If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+a x^2+b x+c=0\), then \(\alpha^{-1}+\beta^{-1}+\gamma^{-1}\) is equal to

If the curves \(\frac{x^2}{4}+\frac{y^2}{9}=1\) and \(\frac{x^2}{16}-\frac{y^2}{k}=1\) cut each other orthogonally, then \(k=\)

If the product of the perpendicular distances from any point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) to its asymptotes is \(\frac{36}{13}\) and its eccentricity is \(\frac{\sqrt{13}}{3}\), then \(a-b=\)

If all possible numbers are formed by using the digits $1, 2, 3, 5, 7$ without repetition and they are arranged in descending order, then the rank of the number $327$ is

Match the items given in List A with those of the items of List B

The correct match is \(\begin{array}{llll}A & B & C & D\end{array}\)

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(a x^2+b x+c=0\). Observe the lists given below
The correct match of List-I from List-II is (i) (ii) (iii) (iv)

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)

A telephonic communication service is working at a carrier frequency of \(20 \mathrm{GHz}\). Only \(20 \%\) of it is utilized for transmission. If each channel requires a bandwidth of \(5 \mathrm{kHz}\), then the number of telephonic channels that can be transmitted simultaneously are

Work done to increase the temperature of one mole of an ideal gas by \(30^{\circ} \mathrm{C}\), if it is expanding under the condition \(V \propto T^{2 / 3}\) is, \((R=8.314 \mathrm{~J} / \mathrm{mol} / \mathrm{K})\)

Under action of force, a \(2 \mathrm{~kg}\) body moves such that its position \(x\) as function of time \(t\) is given by \(x=\alpha t^2 / 2\), where \(x\) is in metre, \(t\) is in seconds and \(\alpha=1 \mathrm{~m} / \mathrm{s}^2\). The work done by the force in the first two seconds is

A plane meets the coordinate axes at \(A, B, C\) respectively such that the centroid of the \(\triangle A B C\) is \((2,3,5)\). Then, the equation of that plane is

When $n=8$, ${\left(\sqrt{3}+i\right)}^n+{\left(\sqrt{3}-i\right)}^n=$

The equation of the circle of radius 5 and touching the co-ordinate axes in third quadrant is