72 Official GATE PH Questions with Verified Solutions & Answer Keys
The eigenvalues of a matrix are $i$, $-2i$ and $3i$. The matrix is...
The eigenvalues and eigenvectors of the matrix are (A) 6, 1 and [1721) (B) 2, 5 and (7(4] (C) 6, 1 and (D) 2, 5 and (4121]...
A vector field is defined everywhere as $\vec{F} = 2x^2\hat{i} + 3y^2\hat{j} + z^2\hat{k}$. The net flux of $\vec{F}$ associated with a cube of side $L$, with one vertex at the ori...
If $\vec{F} = x\hat{i} + y\hat{j}$, then...
Consider a vector $\vec{p} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ in the coordinate system $(\hat{i}, \hat{j}, \hat{k})$. The axes are rotated anti-clockwise about the $Y$-axis by an an...
The points, where the series solution of the Legendre differential equation (1-x)d'y dx? -2xdy + 3₴+1)y=0 will diverge, are located at (A) 0 and 1 (B) 0 and -1 (C) -1 and 1 (D) } a...
Solution of the differential equation x- -+ y=x* , with the boundary condition that y = 1, at x = 1, is (A) y = 5x* - 4 (В) y = x' (C) y = 4x' (D) y= x* 5x...
Three operators X, Y and Z satisfy the commutation relations [X,Y]=ihz, [Y,Z)=ihX and [Z,X]=ihY. The set of all possible eigenvalues of the operator Z, in units of h, is (A) {0, $1...
ae'" An unitary matrix ce' is given, where a,b,c,d,a and B are real. The inverse of the matrix is ae"" -ce'ß (B) ae'" ce'P (A) b d b d ae"a ce"p (C) се"...
The curl of a vector field F is 2^. Identify the appropriate vector field F from the choices given below. (A) F = 2zx + 3zỷ + 5 y2 (B) F = 3zỷ + 5 yz (C) F = 3xỷ + 5y2 (D) F = 2đ +...
A finite wave train, of an unspecified nature, propagates along the positive x axis with a constant speed v and without any change of shape. The differential equation among the fou...
The eigenvalues of the matrix A=° 1 i 0 are (A) real and distinct (B) complex and distinct (C) complex and coinciding (D) real and coinciding...
o, (i =1, 2, 3) represent the Pauli spin matrices. Which one of the following is NOT true ? (A) 0,0, +0,0, = 20 (B) Tr(o,) =0 (C) The eigenvalues of o, are $1 (D) det (o,) =1...
Which one of the functions given below represents the bound state eigenfunction of the operator dx in the region, 0 ≤ x < oo, with the eigenvalue -4 ? (A) A,e2* (B) A, cosh 2x (C) ...
Which one of the following curves gives the solution of the differential equation dx + k,x = k,,where k,,kz and k, are positive constants with initial conditions x =0 at / = 0? (A)...
The value of the integral z-3z+2 - dz, where the contour C is the cirele |2| = 3/2 is (A) 2mie (В) пie (C) -2nie (D) -nie...
A classical particle is moving in an external potential field V (x,y,z) which is invariant under the following infinitesimal transformations x → x = x+бx, y→ y = у+бу, Rz where R,i...
The spin function of a free particle, in the basis in which S, is diagonal, can be written as and 1 with eigenvalues + 2 - and -- , respectively. In the given basis, the normalized...
Consider an anti-symumetric tensor P, with the indices i and j running from 1 lo S. The number of independent components of the rensor is (A) 3 (B) 10 (C) 9 (D) 6 D2 The value oi e...
The cigenvalues of the matrix (A) 5, 2, -2 12 3 0) 3 10 \cdot 1) are (B) - 5, - 1, -1 for x < 3, (C) 5.1, -1 (D) - 5. 1, 1...
The three principal moments of inertia of a methanol (CH,OH) molecule have the property I, = 1, =/ and /, # /. The rotational cnergy eigenvalues are (A) E1I+1) +nmg | (8) 1+1) (C) ...
A 3X3 matrix has elements such that its trace is 11 and its determinant is 36. The eigenvalues of the matrix are all known to be positive integers. The largest eigenvalue of the ma...
