List of top Mathematics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

Consider a volume V enclosed by a closed surface S having unit surface normal \(\hat{n}\). For \(\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}\), the value of the surface integral \(\frac{1}{9} \oint_{S} \mathbf{r} \cdot \hat{n} \,dS\) is
If \(\left(\frac{1-i}{1+i}\right)^{n/2} = -1\), where \(i = \sqrt{-1}\), one possible value of n is
If the system of linear equations
x + my + az = 0
2x + ay + mz = 0
ax + 2y - z = 0
with m and a as non-zero constants, admits a non-trivial solution, then which one of the following conditions is correct?
In Cartesian coordinates, consider the functions \(u(x, y) = \frac{1{2}(x^2 - y^2)\) and \(v(x,y) = xy\). If \((r, \theta)\) are the polar coordinates, the Jacobian determinant \(|\frac{\partial(u,v)}{\partial(r,\theta)}|\) is}
Given a function \(f(x, y) = \frac{x}{a}e^y + \frac{y}{b}e^x\), where \(x = at\) and \(y = bt\) (a and b are non-zero constants), the value of \(\frac{df}{dt}\) at \(t = 0\) is
Two point-particles having masses \(m_1\) and \(m_2\) approach each other in perpendicular directions with speeds \(v_1\) and \(v_2\), respectively, as shown in the figure below. After an elastic collision, they move away from each other in perpendicular directions with speeds \(v'_1\) and \(v'_2\), respectively.
The ratio \(\frac{v'_1}{v_1}\) is
Which one of the following figures represents the vector field \(\mathbf{A} = y\hat{i}\)?
(\(\hat{i}\) is the unit vector along the x-direction)
One of the roots of the equation, \( z^6 - 3z^4 - 16 = 0 \) is given by \( z_1 = 2 \). The value of the product of the other five roots is ...........
The radial component of acceleration in plane polar coordinates is given by:
In the Fourier series expansion of two functions \( f_1(t) = 4t^2 + 3 \) and \( f_2(t) = 6t^3 + 7t \) in the interval \( -\frac{T}{2} \) to \( +\frac{T}{2} \), the Fourier coefficients \( a_n \) and \( b_n \) (\( a_n \) and \( b_n \) are coefficients of \( \cos(n\omega t) \) and \( \sin(n\omega t) \), respectively) satisfy:
The solution \( y(x) \) of the differential equation \( y\frac{dy}{dx} + 3x = 0 \), \( y(1) = 0 \), is described by:
Let \( M \) be a \( 2 \times 2 \) matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are:
The function \( e^{\cos x} \) is Taylor expanded about \( x = 0 \). The coefficient of \( x^2 \) is:
If a function \(y(x)\) is described by the initial-value problem, \(\dfrac{d^2y}{dx^2} + 5\dfrac{dy}{dx} + 6y = 0\), with initial conditions \(y(0) = 2\) and \(\dfrac{dy}{dx}\bigg|_{x=0} = 0\), then the value of \(y\) at \(x = 1\) is ................. . (Round off to 2 decimal places)
Which one of the following statements is correct?
Given, \(\displaystyle \binom{n}{m} = \frac{n!}{m!(n-m)!}\) is the binomial coefficient.
Which one of the following functions has a discontinuity in the second derivative at \(x = 0\), where \(x\) is a real variable?
If \(P\) and \(Q\) are Hermitian matrices, which of the following is/are true?
(A matrix \(P\) is Hermitian if \(P = P^{\dagger}\), where the elements \(p_{ij}^{\dagger} = p_{ji}^{*}\))
Evaluate: \(\displaystyle \lim_{x \to 0^+} x^x\)
The eigenvalues of

\[ \begin{pmatrix} 3 & i & 0 \\ -i & 3 & 0 \\ 0 & 0 & 6 \end{pmatrix} \]

are
The gradient of a scalar field \( S(x, y, z) \) has the following characteristic(s).
If \( \phi(x,y,z) \) is a scalar function which satisfies the Laplace equation, then the gradient of \( \phi \) is

Which one of the following curves correctly represents (schematically) the solution for the equation $\dfrac{df}{dx}+2f=3;\; f(0)=0$ ?

Let matrix $M=\begin{pmatrix} 4 & x \\ 6 & 9\end{pmatrix}$. If $\det(M)=0$, then

The coefficient of $x^3$ in the Taylor expansion of $\sin(\sin x)$ around $x = 0$ is .............. (Specify your answer up to two digits after the decimal point.)
Let $f(x)=3x^6 - 2x^2 - 8$. Which of the following statements is (are) true?