List of top Mathematics Questions

If \( ^nC_{r-1} = 36 \), \( ^nC_r = 84 \), and \( ^nC_{r+1} = 126 \), then the value of \( nr^2 \) is:
Let \( f(x) = x^2 + \frac{1}{x^2} \) and \( g(x) = x - \frac{1}{x} \) for \( x \in \mathbb{R} \setminus \{-1, 0, 1\} \). Then, the local minimum of \( \frac{f(x)}{g(x)} \) is:
The range of the function \( f(x) = \frac{x^2 + x + 1}{x^2 - x + 1} \) is:
The remainder when the polynomial \( 2x^5 - 3x^4 + 5x^3 - 3x^2 + 7x - 9 \) is divided by \( x^2 - x - 3 \) is:
Which of the following quadratic equations whose real roots \( x_1, x_2 \) satisfy the conditions \( x_1^2 + x_2^2 = 5 \), \( 3(x_1^5 + x_2^5) = 11(x_1^3 + x_2^3) \)?
If \( 1, \omega, \omega^2 \) are the cube roots of unity, then evaluate: \[ (2 - \omega)^2 (2 - \omega^2)^2 (2 - \omega^{10})^2 (2 - \omega^{11})^2 \]
Let \(A = \begin{bmatrix}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{bmatrix}\) and \(B = \begin{bmatrix}0 & 0 & n \\ 0 & n & 0 \\ n & 0 & 0\end{bmatrix}\). Then \(A^2 + B^2 + AB =\)
If the complex numbers \( z_1, z_2, 0 \) are vertices of an equilateral triangle, then \( z_1^2 + z_2^2 = \)
Let \( z = x + iy \) be a complex number with \( x, y \in \mathbb{Z} \). Then the area (in square units) of the rectangle whose vertices are the roots of the equation \( \bar{z} \cdot z^3 + z \cdot \bar{z}^3 = 350 \) is:
The domain of the real-valued function \(f(x) = \frac{\sqrt{\log_{0.5}(x-3)}}{\sqrt{x-1}}\) is:
The maximum value of \( f(x) = \frac{x}{1 + 4x + x^2} \) is
The minimum value of \( f(x) = x + \frac{4}{x + 2} \) is
If \( f(x) = \max \{3 - x, 3 + x, 6\} \) is not differentiable at \( x = a \) and \( x = b \), then \( |a| + |b| = \)
At any point \( (x, y) \) on a curve, if the length of the subnormal is \( (x - 1) \) and the curve passes through \( (1, 2) \), then the curve is a conic. A vertex of the curve is:
The function \( y = Ae^x + Be^{-2x} \) satisfies which of the following differential equations?
Let \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) be roots of \( 18x^2 - 9\pi x + \pi^2 = 0 \), \( f(x) = x^2, g(x) = \cos x \). Then
\[ \int_{\alpha}^{\beta} x (g \circ f(x)) \, dx = ? \]
The parametric form of a curve is \( x = \frac{t^3}{t^2 - 1},\ y = \frac{t}{t^2 - 1} \), then \( \int \frac{dx}{x - 3y} = \)
If \( I_n = \int \tan^n x \, dx \), and \( I_0 + I_1 + 2I_2 + 2I_3 + 2I_4 + I_5 + I_6 = \sum_{k=1}^n \frac{\tan^k x}{k} \), then \( n = \)
\( \int_0^{\pi} x \left( \sin^2(\sin x) + \cos^2(\cos x) \right) dx = \)
If $\displaystyle \lim_{x \to \infty} x^n \log_e x = 0$, then $\log_e 12 =$
The x-intercept of a plane $\pi$ passing through the point $(1, 1, 1)$ is $\dfrac{5}{2}$ and the perpendicular distance from the origin to the plane $\pi$ is $\dfrac{5}{7}$. If the y-intercept of the plane $\pi$ is negative and the z-intercept is positive, then its y-intercept is
If the tangent at the point $P$ on the circle $x^2 + y^2 + 6x + 6y = 2$ meets the straight line $5x - 2y + 6 = 0$ at a point $Q$ on the y-axis, then the length of $PQ$ is
The pole of the line $\dfrac{x}{a} + \dfrac{y}{b} = 1$ with respect to the circle $x^2 + y^2 = c^2$ is
Let origin be the centre, $(\pm 3, 0)$ the foci and $\dfrac{3}{2}$ be the eccentricity of a hyperbola. Then the line $2x - y - 1 = 0$
The locus of a variable point whose chord of contact w.r.t. the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ subtends a right angle at the origin is