Consider a cylinder of height h and radius a, closed at both ends, centered at the origin. Let F=ix+ jy+kz be the position vector and îî a unit vector normal to the surface. The su...
The solutions to the differential equation =- X dx y+1 are a family of (A) circles with different radii (B) circles with different centres (C) straight lines with different slopes ...
Consider a counterclockwise circular contour (z| = 1 about the origin. The integral ff (z) dz over this contour is (A) -іл (B) zero (С) іл (D) 2int Common Data for Questions 50 and...
The number of independent components of the symmetric tensor Aij with indices i, j = 1, 2, 3 is...
The eigenvalues of the matrix 0 1 0 1 0 1 0 1 0 are...
The value of the integral C zdz e1 , using the contour C of circle with unit radius 1 z is (A) 0 (B) 1-2 i (C) 1+2 i (D) 2 i...
For a scalar function ߮satisfying the Laplace equation, ߮has...
For the function f (z)= 18 (+^3)(-1)^2 , the residue at the pole z = 1 is (your answer should be an integer) ___________....
The degenerate eigenvalue of the matrix 4 1 1 14 1 1 1 4 3 is (your answer should be an integer) _________________...
In a constant magnetic field of 0.6 Tesla along the z direction, find the value of the path integral ∮AԦ݈݀ \vec\vec\vecԦ in the units of (Tesla m2) on a square loop of side length ...
The matrix − − += 1 1 1 1 3 1 i iA is (A) orthogonal (B) symmetric (C) anti-symmetric (D) unitary...
The value of the integral dze z C z∫ +1 2 , where C is the circle 4=z , is (A) iπ2 (B) i22π (C) i34π (D) i24π...
The solution of the differential equation 02 2 = −yt d y d , subject to the boundary conditions ( ) 1 0=y and ( ) 0= ∞y , is (A) cos t + sin t (B) cosh t + sinh t (C) cos t - sin t...
An operator for a spin-½ particle is given by A= 2ỡ - B, where B= matrices and 2 is a constant. The eigenvalues of A are (A) + 2B/V2 (B) $^B (С) 0, Л В - (*+ý), o denotes Pauli (D)...
Let Vi be the ith component of a vector field V, which has zero divergence. If dj = a/axj, the expression for Eijk Elmk dja,Vm is equal to (A) - đ ,0KV; (B) д,0 V; (C) 037, (D)-0}V...
For the Hamiltonian H = a I +b. where a, € R, b is a real vector, Z is the 2x2 identity matrix, and ® are the Pauli matrices, the ground state energy is (A) | b| (B) 2ao-|b| (C) ao...
Let X be a column vector of dimension n > 1 with at least one non-zero entry. The number of non-zero eigenvalues of the matrix M = XX* is (A) 0 (B) n (C) 1 (D) n-1...
Consider the differential equation dy / dx + y tan(x) = cos(x). If y(O) =0, y(п/3) is _. (up to two decimal places)....
The eigenvalues of a Hermitian matrix are all...
The scale factors corresponding to the covariant metric tensor gij in spherical polar coordinates are (A) 1,r2 ,r' sin? 0 (B) 1,r2, sin? 0 (C) 1,1, 1 (D) 1,r,r sin e...
The absolute value of the integral 57 +-322 22-4 -dz, over the circle |z - 1.5| = 1 in complex plane, is (up to two decimal places)....
A two-state quantum system has energy eigenvalues ‡e corresponding to the normalized states (Wt). Attime t = O, the system is in quantum state ⅖ll4+) + I4-)1. The probability that ...
For the differential equation ( ) 2 2 2 1 0 d y y n n dx x − + = , where n is a constant, the product of its two independent solutions is (a) 1 x (b) x (c) nx (d) 1 1 nx + GATE 201...
During a rotation, vectors along the axis of rotation remain unchanged. For the rotation matrix 0 1 0 0 0 1 1 0 0 − − , the unit vector along the axis of rotati...
The value of the integral ( ) 2 2 cos kx dx x a ∞ −∞ + ∫ , where 0 k > and 0 a > , is (a) ka e a π − (b) 2 ka e a π − (c) 2 ka e a π − (d) 3 2 ka e a π − GATE 2019 GATE - 2019 [SOL...
