If $A=\left[\begin{array}{rrr}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$ then ${\left(A^2\right)}^{-1}=$

Let $a>1$ be a constant. If $f:A\to A$ and $\left(x, y\right)\in f$ satisfy $a^x+a^y=a,$ then $A=$

Let \(\quad f(x)=e^x, \quad g(x)=\sin ^{-1} x \quad\) and \(h(x)=f(g(x))\), then \(\frac{h^{\prime}(x)}{h(x)}\) is equal to

If the tangent drawn at the point \(\mathrm{P}(3 \sqrt{2}, 4)\) on the hyperbola \(\frac{x^2}{9}-\frac{y^2}{16}=1\) meets its directrix at \(Q(\alpha, \beta)\) in fourth quadrant then \(\beta=\)

The number of possible common tangents that can be drawn to the circles $x^2+y^2+4x-6y-3=0$ and $x^2+y^2+4x-2y+1=0$ is

Statement I : The equation of the directrix of the ellipse \(4 x^2+y^2-8 x-4 y+4=0\) is \(3 y=6-4 \sqrt{3}\) Statement II : The equation of the latus rectum of the ellipse \(x^2+4 y^2-4 x-8 y+4=0\) is \(y=2+\sqrt{3}\) Which of the above statement(s) is (are) true?

The number of real roots of \(|x|^2-5|x|+6=0\) is

Two identical bar magnets of magnetic moment \(M\) each, are placed along \(X\) and \(Y\)-axes, respectively at a distance \(d\) from the origin (as shown in the figure). The origin lies on perpendicular bisector of magnet placed on \(X\)-axis and on the magnetic axis of magnet placed on \(Y\)-axis. If the magnitude of total magnetic field at the origin is \(B=\alpha\left[\frac{\mu_0}{4 \pi} \frac{M}{d^3}\right]\), then the value of constant \(\alpha\) will be \(\langle d>>l\), where \(l\) is the length of the bar magnets and direction of \(\mathrm{N}\) to \(\mathrm{S}\) in magnets is opposite with respect to each other)

\(\left|\begin{array}{lll}24 & 25 & 26 \\ 25 & 26 & 27 \\ 26 & 27 & 27\end{array}\right|\) is equal to

\(6 \mu C\) charge is placed at the centre of a cube. What will be the electric flux at each face of the tube? \[ \left[\text { Take } \frac{1}{4 \pi \epsilon_o}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\right] \]

Heat loss takes place from a body maintained at a temperature of \(400^{\circ} \mathrm{C}\) to the surrounding air at \(30^{\circ} \mathrm{C}\) by convection and to the surrounding surfaces at \(30^{\circ} \mathrm{C}\) by radiation. The Newton's cooling coefficient is \(20 \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}\) and the Stefan-Boltzmann constant is \(5.67 \times 10^{-8}\) \(\mathrm{W} / \mathrm{m}^2 \mathrm{~K}^4\). If the rate of heat loss by convection is equal to the rate of heat loss by radiation, the emissivity of the body surface is

Two ions having masses in the ratio \(1: 1\) and charges \(1: 2\) are projected into uniform magnetic field perpendicular to the field with speeds in the ratio \(2: 3\). The ratio of the radii of circular paths along which the two particles move is :

The electric flux from a cube of edge \(l\) is \(\phi\) in an enclosed charge. If the edge of the cube is made \(\frac{2}{3} l\) and the charge enclosed in the cube is doubled, then the electric flux value will be