List of top Mathematics Questions

The order and degree of the following differential equation are, respectively: \[ \frac{d^4y}{dx^4} + 2 \frac{d^2y}{dx^2} + y^2 = 0. \]

Let $f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5$, $f(1) = 0$. Then, $f(x)$ is:

Let $f(x) = |x|$, $x \in \mathbb{R}$. Then, which of the following statements is incorrect?

If $y = a \cos(\log x) + b \sin(\log x)$, then $x^2y'' + xy'1$ is:

If $\tan^{-1} (x^2 - y^2) = a$, where $a$ is a constant, then $\frac{dy}{dx}$ is:

If \[ f(x) = \begin{cases} \frac{\log(1 + ax) + \log(1 - bx)}{x}, & \text{for } x \neq 0 \\ k, & \text{for } x = 0 \end{cases} \] is continuous at $x = 0$, then the value of $k$ is:

\[ \sec^{-1} \left( -\sqrt{2} \right) - \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \] is equal to:

Sum of two skew-symmetric matrices of the same order is always a/an:

If \[ A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \] then $A$ is a/an:

Let both $AB'$ and $B'A$ be defined for matrices $A$ and $B$. If the order of $A$ is $n \times m$, then the order of $B$ is:

Find the foot of the perpendicular drawn from the point \( (1, 1, 4) \) on the line \( \frac{x+2}{5} = \frac{y+1}{2} = \frac{z-4}{-3} \).

Find \( \frac{dy}{dx} \) if \( x^3 + y^3 + x^2 = a^b \), where \( a \) and \( b \) are constants.

Three students run on a racing track such that their speeds add up to 6 km/h. However, double the speed of the third runner added to the speed of the first results in 7 km/h. If thrice the speed of the first runner is added to the original speeds of the other two, the result is 12 km/h. Using the matrix method, find the original speed of each runner.

In a rough sketch, mark the region bounded by \( y = 1 + |x + 1| \), \( x = -2 \), \( x = 2 \), and \( y = 0 \). Using integration, find the area of the marked region.

Evaluate: \[ \int_1^5 \left( |x-2| + |x-4| \right) \, dx \]

Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]

Solve the differential equation: \[ x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x \]

Differentiate \(\frac{\sin x}{\sqrt{\cos x}}\) with respect to x.