If the emf of an AC source is given by \(6 \sin \omega t+4 \sin 2 \omega t \mathrm{~V}\), then the rms value of the emf is

An electric motor exerts a force of \(50 \mathrm{~N}\) on a a cable and pulls it through \(60 \mathrm{~m}\) in \(1 \mathrm{~min}\). The power supplied by the motor is

Six point charges each of the magnitude \(Q\) are placed at the vertices of a regular hexagon of side \(a\) as shown in the figure. Electric field intensity on the line passing through the centre \(O\) and perpendicular to the plane of the figure at a large distance \(x(\gg > a)\) from \(O\) is
\(\left(\text {Let, } \frac{1}{4 \pi \in_0}=k\right)\)

An iron sphere of mass \(20 \times 10^{-3} \mathrm{~kg}\) falls through a viscous liquid with terminal velocity \(0.5 \mathrm{~ms}^{-1}\). The terminal velocity (in \(\mathrm{ms}^{-1}\) ) of another iron sphere of mass \(54 \times 10^{-2} \mathrm{~kg}\) is

If a source is transmitting electromagnetic wave of frequency \(8.2 \times 10^6 \mathrm{~Hz}\), then wavelength of the electromagnetic waves transmitted from the source will be

A given mass of a gas is compressed isothermally until its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be 0.75 of its initial pressure. The ratio of the specific heats of the gas is approximately :

A person holds a ball of mass 0.25 kg in his hand and throws it, so that it leaves his hand with a speed of \(12 \mathrm{~m} \mathrm{~s}^{-1}\). In this process, if his hand moved through a distance of 0.9 m, then the net force acted on the ball is

The shortest distance between the line \(y-x=1\) and the curve \(x=y^2\) is

Match the following
List - I
A) The equation of line passing through \((4,3)\) whose \(\mathrm{X}\)-intercept is twice its Y-intercept
B) The equation of the line passing through the centroid and circumcentre of \(\triangle A B C\) with vertices \(\mathrm{A}(1,1), \mathrm{B}(3,3), \mathrm{C}(6,-6)\)
C) The equation of the line whose \(\mathrm{X}\)-intercept is \((-3 / 5)\)
III) \(x+2 y+\sqrt{2}=0\) and is perpendicular to \(x-y+2=0\)
D) The equation of the line whose distance from the origin is 2 and the normal drawn from the origin makes an angle \(45^{\circ}\) with the positive direction of \(\mathrm{X}\)-axis
List - II
I) \(x+y-2 \sqrt{2}=0\)
II) \(7 x+23 y-8=0\)
IV) \(x+2 y-10=0\)
IV) \(x+2 y-10=0\)
V) \(5 x+5 y+3=0\)
The correct answer is

The equation of tangent of the curve \(y=\sqrt{9-2 x^2}\) at the point where the ordinate and abscissa are equal is

The equation of the plane through the intersection of the planes \(x+2 y+3 z-4=0\) and \(4 x+3 y+2 z+1=0\) and passing through the origin is

If the intercepts made by a variable circle on the X -axis and Y -axis are 8 and 6 units respectively, then the locus of the centre of the circle is

The area of the triangle formed by the lines represented by \(3 x+y+15=0\) and \(3 x^2+12 x y-13 y^2=0\) is