If [x] is the greatest integer $$\le$$ x, then {\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx is equal to :

Let f : R $$\to$$ R be a continuous function. Then \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}} is equal to :

Let {J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A=[a_{ij}{]}_{3\times 3}$$ where {a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.. Then $$\left|adj{A}^{-1}\right|$$ is :

The function f(x), that satisfies the condition $$f(x)=x+\underset{0}{\overset{\pi /2}{\int }}\sin x.\cos yf(y)dy$$, is :

Let the lines (2 $$-$$ i)z = (2 + i)$$\overline{z}$$ and (2 $$+$$ i)z + (i $$-$$ 2)$$\overline{z}$$ $$-$$ 4i = 0, (here i2 = $$-$$1) be normal to a circle C. If the line iz + $$\overline{z}$$ + 1 + i = 0 is tangent to this circle C, then its radius is :

If $$\alpha$$, $$\beta$$ $$\in$$ R are such that 1 $$-$$ 2i (here i2 = $$-$$1) is a root of z2 + $$\alpha$$z + $$\beta$$ = 0, then ($$\alpha$$ $$-$$ $$\beta$$) is equal to :

Let a complex number z, |z| $$\ne$$ 1, satisfy {\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2. Then, the largest value of |z| is equal to ____________.

The least value of |z| where z is complex number which satisfies the inequality \exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1}, is equal to :

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

Let S1, S2 and S3 be three sets defined as S1 = {z$$\in$$C : |z $$-$$ 1| $$\le$$ $$\sqrt{2}$$} S2 = {z$$\in$$C : Re((1 $$-$$ i)z) $$\ge$$ 1} S3 = {z$$\in$$C : Im(z) $$\le$$ 1} Then the set S1 $$\cap$$ S2 $$\cap$$ S3 :

If the equation $$a|z{|}^2+\overline{\overline{\alpha }z+\alpha \overline{z}}+d=0$$ represents a circle where a, d are real constants then which of the following condition is correct?

Let a complex number be w = 1 $$-$$ $$\sqrt{3}$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$ arg(w) = {\pi \over 2}. Then the area of the triangle with vertices origin, z and w is equal to :

If z and $$\omega$$ are two complex numbers such that $$\left|z\omega \right|=1$$ and \arg (z) - \arg (\omega ) = {{3\pi } \over 2}, then \arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right) is : (Here arg(z) denotes the principal argument of complex number z)

Let n denote the number of solutions of the equation z2 + 3$$\overline{z}$$ = 0, where z is a complex number. Then the value of \sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}} is equal to :

Let C be the set of all complex numbers. Let S1 = {z$$\in$$C : |z $$-$$ 2| $$\le$$ 1} and S2 = {z$$\in$$C : z(1 + i) + $$\overline{z}$$(1 $$-$$ i) $$\ge$$ 4}. Then, the maximum value of {\left| {z - {5 \over 2}} \right|^2} for z$$\in$$S1 $$\cap$$ S2 is equal to :

Let C be the set of all complex numbers. Let $$S_1=\{z\in C||z-3-2i{|}^2=8\}$$ $$S_2=\{z\in C|\mathrm{Re}(z)\ge 5\}$$ and $$S_3=\{z\in C||z-\overline{z}|\ge 8\}$$. Then the number of elements in $$S_1\cap S_2\cap S_3$$ is equal to :

The equation \arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4} represents a circle with :

If $${\left(\sqrt{3}+i\right)}^{100}=2^{99}(p+iq)$$, then p and q are roots of the equation :

If S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}, then :

If z is a complex number such that {{z - i} \over {z - 1}} is purely imaginary, then the minimum value of | z $$-$$ (3 + 3i) | is :