In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

If a * b=\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) on the real numbers then * is

If \(\int _{ 0 }^{ a }{ \frac { 1 }{ 4+{ x }^{ 2 } } dx=\frac { \pi }{ 8 } } \) then a is

If v (x, y) = log (e^x + e^y), then \(\frac { { \partial }v }{ \partial x } +\frac { \partial v }{ \partial y } \) is equal to

If \(u(x, y)=e^{x^{2}+y^{2}}\),then \(\frac { \partial u }{ \partial x } \) is equal to

A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
\(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0< x < l \\ 0 & l <x<2l \end{array}\right.\)
The mean and variance of the shorter of the two pieces are respectively.

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² -2t- 8. The time at which the particle is at rest is

If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________

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

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let  \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________

The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________

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

The quadratic equation whose roots are ∝ and β is ___________

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| = _______

If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is

If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then

The number of real numbers in [0, 2π] satisfying sin⁴x - 2sin²x + 1 is

According to the rational root theorem, which number is not possible rational zero of 4x⁷ + 2x⁴ - 10x³ - 5?