List of top Mathematics Questions

Choose the most appropriate option. Solve for \(x>0\)
\[ \log_3\left( \frac{3}{x} \right) + \log_3 x = 1 \]

For \(x>1\), how many roots/solutions of the following equation exist: \[ \log_2\left( \frac{2}{x} \right)\log^2 x + \log^2 x = 1 \]

For any two vectors \( \vec{u} \) and \( \vec{v} \), if \( |\vec{u} + \vec{v}| = |\vec{u} - \vec{v}| \) then the angle between them is equal to

Angles A, B, and C of a triangle \(\Delta ABC\) are in AP and \( b:c = \sqrt{3} : \sqrt{2} \), then angle \( \angle A \) is given by

At what point of the curve \( y^2 = 2x^3 \) is the tangent line perpendicular to the straight line \[ 4x - 3y + 2 = 0? \]

Find the real solution of the system of equations: \[ x^4 + y^4 - x^2 y^2 = 13,\quad x^2 - y^2 + 2xy = 1 \] Satisfying condition: \( xy \geq 0 \)

Find the derivative of \[ y = (1 - x)^m (1 + x)^n \text{ at } x = 0, \text{ where } m, n>0 \]

The straight line \( r = (i - j + k) + \lambda (2i + j - k) \) and the plane \( r \cdot (2i + j - k) = 4 \) are

Let \( a = \cos \theta_1 + i \sin \theta_1 \), \( b = \cos \theta_2 + i \sin \theta_2 \), \( c = \cos \theta_3 + i \sin \theta_3 \) and \( a + b + c = 0 \), then \[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = ? \]

A real solution of the equation \[ \cosh x - 5 \sinh x - 5 = 0 \text{ is} \]

If the expansion of \(\left(x^2 + \frac{2}{x} \right)^n\) has a term independent of \(x\), then \(n\) is

On the ellipse \(9x^2 + 25y^2 = 225\), find the point, the distance from which to the focus \(F_2\) is four times the distance to the focus \(F_1\).

\[ \int_{-1}^1 \frac{x \sin^{-1} x}{\sqrt{1 - x^2}} dx \text{ is equal to} \]

Tangents are drawn from the origin to the curve \(y = \sin x\). Then, the point of contact lie on the curve:

\[ \int \frac{x^2}{(x \sin x + \cos x)^2} dx \text{ is equal to} \]

If \(y^{\frac{1}{m}} + x^{\frac{1}{m}} = 2x\) then

\[ \int_0^1 x e^{2x} dx \text{ is equal to} \]

If \(P(x)\) is a polynomial such that \[ P(x^2 + 1) = \{P(x)\}^2 + 1 \] then \(P'(0)\) is equal to

\[ \int_{-\pi/2}^{\pi/2} |\sin x| dx \text{ equals to} \]

What is the value of \(\tan \left(\frac{\pi}{12}\right)?\)

\[ \lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1 - x}}{x} \]

Choose the most appropriate options.
The period of the function \(f(x) = |\sin x| - |\cos x|\) is

Choose the most appropriate options.
The value of \({}^{40}C_0 + {}^{40}C_1 + {}^{40}C_2 + \ldots + {}^{40}C_{20}\) is

A die is thrown twice and the sum of the numbers appearing is 6. Then, the conditional probability that the number 4 has appeared at least once is

There are 3 true coins and 1 false coin with 'head' on both sides. A coin is chosen at random and tossed 4 times. If 'head' occurs all the 4 times, then the probability that the false coin has been chosen and used is