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GATE ECE 1993 | Question: 1
$1.1$ The eigenvector(s) of the matrix $\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), a \neq 0$, is (are) $(0,0, \alpha)$ $(\alpha, 0,0)$ $(0,0,1)$ $(0, \alpha, 0)$
$1.1$ The eigenvector(s) of the matrix $\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), a \neq 0$, is (are) $(0,0, \alpha)$ $(\alpha, 0,0)$...
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2
GATE ECE 1993 | Question: 2
1.2 The differential equation, $\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+\sin$ $y=0$, is linear non-linear homogeneous of degree two
1.2 The differential equation, $\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+\sin$ $y=0$, is linear non-linear homogeneous of degree two
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3
GATE ECE 1993 | Question: 3
1.3 Simpson's rule for integration gives exact result when $f(x)$ is a polynomial of degree 1 2 3 4
1.3 Simpson's rule for integration gives exact result when $f(x)$ is a polynomial of degree 1 2 3 4
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4
GATE ECE 1993 | Question: 4
1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? $\mathrm{DO} 131=1$ $\mathrm{A}=\mathrm{DIM}^{* * 7} 7$ READ $=15.0$ GOTO $3=10$
1.4 Which of the following is (are) valid FORTRAN 77 statement(s)? $\mathrm{DO} 131=1$ $\mathrm{A}=\mathrm{DIM}^{* * 7} 7$ READ $=15.0$ GOTO $3=10$
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5
GATE ECE 1993 | Question: 5
$1.5$ Fourier series of the periodic function (period $2 \pi$ ) defined by $f(x)=\left\{\begin{array}{ll}0 & -\pi \leq x
$1.5$ Fourier series of the periodic function (period $2 \pi$ ) defined by $f(x)=\left\{\begin{array}{ll}0 & -\pi \leq x
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6
GATE ECE 1993 | Question: 6
1.6 Which of the following improper integrals is (are) convergent? $\int_{0}^{1} \frac{\sin x}{1-\cos x} d x$ $\int_{0}^{\infty} \frac{\operatorname{cox} x}{1+x} d x$ $\int_{0}^{x} \frac{x}{1+x^{2}} d x$ $\int_{0}^{1} \frac{1-\cos x}{x^{5 / 2}} d x$
1.6 Which of the following improper integrals is (are) convergent? $\int_{0}^{1} \frac{\sin x}{1-\cos x} d x$ $\int_{0}^{\infty} \frac{\operatorname{cox} x}{1+x} d x$ $\i...
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7
GATE ECE 1993 | Question: 7
1.7 The function $f(x, y)=x^{2} y-3 x y+2 y+x$, has no local extremum one local minimum but no local maximum one local maximum but no local minimum one local minimum and one local maximum
1.7 The function $f(x, y)=x^{2} y-3 x y+2 y+x$, has no local extremum one local minimum but no local maximum one local maximum but no local minimum one local minimum and ...
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8
GATE ECE 1993 | Question: 8
$2.1 \lim _{x \rightarrow 0} \frac{x\left(e^{x}-1\right)+2(\cos x-1)}{x(1-\cos x)}$ is
$2.1 \lim _{x \rightarrow 0} \frac{x\left(e^{x}-1\right)+2(\cos x-1)}{x(1-\cos x)}$ is
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9
GATE ECE 1993 | Question: 9
2.2 The radius of convergence of the power series $\sum_{0}^{n} \frac{(3 m) !}{(m !)^{3}} x^{i m}$ is
2.2 The radius of convergence of the power series $\sum_{0}^{n} \frac{(3 m) !}{(m !)^{3}} x^{i m}$ is
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10
GATE ECE 1993 | Question: 10
2.3 If the linear velocity $\overrightarrow{\mathrm{V}}$ is given by $\overrightarrow{\mathrm{V}}=x^{2} y \hat{i}+$ $x y z \hat{j}-y z^{2} k$ the anguarl velocity $\vec{w}$ at the point $(1,1,-1)$ is
2.3 If the linear velocity $\overrightarrow{\mathrm{V}}$ is given by $\overrightarrow{\mathrm{V}}=x^{2} y \hat{i}+$ $x y z \hat{j}-y z^{2} k$ the anguarl velocity $\vec{w...
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11
GATE ECE 1993 | Question: 11
2.4 Given the differential equation, $\bar{y}=x-y$ with the initial condition $y(0)=0$. The value of $(0.1)$ calculated numerically upto the third place of decimal by the second order Runge-Kutta method with step size $h=0.1$ is
2.4 Given the differential equation, $\bar{y}=x-y$ with the initial condition $y(0)=0$. The value of $(0.1)$ calculated numerically upto the third place of decimal by the...
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12
GATE ECE 1993 | Question: 12
$2.5$ For $\mathrm{X}=4.0$, the value of 1 in the FORTRAN 77 statement \[ I=-2^{* *} 2+5.0 * X / X * 3+\frac{3}{4} \text { is } \]
$2.5$ For $\mathrm{X}=4.0$, the value of 1 in the FORTRAN 77 statement \[ I=-2^{* *} 2+5.0 * X / X * 3+\frac{3}{4} \text { is } \]
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13
GATE ECE 1993 | Question: 13
2.6 The value of the double integral $\int_{0}^{1} \int_{x}^{1 / x} \frac{x}{1+y^{2}}$ $d x d y$ is
2.6 The value of the double integral $\int_{0}^{1} \int_{x}^{1 / x} \frac{x}{1+y^{2}}$ $d x d y$ is
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14
GATE ECE 1993 | Question: 14
$2.7$ If $\mathrm{A}=\left(\begin{array}{cccc}1 & 0 & 0 & 1 \\ 0 & -1 & 0 & -1 \\ 0 & 0 & i & i \\ 0 & 0 & 0 & -i\end{array}\right)$ the matrix $\mathrm{A}^{4}$, calculated by the use of Cayley - Hamilton theorem or otherwise, is
$2.7$ If $\mathrm{A}=\left(\begin{array}{cccc}1 & 0 & 0 & 1 \\ 0 & -1 & 0 & -1 \\ 0 & 0 & i & i \\ 0 & 0 & 0 & -i\end{array}\right)$ the matrix $\mathrm{A}^{4}$, calculat...
