List of top Mathematics Questions

In \( \triangle ABC \), \( (\cot A + \cot B)(\cot B + \cot C)(\cot C + \cot A) = \)
The product of the slopes of the common tangents drawn to the circles $x^2+y^2+2x-2y-2=0$ and $x^2+y^2-2x+2y+1=0$ is
The area bounded by the curve \( x = \log(|y|) \), the lines \( x = -1 \) and \( x = 0 \) is:
The number of elements in the set $ S = \{x \in \mathbb{Z} : x^2 - 7x + 6 \le 0 \text{ and } x^2 - 3x>0 \} $ is
If \( A + 2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix} \) and \( 2A - B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix} \), then \( \text{Tr}[A] - \text{Tr}[B] \) equals:
If \( \frac{x^4 - 6x^3 + 9x^2 + 5x - 20}{x^2 - x - 2} = f(x) + \frac{a}{x + p} + \frac{b}{x + q} \), then \( 2(a + b) = \)
Evaluate the following expression: \[ \left[\sqrt{2} \left( \cos 56^\circ 15' + i \sin 56^\circ 15' \right)\right]^8 \]
If \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \), then which one of the following is True?
One out of 9 ships is likely to sink when they are set on sail. When 6 ships are set on sail, the probability that exactly 3 of them will not arrive safely is:
In $ \triangle ABC $, if $ r $ is the inradius and $ r_1, r_2, r_3 $ are the ex-radii, then \[ \frac{1}{4} \left[ b^2 \sin 2C + c^2 \sin 2B \right] = \]

If $f(x) = \frac{1 - x + \sqrt{9x^2 + 10x + 1}}{2x}$, then $\lim_{x \to -1^-} f(x) =$

If \( y = x \log \left( \frac{1}{ax} \right) \), then \( x(1 + x) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - y = \)
From a collection of eight cards numbered 1 to 8, if two cards are drawn at random, one after the other with replacement, then the probability that the product of numbers that appear on the cards is a perfect square is:
If \( \lim_{x \to \infty} y(x) = \frac{\pi}{2} \), then the solution of \( x^3 \sin y \frac{dy}{dx} = 2 \) is \( \cos y = \)
If \( -1 \) is a twice repeated root of the equation \( a(x^3 + x^2) + bx + c = 0 \), then the ratio \( a : b : c \) is:
If \[ \frac{x + 2}{x^2 - 3} \text{ is one of the partial fractions of } \frac{3x^3 - x^2 - 2x + 17}{x^4 + x^2 - 12}, \text{ then the other partial fraction of it is:} \]
The coefficient of \( x^3 \) in the expansion of \( (1 - x)^{\frac{3}{2}} \), where \( |x|<1 \), is:
The set \( \{ x \in \mathbb{R} : 4 + 11x - 3x^2>0 \} \) is the interval:
If \( A = \frac{\pi}{24} \), then \[ \frac{\cos A + \cos 3A + \cos 5A + \cos 7A}{\sin A + \sin 3A + \sin 5A + \sin 7A} \]
The equation of a tangent to the circle \(x^2 + y^2 + 2x - 12y - 132 = 0\) which is perpendicular to the line \(12x + 5y + k = 0\) is:
Assertion (A): \(\displaystyle \int_0^{\frac{\pi}{2}} (\sin^6 x + \cos^6 x)\, dx\) lies in the interval \(\left(\frac{\pi}{8}, \frac{\pi}{2}\right)\) Reason (R): \(\sin^6 x + \cos^6 x\) is a periodic function with period \(\dfrac{\pi}{2}\)
If the extremities of a diagonal of a square are \( (1, -2, 3) \) and \( (2, -3, 5) \), then the length of its side is:
If \( \int_0^{2024\pi} \frac{2023^{\sin^2 x}}{2023^{\sin^2 x} + 2023^{\cos^2 x}} dx = k \), then \( \left( \frac{2k}{\pi} + 1 \right) = \)
\( \lim_{x \to -9} \frac{(2.5)^{81 - x^2} - (0.4)^{x + 9}}{x + 9} = \)
In $ \triangle ABC $, if $ r = 1 $, $ R = 4 $, and $ \Delta = 8 $, then \[ \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \]