If \(\alpha, \beta, \gamma\) and \(\delta\) are the roots of the equation \(x^4+3 x^3-6 x^2+2 x-4=0\), then find the equation having roots \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\) and \(\frac{1}{\delta}\)

The locus of the midpoint of the portion of the line \(x \cos \alpha+y \sin \alpha=p\) intercepted by the coordinate axes, where \(p\) is a constant, is

The smallest positive value (in degrees) of \(\theta\) for which \(\tan \left(\theta+100^{\circ}\right)=\tan \left(\theta+50^{\circ}\right) \tan (\theta) \tan \left(\theta-50^{\circ}\right)\) is valid, is

If \(y=\frac{a x+b}{c x+d}\) and \(\frac{d x}{d y}=\frac{a d-b c}{P y^2+Q y+R}\) then \(P+Q+R=\)

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

Let \(f(x)\) be a polynomial and \(a, b\) be distinct real numbers. Then the remainder in the division of \(f(x)\) by \((x-a)(x-b)\) is

If \(f(x)\) defined as given below, is continuous on \(R\), then the value of \(a+b\) is equal to
\[
f(x)=\left\{\begin{array}{cc}
\sin x, & x \leq 0 \\
x^2+a, & 0 < x < 1 \\
b x+3, & 1 \leq x \leq 3 \\
-3, & x>3
\end{array}\right.
\]

If the equation of the circle lying in the first quadrant, touching both the coordinate axes and the line \(\frac{x}{3}+\frac{y}{4}=1\) is \((x-c)^2+(y-c)^2=c^2\), then \(c=\)

A cannon ball is fired from the top of a \(55 \mathrm{~m}\) high cliff with an initial speed of \(50 \mathrm{~m} \mathrm{~s}^{-1}\). The speed of the cannon ball while hitting the ground in \(\mathrm{m} \mathrm{s}^{-1}\) (acceleration due to. gravity \(=10 \mathrm{~ms}^{-2}\) )

Let \(1, \omega\) and \(\omega^2\) be the cube roots of unity. If \(\mathrm{S}\) is the set of all non-singular matrices of the form \(\left[\begin{array}{rrr}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\) where \(a, b, c \in\left\{\omega, \omega^2\right\}\), then the number of elements in \(\mathrm{S}\) is

A body of mass \(0.15 \mathrm{~kg}\) moving with a velocity of \(15 \mathrm{~ms}^{-1}\) comes to rest, when it hits a spring that is fixed at another end. If the force constant of the spring is \(1500 \mathrm{Nm}^{-1}\), then the compression in the spring is

The molecule which has more number of lone pair of electrons than the bond pair of electrons in its central atom is

If a stone thrown vertically upwards from a bridge with an initial velocity of \(5 \mathrm{~m} \mathrm{~s}^{-1}\), strikes the water below the bridge in a time of 3 s, then the height of the bridge above the water surface is
(Acceleration due to gravity \(=10 \mathrm{~m} \mathrm{~s}^{-2}\) )