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16
GATE ECE 1993 | Question: 16
2.9 The differential equation $y+y=0$ is subjected to the boundary conditions \[ y(0)+0 y(\lambda)=0 \] In order that the equation has non-trivial solution (s), the general value of $\eta$ is
2.9 The differential equation $y+y=0$ is subjected to the boundary conditions \[ y(0)+0 y(\lambda)=0 \] In order that the equation has non-trivial solution (s), the gener...
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17
GATE ECE 1993 | Question: 17
2.10 The Laplace transform of the periodic function $f(t)$ described by the curve below, i.e. $f(t)=\left\{\begin{array}{c}\sin t \text { if }(2 n-1) \pi \leq t \leq 2 n \pi(n=1,2,3, \ldots) \\ 0\end{array}\right.$ otherwise is
2.10 The Laplace transform of the periodic function $f(t)$ described by the curve below, i.e. $f(t)=\left\{\begin{array}{c}\sin t \text { if }(2 n-1) \pi \leq t \leq 2 n ...
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20
GATE ECE 1993 | Question: 20
3.3. Although a laser heam is highly directional, its beam width increases with propagation. This increase is due to coherence diffraction polarization interference
3.3. Although a laser heam is highly directional, its beam width increases with propagation. This increase is due to coherence diffraction polarization interference
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22
GATE ECE 1993 | Question: 22
3.5. While you are listening to a programme from a radio, if a near-by electric light bulb is switched on or switched off, you hear a momentary noise in your radio. This is due to electromagnetic radiation emitted by
3.5. While you are listening to a programme from a radio, if a near-by electric light bulb is switched on or switched off, you hear a momentary noise in your radio. This ...
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23
GATE ECE 1993 | Question: 23
3.6. Nuclear fusion reactions require very high temperatures so as to overcome. nuclear forces van der waals forces coulomb forces gravitational forces
3.6. Nuclear fusion reactions require very high temperatures so as to overcome. nuclear forces van der waals forces coulomb forces gravitational forces
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24
GATE ECE 1993 | Question: 24
3.7. In radioactive decay, the disintegration rate of the nuclei is constant at all times inversely proportional to half-life of the nuclei inversely proportional to the number of nuclei at any time directly proportional to the number of nuclei at any time
3.7. In radioactive decay, the disintegration rate of the nuclei is constant at all times inversely proportional to half-life of the nuclei inversely proportional to the ...
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25
GATE ECE 1993 | Question: 25
3.8. In an hydrogen atom $10.2 . \mathrm{eV}$ is given out as radiation when an electron is de-excited to the ground state. The principal quantum number of the excited state is
3.8. In an hydrogen atom $10.2 . \mathrm{eV}$ is given out as radiation when an electron is de-excited to the ground state. The principal quantum number of the excited st...
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26
GATE ECE 1993 | Question: 26
3.9. Typical current voltage characteristic of a solar cell is given in the following figure by curve $A$ curve $B$ curve $C$ curve $D$
3.9. Typical current voltage characteristic of a solar cell is given in the following figure by curve $A$ curve $B$ curve $C$ curve $D$
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27
GATE ECE 1993 | Question: 27
3.10. Consider a solid sphere and a hollow sphere, both of mass $M$, radius $R$ ... a solid sphere is $(2 / 5) \mathrm{MR}^{2}$, and for a hollow sphere $(2 / 3) \mathrm{MR}^{2}$ ]
3.10. Consider a solid sphere and a hollow sphere, both of mass $M$, radius $R$ and initially at rest, which start rolling down the same inclined plane without slipping. ...
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28
GATE ECE 1993 | Question: 28
3.11. An optical fibre consists of a cylindrical dielectric rod of refractive index $\mathrm{n}^{1}$, surrounded by another dielectric of refractive index $n^{2}$, where $n^{2}
3.11. An optical fibre consists of a cylindrical dielectric rod of refractive index $\mathrm{n}^{1}$, surrounded by another dielectric of refractive index $n^{2}$, where ...
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29
GATE ECE 1993 | Question: 29
3.12. For a uniformly charged sphere of radius $R$ and charge density $\rho$, the ratio of magnitude of electric fields at distances $R / 2$ and $2 R$ from the centre, i.e., $\frac{\mathrm{E}(r=\mathrm{R} / 2}{\mathrm{E}(r=2 \mathrm{R})}$ is $\ldots .$
3.12. For a uniformly charged sphere of radius $R$ and charge density $\rho$, the ratio of magnitude of electric fields at distances $R / 2$ and $2 R$ from the centre, i....
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30
GATE ECE 1993 | Question: 30
3.13. A long solenoid of radius $\mathrm{R}$, and having $\mathrm{N}$ turns per unit length carries a time dependent current I $(t)=I_{0} \cos (\omega t)$. The magnitude of induced electric field at a distance $R / 2$ ... $\frac{\mathrm{R}}{2} \mu_{0} N I_{0} \omega \cos (\omega t)$
3.13. A long solenoid of radius $\mathrm{R}$, and having $\mathrm{N}$ turns per unit length carries a time dependent current I $(t)=I_{0} \cos (\omega t)$. The magnitude ...
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