The differential equation associated with the family of concentric circles having their centres at the origin is _________.

The I.F of \(\frac{dy}{dx}-y\) tan x = cos x is _________

The solution of \(\frac{dy}{dx}+y\) cot x = sin 2x is ___________

In (N, *), x * y = max(x, y), x, y \(\in \) N then 7 * (-7)

Define * on Z by a * b = a + b + 1 ∀ a,b \(\in \) Z. Then the identity element of z is ________

The value of \(\int _{ 0 }^{ \frac { \pi }{ 3 } } { tan } x \ dx\) __________

\(\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ 2a }{ f(x) } dx\) if _________

The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is __________

If u = y^x then \(\frac { \partial u }{ \partial y } \) = ............

If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

If a particle moves in a straight line according to s = t³-6t²-15t, the time interval during which the velocity is negative and acceleration is positive is __________

If x = cos θ + i sin θ, then xⁿ + \(\frac { 1 }{ { x }^{ n } } \) is ______

If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if _________

If, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________

Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\) and \(\overset { \rightarrow }{ c } \) be three vectors having magnitudes 1, 1, 2 respectively. If \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } =0\) then the acute angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) is ___________ 

Th point of curve y = 2x² - 6x - 4 at which the targent is parallel to x - axis is __________

For what value of \(\left( \overset { \rightarrow }{ a } \right) \) will the straight lines \(\frac { x+2 }{ a } =\frac { y }{ 3 } =\frac { z-1 }{ 4 } \)and \(\frac { x-3 }{ a } =\frac { y-1 }{ 4 } =\frac { z-7 }{ a } \)be perpendicular?

If θ is the angle between the vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow}{b} \), then sin θ is ___________ 

The equation 7x²- 6\(\sqrt { 3 } \) xy + 13y² - 4\(\sqrt { 3 } \) x - 4y - 12 = 0 represents ____________

In a \(\Delta ABC\)  if C is a right angle, then  \({ tan }^{ -1 }\left( \frac { a }{ b+c } \right) +{ tan }^{ -1 }\left( \frac { b }{ c+a } \right) =\) ________

If ∝, β, ૪ are the roots of 9x³-7x+6 = 0, then ∝ β ૪ is __________

Which of the following is not an elementary transformation?