If a = cos α + i sin α, b = -cos β + i sin β then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________

If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

The area of the parallelogram having diagonals \(\overset { \rightarrow }{ a } =\overset { \wedge }{ 3i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ b } =\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) is _______________

The unit normal vectors to the plane 2x - y + 2z = 5 are ________________

If t₁ and t₂ are the extremities of any focal chord of y² = 4ax then t₁t₂ is ______________

If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is _____________

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

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ 3k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) then \(\overset { \rightarrow }{ a } +\left( -\overset { \rightarrow }{ b } \right) \) will be perpendiculur to \(\overset { \rightarrow }{ c } \) only when t = _________________

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

The area of the circle (x - 2)² + (y - k)² = 25 is _________

The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

Equation of tangent at (-4, -4) on x² = -4y is _____________

If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________ 

If A = [2 0 1] then the rank of AA^T is ______

In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______

If x² - hx - 21 = 0 and x² - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

For real x, the equation \(\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| } \) has ________

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 ________

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

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

The equation of the normal to the circle x² + y² − 2x − 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is

If x³+12x²+10ax+1999 definitely has a positive zero, if and only if

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is