If \(f: R \rightarrow R\) is such that \(f(3)=16, f^{\prime}(3)=4\), then \(\lim _{x \rightarrow 3} \frac{x f(3)-3 f(x)}{x-3}=\)

If \(a , b , c\) are the sides of a triangle ABC and then \(2(\cos A+\cos B +\cos C )=\) \(\left|\begin{array}{lll}b & 1 & a \\ a & 1 & c \\ c & 1 & b\end{array}\right|=0\),

If the circle $x^2+y^2+2kx+4y-4=0$ has it's centre in $4^{\mathrm{th}}$ quadrant and touches the circle $x^2+y^2+6x-2y+6=0$ then, $k=$

If $\cos x+\cos y=p,\sin x+\sin y=q$, then $\cos \left(\frac{x-y}{2}\right)=$

If the equation of the transverse common tangent of the circles \(x^2+y^2-4 x+6 y+4=0\) and \(x^2+y^2+2 x-2 y-2=0\) is \(a x+b y+c=0\), then \(\frac{a}{c}=\)

\(P\) is a point on the segment joining the feet of two vertical poles of heights \(a\) and \(b\). The angles of elevation of the tops of the poles from \(P\) are \(45^{\circ}\) each. Then, the square of the distance between the tops of the poles is

The line passing through \(\left(-1, \frac{\pi}{2}\right)\) and perpendicular to \(\sqrt{3} \sin \theta+2 \cos \theta=\frac{4}{r}\) is :

Identify the correct statements from the folluwing (A) At 298 K, the potential of hydrogen electrode placed in a solution of \(\mathrm{pH}=10\), is -0.59 V (B) The limiting molar conductivity of \(\mathrm{Ca}^{2+}\) and \(\mathrm{Cl}^{-}\)is 119 and \(76 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}\) respectively. The limiting molar conductivity of \(\mathrm{CaCl}_2\) is \(195 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-\frac{1}{2}}\) (C) The correct relationship between \(\mathrm{K}_{\mathrm{c}}\) and \(\mathrm{E}_{\text {cell }}^0\) is \(\mathrm{E}_{\mathrm{ccll}}^{-}=\frac{2.303 \mathrm{RT}}{\mathrm{nF}} \log \mathrm{~K}_{\mathrm{c}}\)

\(\int \frac{3^x}{\sqrt{9^x-1}} d x=\)

For a particle executing SHM, determine the ratio of average acceleration of the particle between extreme position and equilibrium position w.r.t. the maximum acceleration.

The magnetic needle of a vibration magnetometer makes 12 oscillations per minute in the horizontal component of earth's magnetic field. When an external short bar magnet is placed at some distance along the axis of the needle in the same line, it makes 15 oscillations per minute. If the poles of the bar magnet are interchanged, the number of oscillations it makes per minute is

A force of \(4 \mathrm{~N}\) acts on a \(10 \mathrm{~kg}\) body initially at rest. Let \(W_1\) is work done by force during \(0 \leq t \leq\) ls. Likewise \(W_2\) is the work done by force during ls \(\leq t \leq 2 \mathrm{~s}\), where \(t\) is time in second. The ratio \(\frac{W_2}{W_1}\) is

The length of common chord of the circles \(x^2+y^2-6 x-4 y+13-c^2=0\) and \(x^2+y^2-4 x-6 y+13-c^2=0\) is