Let $$f:\mathbb{R}\to \mathbb{R}$$ be a function defined as $$f(x)=a\sin \left(\frac{\pi [x]}{2}\right)+[2-x],a\in \mathbb{R}$$ where $$[t]$$ is the greatest integer less than or equal to $$t$$. If $$\underset{x\to -1}{\lim }f(x)$$ exists, then the value of $$\underset{0}{\overset{4}{\int }}f(x)dx$$ is equal to

Let $$I={\int }_{\pi /4}^{\pi /3}\left(\frac{8\sin x-\sin 2x}{x}\right)dx$$. Then

Let a function $$f:\mathbb{R}\to \mathbb{R}$$ be defined as : $$f(x)=\{\begin{array}{ll}\underset{0}{\overset{x}{\int }}(5-|t-3|)dt,& x>4\\ x^2+bx& ,x\le 4\end{array}$$ where $$\mathrm{b}\in \mathbb{R}$$. If $$f$$ is continuous at $$x=4$$, then which of the following statements is NOT true?

Let $$f(x)=2+|x|-|x-1|+|x+1|,x\in \mathbf{R}$$. Consider $$(\mathrm{S}1):f^{\prime }\left(-\frac{3}{2}\right)+f^{\prime }\left(-\frac{1}{2}\right)+f^{\prime }\left(\frac{1}{2}\right)+f^{\prime }\left(\frac{3}{2}\right)=2$$ $$(\mathrm{S}2):\underset{-2}{\overset{2}{\int }}f(x)\mathrm{d}x=12$$ Then,

$$\underset{0}{\overset{2}{\int }}\left(\left|2x^2-3x\right|+\left[x-\frac{1}{2}\right]\right)\mathrm{d}x$$, where [t] is the greatest integer function, is equal to :

The minimum value of the twice differentiable function $$f(x)=\underset{0}{\overset{x}{\int }}{\mathrm{e}}^{x-\mathrm{t}}{f}^{\prime }(\mathrm{t})\mathrm{dt}-\left(x^2-x+1\right){\mathrm{e}}^x$$, $$x\in \mathbf{R}$$, is :

Let $$I_n(x)={\int }_0^x\frac{1}{{\left(t^2+5\right)}^n}dt,n=1,2,3,\dots .$$ Then :

The integral $$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{1}{3+2\sin x+\cos x}\mathrm{d}x$$ is equal to :

If $$f(\alpha )=\underset{1}{\overset{\alpha }{\int }}\frac{{\log }_{10}\mathrm{t}}{1+\mathrm{t}}\mathrm{dt},\alpha >0$$, then $$f\left({\mathrm{e}}^3\right)+f\left({\mathrm{e}}^{-3}\right)$$ is equal to :

If $$[t]$$ denotes the greatest integer $$\le t$$, then the value of $${\int }_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]\mathrm{d}x$$ is :

Let $$\alpha$$ and $$\beta$$ be the roots of the equation x2 + (2i $$-$$ 1) = 0. Then, the value of |$$\alpha$$8 + $$\beta$$8| is equal to :

Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z $$-$$ 1) $$-$$ arg(z + 1) = {\pi \over 4} intersect :

The number of points of intersection of $$|z-(4+3i)|=2$$ and $$|z|+|z-4|=6$$, z $$\in$$ C, is :

The area of the polygon, whose vertices are the non-real roots of the equation $$\overline{z}=i{z}^2$$ is :

Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right|

Let z1 and z2 be two complex numbers such that $${\overline{z}}_1=i{\overline{z}}_2$$ and \arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi. Then :

Let a circle C in complex plane pass through the points $$z_1=3+4i$$, $$z_2=4+3i$$ and $$z_3=5i$$. If $$z(\ne z_1)$$ is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then $$arg(z)$$ is equal to :

Let $$A=\{z\in C:1\le |z-(1+i)|\le 2\}$$ and $$B=\{z\in A:|z-(1-i)|=1\}$$. Then, B :

The real part of the complex number {{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }} is equal to :