List of top Questions asked in MET

Evaluate \( \int_{0}^{\frac{3\pi}{2}} \sin\left( \left\lfloor \frac{2x}{\pi} \right\rfloor \right) dx \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.
The area bounded by \( y = x|\sin x| \) and X-axis between \( x = 0 \) and \( x = 2\pi \) is
\( \int \frac{dx}{\sin x - \cos x + \sqrt{2}} \) is equal to
The area enclosed by the curves \( |y + x| \le 1 \), \( |y - x| \le 1 \) and \( 2x^2 + 2y^2 = 1 \) is
Evaluate \( \int_{1/n}^{(n-1)/n} \frac{\sqrt{x}}{\sqrt{a-x} + \sqrt{x}} \, dx \).
The largest term in the sequence \( a_n = \frac{n^2}{n^3 + 200} \) is given by
The equation of the tangent to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at $(a, b)$ is
The minimum radius vector of the curve \( \frac{a^2}{x^2} + \frac{b^2}{y^2} = 1 \) is of length
\( f(x)= \begin{cases} |x^3 + x^2 + 3x + \sin x|\left(3 + \sin\frac{1}{x}\right), & x\neq 0 \\ 0, & x=0 \end{cases} \) The number of points, where \( f(x) \) attains its minimum value, is
\( \int \frac{x^2 - 1}{(x^4 + 3x^2 + 1)\tan^{-1}\left(x + \frac{1}{x}\right)} \, dx \) is equal to
\( \int \frac{\cos x}{\sin^2 x (\sin x + \cos x)} \, dx \) is equal to
The triangle formed by the tangent to the curve $f(x)=x^2 + bx - b$ at the point $(1,1)$ and the coordinate axes lies in the first quadrant. If its area is 2, then the value of $b$ is
The function $x^x$ is increasing, when
If \( y = |\sin x|^{|x|} \), then the value of \( \frac{dy}{dx} \) at \( x = -\frac{\pi}{6} \) is
\( \lim_{x \to \frac{\pi}{2}} \frac{(1 - \tan \frac{x}{2})(1 - \sin x)}{(1 + \tan \frac{x}{2})(\pi - 2x)^3} \) is equal to
\( \sum_{m=1}^{n} \tan^{-1}\left(\frac{2m}{m^4 + m^2 + 2}\right) \) is equal to
The length of the longest interval, in which $f(x)=3\sin x - 4\sin^3 x$ is increasing, is
\( f(x)= \begin{cases} \frac{\sin^3(\sqrt{3}) \cdot \log(1+3x)}{(\tan^{-1}\sqrt{x})^2 (e^{5\sqrt{x}}-1)x}, & x\neq 0 \\ a, & x=0 \end{cases} \) is continuous in \( [0,1] \), then \( a \) equals to
If \( \tan^{-1}\left(\frac{a}{x}\right) + \tan^{-1}\left(\frac{b}{x}\right) = \frac{\pi}{2} \), then \( x \) is equal to
\( \cos^{-1}\{\cos 2\cot^{-1}(\sqrt{2}-1)\} \) is equal to
If \( f(x) = \frac{a^x + a^{-x}}{2} \) and \( f(x+y) + f(x-y) = k f(x)f(y) \), then \( k \) is equal to
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = \frac{x}{\sqrt{1+x^2}} \), then \( (f \circ f)(x) \) is
If $a, b, c$ are cube roots of unity, then \[ \begin{vmatrix} e^a & e^{2a} & e^{3a}-1 e^b & e^{2b} & e^{3b}-1 e^c & e^{2c} & e^{3c}-1 \end{vmatrix} \] is equal to
If \( f \circ g = |\sin x| \) and \( g \circ f = \sin^2 \sqrt{x} \), then \( f(x) \) and \( g(x) \) are
The value of the determinant \[ \begin{vmatrix} (a^x + a^{-x})^2 & (a^x - a^{-x})^2 & 1 (b^x + b^{-x})^2 & (b^x - b^{-x})^2 & 1 (c^x + c^{-x})^2 & (c^x - c^{-x})^2 & 1 \end{vmatrix} \] is