List of top Mathematics Questions

\[ \lim_{n\to\infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{4}{n^2}\right)\left(1+\frac{9}{n^2}\right)\cdots\left(1+\frac{n^2}{n^2}\right)\right]^{\frac{1}{n}} = \]
Evaluate \( \int_{-2}^{2} x^4 (4 - x^2)^{\frac{7}{2}} \, dx \):
Area of the region enclosed between the curves \(y^{2}=4(x+7)\) and \(y^{2}=5(2-x)\) is

If the interval in which the real-valued function \[ f(x) = \log\left(\frac{1+x}{1-x}\right) - 2x - \frac{x^{3}}{1-x^{2}} \] is decreasing in \( (a,b) \), where \( |b-a| \) is maximum, then {a}⁄{b} =

If \( M_1 \) is the mean deviation from the mean of the discrete data \( 44, 5, 27, 20, 8, 54, 9, 14, 35 \) and \( M_2 \) is the mean deviation from the median of the same data, then \( M_1 - M_2 = \):
If \( \lim_{x \to 4} \frac{2x^2 + (3+2a)x + 3a{x^3 - 2x^2 - 23x + 60} = \frac{11}{9} \), then find \( \lim_{x \to a} \frac{x^2 + 9x + 20}{x^2 - x - 20} \).}
If \( a, b, c \) are the intercepts made on X, Y, Z-axes respectively by the plane passing through the points \( (1, 0, -2) \), \( (3, -1, 2) \), and \( (0, -3, 4) \), then \( 3a + 4b + 7c = \)?
The system of equations \( x + 3y + 7 = 0 \), \( 3x + 10y - 3z + 18 = 0 \), and \( 3y - 9z + 2 = 0 \) has:
If \( y (\cos x)^{\sin x} = (\sin x)^{\sin x} \), then the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:
The slope of the tangent drawn from the point \( (1,1) \) to the hyperbola \( 2x^2 - y^2 = 4 \) is:
If two dice are thrown, then the probability of getting co-prime numbers on the dice is:

. If a random variable X has the following probability distribution, then the mean of X is:

X = xiP(X = xi)
12k²
2k
3k²

 

If \(6x-5y-20=0\) is a normal to the ellipse \(x^2 + 3y^2 = k\), then \(k =\)
If \( \mathbf{a} = 2\overline{i} - \overline{j}, \overline{b} = 2\overline{j} - \overline{k}, \overline{c} = 2\overline{k} - \overline{i} \) are three vectors and \( \overline{d} \) is a unit vector perpendicular to \( \overline{c} \), If \( \overline{a}, \overline{b}, \overline{d} \) are coplanar vectors, then \( |\overline{d} \cdot \overline{b}| = \):
Define \( f: \mathbb{R} \to \mathbb{R} \) by \[ f(x) = \begin{cases} \frac{1 - \cos 4x}{x^2}, & x < 0 \\ a, & x = 0 \\ \frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x > 0 \end{cases} \] Find the value of \( a \) such that \( f \) is continuous at \( x = 0 \).
The vertices of triangle \( \Delta ABC \) are \( A(2, 3, k) \), \( B(-1, k, -1) \), and \( C(4, -3, 2) \). If \( AB = AC \) and \( k>0 \), then the triangle \( ABC \) is:
The domain of the real valued function: \[ f(x) \;=\; \sqrt[3]{\frac{x \,-\, 2}{2x^2 \,-\, 7x \,+\,5}} \;+\; \log\bigl(x^2 \,-\, x \,-\, 2\bigr). \]
The general solution of the differential equation \[ \frac{dy}{dx} + \frac{\sin(2x + y)}{\cos x} + 2 = 0 \] is:
Evaluate the integral: \[ \int 4\cos^2 x - 5\sin^2 x \cos x \, dx. \]
Evaluate the integral: \[ \int \frac{4\tan^4 x + 3 \tan^2 x - 1}{\tan^2 x + 4} \, dx. \]
Evaluate the integral: \[ \int \left( \sin^3 x + \cos^2 x \right)^2 \, dx \]
Evaluate the integral: \[ I = \int_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \frac{\sin^4(4x)}{1 + e^{4x}} \, dx \]
The area of the region enclosed by the curves \( y^2 = 4(x+1) \) and \( y^2 = 5(x-4) \) is:

If the function \( f(x) \) is given by \[ f(x) = \begin{cases} \frac{\tan(a(x-1))}{\frac{x-1}{x}}, & tif0<x<1 
\frac{x^3-125}{x^2 - 25} , & \text{if } 1 \leq x \leq 4 
\frac{b^x - 1}{x}, & \text{if } x>4 \end{cases} \] is continuous in its domain, then find \( 6a + 9b^4 \).

If the function \( f(x) = x^3 + ax^2 + bx + 40 \) satisfies the conditions of Rolle's theorem on the interval \( [-5, 4] \) and \( -5, 4 \) are two roots of the equation \( f(x) = 0 \), then one of the values of \( c \) as stated in that theorem is: