List of top Mathematics Questions

Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto

Find the intervals in which the function
\( f(x) = x^4 - 4x^3 + 4x^2 + 15 \) is increasing .

If \( |\vec{a}| = 7 \), \( |\vec{b}| = 24 \), \( |\vec{c}| = 25 \) and
\( \vec{a} + \vec{b} + \vec{c} = \vec{0} \),
find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \).

If a line makes angles \( \alpha, \beta, \gamma \) with the x-axis, y-axis, and z-axis respectively,
then prove that: \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]

Evaluate: \[ I = \int_{0}^{\pi/2} \frac{ x \sin x \cos x }{ \sin^4 x + \cos^4 x } \, dx. \]

The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective function $Z = 2ax + by$ occurs at the points A(250, 0) and B(200, 50), then the relation between a and b is:

Assertion (A): The principal value of $\cot^{-1}(\sqrt{3})$ is $\frac{\pi}{6}$.
Reason (R): Domain of $\cot^{-1} x$ is $\mathbb{R} - \{-1, 1\}$.

If $x = a \cos \theta + b \sin \theta$, $y = a \sin \theta - b \cos \theta$, then which one is true?

It is given that \[ X \begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}. \text{ Then matrix X is:} \]

The point which lies in the half-plane $2x + y - 4 \le 0$ is:

The difference of the order and the degree of the differential equation \[ \Big( \frac{d^2 y}{dx^2} \Big)^2 + \Big( \frac{dy}{dx} \Big)^3 + x^4 = 0 \text{ is:} \]

If $(\cos x)^y = (\cos y)^x$, then $\frac{dy}{dx}$ is:

The vector equation of a line which passes through the point (2, -4, 5) and is parallel to the line $\displaystyle \frac{x+3}{3} = \frac{4-y}{2} = \frac{z+8}{6}$ is:

For which value of x, are the determinants \[ \begin{vmatrix} 2x & -3 \\ 5 & x \end{vmatrix} \text{and} \begin{vmatrix} 10 & 1 \\ -3 & 2 \end{vmatrix} \text{ equal?} \]

The value of the cofactor of the element of second row and third column in the matrix \[ \begin{bmatrix} 4 & 3 & 2 \\ 2 & -1 & 0 \\ 1 & 2 & 3 \end{bmatrix} \text{ is:} \]

The value of $\displaystyle \int_{0}^{\pi/6} \sin 3x \, dx$ is:

Let the digits a, b, c be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let y = y(x) be a solution of the differential equation (x cos x)dy +(xy sin x + y cos x – l)dx = 0, 0 < x < \(\frac{π}{2}\). If\(\frac{π}{3}y(\frac{π}{3})=\sqrt3\), then \(| \frac{π}{6 }y“( \frac{π}{6})+2 y'( \frac{π}{6})| \) is equal to______.

The number \(0.4\overline{23}\) converted to a fraction, then it is equal to:

Let tan^-1 x ∈(\(-\frac π2\) \(\frac π2\))  for x ∈ R. Then the number of real solutions of the equation \(\sqrt{1 + \cos (2x)} = \sqrt2 \tan^{-1}(\tan x)\) in the set \((-\frac{3π}{2}, -\frac{π}{2}) ∪ (-\frac{π}{2},\frac{π}{2}) ∪ (\frac{π}{2},\frac{3π}{2})\) is equal to