If the equations $$x^2+2x+3=0$$ and $$a{x}^2+bx+c=0,$$ $$a,b,c\in R,$$ have a common root, then $$a:b:c$$ is

Distance between two parallel planes $$2x+y+2z=8$$ and $$4x+2y+4z+5=0$$ is :

If the lines {{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}} and {{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1} are coplanar, then $$k$$ can have :

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is

The real number $$k$$ for which the equation, $$2x^3+3x+k=0$$ has two distinct real roots in $$\left[0,1\right]$$

The intercepts on $$x$$-axis made by tangents to the curve, $$y=\underset{0}{\overset{x}{\int }}\left|t\right|dt,x\in R,$$ which are parallel to the line $$y=2x$$, are equal to :

The area (in square units) bounded by the curves $$y=\sqrt{x,}$$ $$2y-x+3=0,$$ $$x$$-axis, and lying in the first quadrant is :

Consider : Statement − I : $$\left(p\wedge \sim q\right)\wedge \left(\sim p\wedge q\right)$$ is a fallacy. Statement − II :$$\left(p\to q\right)\leftrightarrow \left(\sim q\to \sim p\right)$$ is a tautology.

Let [$${\epsilon }_0$$] denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then:

Diameter of a plano-convex lens is $$6$$ $$cm$$ and thickness at the center is $$3mm$$. If speed of light in material of lens is $$2\times 10^{8}m/s,$$ the focal length of the lens is

The graph between angle of deviation $$\left(\delta \right)$$ and angle of incidence $$(i)$$ for a triangular prism is represented by

The supply voltage to room is $$120V.$$ The resistance of the lead wires is $$6\Omega$$. A $$60$$ $$W$$ bulb is already switched on. What is the decrease of voltage across the bulb, when a $$240$$ $$W$$ heater is switched on in parallel to the bulb?

This questions has Statement - $$\mathrm{I}$$ and Statement - $$\mathrm{I}$$$$\mathrm{I}$$. Of the four choices given after the Statements, choose the one that best describes into two Statements. Statement - $$\mathrm{I}$$ : Higher the range, greater is the resistance of ammeter. Statement - $$\mathrm{I}$$$$\mathrm{I}$$ : To increase the range of ammeter, additional shunt needs to be used across it.

This question has statement $$\mathrm{I}$$ and statement $$\mathrm{I}$$$$\mathrm{I}$$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - $$\mathrm{I}$$: A point particle of mass $$m$$ moving with speed $$\upsilon$$ collides with stationary point particle of mass $$M.$$ If the maximum energy loss possible is given as f\left( {{1 \over 2}m{v^2}} \right), then f = \left( {{m \over {M + m}}} \right). Statement - $$\mathrm{II}$$: Maximum energy loss occurs when the particles get stuck together as a result of the collision.

If a piece of metal is heated to temperature $$\theta$$ and then allowed to cool in a room which is at temperature $${\theta }_0,$$ the graph between the temperature $$T$$ of the metal and time $$t$$ will be closest to

A uniform cylinder of length $$L$$ and mass $$M$$ having cross-sectional area $$A$$ is suspended, with its length vertical, from a fixed point by a mass-less spring such that it is half submerged in a liquid of density $$\sigma$$ at equilibrium position. The extension $$x_0$$ of the spring when it is in equilibrium is:

The $$\mathrm{I}$$-$$V$$ characteristic of an $$LED$$ is

A diode detector is used to detect an amplitude modulated wave of $$60\%$$ modulation by using a condenser of capacity $$250$$ picofarad in parallel with a load resistance $$100$$ kilo $$ohm.$$ Find the maximum modulated frequency which could be detected by it.

The amplitude of a damped oscillator decreases to $$0.9$$ times its original magnitude in $$5s$$. In another $$10s$$ it will decrease to $$\alpha$$ times its original magnitude, where $$\alpha$$ equals