The sum of the roots of the equation, \({x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,\) is

The vector \(\left( {\hat i \times \vec a.\vec b} \right)\hat i + \left( {\hat j \times \vec a.\vec b} \right)\hat j + \left( {\hat k \times \vec a.\vec b} \right)\hat k\) is equal to

Let \(f : N \rightarrow R\) be a function such that \(f(x+y)=2 f(x) f(y)\) for natural numbers \(x\) and \(y\). If \(f(1)=2\), then the value of \(\alpha\) for which \(\sum \limits_{k=1}^{10} f(\alpha+k)=\frac{512}{3}\left(2^{20}-1\right)\) holds, is

Let f: \(\mathrm{R} \rightarrow \mathrm{R}\) be defined as \(f(x) \rightarrow \frac{\lambda\left|x^{2}-5 x+6\right|}{\mu\left(5 x-x^{2}-6\right)}, x<2\) \(\quad\quad\quad\quad e^{\frac{\tan (x-2)}{x-[x]}}, \quad x>2\) \(\quad\quad\quad\quad \mu \quad\quad\quad\quad x=2\) Where \([x]\) is the greatest integer less than or equal to \(x\). If \(f\) is continuous at \(x=2\), then \(\lambda+\mu\) is equal to:

If the equation of the parabola, whose vertex is at \((5,4)\) and the directrix is \(3 x+y-29=0\), is \(x^{2}+a y^{2}+b x y+c x+d y+k=0\) then \(a + b + c + d + k\) is equal to

Let the curve \(y = y ( x )\) be the solution of the differential equation, \(\frac{ dy }{ dx }=2( x +1) .\) If the numerical value of area bounded by the curve \(y = y ( x )\) and \(x-\) axis is \(\frac{4 \sqrt{8}}{3},\) then the value of \(y (1)\) is equal to

If \(\beta \) is one of the angles between the normals to the ellipse, \(x^2 + 3y^2 = 9\) at the points \(\left( {3\cos \theta ,\sqrt {3\,} \sin \theta } \right)\) and \(\left( { - 3\sin \,\theta ,\sqrt 3 \,\cos \theta } \right); \in \left( {0,\frac{\pi }{2}} \right)\); then \(\frac{{2\,\cot \beta }}{{\sin \,2\theta }}\) is equal to

If the coefficients of \(x\) and \(x^{2}\) in the expansion of \((1+x)^{p}(1-x)^{q}, p, q \leq 15\), are \(-3\) and \(-5\) respectively, then the coefficient of \(x ^{3}\) is equal to \(............\)

Let a circle \(C\) touch the lines \(L_{1}: 4 x-3 y+K_{1}\) \(=0\) and \(L _{2}: 4 x -3 y + K _{2}=0, K _{1}, K _{2} \in R\). If a line passing through the centre of the circle \(C\) intersects \(L _{1}\) at \((-1,2)\) and \(L _{2}\) at \((3,-6)\), then the equation of the circle \(C\) is

The number of bijective functions \(f :\{1,3,5, 7, \ldots \ldots . .99\} \rightarrow\{2,4,6,8, \ldots \ldots, 100\}\), such that \(f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \ldots f(99), \quad\) is

The area of the region bounded by the parabola \((y-2)^{2}=(x-1)\), the tangent to it at the point whose ordinate is \(3\) and the \(\mathrm{x}\)-axis is :

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

For \(n \in N\), if \(\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^1 n=\frac{\pi}{4}\), then \(\mathrm{n}\) is equal to .........