-1: 44
Consider the ordinary differential equation $x^2 \dfrac{d^2y}{dx^2} – 2x \dfrac{dy}{dx} +2y=0$. Given the values of $y(1)=0$ and $y(2)=2$, the value of $y(3)$ (round off to $1$ decimal place), is _________
Consider the ordinary differential equation $x^2 \dfrac{d^2y}{dx^2} – 2x \dfrac{dy}{dx} +2y=0$. Given the values of $y(1)=0$ and $y(2)=2$, the value of $y(3)$ (round of...
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54
GATE2019 CE-2: 1
Euclidean norm (length) of the vector $\begin{bmatrix} 4 & -2 & -6 \end{bmatrix}^T$ is $\sqrt{12}$ $\sqrt{24}$ $\sqrt{48}$ $\sqrt{56}$
Euclidean norm (length) of the vector $\begin{bmatrix} 4 & -2 & -6 \end{bmatrix}^T$ is$\sqrt{12}$$\sqrt{24}$$\sqrt{48}$$\sqrt{56}$
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55
GATE2019 CE-2: 2
The Laplace transform of $\sin h (\text{at})$ is $\dfrac{a}{s^2-a^2} \\$ $\dfrac{a}{s^2 + a^2} \\$ $\dfrac{s}{s^2-a^2} \\$ $\dfrac{s}{s^2+a^2}$
The Laplace transform of $\sin h (\text{at})$ is $\dfrac{a}{s^2-a^2} \\$$\dfrac{a}{s^2 + a^2} \\$$\dfrac{s}{s^2-a^2} \\$$\dfrac{s}{s^2+a^2}$
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56
GATE2019 CE-2: 3
The following inequality is true for all $x$ close to $0$. $2-\dfrac{x^2}{3} < \dfrac{x \sin x}{1- \cos x} <2$ What is the value of $\underset{x \to 0}{\lim} \dfrac{x \sin x}{1 – \cos x}$? $0$ $1/2$ $1$ $2$
The following inequality is true for all $x$ close to $0$.$$2-\dfrac{x^2}{3} < \dfrac{x \sin x}{1- \cos x} <2$$What is the value of $\underset{x \to 0}{\lim} \dfrac{x \si...
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59
GATE2019 CE-2: 26
The probability density function of a continuous random variable distributed uniformly between $x$ and $y$ (for $y>x$) is $\dfrac{1}{x-y}$ $\dfrac{1}{y-x}$ $x-y$ $y-x$
The probability density function of a continuous random variable distributed uniformly between $x$ and $y$ (for $y>x$) is$\dfrac{1}{x-y}$$\dfrac{1}{y-x}$$x-y$$y-x$
2 votes
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61
GATE2019 CE-2: 35
The inverse of the matrix $\begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix}$ is $\begin{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 \end{bmatrix} \\$ ...
The inverse of the matrix $\begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix}$ is $\begin{bmatrix} 10 & -4 & -9 \\ -15 & 4 & 14 \\ 5 & -1 & -6 \end{bmatrix...
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63
GATE2016-2-1
The spot speeds (expressed in km/hr) observed at a road section are $66, 62, 45, 79, 32, 51, 56, 60, 53,$ and $49$. The median speed (expressed in km / hr) is ______ (Note: answer with one decimal accuracy)
The spot speeds (expressed in km/hr) observed at a road section are $66, 62, 45, 79, 32, 51, 56, 60, 53,$ and $49$. The median speed (expressed in km / hr) is ______(Note...
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64
GATE2016-2-2
The optimum value of the function $f(x)=x^2-4x+2$ is $2$ (maximum) $2$ (minimum) $ – 2$ (maximum) $ – 2$ (minimum)
The optimum value of the function $f(x)=x^2-4x+2$ is$2$ (maximum)$2$ (minimum)$ – 2$ (maximum)$ – 2$ (minimum)
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65
GATE2016-2-3
The Fourier series of the function, $\begin{array}{rll} f(x) & =0, & -\pi < x \leq 0 \\ {} & =\pi - x, & 0 < x < \pi \end{array}$ in the interval $[- \pi, \pi ]$ ... $\Sigma_{n-1}^{\infty} \dfrac{(-1)^{n+1}}{2n-1} = \dfrac{\pi}{4}$
The Fourier series of the function,$\begin{array}{rll} f(x) & =0, & -\pi < x \leq 0 \\ {} & =\pi – x, & 0 < x < \pi \end{array}$in the interval $[- \pi, \pi ]$ is$f(x) ...
