If the radical axis of the circles \(x^2+y^2+2 g x+2 f y+c=0\) and \(2 x^2+2 y^2+3 x+8 y+2 c=0\) touches the circle \(x^2+y^2+2 x+2 y+1=0\), then \((4 g-3)(f-2)=\)

Let \(x^2+y^2=16\) be the equation of the auxiliary circle of a hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) and let \((4 \sqrt{2}, 3)\) be a point on the hyperbola. Then, the eccentricity of the hyperbola is

If \(\left[\begin{array}{ccc}5 & a & -7 \\ b & -7 & c \\ -7 & d & -1\end{array}\right]\) is the adjoint of the matrix \(\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{array}\right]\), then \(a+b+c+d=\)

The abscissae of two points \(P, Q\) are the roots of the equation \(2 x^2+4 x-7=0\) and their ordinates are the roots of the equation \(3 x^2-12 x-1=0\). Then the centre of the circle with \(P Q\) as a diameter is

\(\cos ^3 \theta+\cos ^3\left(120^{\circ}+\theta\right)+\cos ^3\left(\theta-120^{\circ}\right)=\)

5 men and 4 women are seated in a row. If the number of arragements in which one particular man and one particular woman are together is ' \(\alpha\) ' and the number of arrangements in which those two are not together is \(\beta\), then \(\alpha: \beta=\)

Which of the following pair of straight lines intersect at right angles?

The mean marks of 25 boys in a class is 61 and the mean marks of 35 girls in the same class is 58 . Then, the mean of all 60 students is

Two balls of charge \(q_1\) and \(q_2\) initially have a velocity of the same magnitude and direction. After a uniform electric field has been applied during a certain time, the direction of the velocity of the first ball changes by \(60^{\circ}\), and the velocity magnitude is reduced by half. The direction of the velocity of the second ball changes thereby by \(90^{\circ}\).
In what proportion will the velocity of the second ball change? Determine the magnitude of the charge-to-mass ratio for the second ball if it is equal to \(k_1\) for the first ball. The electrostatic interaction between the balls should be neglected.

The number of circles that touch all the three lines \(x+y-1=0, x-y-1=0\) and \(y+1=0\) is

Which one of the following coordination complexes exhibit the lowest value of magnetic moment (in BM)?

The ideal solutions formed by mixing two liquids \(\mathrm{A}\) and \(\mathrm{B}\) at \(300 \mathrm{~K}\) in the molar ratio of \(1: 1\) and \(1: 2\) have vapour pressures of \(400 \mathrm{~mm}\) and \(350 \mathrm{~mm}\) respectively. At the same temperature, the vapour pressures of pure liquids \(\mathrm{A}\) and \(\mathrm{B}\) in \(\mathrm{mm}\) respectively are:

The maximum value of $f\left(x\right)=\frac{x}{4+x+x^2}$ on $[-1, 1]$ is