The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________

If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are three non - coplanar vectors, then \(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } +\frac { \overset { \rightarrow }{ b } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } } \) = _____________

The number of normals that can be drawn from a point to the parabola y² = 4ax is __________

The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is ________

The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is _____________

The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is ___________

If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________

The equation of tangent at (1, 2) to the circle x² + y² = 5 is __________

The value of sin 2(tan^-1 0.75) is ___________

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

If A is a non-singular matrix then IA^-1| = ______

If \((2+\sqrt{3})^{x^{2}-2 x+1}+(2-\sqrt{3})^{x^{2}-2 x-1}=\frac{2}{2-\sqrt{3}}\) then x = _________

If f(x) = 0 has n roots, then f'(x) = 0 has __________ roots

If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is ________

Let A be a 3 \(\times\) 3 matrix and B its adjoint matrix If |B| = 64, then |A| = ___________

If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

Area of the greatest rectangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse is

The centre of the circle inscribed in a square formed by the lines x² − 8x − 12 = 0 and y² − 14y + 45 = 0 is

If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is

If cot^-1 2 and cot^-1 3 are two angles of a triangle, then the third angle is

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to 

The polynomial x³ + 2x + 3 has

The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is