If \(z=x+i y\) is a complex number such that \(\bar{z}^{\frac{1}{3}}=a+i b\), then the value of \(\frac{1}{a^2+b^2}\left(\frac{x}{a}+\frac{y}{b}\right)\) is equal to

If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3-3 x^2+3 x+1=\) 0, then \(\alpha^2 \beta^2+\beta^2 \gamma^2+\gamma^2 \alpha^2=\)

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If \(\alpha, \beta, \gamma\) are the roots of the equation \(\left|\begin{array}{ccc}x & 2 & 2 \\ 2 & x & 2 \\ 2 & 2 & x\end{array}\right|=0\) and \(\min (\alpha, \beta, \gamma)=\alpha\), then \(2 \alpha+3 \beta+4 \gamma=\)

Two players A and B are alternately throwing a coin and a die together. A player who first throws head and 6 wins the game. If A starts the game, then the probability that B wins the game is.

The locus of the centre of the circle, which cuts the circle \(x^2+y^2-20 x+4=0\) orthogonally and touches the line \(x=2\), is

If \(A=\left[\begin{array}{ccc}1 & -3 & -5 \\ -2 & 4 & -6 \\ 7 & -11 & 13\end{array}\right]\), then \(\sqrt{|\operatorname{Adj} A|}=\)

The hybridizations of the central atom in the molecules \(\mathrm{BF}_3, \mathrm{BeF}_2, \mathrm{BrF}_3\) are respectively

The area (in sq. units) of the triangle formed by the \(\mathrm{X}\)-axis, the tangent and the normal drawn to the circle \(x^2+y^2=10 x\) at the point \((9,3)\) is

\(\int \frac{\cos 2 x \cdot \sin 4 x}{\cos ^4 x\left(1+\cos ^2 2 x\right)} d x=\)

The equation of the parabola with focus \((1,-1)\) and directrix \(x+y+3=0\), is

The compound that does not undergo haloform reaction is

If A and B are arbitrary constants, then the differential equation having \(\mathrm{y}=\mathrm{Ae}^{\mathrm{x}}+\mathrm{B} \sin 2 \mathrm{x}\) as its general solution is