List of top Mathematics Questions

If \( \omega = \frac{-1 + \sqrt{3}i}{2} \), then \( (3 + \omega + 3\omega^2) \) is:

If \[ A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix}, \quad B = \begin{bmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{bmatrix}, \] then \( AB \) is equal to:

The position vector of A and B are: \[ 2\hat{i} + 2\hat{j} + \hat{k} \quad \text{and} \quad 2\hat{i} + 4\hat{j} + 4\hat{k} \] The length of the internal bisector of \( \triangle AOB \) is:

If the coordinates at one end of a diameter of the circle \( x^2 + y^2 - 8x - 4y + c = 0 \) are \( (-3, -2) \), then the coordinates at the other end are:

If \( \tan \theta = \sqrt{n} \) for some non-square natural number \( n \), then sec \( 2\theta \) is:

If \( z = x + iy \), \( z^{1/3} = a - ib \), then \( \frac{x}{a} = \frac{y}{b} = k \) is equal to:

The system of linear equations: \[ x + y + z = 0, \quad 2x + y - z = 0, \quad 3x + 2y = 0 \] \text{has:}

If \((\displaystyle\int_0^axdx)\le(a+4)\), then

In order to solve the differential equation $x \cos x \frac{d y}{d x}+y(x \sin x+\cos x)=1$ the integrating factor is:

Equation of two straight lines are $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ .Then

A bag contains $2n$ coins out of which $n-1$ are unfair with heads on both sides and the remaining are fair. One coin is picked from the bag at random and tossed. If the probability that head falls in the toss is $\frac{41}{56}$, then the number of unfair coins in the bag is

The equation of the curve passing through the point $\left(a, -\frac{1}{a}\right)$ and satisfying the differential equation $y-x \frac{dy}{dx}=a\left(y^{2}+\frac{dy}{dx}\right)$ is

Consider $\frac{x}{2}+\frac{y}{4} \ge1,$ and $\frac{x}{3}+\frac{y}{4} \le1, x, y \ge0.$ Then number of possible solutions are :

If $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are in A.P. where $a_{i}>0$ for all $i$, then $\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots .+$ $\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}$ is?

The locus of the mid-point of a chord of the circle $x^2+ y^2 = 4$, which subtends a right angle at the origin is

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

If \( y(x) = 2e^{2x} + e^{\beta x}, \beta \neq 2 \), is a solution of the differential equation \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0, \] satisfying \( \frac{dy}{dx} (0) = 5 \), then \( y(0) \) is equal to

Let \( M \) and \( N \) be any two \( 4 \times 4 \) matrices with integer entries satisfying

\[ MN = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} = 2 \]

Then the maximum value of \( \det(M) + \det(N) \) is ...............

Let \( M \) be a \( 3 \times 3 \) matrix with real entries such that

\[ M^2 = M + 2I, \quad \text{where} \quad I \text{ denotes the } 3 \times 3 \text{ identity matrix.} \]

If \( \alpha, \beta, \gamma \) are eigenvalues of \( M \) such that \( \alpha\beta\gamma = -4, \) then \( \alpha + \beta + \gamma \) is equal to .............

If

\[ \begin{pmatrix} z \\ y \end{pmatrix} \]

is an eigenvector corresponding to a real eigenvalue of the matrix

\[ \begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & -4 \\ 0 & 1 & 3 \end{pmatrix} \]

then \( z - y \) is equal to .................

Let

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & x^2 - y^2 \neq 0 \\ 0, & x^2 - y^2 = 0 \end{cases} \]

Then the directional derivative of \( f \) at \( (0, 0) \) in the direction of \( \frac{4}{5} \hat{i} + \frac{3}{5} \hat{j} \) is ............