A projectile of mass 1kg is launched at an angle of 0 30 from the horizontal direction at 0 t = and takes time T before hitting the ground. If its initial speed is 1 10 ms−, the va...
A real, invertible 3x3 matrix M has eigenvalues 2,, (i =1,2,3) and the corresponding eigenvectors are |e;), (i=1,2,3) respectively. Which one of the following is correct? M/e) le),...
Let â and ât, respectively denote the lowering and raising operators of a one- dimensional simple harmonic oscillator. Let n) be the energy eigenstate of the simple harmonic oscill...
For a complex variable z and the contour c: z = 1 taken in the counter clockwise 2 direction, + 3 dz =...
The product of eigenvalues of 0 0 0 - 0 1) 0 0, (A) (B) THON (A) (8) (D) the three-dimensional real vector space. Which one of the following is correct? S is an orthonormal set S i...
Let u" denote the 4-velocity of a relativistic particle whose square U"U, = 1. If E upo is the Levi-Civita tensor then the value of Eupol"u' uPu" is _...
The value of lit, I (x) sin. dr is._ Consider the Hamiltonian A = , + Al' where 0 0 i, = and ' is the time independent perturbation given by E) Ê' = k , where k > 0. If, the maximu...
(A) Choose the graph that best describes the variation of dielectric constant (Er) with temperature (I) in a ferroelectric material. (Ic is the Curie temperature) ^ Er (B) ^ Er Tc ...
P and Q are two Hermitian matrices and there exists a matrix R, which diagonalizes both of them, such that RPR-1 = S, and RQR-1 = Sz, where S1 and Sz are diagonal matrices. The cor...
For the Op-Amp circuit shown below, choose the correct output waveform corresponding to the input Vin0 = 1.5 sin 20𝜋t (in Volts). The saturation voltage for this circuit is V01 = ...
What is the maximum number of free independent real parameters specifying an n-dimensional orthogonal matrix?...
A power supply has internal resistance RS and open load voltage VS = 5 V. When a load resistance R is connected to the power supply, a voltage drop of V = 4 V is measured across th...
In cylindrical coordinates (s, 𝜑, z), which of the following is a Hermitian operator?...
A particle of mass m in the x-y plane is confined in an infinite two-dimensional well with vertices at (0, 0), (0, L), (L, L), (L, 0). The eigenfunctions of this particle are 𝜓0,0...
Consider a two dimensional Cartesian coordinate system in which a rank 2 contravariant tensor is represented by the matrix (0 1 1 0). The coordinate system is rotated anticlockwise...
A 4 × 4 matrix M has the property M† = -M and M4 = 𝟏, where 𝟏 is the 4 × 4 identity matrix. Which one of the following is the CORRECT set of eigenvalues of the matrix M?...
Consider the vector field V⃗ consisting of the velocities of points on a thin horizontal disc of radius R = 2 m, moving anticlockwise with uniform angular speed \omega = 2 rad/sec ...
A\alpha and B\beta (\alpha, \beta = 1,2,3, ⋯ , n) are contravariant and covariant vectors, respectively. By convention, any repeated indices are summed over. Which of the following...
Which of the following option(s) is/are correct for the ground state of a hydrogen atom?...
The figure shows a system of two equal masses m and three massless horizontal springs with spring constants k1 , k2 , k1. Ignore gravity. The masses can move only in the horizontal...
A two -level quantum system has energy eigenvalues E1 and E2. A perturbing potential H' = 𝜆Δ𝜎x is introduced, where Δ is a constant having dimensions of energy, 𝜆 is a small dim...
A system of three non -identical spin 1 2 particles has the Hamiltonian H = A \hbar2 ( S 1 + S 2) ⋅ S 3, where S 1, S 2 and S 3 are the spin operators of particles labelled 1, 2 an...
In coordinates (t , x), a contravariant second rank tensor A has non-zero diagonal components Att = P and Axx = Q, with all other components vanishing, and P, Q being real constant...
Consider the integral I = 1 2𝜋i ∮ z4 - 1 (z - a b) (z - b a) dz, where z is a complex variable and a, b are positive real numbers. The i ntegral is taken over a unit circle with c...
A linear magnetic material in the form of a cylinder of radius R and length L is placed with its axis parallel to the z axis. The cylinder has uniform magnetization Mk̂. Which of t...