0 votes
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66
GATE2016-2-4
$X$ and $Y$ are two random independent events. It is known that $P(X)=0.40$ and $P(X \cup Y^C)=0.7$. Which one of the following is the value of $P(X \cup Y)$? $0.7$ $0.5$ $0.4$ $0.3$
$X$ and $Y$ are two random independent events. It is known that $P(X)=0.40$ and $P(X \cup Y^C)=0.7$. Which one of the following is the value of $P(X \cup Y)$?$0.7$$0.5$$0...
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67
GATE2016-2-5
What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \dfrac{xy}{x^2+y^2}$? $1$ $-1$ $0$ Limit does not exist
What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \dfrac{xy}{x^2+y^2}$?$1$$-1$$0$Limit does not exist
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68
GATE2016-2-25
A circular curve of radius R connects two straights with a deflection angle of $60^{\circ}$. The tangent length is $0.577 \: R$ $1.155 \: R$ $1.732 \:R$ $3.464 \: R$
A circular curve of radius R connects two straights with a deflection angle of $60^{\circ}$. The tangent length is$0.577 \: R$$1.155 \: R$$1.732 \:R$$3.464 \: R$
0 votes
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69
GATE2016-2-26
Consider the following linear system. $x+2y-3z=a$ $2x+3y+3z=b$ $5x +9y-6z=c$ This system is consistent if $a, b$ and $c$ satisfy the equation $7a-b-c=0$ $3a+b-c=0$ $3a-b+c=0$ $7a-b+c=0$
Consider the following linear system.$x+2y-3z=a$$2x+3y+3z=b$$5x +9y-6z=c$This system is consistent if $a, b$ and $c$ satisfy the equation$7a-b-c=0$$3a+b-c=0$$3a-b+c=0$$7a...
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71
GATE2016-2-28
The angle of intersection of the curves $x^{2}=4y$ and $y^{2} = 4x$ at point $(0,0)$ is $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$
The angle of intersection of the curves $x^{2}=4y$ and $y^{2} = 4x$ at point $(0,0)$ is$0^{\circ}$$30^{\circ}$$45^{\circ}$$90^{\circ}$
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72
GATE2016-2-29
The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.
The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.
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73
GATE2016-2-30
The quadratic approximation of $f(x)=x^3 – 3x^2 -5$ at the point $x=0$ is $3x^2 -6x-5$ $-3x^2-5$ $-3x^2+6x-5$ $3x^2-5$
The quadratic approximation of $f(x)=x^3 – 3x^2 -5$ at the point $x=0$ is$3x^2 -6x-5$$-3x^2-5$$-3x^2+6x-5$$3x^2-5$
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74
GATE2016-1-1
Newton-Raphson method is to be used to find root of equation $3x-e^x+ \sin x=0$.If the initial trial value for the root is taken as $0.333$, the next approximation for the root would be _______ (note: answer up to three decimal)
Newton-Raphson method is to be used to find root of equation $3x-e^x+ \sin x=0$.If the initial trial value for the root is taken as $0.333$, the next approximation for th...
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76
GATE2016-1-3
If the entries in each column of a square matrix $M$ add up to $1$, then an eigenvalue of $M$ is $4$ $3$ $2$ $1$
If the entries in each column of a square matrix $M$ add up to $1$, then an eigenvalue of $M$ is$4$$3$$2$$1$
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80
GATE2016-1-27
The value of $\int_0^{\infty} \dfrac{1}{1+x^2} dx + \int _0^{\infty} \dfrac{\sin x}{x} dx$ is $\dfrac{\pi}{2} \\$ $\pi \\$ $\dfrac{3 \pi}{2} \\$ $1$
The value of $\int_0^{\infty} \dfrac{1}{1+x^2} dx + \int _0^{\infty} \dfrac{\sin x}{x} dx$ is$\dfrac{\pi}{2} \\$$\pi \\$$\dfrac{3 \pi}{2} \\$$1$
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