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$1.1$ 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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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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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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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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$1.5$ Fourier series of the periodic function (period $2 \pi$ ) defined by $f(x)=\left\{\begin{array}{ll}0 & -\pi \leq x 0 answers 6 1.6 Which of the following improper integrals is (are) convergent?$\int_{0}^{1} \frac{\sin x}{1-\cos x} d x\int_{0}^{\infty} \frac{\operatorname{cox} x}{1+x} d x\i...
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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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$2.1 \lim _{x \rightarrow 0} \frac{x\left(e^{x}-1\right)+2(\cos x-1)}{x(1-\cos x)}$ is
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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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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 $\vec{w... 0 answers 11 2.4 Given the differential equation,$\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... 0 answers 12$2.5$For$\mathrm{X}=4.0$, the value of 1 in the FORTRAN 77 statement $I=-2^{* *} 2+5.0 * X / X * 3+\frac{3}{4} \text { is }$ 0 answers 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 0 answers 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$\mathrm{A}^{4}$, calculat... 0 answers 15 2.8 Given,$\mathrm{V}=x \cos ^{2} y \hat{i}+x^{2} e^{x} \hat{j}+z \sin ^{2} y \hat{k}$and$S$the surface of a unit cube with one corner at the origin and edges paralle... 0 answers 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 gener... 0 answers 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 ...
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3.1 Two particles of masses $M_{1}$ and $M_{2}\left(M_{1}>M_{2}\right)$ attract each other with a force inversely proportional to the square of the distance between them....
3.4 A plane electromagnetic wave of the form $\vec{E}=\hat{y} E_{0}\left[\cos 2 \pi\left(5 \times 10^{14} \sec ^{-1}\right) t-\left(2.5 \times 10^{6} \mathrm{~m}^{-1}\rig... 0 answers 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 ... 0 answers 23 3.6. Nuclear fusion reactions require very high temperatures so as to overcome. nuclear forces van der waals forces coulomb forces gravitational forces 0 answers 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 ... 0 answers 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 st... 0 answers 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$0 answers 27 3.10. Consider a solid sphere and a hollow sphere, both of mass$M$, radius$R$and initially at rest, which start rolling down the same inclined plane without slipping. ... 0 answers 28 3.11. An optical fibre consists of a cylindrical dielectric rod of refractive index$\mathrm{n}^{1}$, surrounded by another dielectric of refractive index$n^{2}$, where ... 0 answers 29 3.12. For a uniformly charged sphere of radius$R$and charge density$\rho$, the ratio of magnitude of electric fields at distances$R / 2$and$2 R$from the centre, i.... 0 answers 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 ... 1 answer 31 Suppose$a, b, c$are in$\text{A.P.}$and$a^{2}, b^{2}, c^{2}$are in$\text{G.P.}$If$a < b < c$and$a + b + c = \frac{3}{2},$then the value of$a$is$\frac{1}{2 \...
If $(^{n}C_{0} + ^{n}C_{1}) (^{n}C_{1} + ^{n}C_{2}) \dots (^{n}C_{n-1} + ^{n}C_{n}) = k \; ^{n}C_{0} \; ^{n}C_{1} \dots ^{n}C_{n-1},$ then $k$ is equal to $\frac{(n+1)^{n... 1 answer 33 Suppose$f : \mathbb{R} \rightarrow \mathbb{R}$is a continuous function such that$f(x) = \frac{2 – \sqrt{x+4}}{\sin 2x}$for all$x \neq 0.$Then the value of$f(0)$... 1 answer 34 The number of distinct even divisiors of $$\prod_{k=1}^{5} k!$$ is$24326472$1 answer 35 For a cyclic group$\text{G}$of order$12$, the number of subgroups of$\text{G}$is$26811$1 answer 36 A binary tree starts with a single root node at the top of the tree. Each node can have either a left child or a right child, or both, or neither. The children of a node ... 1 answer 37 A student has an average score of$80$from her first four Mathematics tests, and$88$from her first five Physics tests. How much must she score in her upcoming tests to... 2 answers 38 The roots of the polynomial$p(x)=x^{4}-2x^{3}-2x^{2}+8x-8$are:$1, -1, 2, 2+3 i1+i, 1-i, 2, -21, -1+i, 2, 2+3 i1+i, -1+i, 2, -2$1 answer 39 There are two longest subsequences, not necessarily contiguous, common to the strings$\text{“ARTIFICIAL"}$and$\text{“INTELLIGENCE".}$They are$\text{“IIC"}$and... 1 answer 40 Consider the following code, in which$\text{A}$is an array indexed from$0.\$ Function foo (A , n) { m = A [0] ; x = 0 ; For i = 0 to n – 1 { x = x + A [I] ; if (m < x...